THE METAPHYSICAL
PRINCIPLES OF THE
INFINITESIMAL
CALCULUS
Translators
Michael Allen
Henry D. Fohr
Editor
Samuel D. Fohr
SOPHIA PERENNIS
Second
Impression 2004
Library of Congress Cataloging-in-Publication
Data
Guénon, René
[Principes du calcul infinitésimal. English]
The Metaphysical Principles of the infinitesimal calculus / René Guénon ;
translated by Henry D. Fohr, Michael Allen ; edited by Samuel D. Fohr
p.
cm. — (Collected works of René Guénon)
Includes index.
THE PUBLISHER
GIVES SPECIAL THANKS TO
HENRY D. AND JENNIE L. FOHR
FOR MAKING THIS EDITION POSSIBLE
CONTENTS
Editorial
Note xiii
Preface
1
Infinite
and Indefinite 7
The
Contradiction of‘Infinite Number’ 15
The
Innumerable Multitude 19
The
Measurement of the Continuous 25
Questions
Raised by the Infinitesimal Method 31
‘Well-Founded
Fictions’ 35
‘Degrees
of Infinity’ 41
‘Infinite
Division’ or Indefinite Divisibility 47
Indefinitely
Increasing; Indefinitely Decreasing 54
Infinite
and Continuous 60
The
‘Law of Continuity’ 64
The
Notion of the Limit 69
Continuity
and Passage to the Limit 74
‘Vanishing
Quantities’ 78
Zero is
not a Number 83
The
Notation of Negative Numbers 89
Representation
of the Equilibrium of Forces 95
Variable
and Fixed Quantities 100
Successive
Differentiations 103
Various Orders of Indefinitude 106
21 The
Indefinite is Analytically Inexhaustible ill
22 The
Synthetic Character of Integration 115
23 The
Arguments of Zeno of Elea 120
24 The
True Conception of‘Passage to the Limit’ 124
25 Conclusion
128
Index
131
The
past century has witnessed an erosion of earlier cultural
values as well as a blurring of the distinctive characteristics of the world’s
traditional civilizations, giving rise to philosophic and moral relativism,
multiculturalism, and dangerous fundamentalist reactions. As early as the
1920s, the French metaphysician René Guénon (1886-1951) had diagnosed
these tendencies and presented what he believed to be the only possible
reconciliation of the legitimate, although apparently conflicting, demands of
outward religious forms, ‘exoterisms’, with their essential core, ‘esoterism’.
His works are characterized by a foundational critique of the modern world
coupled with a call for intellectual reform; a renewed examination of metaphysics,
the traditional sciences, and symbolism, with special reference to the
ultimate unanimity of all spiritual traditions; and finally, a call to the work
of spiritual realization. Despite their wide influence, translation of Guénon’s works
into English has so far been piecemeal. The Sophia Perennis edition is
intended to fill the urgent need to present them in a more authoritative and
systematic form. A complete list of Guénon’s works, given in the order
of their original publication in French, follows this note.
Guénon’s early
and abiding interest in mathematics, like that of Plato, Pascal, Leibnitz, and
many other metaphysicians of note, runs like a scarlet thread throughout his
doctrinal studies. In this late text published just five years before his
death, Guénon devotes
an entire volume to questions regarding the nature of limits and the infinite
with respect to the calculus both as a mathematical discipline and as symbolism
for the initiatic path. This book therefore extends and complements the
geometrical symbolism he employs in other works, especially The Symbolism of
the Cross, The Multiple States of the Being, and Symbols of Sacred
Science.
According to Guénon, the concept ‘infinite
number’ is a contradiction in terms. Infinity is a metaphysical concept at a
higher level
of reality than that of
quantity, where all that can be expressed is the indefinite, not the infinite.
But although quantity is the only level recognized by modern science, the
numbers that express it also possess qualities, their quantitative aspect being
merely their outer husk. Our reliance today on a mathematics of approximation
and probability only further conceals the ‘qualitative mathematics’ of the
ancient world, which comes to us most directly through the Pythagorean-Platonic
tradition.
Guenon
often uses words or expressions set off in ‘scare quotes’. To avoid clutter,
single quotation marks have been used throughout. As for transliterations,
Guenon was more concerned with phonetic fidelity than academic usage. The
system adopted here reflects the views of scholars familiar both with the
languages and Guénon’s writings. Brackets indicate editorial
insertions, or, within citations, Guénon’s additions. Wherever
possible, references have been updated, and English editions substituted.
The
translation in its final form is based on the work of the mathematician
Michael Allen, who had before him an earlier version by Henry Fohr edited by
his son Samuel Fohr. Reference was also made to submissions by Richard Pickrell
and Fatima Casewit. The text was reviewed by mathematician and traditionalist
author Dr. Wolfgang Smith, and the entire text checked for accuracy and further
revised by Patrick Moore and Marie Hansen. For help with proofing and selected
chapters thanks go to Cecil Bethell (who also provided the index), John
Champoux, Allan Dewar, and John Ahmed Herlihy. Latin translations were provided
by David Matz. Cover design by Michael Buchino and Gray Henry, based on a
drawing by Guénon’s friend
and collaborator Ananda K. Coomaraswamy.
THE WORKS
OF RENÉ GUÉNON
Introduction
to the Study of the Hindu Doctrines (1921)
Theosophy:
History of a Pseudo-Religion (1921)
The
Spiritist Fallacy (1923)
East
and West (1924)
Man
and His Becoming according to the Vedanta
(1925)
The
Esoterism of Dante (1925)
The
Crisis of the Modern World (1927)
The
King of the World (1927)
Spiritual Authority and
Temporal
Power (1929)
The
Symbolism of the Cross (1931)
The
Multiple States of the Being (1932)
The
Reign of Quantity and the Signs of the Times
(1945)
Perspectives
on Initiation (1946)
The
Great Triad (1946)
The
Metaphysical Principles of the Infinitesimal Calculus
(1946)
Initiation and Spiritual
Realization
(1952)
Insights into Christian
Esoterism
(1954)
Symbols
of Sacred Science (1962)
Studies
in Freemasonry and the Compagnonnage (1964)
Studies
in Hinduism (1966)
Traditional Forms and
Cosmic
Cycles
(1970)
Insights
into Islamic Esoterism and Taoism (1973)
Reviews
(1973)
Miscellanea
(1976)
Although
the present study might, at least at first glance,
appear to have only a rather ‘specialist’ character, the undertaking seemed
worthwhile in order to clarify and explain more thoroughly various notions to
which we have had recourse on various occasions when we have made use of
mathematical symbolism, and this reason alone would suffice to justify it.
However, we should add that there are still other, secondary reasons,
concerning especially what one could call the ‘historical’ aspect of the
question; the latter, indeed, is not entirely devoid of interest from our point
of view inasmuch as all the discussions that have arisen on the subject of the
nature and value of the infinitesimal calculus offer a striking example of
that absence of principles which characterizes the profane sciences, that is,
the only sciences that the moderns know and even consider possible. We have
already often noted that most of these sciences, even insofar as they still
correspond to some reality, represent no more than simple, debased residues of
some of the ancient, traditional sciences: the lowest part of these sciences,
having ceased to have contact with the principles, and having thereby lost its
true, original significance, eventually underwent an independent development
and came to be regarded as knowledge sufficient unto itself, although in truth
it so happens that its own value as knowledge is thereby reduced to almost
nothing. This is especially apparent with the physical sciences, but as we have
explained elsewhere,[1] in
this respect modern mathematics itself is no exception if one compares it to
what was for the ancients the science of numbers and geometry; and when we
speak here of the ancients one must understand by that even those of‘classical’
antiquity, as the least study of Pythagorean and Platonic theories suffices to
show, or at least should show were it not necessary to take into account the
extraordinary
incomprehension of those
who claim to interpret them today. Were this incomprehension not so complete,
how could one maintain, for example, a belief in the empirical’ origin of the
sciences in question? For in reality they appear on the contrary all the more
removed from any ‘empiricism’ the further back one goes in time, and this is
equally the case for all other branches of scientific knowledge.[2]
Mathematicians
of modern times, and more particularly still those who are our contemporaries,
seem to be ignorant of what number truly is; and by this we do not mean to
speak solely of number in the analogical and symbolic sense as understood by
the Pythagoreans and Kabbalists, which is all too obvious, but—and this might
seem stranger and almost paradoxical—even of number in its simply and strictly
quantitative sense. Indeed, their entire science is reduced to calculation in
the narrowest sense of the word,[3]
that is, to a mere collection of more or less artificial procedures,
which are in short only valuable with respect to the practical applications to
which they give rise. Basically this amounts to saying that they replace number
with the numeral; and furthermore, this confusion of the two is today so
widespread that one could easily find it at any moment, even in the expressions
of everyday language.[4] Now
a numeral is, strictly speaking, no more than the clothing of a number; we do
not even say its body, for it is rather the geometric form that can, in certain
respects, legitimately be considered to constitute the true body of a number,
as the theories of the ancients on polygons and polyhedrons show when seen in
the light of the symbolism of numbers; and this, moreover, is in accordance
with the fact that all ‘embodiment’ necessarily implies a ‘spatialization’. We
do not mean to say, however, that numerals themselves are entirely arbitrary
signs, the form of which has been determined only by the fancy of one or more
individuals; there must be both numerical
and alphabetical
characters—the two of which, moreover, are not distinguished in some languages[5]—and
one can apply to the one as well as to the other the notion of a hieroglyphic,
that is to say an ideographic or symbolic origin, and this holds for all
writing without exception, however obscured this origin might be in some cases
due to more or less recent distortions or alterations.
What
is certain is that in their notation mathematicians employ symbols the meaning
of which they no longer understand, and which are like vestiges of forgotten
traditions; and what is more serious, not only do they not ask themselves what
this meaning might be, it even seems that they do not want them to have any
meaning at all. Indeed, they tend more and more to regard all notation as
simple ‘convention’, by which they mean something set out in an entirely
arbitrary manner, but this is a true impossibility, for one never establishes a
convention without having some reason for doing so, and for doing precisely
that rather than anything else; it is only to those who ignore this reason that
the convention can appear as arbitrary, just as it is only to those who ignore
the cause of an event that it can appear ‘fortuitous’. This is indeed what
occurs here, and one can see in it one of the more extreme consequences of the
absence of principles, which can even cause the science—or what is so called,
for at this point it no longer merits the name in any respect—to lose all
plausible significance. Moreover, by the very fact of the current conception of
science as exclusively quantitative, this ‘conventionalism’ has gradually
spread from mathematics to the more recent theories of the physical sciences,
which thus distance themselves further and further from the reality they intend
to
explain; we have
emphasized this point sufficiently enough in another work to be able to
dispense with further remarks in this regard, and all the more so since we now
intend to occupy ourselves more particularly with mathematics alone. From this
viewpoint we will only add that when one completely loses sight of the meaning
of a notation it becomes all too easy to pass from a legitimate and valid use
of it to one that is illegitimate and in fact no longer corresponds to
anything, and which can sometimes even be entirely illogical. This may seem
rather extraordinary when it is a question of a science like mathematics which
should have particularly close ties with logic, yet it is nevertheless all too
true that one can find multiple illogicalities in mathematical notions as they
are commonly envisaged in our day.
One
of the most remarkable examples of these illogical notions, and which we shall
consider first and foremost, even though it is certainly not the only one we
shall encounter in the course of our exposition, is that of the so-called
mathematical or quantitative infinite, which is the source of almost all the
difficulties that can be raised against the infinitesimal calculus, or, perhaps
more precisely, against the infinitesimal method, for we here have something
that, whatever the ‘conventionalists’ might think, goes beyond the range of a
simple ‘calculation’ in the ordinary sense of the word; and this notion is the
source of all difficulties without exception, save those that proceed from an
erroneous or insufficient conception of the notion of the ‘limit’, which is
indispensable if the rigor of the infinitesimal method is to be justified and
made anything more than a simple method of approximation. As we shall see,
moreover, there is a distinction to be made between cases in which the
so-called infinite is only an absurdity pure and simple, that is, an idea
contradictory in itself, such as that of an ‘infinite number’, and cases in
which it is only employed in an improper way in the sense of indefinite; but it
should not be believed because of this that the confusion of the infinite and
the indefinite can itself be reduced to a mere question of words, for it rests
quite truly with the ideas themselves. What is singular is that this confusion,
which had it once been dispelled would have cut short so many discussions, is
found in the writings of Leibnitz himself, who is generally regarded as the
inventor of the
infinitesimal calculus,
although we would rather call him its ‘for- mulator’, for his method
corresponds to certain realities that, as such, have an existence independent
of those who conceive of them and who express them more or less perfectly;
realities of the mathematical order, like all other realities, can only be
discovered and not invented, while on the contrary it is indeed a question
of‘invention’ when, as occurs all too often in this field, one allows oneself
to be swept away by the ‘game’ of notation into the realm of pure fantasy. But
it would assuredly be quite difficult to make some mathematicians understand
this difference, since they willingly imagine that the whole of their science
is and must be no more than a ‘fabrication of the human mind’, which, if we
had to believe them, would certainly reduce their science to a trifling thing
indeed. Be that as it may, Leibnitz was never able to explain the principles of
his calculus clearly, and this shows that there was something in it that was
beyond him, something that was as it were imposed upon him without his being
conscious of it; had he taken this into account, he most certainly would not
have engaged in any dispute over ‘priority’ with Newton. Besides, these sorts
of disputes are always completely vain, for ideas, insofar as they are true,
are not the property of anyone, despite what modern ‘individualism’ might have
to say; it is only error that can properly be attributed to human individuals.
We shall not elaborate further on this question, which could take us quite far
from the object of our study, although in certain respects it would perhaps not
be profitless to make it clear that the role of those who are called ‘great
men’ is to a great extent often a role of ‘reception’, though they are
generally the first to delude themselves as to their own ‘originality’.
What
concerns us more directly for the moment is this: if we must point out such
deficiencies in Leibnitz—deficiencies all the more serious in that they bear
above all on questions of principles— what could be said of those found in
other modern philosophers and mathematicians, to whom Leibnitz is certainly
superior in spite of everything? This superiority he owes on the one hand to
the studies he made of the Scholastic doctrines of the Middle Ages, even though
he did not always fully understand them, and on the other hand to certain
esoteric data, principally of a Rosicrucian origin or
inspiration,[6]
data obviously very incomplete and even fragmentary, which he moreover
sometimes applied quite poorly, as we shall presently see in some examples. It
is to these two ‘sources’, to speak as the historians do, that one can
definitively relate nearly all that is really valid in his theories, and this
also allowed him to react, albeit imperfectly, against the Cartesianism which,
in the double domain of philosophy and science, represented the whole ensemble
of the tendencies and conceptions that are most specifically modern. This
remark suffices, in short, to explain in a few words all that Leibnitz was, and
if one seeks to understand him, one must never lose sight of this general
information, which we have for this reason deemed worthwhile to set forth at
the outset; but it is time to leave these preliminary considerations in order
to enter into the examination of the very questions that will allow us to
determine the true significance of the infinitesimal calculus.
1
AND INDEFINITE
Proceeding in a manner inverse to
that of profane science, and in accordance with the unchanging perspective of
all traditional science, we must before all else set forth the principle that
will allow us almost immediately to resolve the difficulties to which the
infinitesimal method has given rise, without letting ourselves be led astray by
potentially interminable discussions, as indeed happens in the case of those
modern philosophers and mathematicians who, by the very fact that they lack
this principle, have never provided a satisfactory and definitive solution to
these difficulties. This principle is the very idea of the Infinite, understood
in its only true sense, which is the purely metaphysical sense, and on this
subject, moreover, we have only summarily to recall what we have already
expressed more completely elsewhere:1 the Infinite is properly that
which has no limits, for ‘finite’ is obviously synonymous with ‘limited’; one
cannot then correctly apply this term to anything other than that which has
absolutely no limits, that is to say the universal All, which includes in
itself all possibilities and consequently cannot be limited by anything in any
way; the Infinite, thus understood, is metaphysically and logically necessary,
for not only does it not imply any contradiction, not enclosing within itself
anything negative, but it is on the contrary its negation that would be
contradictory. Furthermore, there can obviously be only one Infinite, for two
supposedly distinct infinites would limit and therefore inevitably exclude one
another; consequently, every time the term ‘infinite’ is
used in any sense other
than that which we have just mentioned, we can be assured a priori that
this use is necessarily improper, for it amounts in short either to ignoring
the metaphysical Infinite altogether, or to supposing another Infinite
alongside it.
It
is true that the Scholastics admitted what they called the infinitum
secundum quid [the infinite in a certain respect], and that they carefully
distinguished it from the infinitum absolutum [the absolute infinite],
which alone is the metaphysical Infinite; but we can see here only an
imperfection in their terminology, for although this distinction allowed them
to escape the contradiction of a plurality of infinites understood in the
proper sense, the double use of the word infinitum nonetheless certainly
risked causing multiple confusions, and besides, one of the two meanings was
then altogether improper, for to say that something is infinite only in a
certain respect—and this is the exact significance of the expression infinitum
secundum quid—is to say that in reality it is not infinite at all.[7]
Indeed, it is not because a thing is not limited in a certain sense or in
a certain respect that one can legitimately conclude that it is limited in no
way at all, the latter being necessary for it to be truly infinite; not only
can it be limited in other respects at the same time, but we can even say that
it is of necessity so, inasmuch as it is a determined thing, which, by its very
determination, does not include every possibility, and this amounts to saying
that it is limited by that which lies outside of it; if, on the contrary, the
universal All is infinite, this is precisely because there is nothing that lies
outside of it.[8]
Therefore every determination, however general one supposes it to be and
however far one extends the term, necessarily excludes the true notion of the
infinite;[9] a
determination, whatever it might be, is always a limitation, since its
essential character is to define a certain
domain of possibilities in
relation to all the rest, and by that very fact to exclude all the rest. Thus
it is truly nonsense’ to apply the idea of the infinite to any given
determination, as for example, in the instance we are considering more
particularly here, to quantity or to one or another of its modes. The idea of a
‘determined infinite’ is too manifestly contradictory for us to dwell upon any
longer, although this contradiction has most often escaped the profane thought
of the moderns; and even those whom one might call ‘semiprofane’,[10]
like Leibnitz, were unable to perceive it clearly. In order to bring out the
contradiction still further we could say in other fundamentally equivalent
terms that it is obviously absurd to wish to define the Infinite, since a
definition is in fact nothing other than the expression of a determination, and
the words themselves show clearly enough that what is subject to definition can
only be finite or limited. To seek to place the Infinite within a formula, or,
if one prefer, to clothe it in any form whatsoever is, consciously or unconsciously,
to attempt to fit the universal All into one of its minutest parts, and this is
assuredly the most manifest of impossibilities.
What
we have just said suffices to establish, without leaving room for the slightest
doubt and without necessitating any other considerations that there cannot be
a mathematical or quantitative infinite, and that this expression does not
even have any meaning, because quantity is itself a determination. Number,
space, and time, to which some people wish to apply the notion of this
so-called infinite, are determined conditions, and as such can only be finite;
they are but certain possibilities, or certain sets of possibilities, beside
and outside of which there exist others, and this obviously implies their
limitation. In this instance still more can be said: to conceive of the
Infinite quantitatively is not only to limit it, but in addition it is to
conceive of it as subject to increase and decrease, which is no less absurd;
with similar considerations one quickly finds oneself
envisaging not only
several infinites that coexist without confounding or excluding one another,
but also infinites that are larger or smaller than others; and finally, the
infinite having become so relative under these conditions that it no longer
suffices, the ‘transfinite’ is invented, that is, the domain of quantities
greater than the infinite. Here, indeed, it is properly a matter of
‘invention’, for such conceptions correspond to no reality. So many words, so
many absurdities, even regarding simple, elementary logic, yet this does not
prevent one from finding among those responsible some who even claim to be
‘specialists’ in logic, so great is the intellectual confusion of our times!
We
should point out that just now we did not merely say ‘to conceive of a
quantitative infinite’, but ‘to conceive of the Infinite quantitatively’, and
this calls for a few words of explanation. By this expression we wanted to allude
more particularly to those who are called ‘infinitists’ in contemporary
philosophical jargon; indeed, all the discussions between ‘finitists’ and
‘infinitists’ clearly show that at least both have in common this completely
false idea that the metaphysical Infinite is akin to the mathematical
infinite, if they do not purely and simply identify the two.[11]
Thus they all equally ignore the most elementary principles of metaphysics,
since it is on the contrary precisely the conception of the true, metaphysical
Infinite that alone allows us to reject absolutely every ‘particular infinite’,
if one may so express it, such as the so-called quantitative infinite, and to
be assured in advance that, wherever it is encountered, it can only be an
illusion; we shall then only need to ask what could have brought about this
illusion in order to be able to replace it with a notion closer to the truth.
In short, every time it is a question of a particular thing, of a determined
possibility, we can be certain a priori that it is limited by that very
fact, and, we can say, limited by its very nature, and this holds equally true
in the case where, for whatever
reason, we cannot actually
reach its limits; but it is precisely this impossibility of reaching the limits
of certain things, and sometimes even of conceiving of them clearly, that
causes the illusion that these things have no limits, at least among those for
whom the metaphysical principle is lacking; and, let us say it again, it is
this illusion and nothing more that is expressed in the contradictory assertion
of a ‘determined infinite’.
In
order to rectify this false notion, or rather to replace it with a true
conception of things,[12] we
must here introduce the idea of the indefinite, which is precisely the idea of
a development of possibilities the limits of which we cannot actually reach;
and this is why we regard the distinction between the Infinite and the
indefinite as fundamental to all questions in which the so-called mathematical
infinite appears. Without doubt this is what corresponds in the intention of
its authors to the Scholastic distinction between the infinitum absolutum
and the infinitum secundum quid. It is certainly unfortunate that
Leibnitz, who had borrowed so much from Scholasticism, had neglected or not
been aware of this, for however imperfect the form in which it was expressed,
it would have allowed him to respond quite easily to certain objections raised
against his method. In contrast to this, it seems that Descartes had indeed
tried to establish the distinction in question, but he was very far from having
expressed or even conceived of it with sufficient precision, since according to
him the indefinite is that of which we do not perceive the limits, and which
in reality could be infinite, although we could not affirm it to be so, whereas
the truth is that we can on the contrary affirm that it is not so and that it
is by no means necessary to perceive its limits in order to be certain that
they exist. One can
thus
see how vague and confused are all such explanations, and always as a result of
the same lack of principle. Descartes indeed said: ‘And for us, seeing things
in which, in a certain sense,[13] we
note no limits, we cannot ascertain thereby that they are infinite, but we
shall only consider them to be indefinite.’[14]
And he gives as examples the extension and divisibility of bodies; he does not
contend that these things are infinite, but he does not seem to want to deny it
formally either, and all the more so since he had just declared that he did
not wish to ‘entangle himself in disputes over the infinite,’ which is rather
too easy a way to brush aside the difficulties, even if he does say a little
later that ‘although we shall observe properties that seem to us not to have
any limits, we do not fail to recognize that this proceeds from our lack of
understanding and not from their nature.’[15]
In short, he wishes with good reason to reserve the name infinite for what has
no limits; but on the one hand he appears not to know with the absolute certitude
that is implied in all metaphysical knowledge, that what has no limits cannot
be anything but the universal All, and on the other hand the very notion of the
indefinite needs to be much more precise; had it been so, a great number of
subsequent confusions would doubtless not have been as readily produced.[16]
We say that the indefinite cannot be infinite because it
always implies a certain determination, whether it is a question of extension,
duration, divisibility, or some other possibility; in a word, whatever the
indefinite may be, and according to whatever aspect it is considered, it is
still of the finite and can only be of the finite. No
doubt, its limits may be
extended until they are found to be out of our reach, at least insofar as we
seek to reach them in a certain manner that we can call ‘analytical’, as we
shall explain more thoroughly in what follows; but they are by no means
abolished thereby, and in any case, if limitations of a certain order can be
abolished, others possessing the same nature as the first will still remain,
for it is by virtue of its nature, and not simply by some more or less exterior
or accidental circumstances, that every particular thing is finite, whatever
the degree to which certain limits can be extended. In this regard one might
point out that the sign oo, by which mathematicians represent their so-called
infinite, is itself a closed figure, therefore visibly finite, just like the
circle, which some people have wished to make a symbol of eternity, while it can
in fact only be a figure of a temporal cycle, indefinite merely in its order,
that is to say, of what is properly called perpetuity;[17]
and it is easy to see that this confusion of eternity with perpetuity, so
common among modern Westerners, is closely related to that of the Infinite and
the indefinite.
In
order to better understand the idea of the indefinite and the manner in which
it is formed from the finite taken in its ordinary sense, one can consider an
example such as that of the sequence of numbers: here, it is obviously never
possible to stop at a determined point, since after every number there is
always another that can be obtained by adding a unit; consequently, the
limitation of this indefinite sequence must be of an order other than that which
applies to a definite set of numbers taken between any two deter- T mined
numbers; it must derive not from particular properties of certain numbers, but
rather from the very nature of number in all its generality, that is to say
from the determination that, essentially constituting this nature, makes number
at once what it is and not anything else. One could make exactly the same
observation if it were no longer a question of number but of space or time
likewise
considered
in every possible extension to which they are subject.[18]
Any such extension, as indefinite as one conceives it to be and as it in
fact is, will never in any way take us out of the finite. Indeed, whereas the
finite necessarily presupposes the Infinite—since the latter is that which
comprehends and envelops all possibilities—the indefinite on the contrary
proceeds from the finite, of which it is in reality only a development and to
which it is consequently always reducible, for it is obvious that whatever
process one might apply, one cannot derive from the finite either anything more
or anything other than that which was already potentially contained therein. To
take again the example of the sequence of numbers, we can say that this
sequence, with all the indefinitude it implies, is given to us by its law of
formation, since it is from this very law that its indefinitude immediately
results; now this law consists in the following, that given any number, one can
form the next by adding a unit. The sequence of numbers is therefore formed by
successive additions of the unit to itself, indefinitely repeated, which is
basically only the indefinite extension of the process of formation for any
arithmetical sum; and here one can see quite clearly how the indefinite is
formed starting from the finite. This example, moreover, owes its particular
clarity to the discontinuous character of numerical quantity; but, to take
things in a more general fashion applicable to all cases, it would suffice to
insist on the idea of‘becoming’ that is implied by the term ‘indefinite’, and
this we expressed above in speaking of the development of possibilities, a
development that in itself and in its whole course always consists of something
unfinished;[19]
the importance of the consideration of‘variables’ as they concern the
infinitesimal calculus will give to this last point its full significance.
2
‘infinite
number’
As we will see yet more clearly in the following, there are
some cases in which it suffices to replace the idea of the so-called infinite
with that of the indefinite in order to dispel all difficulties immediately;
but there are others in which even this is not possible, because it is a
question of something clearly determined—‘fixed’, so to speak, by
hypothesis—which, as such, cannot be called indefinite, according to our last
remarks above. Thus, for example, one can say that the sequence of numbers is
indefinite, but not that a certain number, however great one supposes it to be
and whatever position it occupies in the sequence, is indefinite. The idea of
an ‘infinite number’, understood as ‘the greatest of all numbers’, or ‘the
number of all numbers’, or, again, ‘the number of all units’, is in itself a
truly contradictory idea, the impossibility of which would remain even were one
to renounce the unjustifiable use of the word ‘infinite’. There cannot be a
number greater than all others, for however great a number might be, one can always
form a greater one from it by adding a unit, in accordance with the law of
formation which we set forth above. This amounts to saying that the sequence
of numbers cannot have a final term, and it is precisely because it does not
‘terminate’ that it is truly indefinite; as the number of all the terms of the
sequence could itself only be the last of them, it can be said that the
sequence is not ‘numerable’, and this is an idea we shall have to return to
more fully in what follows.
The
impossibility of an ‘infinite number’ can be established further by various
arguments. Leibnitz, who at least recognized this
quite clearly,[20]
used one that consisted in comparing the sequence of even numbers to that of
whole numbers: to every number there corresponds another number equal to its
double, such that one can make the two sequences correspond term by term, with
the result that the number of terms must be the same in both; but there are
obviously twice as many whole numbers as there are even, since the even numbers
alternate by twos in the sequence of whole numbers; one thus ends up with a
manifest contradiction. One can generalize this argument by taking, instead of
the sequence of even numbers, that is, multiples of two, that of multiples of
any number whatsoever, and the reasoning will be identical; or again, in the
same way one could take the sequence of the squares of whole numbers,[21] or
more generally that of their powers of any exponent. Whatever the case, the
conclusion will always be the same: a sequence containing only a part of the
whole numbers will have the same number of terms as another containing all of
them, which would amount to saying that the whole is not greater than its part,
and, as soon as one allows that there is a number of all numbers, this
contradiction will be inescapable. Nevertheless, some have thought to avoid it
by supposing at the same time that there are numbers for which multiplication
by a certain number or elevation to a certain power is not possible, precisely
because such operations would yield a result exceeding the so-called ‘infinite
number’; there are even those who have indeed been led to envisage numbers said
to be ‘greater than infinite’, whence such theories as that of Cantor’s
‘transfinite’, which may be quite ingenious, but are no longer logically valid:[22] is
it even
conceivable that one could
dream of calling a number ‘infinite’ when it is on the contrary so ‘finite’
that it is not even the greatest of all numbers? Moreover, with such theories
there would be numbers to which none of the rules of ordinary calculation would
apply any longer, or, in short, numbers that would no longer truly be numbers
but merely called such by convention.4 This inevitably occurs when,
seeking to conceive of an ‘infinite number’ otherwise than as the greatest of
all numbers, one envisages various ‘infinite numbers’, supposedly unequal to
each other, to which we attribute properties that no longer have anything in
common with those of ordinary numbers; thus one escapes one contradiction only
to fall into others, and all this is at bottom only the product of the most
meaningless ‘conventionalism’ imaginable.
Thus, the idea of a so-called ‘infinite number’, whatever
manner in which it is presented and whatever name by which one wishes to designate
it, always comprises contradictory elements; moreover, one has no need of such
an absurd supposition from the moment one forms a proper conception of what the
indefinitude of number really is, and when one further recognizes that number,
despite its indefinitude, is by no means applicable to all that exists. We need
not dwell upon this last point here, as we have already sufficiently explained
it elsewhere. Number is only a mode of quantity, and quantity itself only a
category or special mode of being, not coextensive with it, or, more precisely
still, quantity is only a condition proper to one certain state of existence in
the totality of universal existence; but this is precisely the point that most
moderns have difficulty understanding, habituated as they are to wanting to
reduce everything to quantity and even to evaluating everything
the infinite are given, the
infinitesimal limits will also be given (I do not say the ultimate limits)
which follow upon them], which, though he never explained himself more clearly,
seems to indicate that he supposed that in a numerical sequence there could be
terms ‘beyond the infinite’.
4. One can by no means say that here it is a question of an
analogical use of the idea of number, for this would imply transposition to a
domain other than that of quantity; on the contrary, considerations of this
sort always refer exclusively to quantity understood in its most literal sense.
numerically.[23]
However, in the very domain of quantity there are things that escape number, as
we shall see when we come to the subject of continuity; and even without
departing from the sole consideration of discontinuous quantity, one is already
forced to admit, at least implicitly, that number is not applicable to everything,
when one recognizes that the multitude of all numbers cannot constitute a
number, which, moreover, is finally only an application of the incontestable
truth that what limits a certain order of possibilities must necessarily be
beyond and outside that which it limits.[24]
Only it must be understood that such a multitude, be it discontinuous, as in
the case of the sequence of numbers, or continuous—a subject we shall have to
return to shortly—can in no wise be called infinite, and in such cases there
can never be anything but the indefinite; and it is this notion of multitude
that we are now going to examine more closely.
As we have seen, Leibnitz by no means admits ‘infinite number’,
since on the contrary he expressly declares that this would imply contradiction
in whatever sense one took it; on the other hand, he does admit what he calls
an ‘infinite multitude’, though without making it clear—as the Scholastics
would at least have done—that in any case it can only be an infinitum
secundum quid, the sequence of numbers being, for him, an example of such a
multitude. From another point of view, however, in the quantitative domain, and
even in that of continuous magnitude, the idea of the infinite always appears
to him as suspect of at least possible contradiction, for, far from being an
adequate idea, it inevitably entails a certain amount of confusion, and we
cannot be certain that an idea implies no contradiction unless we distinctly
conceive all of its elements;[25]
this hardly allows according this idea a ‘symbolic’—we would rather say
‘representative’—character, and as we shall see later, this is why he
never dared to give a
clear verdict on the reality of the ‘infinitely small’; but this very
perplexity, this doubtful attitude, brings out even better the lack of
principle that led him to admit that one could speak of an ‘infinite
multitude’. From this one might also wonder if he did not think that in order
to be ‘infinite’, as he calls it, such a multitude must not only be
‘numerable’, which is obvious, but that it must not even be quantitative at
all, taking quantity in all its extension and in all its modes; this would be
true in certain cases, but not in all; however it may be, it remains a point on
which he never clearly explained himself.
The
idea of a multitude that surpasses all number, and that consequently is not a
number, seems to have astonished most of those who have discussed the
conceptions of Leibnitz, be they ‘finitists’ or ‘infinitists’; it is
nevertheless far from unique to Leibnitz, as they have generally seemed to
believe, and, on the contrary, was quite common among the Scholastics.[26]
This idea was applied specifically to everything that is neither a number nor
‘numerable’, that is, all that does not relate to the domain of discontinuous
quantity, whether it be a question of things belonging to other modes of
quantity, or of what is entirely outside of the quantitative domain, for it
concerned an idea belonging to the order of ‘transcendentals’, or general modes
of being, which, contrary to its special modes like quantity, are coextensive
with it.[27]
This also allows one to speak of the multitude of divine attributes for
example, or again of the multitude of angels, that is, of beings belonging to
states that are not
subject to quantity, where,
consequently, there can be no question of number; it is also this that allows
one to speak of the states of being or degrees of existence as multiple or as
constituting an indefinite multitude, even though quantity is only one special
condition of a single one of them. On the other hand, since the idea of multitude,
contrary to that of number, is applicable to all that exists, there must
necessarily be multitudes of a quantitative order, notably in the domain of
continuous quantity, and this is why we said just now that it would not be
correct to consider every case of the so- called ‘infinite multitude’, that is,
that which surpasses all number, as entirely escaping the domain of quantity.
Furthermore, number itself can also be regarded as a species of multitude, but
on the added condition that it be a ‘multitude measured by the unit’, according
to the expression of Saint Thomas Aquinas; all other sorts of multitude, being
‘innumerable’, are ‘non-measured’, which is not to say they are infinite, but merely
that they are indefinite.
While
on the subject, it is appropriate to note a rather singular fact: for Leibnitz,
this multitude, which does not constitute a number, is nonetheless a ‘result
of units’.[28]
How should we understand this, and indeed, what are the units in question? The
word unit can be taken in two completely different senses:[29]
on the one hand, there is the arithmetical or quantitative unit, which is the
first element of number, its point of departure, and, on the other hand, there
is what is analogously designated as metaphysical Unity, which is identified
with pure Being itself; we see no other possible meaning outside of these; but
furthermore, whenever one speaks of‘units’ in the plural, this can obviously
only be understood in the quantitative sense. If this is so, however, then the
sum of these units cannot be anything other than a number, and can in no way
transcend number; it is true that Leibnitz said ‘result’ and not ‘sum’, but
this distinction, even if it is intentional, nonetheless remains an
unfortunate obscurity. Besides, he declares elsewhere that multitude, without
being a number, is nevertheless conceived by analogy
with number: ‘When there
are more things,’ he says, ‘than can be comprehended by any number, we yet
attribute to them analogically a number that we call infinite,’ although this
would only be a ‘manner of speaking’, a modus loquendi,[30]
and even, in this form, a most incorrect manner of speaking, since in reality
the thing in question is not a number at all; but whatever the imperfections of
expression and the confusions to which they might give rise, we must in any
case admit that an identification of multitude with number was assuredly not at
the root of his thought.
Another
point to which Leibnitz seems to attach great importance is that the
‘infinite’, such as he conceives of it, does not constitute a whole;[31]
this is a condition he regards as necessary if the idea is to escape
contradiction, but here we have another rather obscure point. One might well wonder
what sort of ‘whole’ is in question here, and it is first of all necessary to
put aside entirely the idea of the universal All, which is on the contrary, as
we have said from the beginning, the metaphysical Infinite itself, the only
true Infinite, which could by no means be in question here; indeed, whether it
is a question of continuous or discontinuous, the ‘indefinite multitude’ that
Leibnitz envisages in any case only makes sense in a restricted and contingent
domain of a cosmological and not metaphysical order. It is obviously a
question, moreover, of a whole conceived as composed of parts, whereas, as we
have explained elsewhere,[32]
the universal All is properly ‘without parts’, by very reason of its infinity,
since these parts are necessarily relative and finite and thus could
not have any real
connection with it, which amounts to saying that for it they do not exist. So,
as regards the question posed, we must confine ourselves to the consideration
of a particular whole; but here again, and precisely in what concerns the mode
of composition of such a whole and its relation with its parts, there are two
cases to consider, corresponding to two very different senses of the same word
‘whole’. First, there is the whole that is nothing more or other than the
simple sum of its parts, of which it is composed in the manner of an
arithmetical sum, which Leibnitz says is obviously fundamental, for this mode
of formation is precisely that which is proper to number, and he does not allow
us to go beyond number; but in fact this notion, far from representing the only
way in which a whole can be conceived, is not even that of a true whole in the
most rigorous sense of the term. Indeed, a whole that is thus only the sum or
result of its parts and which consequently is logically posterior to them, is,
as such, nothing other than an ens rationis [a being of reason or of
the mind], for it is ‘one’ and ‘whole’ only in the measure that we conceive it
as such; in itself it is strictly speaking only a ‘collection’, and it is we
who, by the manner in which we envisage it, confer upon it in a certain
relative sense the character of unity and totality. On the contrary, a true
whole possessing this character by its very nature, must be logically anterior
to its parts and independent of them: such is the case with a continuous set,
which we can divide into parts arbitrarily, that is, into parts of any size,
without in the least presupposing the actual existence of these parts; here, it
is we who give a reality to the parts as such, by an ideal or effective
division, and this case is thus the exact inverse of the preceding.
Now,
the whole question comes back in short to knowing whether, when Leibnitz says
that ‘the infinite is not a whole,’ he excludes this second sense as well as the
first; it seems that he does, and this is probable since it is the only case in
which a whole would truly be ‘one’, and since the infinite, according to him,
is nec unum, nec totum [neither one nor a whole]. What further confirms
this is that this latter, and not the former, is what applies to a living being
or an organism when it is considered from the point of view of totality; now
Leibnitz says: ‘Even the Universe is not a whole, and it must not be conceived
of as an animal with God for its soul, as the
ancients thought.’[33]
However, if this is so, one does not really see how the ideas of the infinite
and the continuous can be connected, as he most often takes them to be, since
the idea of the continuous is, at least in a certain sense, linked precisely to
this second conception of totality; but this is a point that will be better
understood in the light of what is to follow. In any case, what is certain is
that if Leibnitz had conceived of a third sense of the word ‘whole’, a purely
metaphysical sense superior to the other two, namely the idea of the universal
All as we set it forth at the very beginning, he would not have been able to
say that the idea of the infinite excludes totality, for he declares moreover:
‘The real infinite is perhaps the absolute itself, which is not composed of
parts, but having parts, comprehends them by eminent reason, as to the degree
of its perfection.’[34]
Here, one could say, there is at the very least a ‘glimmer’, for this
time, almost by exception, he takes the word ‘infinite’ in its true sense,
although it would be erroneous to say that this infinite ‘has parts’, however
one wishes to understand this; but it is then strange that he again expresses
his thought only in a doubtful and perplexing form, as if he were not exactly
settled as to the significance of the idea; and indeed perhaps he never was,
for otherwise one could not explain why he so often turned away from its proper
meaning, and why, when he speaks of the infinite, it is sometimes so difficult
to know whether his intention was to take this term rigorously, albeit wrongly,
or whether he had in view only a simple ‘manner of speaking’.
OF THE CONTINUOUS
Until now, when speaking of number
we have had in view whole number exclusively,[35]
and logically this was so of necessity, since we were regarding numerical
quantity strictly as discontinuous quantity: between two consecutive terms in
the sequence of whole numbers there is always a perfectly definite interval,
marked by the difference of a unit existing between these two numbers, which,
when one keeps to the consideration of whole number, is in no way reducible. In
reality, moreover, it is whole number alone that is true number, what one might
call pure number; and the sequence of whole numbers, starting from the unit,
continues increasing indefinitely without ever arriving at a final term, the
supposition of which, as we have seen, would be contradictory; but it goes
without saying that the sequence develops entirely in a single direction, and
so the other, opposite direction —that of indefinite decrease — cannot be
represented by it, although from another point of view there is a certain
correlation and a sort of symmetry between the considerations of indefinitely
increasing and indefinitely decreasing quantities, as we shall demonstrate
further on. However, people have not stopped at whole number, but have been led
to consider various kinds of number; it is usually said that these are
extensions
or generalizations of the
idea of number, and this is true after a certain fashion; but at the same time
these extensions are also distortions, and this modern mathematicians seem too
easily to forget, since their ‘conventionalism’ leads them to misunderstand the
origin and raison d’être of these numbers. In fact, numbers
other than whole numbers always appear above all as the representation of the
results of operations that would be impossible were one to keep to the point of
view of pure arithmetic, which in all rigor is the arithmetic of whole numbers
alone: thus a fractional number, for example, is no more than the
representation of the result of a division that cannot in fact be made, that
is, one that must be declared arithmetically impossible, and this, moreover,
is implicitly recognized when it is said, according to ordinary mathematical
terminology, that one of the two numbers in question is not divisible by the
other. Here we should point out that the definition commonly given to
fractional numbers is absurd; fractions can in no way be ‘parts of a unit’, as
is said, for the true arithmetical unit is necessarily indivisible and without
parts; and from this results the essential discontinuity of number, which is
formed from the unit; but let us see whence this absurdity arises.
Indeed,
one does not arbitrarily consider the results of the aforementioned operations
thus, instead of regarding them purely and simply as impossible; generally
speaking, it is in consequence of the application made of number—discontinuous
quantity—to the measurement of magnitudes belonging to the order of continuous
quantity, as, for example, spatial magnitudes. Between these two modes of
quantity is a difference of nature such that a correspondence between the two
cannot be perfectly established; to remedy this to a certain point, at least
insofar as it is possible, one seeks to reduce, as it were, the intervals of
this discontinuity constituted by the sequence of whole numbers, by introducing
other numbers between its terms, and fractional numbers first of all, which would
be meaningless apart from this consideration. It is then easy to understand
that the absurdity we just pointed out concerning the definition of fractions
arises quite simply from a confusion of the arithmetical unit with what are
called ‘units of measurement’, units that are such only by convention, and that
in reality are magnitudes
of another sort than
number, notably geometric magnitudes. The unit of length, for example, is only
a certain length chosen for reasons foreign to arithmetic, and the number 1 is
made to correspond to it in order to be able to measure all other lengths by
reference thereto; but all length, even when so represented by the unit, is by
its very nature as continuous magnitude no less always and indefinitely
divisible. Comparing it to other lengths that are not exact multiples of it,
one might thus have to consider parts of this unit of measurement, which would
in no way be parts of the arithmetical unit on that account; and it is only
thus that the consideration of fractional numbers is really introduced, as a
representation of the ratios of magnitudes that are not exactly divisible by
one another. The measurement of a magnitude is indeed no more than the
numerical expression of its ratio to another magnitude of the same kind taken
as the unit of measurement, or, basically, as the term of comparison; and this
is why the ordinary method of measuring geometric magnitudes is essentially
founded on division.
It
must be said, moreover, that in spite of this method something of the discontinuous
nature of number is always bound to remain, preventing one from thus obtaining
a perfect equivalent to the continuous; reduce the intervals as much as one
likes—which finally is to say, reduce them indefinitely, rendering them smaller
than any quantity that can be given in advance—but they will never be done away
with entirely. To make this clearer, let us take the simplest example of a
geometric continuum, a straight line: we shall consider half a straight line,
extending indefinitely in a certain direction,[36]
and let us agree to make each of its points correspond to a number
expressing the distance of the point from the origin, represented by zero, as
its distance from itself is obviously nothing; starting from this origin, the
whole numbers will then correspond to the successive extremities of all
segments equal to each other and to the unit of length; the points contained
between these will be representable
only by fractional
numbers, since their distances from the origin are not exact multiples of the
unit of length. It goes without saying that, taking fractional numbers with
greater and greater denominators, hence smaller and smaller differences, the
intervals between the points to which these numbers correspond will be reduced
in the same proportion; in this way the intervals can be decreased indefinitely,
theoretically at any rate, since the possible denominators of the fractional
numbers are themselves whole numbers, the sequence of which increases
indefinitely.[37] We
say theoretically because in fact the multitude of fractional numbers is
indefinite, and one could never use them all, but let us suppose that ideally
all the possible fractional numbers could be made to correspond to the points
on the half of the line in consideration. Despite the indefinite decrease of
the intervals, a multitude of points to which no number will correspond will
still remain on this line. At first this might seem strange and even
paradoxical, but it is nevertheless easily demonstrated, for such a point can
be obtained by means of a very simple geometric construction. Let us construct
a square having for its side the line segment with extremities at the points o
and i, and let us draw the diagonal of the square starting from the origin,
then a circle having for its center the origin and for its radius this
diagonal; the point at which this circle cuts the straight line cannot be represented
by any whole or fractional number, since its distance from the origin is equal
to the diagonal of the square, which is incommensurable with its side, that
is, with the unit of length. Thus, the multitude of fractional numbers, despite
an indefinite decrease of their differences, still does not suffice to fill, so
to speak, the intervals between the points contained in the line,[38]
which amounts to saying that this multitude is not a real and adequate
equivalent to linear continuity; in order to express the measurement of certain
lengths, one is thus forced to introduce still other kinds of numbers, what are
called incommensurable numbers, that is, those having no
common measure with the
unit. Such are the irrational numbers, which represent the results of
arithmetically impossible extractions of roots, as, for example, the square
root of a number that is not a perfect square; thus in the preceding example,
the ratio of the diagonal of the square to its side, and consequently the
point having-a distance from the origin equal to this diagonal, can be
represented only by the irrational number ^2, which is indeed incommensurable,
for there exists no whole or fractional number the square of which is equal to
2; and besides these irrational numbers there are still other incommensurable
numbers, the geometrical origin of which is obvious, as, for example, the
number it, which represents the ratio of the circumference of a
circle to its diameter.
Without
entering further into the question of the ‘composition of the continuous’, it
will thus be seen that number, however far the notion is extended, is never
perfectly applicable to it; finally this application always amounts to
replacing the continuous with a discontinuity, the intervals of which can be
very small, and can even become smaller and smaller still by an indefinite
series of successive divisions, but without ever being done away with, for in
reality there is no ‘final term’ to which the divisions might lead, since a
continuous quantity, however small it might be, will always remain indefinitely
divisible. It is to these divisions of the continuous that the consideration of
fractional numbers properly corresponds; but, and this is particularly
important to note, a fraction, however minute it might be, is always a
determined quantity, and however small one supposes the difference between two
fractions there is always an equally determined interval. Now the property of
indefinite divisibility that characterizes continuous magnitudes obviously
demands that one always be able to take elements as small as one wishes, and
that the intervals existing between these elements can likewise be rendered
less than any given quantity; but—and it is here that we see the insufficiency
of fractional numbers, and even, we can say, of number altogether—in order that
there really be continuity, these elements and these intervals must not be
conceived of as something determined. Consequently, the most perfect representation
of continuous quantity will be obtained by the consideration not of fixed and
determined magnitudes such as those just
discussed, but on the
contrary of variables, for then their variability can itself be regarded as
accomplished in a continuous fashion; and these quantities must be capable of
indefinite decrease by virtue of their variability, without ever canceling
themselves out or reaching a ‘minimum’, which would be no less contradictory
than ‘final terms’ of the continuous: here, precisely, as we shall see, is the
true notion of infinitesimal quantities.
QUESTIONS
RAISED BY
THE INFINITESIMAL
METHOD
When Leibnitz first presented the
infinitesimal method,[39]
and even again in several other works that followed,[40]
he particularly emphasized the uses and applications of the new calculus, in
keeping with the modern tendency to attribute more importance to the practical
applications of science than to science itself, as such; it would be difficult
to say whether this tendency truly existed in Leibnitz, or whether this manner
of presenting his method was only a sort of concession on his part. Be that as
it may, in order to justify a method it certainly does not suffice to show the
advantages it might have over other, previously accepted methods, or the
conveniences it might furnish practically for calculation, nor even the results
it might in fact have given; and the adversaries of the infinitesimal method
did not fail to make use of this, and it was only their objections that
persuaded Leibnitz to explain the principles, and even the origins, of his
method. It is very possible, moreover, that on this last point he might never
have spoken at all, but ultimately this is of little importance, for very
often the occasional causes of a discovery
are in themselves only
rather insignificant circumstances; at any rate, of what he wrote on the
subject,[41]
all that interests us is the fact that he passed from a consideration of the
‘assignable’ differences existing between numbers to a consideration of the
‘unassignable’ differences that can be conceived of between geometric
magnitudes by reason of their continuity, and that he also attached great importance
to this order, as being so to speak ‘demanded by the nature of things’. From
this it follows that for him infinitesimal quantities do not naturally appear
directly to us, but only as a result of passing from a consideration of the
variability of discontinuous quantity to that of continuous quantity, and from
the application of the first to the measurement of the second.
What
exactly is the meaning of these infinitesimal quantities Leibnitz was
reproached for using without having first defined what he meant by them, and
did this meaning allow him to regard his calculus as absolutely rigorous, or
on the contrary merely as a method of approximation? To respond to these two
questions would, by that very fact, be to resolve the most important objections
raised against him; but unfortunately he himself never responded very clearly,
and even his various attempts to do so do not always seem in complete accord
with one another. In this connection it is worth noting that generally speaking
Leibnitz was in the habit of explaining the same thing differently according to
the audience he was addressing; we would certainly not hold this behavior
against him, which is irritating only for systematic minds, for in principle
he was only conforming to an initiatic and, more particularly, Rosicrucian
precept according to which it is fitting to speak to each in his own language;
only he sometimes happened to apply the precept rather poorly. Indeed, if it is
obviously possible to clothe the same truth in different expressions, it is
understood that this be done without ever distorting or diminishing it, being
always careful to refrain from any manner of speaking that could give rise to
false conceptions; in this regard Leibnitz failed in a number of instances.[42]
Thus, he pushed
the idea of
‘accommodation’ to the point of sometimes seeming to justify those who wished
to see in his calculus merely a method of approximation, for at times he
presented it as being no more than a sort of abridged version of the ancients’
‘method of exhaustion’, useful for facilitating calculations but yielding
results that have to be verified by this other method if a rigorous
demonstration is desired; and it is nevertheless quite certain that this was
not fundamentally what he thought, but that, in reality, he saw in it much more
than a simple expedient intended to shorten calculations.
Leibnitz frequently declared that infinitesimal quantities
cannot but be ‘incomparable’, but as to the precise meaning in which this word
is to be understood, he gave an explanation that is not only rather
unsatisfying, but even most regrettable, for it could not but provide
ammunition to his adversaries, who, moreover, did not fail to avail themselves
of it; here, again, he was certainly not expressing what he truly thought, and
we can see in this another example of an excessive ‘accommodation’, yet more
serious than the first, that would substitute erroneous views for ‘adapted’
expressions of the truth. Leibnitz writes:
One need not take the infinite here
rigorously, but only in the manner in which one says in optics that the rays of
the sun come from an infinitely distant point, and may thus be treated as
parallel. And when there are several degrees of the infinite or of the
infinitely small, this is like the terrestrial globe being regarded as a point
with respect to the distance of the fixed stars, and a ball we might take in
hand being again a point in comparison with the semi-diameter of the
terrestrial globe, such that the distance of the fixed stars is like an
infinite infinitude with respect to the diameter of the ball. For instead of
the infinite or the infinitely small, one takes quantities as great or as small
as is necessary for the error to be less than a given error, such that one
differs from the style of Archimedes only in expression, which in our method is
more direct, and more conformable with the art of invention.5
did have some theoretical idea of the
nature of the ‘gift of tongues’, he was nevertheless far from having received
it effectively.
5. ‘Mémoire de
M.G.G. Leibnitz touchant son sentiment sur le Calcul différentiel’ in
the Journal de Trévoux, 1701.
It was unfailingly pointed
out to Leibnitz that however small the terrestrial globe might be with respect
to the heavens, or a grain of sand in relation to the terrestrial globe, they
are nonetheless fixed and determined quantities, and if one of these quantities
can be regarded as practically negligible in comparison with the other, this is
nevertheless only a simple approximation; his reply was that he had only wished
to ‘avoid the subtleties’ and to ‘make the reasoning evident to all,’[43]
which fully confirms our interpretation, and which, furthermore, is already a
sort of manifestation of the ‘popularizing’ tendency of modern scholars. What
is most extraordinary is that he was able to write afterwards: ‘At any rate,
there was not the slightest thing that should have caused anyone to imagine
that I indeed meant a very small, but always fixed and determined, quantity,’
to which he added: ‘Besides, I had already written some years ago to Bernoulli
of Groningen that the infinites and infinitely small might be taken for
fictions, similar to imaginary roots,[44]
without thereby harming our calculus, these fictions being useful and founded
in reality.’[45]
Moreover, it seems that he never did understand exactly in what respect his
comparison was flawed, for he presents it again in the same terms about ten
years later;[46]
but, at any rate since he expressly declared that his intention had not been to
present the infinitesimal quantities as determined, we must conclude from this
that, for him, the meaning of the comparison amounts to the following: a grain
of sand, though not infinitely small, can, however, without appreciable
disadvantage, be considered as such in relation to the earth, and thus there is
no need to envisage the infinitely small ‘rigorously’—they may even be regarded
as mere fictions if one so desires; but however one takes them, such a
consideration is nonetheless manifestly unsuitable to give any other idea of
the infinitesimal calculus than that of a simple calculus of approximation,
which would assuredly have been insufficient in the eyes of Leibnitz himself.
‘well-founded
fictions’
The thought most characteristic of
Leibnitz, although he does not always affirm it with the same force, and on
which he sometimes even seems, albeit exceptionally, not to wish to deliver a
categorical verdict, is that basically infinite and infinitely small quantities
are only fictions; but, he adds, they are ‘well-founded fictions’, and by this
he does not simply mean that they are useful for calculation,[47] or
even for ‘finding real truths’, although sometimes he does also insist on this
usefulness; but he constantly repeats that these fictions are ‘founded in
reality’, that they are fundamentum in re, which obviously implies
something of a more than purely utilitarian value; and for him this value
itself must after all be explained by the basis these fictions have in reality.
In any case, he believes that for the method to be reliable, it suffices to
envisage, not infinite and infinitely small quantities in the rigorous sense of
these expressions, since this would have no corresponding reality, but simply
quantities as great or as small as one likes, or as is necessary in order for
the error to be rendered less than any given quantity. It is still necessary
to examine whether it is true that, as he declares, this error is thereby null,
that is, whether this manner of envisaging the infinitesimal calculus gives
him a perfectly rigorous foundation, but we shall have to return to this
question later. However it might be with
respect
to this last point, for him statements concerned with the infinite and
infinitely small quantities fall under the category of assertions that
according to him are only toleranter verae [reasonably true], or
‘tolerable’, and must be ‘redressed’ by an explanation, as when one regards
negative quantities as ‘less than zero’, or as in a number of other cases in
which the language of geometry implies ‘a certain figurative and cryptic manner
of speaking’;[48]
the word ‘cryptic’ would seem to be an allusion to the symbolic and profound
meaning of geometry, but this is not at all what Leibnitz had in mind, and
perhaps as is so often the case with him in so speaking he had only the memory
of some esoteric notion, more or less poorly understood.
As for the sense in which one should understand the statement
that infinitesimal quantities are ‘well-founded fictions’, Leibnitz declared
that ‘the infinites and infinitely small are founded in such a way that within
the realm of geometry, and even in nature, they may be treated as if they were
perfectly real’;[49]
indeed, for him, everything that exists in nature in some way implies the
consideration of the infinite, or at least of what he believed could be called
such. As he said, ‘the perfection of the analysis of transcendentals, or of
geometry involving the consideration of some infinite would without doubt be
all the more important on account of the applications one can make of it to the
operations of nature, which introduces the infinite in all that it does’;[50]
but perhaps this is only because we cannot have adequate ideas of it, and
because it always introduces elements we cannot perceive with complete
distinctness. If this is so, then it is necessary not to take too literally
such assertions as the following for example: ‘Since our method is properly
that part of general mathematics that treats of the infinite, one has great
need of it in applying mathematics to physics, for as a rule the character of
the infinite Author enters into the operations of nature.’[51]
But if by this
even Leibnitz only means
that the complexity of natural things goes incomparably beyond the limits of
distinct perception, it nonetheless remains that the infinite and infinitely
small quantities must have their fundamentum in re', and this foundation
is found in the nature of things, at least as conceived by him, and is none
other than what he calls the ‘law of continuity’, which we shall have to
examine a little later, and which he regards, rightly or wrongly, as being in
short only a particular case of a certain ‘law of justice’, which is itself
ultimately connected to the idea of order and harmony, and which equally finds
its application every time a certain symmetry must be observed, as, for
example, in the case of combinations and permutations.
Now,
if the infinite and infinitely small quantities are only fictions, and even
admitting that they really are ‘well-founded’, one might ask oneself this: why
use such expressions, which, even if they can be regarded as toleranter
verae, are nonetheless incorrect? Here is something which presages, one
might say, the ‘conventionalism’ of modern science, though with the notable
difference that the latter is no longer in any way preoccupied with knowing
whether the fictions to which it has recourse are ‘well-founded’ or not, or,
according to another expression of Leibnitz, whether they can be interpreted sano
sensu [in
a reasonable way], or even whether they have any meaning at all. Moreover,
since one can do without these fictional quantities and be content with
envisaging in their place quantities that can simply be rendered as great or as
small as one T. likes, and which, for that reason, can be said to be
indefinitely great or indefinitely small, it would no doubt have been better to
do so from the start and thus avoid introducing fictions that, whatever might
be their fundamentum in re, are, ultimately, of no practical use, not
only with regard to calculation, but even regarding the infinitesimal method
itself. The expressions ‘indefinitely great’ and ‘indefinitely small’, or what
amounts to the same but is perhaps more precise, ‘indefinitely increasing’ and
‘indefinitely decreasing’, not only have the advantage of being the only ones
that are rigorously exact; they also show clearly that the quantities to which
they are applied can only be variable, and not determined, quantities. As a
mathematician has rightly said, ‘the infinitely small is not a very
small quantity, having an actual value capable of being
determined; its character is to be eminently variable, and to be able to take
on a value less than that of any other one might wish to specify; it would be
much better to call them indefinitely small.’[52]
The use of these terms would have prevented many difficulties
and disputes, and there is nothing surprising about this, since it is not a
simple question of words, but the replacement of a false idea with a true one,
of a fiction with a reality; notably, it would have prevented anyone from
taking the infinitesimal quantities to be fixed and determined quantities, for
as we said above the word ‘indefinite’ always carries with it the idea of
‘becoming’, and consequently of change, or, when it is a question of
quantities, of variability; and, had Leibnitz made a habit of using these
terms, he would doubtless not have allowed himself to be so easily drawn into
the unfortunate comparison concerning the grain of sand. What is more, reducing
infinite parva ad indefinite parva [the infinitely small to the
indefinitely small] would at any rate have been clearer than reducing them ad
incomparabiliter parva [to the incomparably small]; precision would thereby
have been gained without any loss of exactitude—quite the contrary.
Infinitesimal quantities assuredly are ‘not comparable’ to ordinary quantities,
but this can be understood in more than one way, and indeed it has often
enough been taken in other senses than were intended. It is better to say that
they are ‘unassignable’, to use another expression of Leibnitz, for it seems
that this term can be rigorously understood only of quantities that are capable
of becoming as small as one likes, that is, smaller than any given quantity,
and consequently to which one can by no means ‘assign’ a determined value,
however small it might be, and this is indeed the sense of indefinite parva.
Unfortunately, it is next to impossible to know whether, in Leibnitz’s thought,
‘incomparable’
and ‘unassignable’ are truly and completely synonymous; but
in any case, it is at the very least certain that a truly ‘unassignable’
quantity, by reason of the possibility of indefinite decrease that it implies,
will thereby be ‘incomparable’ with respect to any given quantity, and, to
extend the idea to different orders of the infinitesimal, even with respect to
any quantity in relation to which it can decrease indefinitely, as long as the
latter is regarded as possessing at least a relative fixity.
If there is one point on which everyone can easily agree,
even without going more deeply into questions of principles, it is that the
notion of the indefinitely small, at least from the purely mathematical point
of view, is perfectly sufficient for infinitesimal analysis, and the
‘infinitists’ themselves recognize this without great difficulty.[53]
In this respect one can thus be content with a definition such as that
given by Carnot: ‘What is an infinitely small quantity in mathematics? Nothing
other than a quantity that can be rendered as small as one likes, without one’s
being obliged on that account to vary those to which one wants to relate it.’[54]
But as for the true significance of infinitesimal quantities, the entire matter
is not limited to this; for the calculus it matters little that the infinitely
small are only fictions, since one can be content with a consideration of the
indefinitely small, which raises no logical difficulty; furthermore, since for
the metaphysical reasons set out at the beginning we cannot admit a
quantitative infinite, whether infinitely great or infinitely small,[55] or
indeed any infinite of a determined and relative order, it is quite certain
that these can only be fictions and nothing else; but if rightly
or wrongly these fictions were introduced into the
infinitesimal calculus in the beginning, this is because according to
Leibnitz’s intention they nevertheless correspond to something, however faulty
the manner in which they expressed it. Since we are here concerned with
principles and not merely with a method of calculation in and of itself (which
would be without interest for us) we should therefore ask what exactly is the
value of these fictions, not only from the logical point of view, but also from
the ontological point of view, whether they are as ‘well-founded’ as Leibnitz
believed, and whether we can even say with him that they are toleranter
verae, and at the very least accept them as such modo sano
sensu intelligantur
[understood in a reasonable way]. To answer these questions it will be necessary
for us to examine more closely his conception of the ‘law of continuity’, for
it was here that he thought to find the fundamentum in re of the
infinitely small.
‘degrees
of infinity’
We have not yet had occasion in the
preceding pages to see all the confusions that are inevitably introduced when
the idea of the infinite is taken otherwise than in its one true and properly
metaphysical sense; more than one example could be found, notably, in
Leibnitz’s long discussion with Jean Bernoulli on the reality of infinite and
infinitely small quantities, which moreover never came to any definitive
conclusion; nor, indeed, could it have done so, given the continual confusion
on both sides, and the lack of principles from which this confusion proceeded;
moreover, whatever the order of ideas in question, ultimately it is always the
lack of principles which alone renders questions insoluble. One might well be
astonished to learn, among other things, that Leibnitz distinguished between
‘infinite’ and ‘interminable’, and that he had thus not absolutely rejected
the idea—nonetheless manifestly contra-dictory—of a ‘terminating infinite’, and
went so far as to ask himself ‘whether it might be possible for there to exist,
for example, an infinite straight line that might nevertheless terminate at
both ends.’[56] No
doubt he is reluctant to admit this possibility, ‘all the more so since it
seems to me,’ he says elsewhere, ‘that the infinite, taken rigorously, must
have its source in the interminable, without which I see no means of finding a
proper foundation for distinguishing it from the finite.’[57]
But even if one puts it more affirmatively (which he did not do) and says
that ‘the infinite has its source in the interminable,’ one still does not take
them to be absolutely identical, but rather as distinguished from one another
to a certain degree; and as long as that is
so, one risks finding
oneself checked by a crowd of strange and contradictory ideas. It is true that
Leibnitz declares that he would not willingly admit these ideas without first
being ‘forced by indubitable demonstrations’, but it is already serious enough
to attribute a certain degree of importance to them, and even to be able to
envisage them other than as pure impossibilities; as for the idea of a sort of
‘terminating eternity’, to take one example from those he sets forth in this
connection, we can see in it only the product of a confusion between the
notions of eternity and duration, which is absolutely unjustifiable with
respect to metaphysics. We readily grant that the time in which we pass our
corporeal lives really is indefinite, which is in no way incompatible with its
‘terminating at both ends’, which is to say, in conformity with the traditional
cyclic conception, that it has both a beginning and an end; we also grant that
there exist other modes of duration, such as that which the Scholastics call aevum,
the indefinitude of which is, if one may so express it, indefinitely greater
than that of this time; but all these modes, in all their possible extension,
are nonetheless only indefinite, since it is always a question of particular
conditions of existence proper to this or that state; and, precisely insofar as
each is a kind of duration— which implies succession—not one can be identified
with or assimilated to eternity, with which it has no more connection than
does the finite, whatever its mode, nor again with the true Infinite, for the
notion of a relative eternity has no more meaning than that of a relative
infinite. In all of this we have only various orders of indefinitude, as will
be seen more clearly later on, but Leibnitz, for want of having made the
necessary and essential distinctions, and above all for not having laid down
before all else the principle that alone would have prevented him from going
astray, found himself very much at a loss to refute Bernoulli’s opinions;
indeed, so equivocal and hesitant were Leibnitz’s responses that Bernoulli even
took him to be much closer than was really the case to his own ideas about the
‘infinity of worlds’ and the different ‘degrees of infinity’.
This
notion of the so-called ‘degrees of infinity’ amounts in short to supposing
that there can exist worlds incomparably greater and incomparably smaller than
our own, the corresponding parts of each being in equal proportion to one
another, such that the
inhabitants of any one of
these worlds would have just as much reason to call theirs infinite as we
would with respect to ours; for our part we would rather say they would have
just as little reason. Such a manner of envisaging things would not appear
absurd a priori without the introduction of the idea of the infinite,
which is certainly nothing to the purpose, for however great one imagines them
to be, each of these worlds is nonetheless limited; how then can they be called
infinite? The truth is that none of them can really be so, if only because they
are conceived as multiple, for here we return to the contradiction of a
plurality of infinites; and besides, even if it happens that some or even many
consider our world to be infinite, this assertion nonetheless can offer no
acceptable meaning. Moreover, one might wonder if they really are different
worlds, or if, quite simply, they are not rather more or less extended parts of
the same world, since by hypothesis they must all be subject to the same
conditions of existence—notably to spatiality—and simply developed on an
enlarged or diminished scale. It is in a completely different sense that one
can truly speak, not of an infinity, but of an indefinitude of worlds, since
apart from the conditions of existence such as space and time, which are proper
to our world considered in all the extension of which it is susceptible, there
is an indefinitude of others, equally possible; a world, or, in short, a state
of existence, is thus defined by the totality of the conditions to which it is
subject; but, by the very fact that it will always be conditioned, that is,
determined and limited, and hence unable to contain all possibilities, it can
never be regarded as infinite, but only indefinite.[58]
Fundamentally,
the consideration of‘worlds’ in the sense understood by Bernoulli,
incomparably larger or smaller in relation to one another, is not very
different from what Leibnitz resorted to when he envisaged ‘the firmament with
respect to the earth, and the earth with respect to a grain of sand,’ and the
latter with respect to ‘a particle of magnetic material passing through a
lens.’ Only here Leibnitz does not claim to speak of gradus infinitatis
[grade of infinity] in the strict sense; on the contrary, he even means to
show that ‘one need not take the infinite rigorously,’ and he is content to
envisage ‘incomparables’, to
which no logical objection can be raised. The shortcoming of his comparison is
of quite another order, and as we have already said, lies in the fact that it
is only capable of giving an inexact, or even completely false, idea of the infinitesimal
quantities as they figure in the calculus. In what follows we shall have
occasion to substitute for this consideration that of the true multiple degrees
of indefinitude, taken in increasing as well as decreasing order; for the
moment, therefore, we shall not dwell further on it.
In
short, the difference between Bernoulli and Leibnitz is that for the first,
even though he presents them only as a probable conjecture, it is truly a
question of ‘degrees of infinity’, while the second, doubting their probability
and even their possibility, limits himself to replacing them with what could be
called ‘degrees of incomparability’. Aside from this difference, which is
moreover assuredly extremely important, they share in common the notion of a
series of worlds that are similar, but on different scales. This notion is not
without a certain incidental connection with discoveries made in the same
period with the microscope, and with certain views that arose as a
consequence—although later observations were in no way to justify them—such as
the theory of the ‘encasement of embryos’; now it is not true of embryos that
every part of the living being must be actually and physically ‘preformed’, and
the organization of a cell bears no resemblance to that of the entire body of
which it is an element. There seems to be no doubt that this was in fact the
origin of Bernoulli’s notion, at any rate; indeed, among other things highly
significant in this regard, he says that the particles of a body coexist in the
whole ‘in the same way that, in accordance with Harvey and others, though not
with Leeuwenhoeck, there exist within an animal innumerable ovules, within each
ovule one or several animalcules, within each animalcule again innumerable
ovules, and so on to infinity.’[59] As
for Leibnitz, his was likely a completely different point of departure; thus,
the idea that all the stars that we can see can only be components of the body
of an incomparably greater being, recalls the Kabbalistic conception of the
‘Great Man’, but singularly
materialized and
‘spatialized’ through a sort of ignorance of the true analogical value of
traditional symbolism; likewise, the idea of the ‘animal’, that is, the living
being, subsisting corporeally after death, but ‘in miniature’, is obviously
inspired by the traditional Judaic concept of the luz or ‘kernel of
immortality’,[60]
which Leibnitz equally distorted by connecting it with the notion of worlds
incomparably smaller than our own, saying, ‘nothing prevents animals from being
transferred to such worlds after death; indeed, I think that death is no more
than a contraction of the animal, just as generation is no more than an
evolution,’[61]
this last word being taken here simply in its etymological sense of
‘development’. All this is fundamentally only an example of the dangers that
exist when one wishes to make traditional notions agree with the views of
profane science, which can only be done to the detriment of the former; these
notions are most clearly independent of the theories brought about by microscopic
observations, and in comparing and muddling them, Leibnitz was already acting
as would the occultists later on, for they particularly delighted in these
sorts of unjustified comparisons. Moreover, the superposition of ‘incomparables’ of
different orders seemed to him in conformity with his notion of the ‘best of
worlds’, furnishing a means of investing it with ‘as much being or reality as
possible’, to quote from his definition; and as we have already pointed out
elsewhere,[62]
this idea of the ‘best of worlds’ is also derived from yet another ill-applied
traditional doctrine, this one borrowed from the symbolic geometry of the
Pythagoreans. According to this geometry, of all lines of equal length, the
circumference of a circle is that which encloses the maximum surface area, and
of all bodies of equal surface area, the sphere is likewise that which contains
the maximum volume, and this is one of the reasons why these figures were
regarded as the most perfect. But if in this respect there is a maximum, there
is nonetheless no minimum, that
is, there exist no figures
enclosing a surface area or a volume less than all others, and this is why
Leibnitz was led to think that, although there is a ‘best of worlds’, there is
no ‘worst of worlds’, that is, a world containing less being than any other
possible world. Moreover, we know that this notion of the ‘best of worlds’,
like that of ‘incomparables’, is linked to the well-known
comparisons involving the ‘garden full of plants’ and the ‘pond filled with
fish’, where ‘each twig of the plant, each member of the animal, each drop of
its humors, is again such a garden or such a pond’;[63]
and this naturally brings us to another, related question, that of the
‘infinite division of matter’.
‘infinite
division’
OR INDEFINITE
DIVISIBILITY
For Leibnitz, not only is matter
divisible, but all its parts are ‘actually sub-divided without end,... each
part into parts, each having some movement of its own’;[64]
and he emphasizes this point above all in order to offer theoretical support to
the concept we last explained: ‘It follows from the actual division that in
every part of matter, however small it might be, there is as it were a world
consisting of innumerable creatures.’[65]
Bernoulli likewise supposes this actual division of matter in partes numéro infinitas
[into infinitely many parts], but he draws from it conclusions Leibnitz did not
accept: ‘If a finite body,’ he says, ‘has parts infinite in number, I have
always believed, and still do, that the smallest of these parts must have an
unassignable, or infinitely small, ratio to the whole’;[66]
to which Leibnitz responds: ‘Even if one agrees that there is no portion
of matter that is not actually divided, one does not, however, arrive at
indivisible elements, or at parts smaller than all others, or infinitely small,
but only at ever smaller parts, which, however, are ordinary quantities, just
as in augmentation one arrives at ever greater quantities.’[67]
Thus it is the existence of minimae por- tiones [smallest parts], or
of‘final elements’, that Leibnitz contests;
for Bernoulli, on the
contrary, it seems clear that actual division implies the simultaneous
existence of all the elements in question, just as, if an ‘infinite’ sequence
be given, all of its constituent terms must be given simultaneously, which
implies the existence of a terminus infinitesimus [infinitesimal
limit]. But for Leibnitz the existence of this limit is no less contradictory
than that of an ‘infinite number’, and the notion of a smallest of numbers, or
a fractio omnium infima [a part smaller than all others], no less absurd
than that of a greatest of numbers. What he considers to be the ‘infinity’ of a
sequence is characterized by the impossibility of arriving at a final term, and
matter would likewise not be ‘infinitely’ divided if this division could ever
be completed and end at ‘final elements’; and it is not only that we could not
in fact ever arrive at these final elements, as Bernoulli concedes, but that
they should not exist in nature at all. There are no indivisible corporeal
elements, or ‘atoms’ in the proper sense of the word, any more than there are
indivisible fractions that cannot yield ever smaller fractions in the numerical
order, or, in the geometric order, linear elements that cannot be divided into
ever smaller elements.
In
all of this Leibnitz basically takes the word ‘infinite’ in exactly the same
sense as he does when speaking of an ‘infinite multitude’; for him, to say of
any sequence, including that of the whole numbers, that it is infinite is not
to say that it must come to a terminus infinitesimus or an ‘infinite
number’, but on the contrary that it must have no final term, since its terms
are plus quam numéro desig- naripossint
[more than can be numbered], that is, they constitute a multitude that
surpasses all number. Similarly, if one can say that matter is infinitely
divided, this is because any one of its portions, however small, always
encloses such a multitude; in other words, matter does not have partes
minimae [smallest parts] or simple elements, it is essentially a
composite: ‘It is true that simple substances, that is, those that do not exist
by aggregation, really are indivisible, but they are immaterial, and are only
principles of action.’5 It is in the sense of an innumerable
multitude—which, moreover, is the sense Leibnitz most commonly employs—that the
idea of the
so-calledinfinite can be
applied to matter, to geometric extension, and in general to the continuous,
taken in relation to its composition; besides, this sense is not exclusive to
the infinitum continuum [continuous infinite] but extends to the infinitum
discretum [discrete infinite] as well, as we have seen both in the
example of the multitude of all the numbers and in that of the ‘infinite
sequence’. This is why Leibnitz was able to say that a magnitude is infinite
insofar as it is ‘inexhaustible’, which means that ‘one can always take a
magnitude as small as one likes’, and, ‘it remains true, for example, that 2 is
as much as ¥1 + ¥2 + ¥4 + ¥8 + ¥16 + ¥32 +..., which is an infinite series,
comprised at once of all fractions with a numerator of 1 and denominators in
double geometric progression, although only ordinary numbers are ever used,
that is, one never introduces any infinitely small fraction, or one with an
infinite number for its denominator.’6 Moreover, what was just said
allows us to understand how Leibnitz, while affirming that the infinite, as he
understands it, is not a whole, nevertheless could apply this idea to the
continuous: a continuous set, as any given body, indeed constitutes a whole,
even what we above called a true whole, logically anterior to its parts and
independent of them, but it is obviously always finite as such; it is therefore
not with respect to the whole that Leibnitz is able to call it infinite, but
only with respect to its parts into which it can be divided, and only insofar
as the multitude of these parts effectively surpasses every assignable number.
This is what one might call an analytical conception of the infinite, since in
fact, it is only analytically that the multitude in question is inexhaustible,
as we shall explain later.
If
we now question the worth of the idea of ‘infinite division’, we must recognize
that, as with the ‘infinite multitude’, it contains a certain portion of truth,
though its manner of expression is anything but safe from criticism. First of
all, it goes without saying that, in accordance with all that we have explained
so far, there can be no question of infinite division, but only of indefinite
division; and on the other hand it is necessary to apply this idea not to
matter in general, which would perhaps have no meaning, but only to bodies, or
to corporeal matter if one
insists on speaking of ‘matter’ here, in spite of the extreme obscurity of the
notion, and the many equivocations to which it gives rise.[68]
In fact, it is to extension that divisibility properly pertains, not to
matter, in whatever sense this is understood, and the two could only be
confused were one to adopt the Cartesian concept, according to which the nature
of bodies consists essentially and uniquely in extension, a concept, moreover,
that Leibnitz also did not admit. If, then, all bodies are necessarily divisible,
this is because they possess extension, and not because they are material. Now
let us again recall that extension, being something determined, cannot be
infinite; hence, it obviously cannot imply any possibility more infinite than
itself; but as divisibility is a quality inherent to the nature of extension,
its limitations can only come from this nature itself; as long as there is
extension, it is always divisible, and one can thus consider its divisibility
to be truly indefinite, its indefinitude being conditioned, moreover, by that
of extension. Consequently, extension as such cannot be composed of
indivisible elements, for these elements would have to be extensionless to be
truly indivisible, and a sum of elements with no extension can no more
constitute an extension than a sum of zeros can constitute a number; this is
why, as we have explained elsewhere,[69]
points are not the elements or parts of a line; the true linear elements are
always distances between points, which latter are only their extremities.
Moreover, Leibnitz himself envisaged things thus in this regard, and according
to him, this is precisely what marks the fundamental difference between his
infinitesimal method and Cavalieri’s ‘method of indivisibles’, namely, that he
does not consider a line to be composed of points, or a surface of lines, or a
volume of surfaces: points, lines, and surfaces are here only limits or extremities,
not constituent elements. It is indeed obvious that points, multiplied by any
quantity at all, can never produce length, since, rigorously speaking, they are
null with respect to length; the true elements of a magnitude must always be of
the same nature as the magnitude, although incomparably less: this leaves no
room for
‘indivisibles’, and what
is more, it allows us to observe in the infinitesimal calculus a certain law
of homogeneity, which implies that ordinary quantities and infinitesimal
quantities of various orders, although incomparable among themselves, are
nonetheless magnitudes of the same species.
From
this point of view one can say in addition that the part, whatever it be, must
always preserve a certain ‘homogeneity’ or conformity of nature with the whole,
at least insofar as the whole is considered able to be reconstituted by means
of its parts, by a procedure comparable to that used in the formation of an
arithmetical sum. Moreover, this is not to say that no simple thing exists in
reality, for composites can be formed, starting from their elements, in a way
completely different from this; but then, to speak truly, these elements are no
longer properly ‘parts’, and as Leibnitz recognized, they can in no way be of a
corporeal order. What is indeed certain is that one cannot arrive at simple,
that is, indivisible, elements without departing from the special condition
that is extension; the latter could not be resolved into such elements without
ceasing to be as extension. It immediately follows that there cannot exist
indivisible corporeal elements, as this notion implies a contradiction; for
indeed, such elements would have to be without extension, and then they would
no longer be corporeal, for by very definition the word ‘corporeal’ necessarily
entails extension, although this is not the whole nature of bodies; thus,
despite all the reservations we must make in other regards, Leibnitz is at
least entirely right in his position against atomism.
But
until now we have spoken only of divisibility, that is to say the possibility
of division; must we go further and admit with Leibnitz an ‘actual division’?
This idea is also not exempt from contradiction, for it amounts to supposing
an entirely realized indefinite and on that account is contrary to the very
nature of indefinitude, which, as we have said, is always a possibility in the
process of development, hence essentially implying something unfinished, not
yet completely realized. Moreover, there is in fact no reason to make such a
supposition, for when presented with a continuous set we are given the whole,
not the parts into which it can be divided, and it is only we who conceive that
it is possible for us to divide this whole
into parts capable of
being rendered smaller and smaller so as to become less than any given
magnitude, provided the division be carried far enough; in fact, it is
consequently we who realize the parts, to the extent that we effectuate the
division. Thus, what exempts us from having to suppose an ‘actual division’ is
the distinction we established earlier on the subject of the different ways of
envisaging a whole: a continuous set is not the result of the parts into which
it is divisible but is on the contrary independent of them, and, consequently,
the fact that it is given to us as a whole by no means implies the actual
existence of those parts.
Likewise,
from another point of view and passing on to a consideration of the
discontinuous, we can say that if an indefinite numerical sequence is given,
this in no way implies that all the terms it contains are given distinctly,
which is impossible precisely inasmuch as it is indefinite; in reality, to
give such a sequence is simply to give the law that enables one to calculate
the term occupying a determined position, or, for that matter, any position
whatsoever within the sequence.[70] If
Leibnitz had given this answer to Bernoulli, their discussion on the existence
of the terminus infinitesimus would thereby have been brought to an
immediate close; but he would not have been able to do so without logically
being led to renounce his idea of ‘actual division’, unless he were to deny all
correlation between continuous and discontinuous modes of quantity.
Be
that as it may, as far as the continuous is concerned at any rate, it is
precisely in the ‘indistinction’ of its parts that we can see the
root of the idea of the
infinite such as it was understood by Leibnitz, since, as we said earlier, this
idea always carries with it a certain amount of confusion; but this
‘indistinction’, far from presupposing a realized division, tends on the
contrary to exclude it, even apart from the completely decisive reasons we have
just noted. Therefore, even if Leibnitz’s theory is right insofar as it is
opposed to atomism, it must be corrected elsewhere if it is to correspond to
truth; the ‘infinite division of matter’ must be replaced by the ‘indefinite
divisibility of extension’; here, in its briefest and most precise expression,
is the conclusion to which all the considerations we have just set forth
ultimately lead.
increasing;
Before continuing the examination of
questions properly relating to the continuous, we must return to what was said
above about the non-existence of afractio omnium infima, which will
allow us to see how the correlation or symmetry that exists in certain respects
between indefinitely increasing and indefinitely decreasing quantities can be
represented numerically. We have seen that in the domain of discontinuous
quantity, as long as it is only the sequence of whole numbers that needs to be
considered, these numbers must be regarded as increasing indefinitely starting
from the unit, but that there can obviously be no question of an indefinite
decrease since the unit is essentially indivisible; were the numbers to be
taken in the decreasing direction, one would necessarily find oneself stopped
at the unit itself, so that the representation of the indefinite by whole
numbers is limited to a single direction, that of indefinite increase. On the
other hand, when it is a question of continuous quantity, one can envisage
indefinitely decreasing quantities as well as indefinitely increasing ones; and
the same occurs in discontinuous quantity itself as soon as, in order to
express this possibility, the consideration of fractional numbers is
introduced. Indeed, one can envisage a sequence of fractions continuing to
decrease indefinitely; that is, however small a fraction might be, a smaller
one could always be formed, and this decrease can no more arrive at afractio
minima [smallest fraction] than can the increase of whole numbers at a numerus
maximus [greatest number].
If
we wish to use a numerical representation in order to make evident the
correlation between the indefinitely increasing and the indefinitely
decreasing, it suffices to consider the sequence of whole numbers together with
that of their inverses; a number is said to be the inverse of another when the
product of the two is equal to the unit, and for this reason the inverse of the
number n is represented by the notation Vn. Whereas the sequence
of whole numbers goes on increasing indefinitely starting from the unit, the
sequence of their inverses decreases indefinitely, starting from the same unit,
which is its own inverse, and which is therefore the common point of departure
for the two sequences; to each number in one sequence there thus corresponds a
number in the other, and inversely, such that the two sequences are equally
indefinite, and in exactly the same way, though in contrary directions. The
inverse of a number is obviously as small as the number itself is great, since
their product always remains constant; however great a number n might
be, the number n +1 will be greater still by virtue of the very law of
formation for the indefinite sequence of whole numbers, and similarly, as
small as a number Vn might be, the number V^+i) will be smaller still;
and this clearly proves the impossibility of any ‘smallest of numbers’, which
notion is no less contradictory than is that of a ‘greatest of numbers’, for,
if it is impossible to stop at a determined number in the increasing direction,
it will be no more possible to stop in the decreasing direction. Moreover,
since this correlation which is found in numerical discontinuity occurs first
of all as a consequence of the application of this discontinuity to the continuous,
as we said concerning fractional numbers, the introduction of which it
naturally supposes, it can only express the correlation that exists within the
continuous itself between the indefinitely increasing and the indefinitely
decreasing in its own way, which is necessarily conditioned by the nature of
number. Therefore, whenever continuous quantities are considered capable of
becoming as great or as small as one likes, that is, greater or smaller than
any determined quantity, one can always observe a symmetry and, in a manner of
speaking, a parallelism presented by these two inverse kinds of variability.
This remark will subsequently help us to understand better the possibility of
different orders of infinitesimal quantities.
It would be good to point out that although the symbol Vn evokes
the idea of fractional numbers, and although it is in fact incontestably
derived from them, the inverses of the whole numbers need not be defined here
as such, and this in order to avoid the difficulty presented by the ordinary
notion of fractional numbers from the strictly arithmetical point of view, that
is, the conception of fractions as parts of the unit’. Indeed, it suffices to
consider the two sequences to be constituted by numbers respectively greater
and smaller than the unit, that is, as two orders of magnitude that have their
common limit in the latter, and that at the same time both can be regarded as
issuing from this unit, which is truly the primary source of all numbers; what
is more, if one wished to consider the two indefinite sets as forming a single
sequence, one could say that the unit occupies the exact mid-point within this
sequence, since, as we have seen, there are exactly as many numbers in the one
set as in the other. Moreover, if, to generalize further, instead of
considering only the sequence of whole numbers and their inverses, one wished
to introduce fractional numbers properly speaking, nothing would be changed as
far as the symmetry of increasing and decreasing quantities is concerned: on
one side one would have all the numbers greater than the unit, and on the
other all those smaller than the unit; here, again, for any number a/b
> i, there will be a corresponding number ^/a < i in the other
group, and reciprocally, such that (a/b) (b/a) = i, just as
earlier we had (n) (Vn) = i, and there will thus be exactly the same
number of terms in each of these two indefinite groups separated by the unit;
it must moreover be understood that when we say ‘the same number of terms’, we
simply mean that the two multitudes correspond term by term, and not that they
can themselves on that account be considered ‘numerable’. Any two inverse
numbers multiplied together always produce again the unit from which they
proceeded; one can say further that the unit, occupying the mid-point between
the two groups, and being the only number that can be regarded as belonging to
both at once[71] —
although in reality it
would be more correct to say that it unites rather than separates them—corresponds
to the state of perfect equilibrium, and contains in itself all numbers which
issue from it in pairs of inverse or complementary numbers, each pair by virtue
of this complementarity constituting a relative unity in its indivisible
duality;[72]
but we shall return a little later to this last consideration and to the
consequences it implies.
Instead
of saying that the series of whole numbers is indefinitely increasing and that
of their inverses indefinitely decreasing, one could also say, in the same sense,
that the numbers thus tend on the one hand toward the indefinitely great and on
the other toward the indefinitely small, on condition that we understand by
this the actual limits of the domain in which these numbers are considered, for
a variable quantity can only tend toward a limit. The domain in question is, in
short, that of numerical quantity, taken in every possible extension;[73]
this again amounts to saying that its limits are not determined by such and
such a particular number, however great or small it might be supposed, but by
the very nature of number as such. By the very fact that number, like
everything else of a determined nature, excludes all that it is not, there can
be no question of the infinite; moreover, we have just said that the indefinitely
great must inevitably be conceived of as a limit, although it is in no way a terminus
ultimus [ultimate limit] of the series of numbers, and in this connection
one can point out that the expression ‘tend toward infinity’, frequently
employed by mathematicians in the sense of „ ‘increase indefinitely’, is again
an absurdity, since the infinite obviously implies the absence of any limit,
and that consequently there is nothing toward which it is possible to tend.
What is also rather remarkable is that certain mathematicians, while
recognizing the inaccuracy and improper character of the expression ‘tend
toward
infinity’,
on the other hand feel no scruple at all about taking the expression ‘tend
toward zero’ in the sense of ‘decrease indefinitely’; zero, however, or the
‘null quantity’, is, with respect to decreasing quantities, exactly the same as
the so-called ‘quantitative infinite’ is with respect to increasing quantities;
but we shall have to return to these questions later, particularly when we come
to the subject of zero and its different meanings.
Since the sequence of numbers in its entirety is not
‘terminated’ by a given number, it follows that there is no number however
great that could be identified with the indefinitely great in the sense just
understood; and, naturally, the same is true for the indefinitely small. One
can only regard a number as practically indefinite, if one may so express it,
when it can no longer be expressed by language or represented by writing, which
in fact inevitably occurs the moment one considers numbers that go on
increasing or decreasing; here we have a simple matter of‘perspective’, if one
wishes, but all in all even this is in keeping with the character of the
indefinite, insofar as the latter is ultimately nothing other than that of
which the limits can be, not done away with, since this would be contrary to
the very nature of things, but simply pushed back to the point of being
entirely lost from view. In this connection some rather curious questions
should be considered; thus, one could ask why the Chinese language
symbolically represents the indefinite by the number ten thousand; the
expression ‘the ten thousand beings’, for example, means all beings, which
really make up an indefinite or ‘innumerable’ multitude. What is quite
remarkable is that it is precisely the same in Greek, where a single word
likewise serves to express both ideas at once, with a simple difference in
accentuation, obviously only a quite secondary detail, and doubtless only due
to the need to distinguish the two meanings in usage: pvpioi, ‘ten thousand’;
pvpioi, ‘an indefinitude’. The true reason for this is the following: the
number ten thousand is the fourth power of ten; now, according to the
formulation of the Tao Te Ching, ‘one produced two, two produced three,
three produced all numbers,’ which implies that four, produced immediately
after three, is in a way equivalent to the whole set of numbers, and this
because, when one has the quaternary by adding the first four numbers, one also
has the denary,
which
represents a complete numerical cycle: i + 2+ 3 + 4 = 10, which, as we have
already said on other occasions, is the numerical formula of the Pythagorean Tetraktys.
One can further add that this representation of numerical indefinitude has its
correspondence in the spatial order: it is common knowledge that raising a
number from one degree to the next highest power represents in this order the
addition of a dimension; now, our space having only three dimensions, its
limits are transcended when one goes beyond the third power, which, in other
words, amounts to saying that elevation to the fourth power marks the very
term of its indefinitude, since, as soon as it is effected, one has thereby
departed from space and passed on to another order of possibilities.
10
THE CONTINUOUS
The idea of the infinite as
Leibnitz most often understood it, which, let us never forget, was merely that
of a multitude surpassing all number, sometimes appears under the aspect of a
‘discontinuous infinite’, as in the case of so-called infinite numerical
sequences; but its most usual aspect, and also its most important one as far as
the significance of the infinitesimal calculus is concerned, is that of the
‘continuous infinite’. In this regard it is useful to recall that when
Leibnitz, beginning the research that at least according to what he himself
said, would lead to the discovery of his method, was working with sequences of
numbers, he at first considered only differences that are ‘finite’ in the
ordinary sense of the word; infinitesimal differences appeared to him only when
there was a question of applying numerical discontinuity to the spatial
continuum. The introduction of differentials was therefore justified by the
observation of a certain analogy between the respective kinds of variability
within these two modes of quantity; but their infinitesimal character arose
from the continuity of the magnitudes to which they had to be applied, and
thus, for Leibnitz, a consideration of the ‘infinitely small’ is closely
linked to that of the ‘composition of the continuous’.
Taken ‘rigorously’,
‘infinitely small’, would be partes minimae of the continuous, as
Bernoulli thought; but clearly the continuous, insofar as it exists as such, is
always divisible, and consequently it could not have partes minimae.
‘Indivisibles’ cannot even be said to be parts of that with respect to which
they are indivisible, and ‘minimum’ can be understood here only as a limit or
extremity, not as
an element: ‘Not only is a line less than any surface,’
Leibnitz says, ‘it is not even part of a surface, but merely a minimum or an
extremity’;[74]
and from his point of view this assimilation between extremum and minimum
can be justified by the ‘law of continuity’, in that according to him it
permits ‘passage to the limit’, as we shall see later. As we have said already,
the same holds for a point with respect to a line, as well as for a surface
with respect to a volume; on the other hand, the infinitesimal elements must be
parts of the continuous, without which they could not even be quantities; and
they can be so only on condition of not truly being ‘infinitely small’, for
then they would be nothing other than partes minimae [smallest parts] or
‘final elements’, of which the very existence implies a contradiction in
regard to the continuous. Thus the composition of the continuous prevents
infinitely small quantities from being anything more than simple fictions; but
from another point of view, it is nevertheless precisely the existence of this
continuity that makes them ‘well-founded fictions’, at least in Leibnitz’s
eyes: if‘within the realm of geometry they may be treated as if they were
perfectly real,’ this is because extension, which is the object of geometry, is
continuous; and, if it is the same with nature, this is because bodies are
likewise continuous, and also because there is also continuity in all phenomena
such as movement, of which these bodies are the seat, and which are the objects
of mechanics and physics. Moreover, if bodies are continuous, this is because
they are extended and participate in the nature of extension; and similarly,
the continuity of movement, as well as of the various phenomena more or less
directly connected to it, derives essentially from its spatial character. Thus
the continuity of extension is ultimately the true foundation of all other
continuity that is observed in corporeal nature; and this, moreover, is why in
introducing an essential distinction that Leibnitz did not make in this regard,
we specified that in reality one must attribute
the property of
‘indefinite divisibility’ not to ‘matter’ as such, but rather to extension.
Here
we need not examine the question of other possible forms of continuity,
independent of its spatial form; indeed, one must always return to the latter
when considering magnitudes, and its consideration thus suffices for all that
pertains to infinitesimal quantities. We should, however, include together with
it the continuity of time, for contrary to the strange opinion of Descartes on
the subject, time really is continuous in and of itself, and not merely with
respect to its spatial representation in the movement used to measure it.2
In this regard, one could say that movement is as it were doubly continuous,
for it is so in virtue both of its spatial and of its temporal condition; and
this sort of combination of space and time, from which movement results, would
not be possible were the one discontinuous and the other continuous. This
consideration also allows the introduction of continuity into various
categories of natural phenomena that pertain more directly to time than to
space, although occurring in both, as, for example, with any processes of organic
development. As for the composition of the temporal continuum, moreover, one
could repeat everything said concerning the composition of the spatial
continuum, and in virtue of this sort of symmetry which, as we have seen,
exists in certain respects between space and time, one will arrive at strictly
analogous conclusions; instants conceived of as indivisible are no more parts
of duration than are points of extension, as Leibnitz likewise recognized, and
here again we have a thesis with which the Scholastics were quite familiar; in
short, it is a general characteristic of all continuity that its nature
precludes the existence of‘final elements’.
All
that we have said up to this point sufficiently shows in what sense one may
understand that from Leibnitz’s point of view, the continuous necessarily
embraces the infinite; but we cannot, of course, suppose that there is any
question of an ‘actual infinity’, as if all possible parts are effectively
given whenever a whole is given; nor is there any question of a true infinity,
which any determination whatsoever would exclude, and which consequently cannot
be
implied by the
consideration of any particular thing. Here, however, as in every case in which
the idea of an alleged infinite presents itself, different from the true
metaphysical Infinite, but in itself representing something other than a pure
and simple absurdity, all contradiction disappears, and with it all logical
difficulty, if one replaces the so-called infinite with the indefinite, and if
one simply says that all continuity, when taken with respect to its elements,
embraces a certain indefinitude. It is also for lack of having made this
fundamental distinction between the Infinite and the indefinite that some
people have mistakenly believed it impossible to escape the contradiction of a
determined infinite except by rejecting the continuous altogether and replacing
it with the discontinuous; thus Renouvier, who rightly denied the mathematical
infinite, but to whom the idea of the metaphysical Infinite was nevertheless
completely foreign, believed that the logic of his ‘finitism’ obliged him to
go so far as to accept atomism, thus falling prey to a concept no less
contradictory than the one he wished to avoid, as we saw earlier.
continuity’
Whenever there exists a
continuum, we can say with Leibnitz that there is something of the continuous
in its nature, or, if one prefer, that there must be a certain ‘law of
continuity’ applying to all that presents the characteristics of the
continuous; this is obvious enough, but it by no means follows that such a law
must then be applicable to absolutely everything, as he claims, for, if the
continuous exists, so does the discontinuous, even in the domain of quantity;[75]
number, indeed, is essentially discontinuous, and it is this very discontinuous
quantity, and not continuous quantity, that is really the first and fundamental
mode of quantity, what one might properly call pure quantity, as we have said
elsewhere.2 Moreover, nothing allows us to suppose a priori
that, outside of pure quantity, a continuity of some kind exists everywhere,
and, to tell the truth, it would be quite astonishing if, among all possible
things, number alone had the property of being essentially discontinuous; but
our
intention is not to
determine the bounds within which a ‘law of continuity’ truly is applicable, or
what restrictions should be brought to bear on all that goes beyond the domain
of quantity understood in its most general sense. We shall limit ourselves to
giving one very simple example of discontinuity, taken from the realm of
natural phenomena: if it takes a certain amount of force to break a rope, and
one applies to the rope a slightly lesser force, what will result is not a
partial rupture, that is, the rupture of some part of the strands making up the
rope, but merely tension, which is something completely different; if one
augments the force in a continuous way, the tension will also increase
continuously, but there will come a moment when the rupture will occur, and
then, suddenly and as it were instantaneously, there will be an effect of quite
another nature than the preceding, which manifestly implies a discontinuity;
and thus it is not true to say, in completely general terms and without any
sort of restriction, that natura non facit saltus [nature does not make
leaps].
However
that may be, it is at any rate sufficient that geometric magnitudes should be
continuous, as indeed they are, in order that one always be able to take from
them elements as small as one likes, hence elements that are capable of
becoming smaller than any assignable magnitude; and as Leibnitz said, ‘a
rigorous demonstration of the infinitesimal calculus no doubt consists in
this,’ which applies precisely to these geometric magnitudes. The ‘law of continuity’
can thus serve as the. fundamentum in re of these fictions that are the
infinitesimal quantities, and, moreover, as well as the other fictions of
imaginary roots (since Leibnitz linked the two in this respect), but for all
that without it being necessary to see in it ‘the touchstone of all truth’, as
he would perhaps have wished. Furthermore, even if one does admit a ‘law of
continuity’, though of course still maintaining certain restrictions as to its
range, and even if one recognizes that this law can serve to justify the
foundation of the infinitesimal calculus, modo sano sensu intelligantur, it
by no means follows that one must conceive of it exactly as Leibnitz did, or
that one must accept all the consequences he attempted to draw from it; it is
this conception and these consequences that we must now examine a little more
closely.
In
its most general form, this law finally amounts to the following, which
Leibnitz stated on many occasions in different terms, but always with
fundamentally the same meaning: whenever there is a certain order to principles
understood here in the relative sense of whatever is taken as starting-point,
there must always be a corresponding order to the consequences drawn from
them. As we have already pointed out, this is then only a particular case of
the ‘law of justice’, or of order, which postulates ‘universal
intelligibility’. For Leibnitz it is therefore fundamentally a consequence or
application of the ‘principle of sufficient reason’, if not this principle
itself insofar as it applies more particularly to combinations and variations
of quantity. As he says, ‘continuity is an ideal thing [which is moreover far
from as clear a statement as one might desire], but the real is nevertheless
governed by the ideal or abstract... because all is governed by reason.’[76]
There is assuredly a certain order in things, which is not in question, but
this order can be conceived of quite differently from the manner of Leibnitz,
whose ideas in this regard were always influenced more or less directly by his
so-called ‘principle of the best’, which loses all meaning as soon as one has
understood the metaphysical identity of the possible with the real;[77]
what is more, although he was a declared adversary of narrow Cartesian rationalism,
when it comes to his conception of ‘universal intelligibility’, one could
reproach him for having too readily confused ‘intelligible’ with ‘rational’;
but we shall not dwell further on these considerations of a general order, for
they would lead us far afield from our subject. In this connection we will only
add that one might well be astonished that, after having affirmed that
‘mathematical analysis need not depend on metaphysical controversies’—which is
quite contestable, moreover, since it amounts to making of mathematics a
science entirely ignorant of its own principles, in accordance with the purely
profane point of view; besides, incomprehension alone can give birth to
controversies in the metaphysical domain—after such an assertion Leibnitz
himself, in support of his ‘law of causality’, to which he links this
mathematical analysis, finally comes to
invoke an argument no
longer metaphysical indeed, but definitely theological, which could in turn
lead to many other controversies. ‘It is because all is governed by reason,’ he
says, ‘and because otherwise there would be neither science nor rules, which
would not conform to the nature of the sovereign principle,’[78] to
which one could respond that in reality reason is only a purely human faculty,
of an individual order, and that, even without having to go back to the
‘sovereign principle’, intelligence understood in its universal sense, that is,
as the pure and transcendent intellect, is something completely different from
reason, and cannot be likened to it in any way, such that if it is true that
nothing is ‘irrational’, there are nevertheless many things that are
‘supra-rational’, but which on that account are no less ‘intelligible’.
Let
us now move on to a more precise statement of the ‘law of continuity’, a
statement that relates more directly to the principles of the infinitesimal
calculus than the preceding: ‘If with respect to its data one case approaches
another in a continuous fashion and finally disappears into it, it necessarily
follows that the results of the cases equally approach in a continuous fashion
their sought-out solutions, and that they must finally terminate in one another
reciprocally.’[79]
There are two things here, which it is important to distinguish: first, if the difference
between the two cases diminishes to the point of becoming less than any
assignable magnitude in datis [in the given], the same must hold in
quaesitis [in what is sought];
this, in short, is only an
application of the more general statement, and this part of the law raises no
objections as soon as it is admitted that continuous variations exist and that
the infinitesimal calculus is properly linked precisely to the domain in which
such variations are effected, namely the geometric domain, but must it be
further admitted that casus in casum tandem evanescat [one case finally
disappears into the other], and that consequently eventus casuum tandem in
se invicem desinant [the outcomes of the cases finally end in each other]?
In other words, will the difference between the two cases ever become
rigorously null, in consequence of their continuous and indefinite decrease, or
again, if one prefer, will their decrease, though indefinite, ever come to an
end? This is fundamentally the question of knowing whether, within a
continuous variation, the limit can be reached, and on this point we will first
of all make this remark: as the indefinite always includes in a certain sense
something of the inexhaustible, insofar as it is implied by the continuous, and
as Leibnitz moreover did not suppose that the division of the continuous could
ever arrive at a final term, nor even that this term could really exist, is it
completely logical and coherent on his part to maintain at the same time that a
continuous variation, which is effected per infinitos gradus intermedios
[by infinite intermediary steps],[80]
could reach its limit? This is certainly not to say that such a limit can in no
way be reached, which would reduce the infinitesimal calculus to no more than a
simple method of approximation; but if it is effectively reached, this must not
be within the continuous variation itself, nor as a final term in the
indefinite sequence of gradus mutationis [degrees of change].
Nevertheless, it is by this ‘law of continuity’ that Leibnitz claims to justify
the ‘passage to the limit’, which is not the least of the difficulties to which
his method gives rise from the logical point of view, and it is precisely here
that his conclusions become completely unacceptable; but to make this aspect of
the question entirely understandable, we must begin by clarifying the
mathematical notion of the limit itself.
12
OF THE LIMIT
The notion of the limit is
one of the most important we have to examine here, for the value of the
infinitesimal method, at least insofar as its rigor is concerned, depends
entirely upon it; one could even go so far as to say that, ultimately, ‘the
entire infinitesimal algorithm rests solely on the notion of the limit, for it
is precisely this rigorous notion that serves to define and justify all the
symbols and formulas of the infinitesimal calculus.’[81]
Indeed, the object of this calculus ‘amounts to calculating the limits of
ratios and the limits of sums, that is, to finding the fixed values toward
which the ratios or sums of variable quantities converge, inasmuch as these
quantities decrease indefinitely according to a given law.’[82]
To be even more precise, let us say that of the two branches into which the
infinitesimal calculus may be divided, the differential calculus consists in
calculating the limits of ratios, of which the two terms decrease indefinitely,
at the same time following a certain law in such a way that the ratio itself
always maintains a finite and determined value; and the integral calculus consists
in calculating the limits of sums of elements, of which the multitude increases
indefinitely as the value of each element decreases indefinitely, for both of
these conditions must be united in order for the sum itself always to remain a
finite and determined quantity. This being granted, one can say in a general
way that the limit of a variable quantity is another quantity considered to be
fixed, which the variable quantity is supposed to approach through the values
it successively takes on
in the course of its
variation, until it differs from the fixed quantity by as little as one likes,
or in other words, until the difference between the two quantities becomes less
than any assignable quantity. The point which we must emphasize most
particularly, for reasons that will be better understood in what follows, is
that the limit is essentially conceived as a fixed and determined quantity;
even though it will not be given by the conditions of the problem, one should
always begin by supposing it to have a determined value, and continue to regard
it as fixed until the end of the calculation.
But the conception of the limit in and of itself is one
thing, and the logical justification of the passage to the limit’ quite
another; Leibnitz believed that
what in general justifies
this ‘passage to the limit’ is that the same relations that exist among several
variable magnitudes also subsist among their fixed limits when their
variations are continuous, for then they will indeed reach their respective
limits; this is another way of putting the principle of continuity.[83]
But
the entire question is precisely that of knowing whether a variable quantity,
which approaches its fixed limit indefinitely and which, consequently, can
differ from it by as little as one likes, according to the very definition of a
limit, can effectively reach this limit precisely as a consequence of this
variability, that is, whether a limit can be conceived as the final term in a
continuous variation. We shall see that in reality this solution is
unacceptable; but putting aside the question, to return to it later, we will
only say for now that the true notion of continuity does not allow
infinitesimal quantities to be considered as if they could ever equal zero, for
they would then cease to be quantities; now, Leibnitz himself held that they
must always preserve the character of true quantities, even when they are
considered to be ‘vanishing’. An infinitesimal difference can therefore never
be strictly null; consequently, a variable, insofar as it is regarded as such,
will always really differ from its limit, and could not reach this limit
without thereby losing its variable character.
On this point, aside from one slight reservation, we can thus
entirely accept the considerations a previously cited mathematician sets forth
in these terms:
What characterizes a limit as we have
defined it is that the variable can approach it as much as one might wish,
while nonetheless never being able to strictly reach it; for in order that the
variable in fact reach it, a certain infinity would have to be realized, which
is necessarily ruled out.... And one must also keep to the idea of an
indefinite, that is to say an even greater, approximation.[84]
Instead
of speaking of ‘the realization of a certain infinity’, which has no meaning
for us, we will simply say that a certain indefinitude would have to be
exhausted precisely insofar as it is inexhaustible, but that at the same time
the possibilities of development contained within this very indefinitude allow
the attainment of as close an approximation as might be desired, ut error
fiat minor dato [that the error may become smaller than any given
error], according to an expression of Leibnitz, for whom ‘the method is
certain’ as soon as this result is attained.
The distinctive feature of the limit,
and that which prevents the variable from ever exactly reaching it, is that its
definition is different from that of the variable; and the variable, for its
part, while approaching the limit more and more closely, never reaches it,
because it must never cease to satisfy its original definition, which, as we
have said, is different. The necessary distinction between the two definitions
of the limit and the variable is met with everywhere.... This fact, that the
two definitions, although logically distinct, are nevertheless such that the
objects they define can come closer and closer to one another,[85]
explains what
might at first seem strange, that is,
the impossibility of ever making coincide two quantities over which one has the
authority to diminish the difference until it becomes so small as to pass
beyond expressibility.[86]
There is hardly any need
to say that in virtue of the modern tendency to reduce everything exclusively
to the quantitative, some people have not failed to find fault with this
conception of the limit for introducing a qualitative difference into the
science of quantity itself; but if it must be discarded for this reason, it
would likewise be necessary to ban from geometry entirely—among other
things—the consideration of similarity, which is also purely qualitative, since
it concerns only the form of figures, abstracting them from their magnitudes,
and hence from tjieir properly quantitative element, as we have already
explained elsewhere. In this connection, it would also be good to note that one
of the chief uses of the differential calculus is to determine the directions
of the tangents at each point on a curve, the totality of which defines the
very form of the curve, and that in the spatial order direction and form are
precisely elements of an essentially qualitative character.[87]
What is more, it is no solution to claim to purely and simply do away with the
‘passage to the limit’ on the pretext that the mathematician can dispense with
actually passing to it without in any way hindering him from pushing his
calculation to its end; this may be true, but what matters is this: under these
conditions, up to what point would one have the right to consider this calculus
to rest on rigorous reasoning, and even if ‘the method is thus certain’, will
it not be so only as a simple method of approximation? One could object that
the conception we just explained also makes the ‘passage to the limit’
impossible, since the character of this limit is precisely such as to prevent
its ever being reached; but this is true only in a certain sense, and only insofar
as one considers variable quantities as such, for we did not say that the limit
could in no way be reached, but—and it is essential that this be made
clear—that it could not be reached within the
variation, and as a term
of the latter. The only true impossibility is the notion of a ‘passage to the
limit’ constituting the result of a continuous variation; we must therefore
replace it with another notion, and this we shall do more explicitly in what
follows.
PASSAGE TO THE LIMIT
We can now return to our examination of the
‘law of continuity’, or, to be more exact, to the aspect of the law that we
had to momentarily lay aside, and which is precisely that aspect by which
Leibnitz believed ‘passage to the limit’ could be justified. For him what
follows/from it is
that with continuous
quantities, the extreme exclusive case may be treated as inclusive, and that
such a case, although totally different in nature, is thus as if contained in a
latent state in the general law of the other cases.[88]
Although
Leibnitz himself does not appear to have suspected it, it is precisely here
that the principal logical error in his conception of continuity lies, which
one may quite easily recognize in the consequences he draws from it and in the
ways in which he applies it. Here are a few examples:
In accordance with my law of
continuity, one is allowed to consider rest to be an infinitely small motion, that
is, to be equivalent to a species of its contradictory, and coincidence to be
an infinitely small distance, equality the last of inequalities, etc.[89]
[Or again]: In accordance with this law of continuity, which excludes all
sudden changes, the case of rest can be regarded as a
special
case of motion, namely as a vanishing or minimum motion, and the case of
equality as a case of vanishing inequality. It follows that the laws of motion
must be established in such a way that there be no need for special rules for
bodies in equilibrium and at rest, but that the latter should themselves arise
from the rules concerning bodies in disequilibrium and in motion; or, if one
does wish to set forth particular rules for rest and equilibrium, one must take
care that they not be such as to disagree with the hypothesis that holds rest
to be an incipient motion or equality the final inequality.[90]
Let us add one more
quotation on the subject, in which we find a new example, of a somewhat
different kind from the preceding, but no less contestable from the logical
point of view:
Although
it is not rigorously true that rest is a species of motion, or that equality is
a species of inequality, just as it is not true that the circle is a species of
regular polygon, one can nevertheless say that rest, equality, and the circle
are the terminations of motion, inequality, and the regular polygon, which, by
continual change, arrive at the former by vanishing. And although these terminations
are exclusive, that is, not rigorously included within the varieties they
limit, they nevertheless have the same properties as they would if they were so
included, in accordance with the language of infinites or infinitesimals, which
takes the circle, for example, as a regular polygon with an infinite number of
sides. Otherwise the law of continuity would be violated, that is to say that
because one passes from polygons to the circle by a continual change, without
any break, there must likewise be no break in the passage from the attributes
of polygons to those of the circle.[91]
It is worth pointing out that, as is indicated in the
beginning of the last passage cited above, Leibnitz considers these assertions
to be
of
the same kind as those that are merely toleranter verae, which, he says
elsewhere,
above all serve the art of
invention, although, in my opinion, they contain something of the fictional and
imaginary which however can easily be rectified by reducing them to ordinary
expressions, in order that they not produce error.[92]
But
are they not precisely that already, and in reality do they not rather contain
contradictions pure and simple? No doubt Leibnitz recognized that the extreme
case, or ultimus casus, is exclusivus, which obviously implies
that it falls outside of the series of cases that are naturally included in the
general law; but then with what right can it be included in this law in spite
of it, and be treated ut inclusivum [as inclusive], that is, as if it
were only one particular case Contained within the series? It is true that the
circle is the limit of a regüla^ polygon with an indefinitely
increasing number of sides, but its definition is essentially other than that
of polygons; and in such an example one can see quite clearly that there exists
a qualitative difference between the limit itself and that of which it is the
limit, as we have said before. Rest is in no way a particular case of motion,
nor equality a particular case of inequality, nor coincidence a particular
case of distance, nor parallelism a particular case of convergence; besides,
Leibnitz does not suppose that they are so in a rigorous sense, but he
nonetheless maintains that they can in some way be regarded as such, with the
result that ‘the genus terminates in the opposed quasi-species,’ and that
something can be ‘equivalent to a species of its contradictory.’[93]
Moreover, let us note in passing that Leibnitz’s notion of‘virtuality’ seems to
be linked to this same order of ideas, as he gives it the special sense of
potentiality viewed as incipient actuality,[94]
which again is no less contradictory than the other examples just cited.
Whatever
the point of view from which things are envisaged, it is not in the least clear
that a certain species could be a ‘borderline case’ of the opposite species or
genus, for it is not in this way that opposed things limit each other
reciprocally, but definitely to the contrary in that they exclude one another,
and it is impossible for one contradictory to be reduced to another; for
example, can inequality have any significance apart from the degree to which
it is opposed to equality and is its negation? We certainly cannot say that
assertions such as these are even toleranter verae, for even if one does
not accept the existence of absolutely separate genuses, it is nonetheless true
that any genus, defined as such, can never become an integral part of another
equally defined genus when the definition of this latter does not include its
own, even if it does not exclude it formally as in the case of contradictories;
and if a connection can be established between different genuses, this is not
in virtue of that in which they effectively differ, but only in virtue of a
higher genus, which includes both. Such a conception of continuity, which ends
up abolishing not only all separation, but even all effective distinction, in
allowing direct passage from one genus to another without reducing the two to a
higher or more general genus, is in fact the very negation of every true
logical principle; and from this to the Hegelian affirmation of the ‘identity
of contradictories’ is then but one step which is all too easy to take.
‘vanishing
quantities’
For Leibnitz, the justification for
‘passage to the limit’ ultimately consists in the fact that the particular case
of the ‘vanishing quantities’, as he says, must in a certain sense be included
within the general rule by virtue of continuity; moreover, these vanishing
quantitiesvannot be regarded as ‘absolute nothings’, or as pure zeros, for by
Telson of the same continuity they maintain among themselves determined
ratios—and generally differ from unity—in the very instant in which they
vanish, which implies that they are still real quantities, although
‘unassignable’ with respect to ordinary quantities.[95]
However, if these vanishing quantities—or the infinitesimal quantities, which
amounts to the same thing—are not ‘absolute nothings’, even when it is a
question of differentials of orders higher than the first, they must still be
considered ‘relative nothings’, which is to say that, while retaining the
character of real quantities, they can and must be negligible with regard to
ordinary quantities, with which they are ‘incomparable’;[96]
but multiplied by ‘infinite’ quantities, or quantities incomparably greater
than ordinary ones, they again produce these ordinary quantities, which could
not be so
if they were absolutely
nothing. In light of the definitions we presented earlier, one can see that
the consideration of the ratios of vanishing but still determined quantities
refers to the differential calculus, while the consideration of the
multiplication of these quantities by ‘infinite’ quantities, yielding ordinary
quantities, refers to the integral calculus. The difficulty in all this is to
admit that quantities that are not absolutely null must nonetheless be treated
in the calculus as if they were, which risks giving the impression that it is
merely a question of simple approximation; again, in this regard Leibnitz
sometimes seems to invoke the ‘law of continuity’, by which the ‘borderline
case’ finds itself included within the general rule, as if this were the only
postulate his method required; this argument is quite unclear, however, and one
should rather return to the notion of‘incomparables’ as he himself often does,
moreover, in order to justify the elimination of infinitesimal quantities from
the results of the calculus.
Indeed,
Leibnitz considers as equal not only those quantities of which the difference
is null, but even those of which the difference is incomparable with respect to
the quantities themselves; this notion of‘incomparables’ is, for him, the
foundation not only for the elimination of infinitesimal quantities, which
thus disappear in the face of ordinary quantities, but also for the distinction
between different orders of infinitesimal or differential quantities, the
quantities of each order being incomparable with respect to those of the preceding,
as those of the first order are with respect to ordinary quantities, but
without ever arriving at ‘absolute nothings’. T call two magnitudes
incomparable,’ says Leibnitz, ‘when one, despite multiplication by any finite
number whatsoever, can nonetheless not exceed the other, in the same way that
Euclid treated it in the fifth definition of his fifth book.’[97]
However, there is nothing there to indicate whether this definition should be
understood of fixed and determined, or of variable, quantities; but one can
admit that in all its generality it must apply without distinction to both
cases; the entire question would then be one of knowing whether two fixed
quantities, however different they might be within the scale of magnitudes,
could ever be regarded as
truly ‘incomparable’, or whether they would only be so relative to the means of
measurement at our disposal. But we shall not dwell further on this point,
since Leibnitz himself declared elsewhere that this is not the case with
differentials,[98]
from which it is necessary to conclude, not only that the comparison of the
grain of sand is in itself manifestly faulty, but also that it fundamentally
does not answer, even in his own thought, to the true notion of‘incomparables’, at
least insofar as this notion must be applied to the infinitesimal quantities.
Some people, however, have believed that the infinitesimal
calculus can be rendered perfectly rigorous only on the condition that the
infinitesimal quantities be regarded as null, and at the same time they have
wrongly thought that one can suppose an error to be null as long as one c^n
also suppose it to be as small as one likes; wrongly, we say, for that would be
the same as to admit that a variable, as such, could reach its limit. Here is
what Carnot has to say on the subject:
There are those who believe they have sufficiently
established the principle of infinitesimal analysis with the following
reasoning: it is obvious, they say, and universally acknowledged, that the
errors to which the procedure of infinitesimal analysis would give rise— if
there were any—could always be supposed as small as one might wish; it is also
obvious that any error one is free to suppose as small as one likes is null,
for since one can suppose it to be as small as one wishes, one can suppose it
to be zero; therefore, the results of the infinitesimal analysis are rigorously
exact. This argument, plausible at first sight, is nevertheless anything but
valid, for it is false to say that because one is free to render an error as
small as one likes one can thus render it absolutely null.... One is faced with
the necessary alternative either of committing an error, however slight one
might suppose it to be, or of falling back on a formula that says nothing, and
such is precisely the crux of the difficulty with the infinitesimal analysis.[99]
It
is certain that any formula in which a ratio appears in the form °/o ‘says
nothing’, and one could even say that it has no meaning in and of itself; it is
only in virtue of a convention—justified, moreover—that one can give any sense
to the expression °/o, regarding it as a symbol of indeterminacy;[100]
but this very indeterminacy then means that the ratio in this form can be
equal to anything, whereas on the contrary it must maintain a determined value
in every particular case; it is the existence of this determined value that
Leibnitz puts forward,[101]
and in itself this argument is completely unassailable.[102]
However, it is quite necessary to recognize that the notion of ‘vanishing
quantities’ has ‘the tremendous drawback of considering quantities in that
state in which they so to speak cease to be quantities’, to use Lagrange’s
expression; but contrary to what Leibnitz thought, there is no need to consider
them precisely in the instant in which they vanish, nor even to suppose that
they really could vanish, for in that case they would indeed cease to be
quantities. Moreover, this essentially supposes that strictly speaking there is
no ‘infinitely small’ quantity, for this ‘infinitely small’ quantity—or at
least what would be called such in Leibnitz’s language—could only be zero, just
as an ‘infinitely great’ quantity, taken in the same sense, could only be an
‘infinite number’; but in reality zero is not a number, and ‘null quantities’
have no more existence than do ‘infinite quantities’. The mathematical zero, in
its rigorous and strict sense, is but a negation, at least as far as its quantitative
aspect is concerned, and one cannot say that the absence of quantity itself
constitutes a quantity; we shall return to this point shortly, in order to
develop more completely the consequences that result from it.
In
sum, the expression ‘vanishing quantities’ has above all the drawback of
producing an equivocation, and of leading to the belief that infinitesimal
quantities can be considered as quantities that are effectively annulled, for
without altering the meaning of these words, it is difficult to understand how,
when it is a question of quantities, ‘to vanish’ could mean anything other than
to be annulled. In reality, these infinitesimal quantities, understood as
indefinitely decreasing quantities, which is their true significance, can never
be called ‘vanishing’ in the proper sense of the word. It would most certainly
have been preferable had the notion never been introduced, as it is
fundamentally bound up with Leibnitz’s conception of continuity, and, as such,
inevitably contains the same element of contradiction inherent in the
illogicality of this latter. Now if an error, despite being able to be rendered
as small as one likes, can never become absolutely null, how can the
infinitesimal cdçulus be truly rigorous, and if the error
is in fact only practically negligible, would it not be necessary to conclude
that the calculus is thus reduced to a simple method of approximation, or at
least, as Carnot says, of ‘compensation’? This is a question that we must
resolve in what follows; but as we have here been brought to speak of zero and
of the so-called ‘null quantity’ it will be worthwhile to deal with this other
subject first, the importance of which, as we shall see, is far from
negligible.
The
indefinite decrease of numbers can no more end in a‘null
number’ than their indefinite increase can in an ‘infinite number’, and for the
same reason, since each of these numbers must be the inverse of the other;
indeed, in accordance with what was said earlier on the subject of inverse
numbers, as each of the two sets—the one increasing, the other decreasing—is
equally distant from the unit, the common point of departure for both, and as
there must further necessarily be as many terms in the one as in the other,
their final terms—namely, the ‘infinite number’ and the ‘null number’— if they
existed, would themselves have to be equally distant from the unit, and thus
the inverses of one another.[103]
Under these conditions, if the sign oo is in reality only a symbol for
indefinitely increasing quantities, then logically the sign o should likewise
be able to be taken as a symbol for indefinitely decreasing quantities, in
order to express in notation the symmetry that, as we have said, exists between
the two; but unfortunately this sign o already has quite another significance,
for it originally served to designate the complete absence of quantity, whereas
the sign oo has no real sense that
would correspond to the
former. Here, as with the vanishing quantities’, we have yet another source of
confusion, and in order to avoid this it would be necessary to create another
symbol, apart from zero, for indefinitely decreasing quantities, since these
quantities are characterized precisely by the fact that they can never be
annulled, despite any variation they might undergo; at any rate, with the
notation currently employed by mathematicians, it seems almost impossible to
prevent confusions from arising.
If
we emphasize the fact that zero, insofar as it represents the complete absence
of quantity, is not a number and cannot be considered as such—even though this
might appear obvious enough to those who have never had occasion to take
cognizance of certain disputes—this is because, as soon as one admits the
existence of a null number’, which would have to be the ‘smallest of numbers’,
one is inevitably led by way of correlation to suppose as its inverse an ‘infinite
number’, in the sense of the ‘greatest of numbers’. If, therefore, one accepts
the postulate that zero is a number, the arguments in favor of an ‘infinite
number’ follow in a perfectly logical manner;[104]
but it is precisely this postulate that we must reject, for if the consequences
deduced from it are contradictory—and we have seen that the existence of an
‘infinite number’ is indeed so—then the postulate in itself must already imply
contradiction. Indeed, the negation of quantity can in no way be assimilated to
a particular quantity; the negation of number or of magnitude can in no sense
and to no degree constitute a species of number or magnitude; to claim the
contrary would be to maintain that a thing could be ‘equivalent to a species of
its contradictory,’ to use Leibnitz’s expression, and would be as much as to
say immediately that the negation of logic is itself logic.
It
is therefore contradictory to speak of zero as a number, or to suppose that a
‘zero in magnitude’ is still a magnitude, from which would inevitably result
the consideration of as many distinct zeros as there are different kinds of
magnitude; in reality, there can only be zero pure and simple, which is none
other than the negation of
quantity, whatever the
mode envisaged.[105]
When such is accepted as the true sense of the arithmetical zero, taken
‘rigorously’, it becomes obvious that this sense has nothing in common with the
notion of indefinitely decreasing quantities, which are always quantities; they
are never an absence of quantity, nor again are they anything that is as it
were intermediate between zero and quantity, which would be yet another
completely unintelligible conception, and which in its own order would recall
that of Leibnitzian ‘virtuality’, which we had occasion to mention earlier.
We
can now return to the other meaning that zero actually has in common notation,
in order to see how the confusions we spoke of were introduced. We said earlier
that in a way a number can be regarded as practically indefinite when it is no
longer possible for us to express or represent it distinctly in any way; such a
number, whatever it might be, can only be symbolized in the increasing order by
the sign oo, insofar as this represents the indefinitely great; it is therefore
not a question of a determined number, but rather of an entire domain, and this
is necessary moreover if it is to be possible to envisage inequalities and
even different orders of magnitude within the indefinite. Mathematical notation
lacks a symbol for the corresponding domain in the decreasing order, what might
be called the domain of the indefinitely small; but since a number belonging to
this domain is, in fact, negligible in calculations, it is in practice
habitually considered to be null, even though this is only a simple
approximation resulting from the inevitable imperfection of ’ our means of
expression and measurement, and it is doubtless for this reason that it came to
be represented by the same symbol o that also represents the rigorous absence
of quantity. It is only in this
sense that the sign o
becomes in a way symmetrical to the sign oo and that the two can be placed
respectively at the two extremities of the sequence of numbers as we envisaged
it earlier, with the whole numbers and their inverses extending indefinitely in
the two opposite directions of increase and decrease. This sequence then
presents itself in the following form: o ... ¥4, ¥3, ¥2,1, 2, 3, 4 ... 00; but
we must take care to recall that o and 00 represent not two determined numbers
terminating the series in either direction, but two indefinite domains, in
which on the contrary there can be no final terms, precisely by reason of their
indefinitude; moreover, it is obvious that here zero can be neither a ‘null
number’, which would be a final term in the decreasing direction, nor again a
negation or absence of quantity, which would have no place in this sequence of
numerical quantities.
As
we explained previously, any two numbers in the sequence that are equidistant
from the central unit are inverses or comple- mentaries of one another, thus
producing the unit when multiplied together: (¥n)(n) = i, such that for the two
extremities of the sequence, one would be led to write (o)(oo) =1 as well; but,
since the signs o and 00, the two factors of this product, do not represent
determined numbers, it follows that the expression (o)(oo) itself constitutes a
symbol of indeterminacy, or what one would call an ‘indeterminate form’, and
one must therefore write (o)(oo) = n, where n could be any number;[106] it
is no less true that in any case one will thus be brought to ordinary finitude,
the two opposed indefinites so to speak neutralizing one another. Here, once
again, one can clearly see that the symbol 00 most emphatically does not represent
the Infinite, for the Infinite, in its true sense, can have neither opposite
nor complementarity, nor can it enter into correlation with anything at all, no
more with zero, in whatever sense it might be understood, than with the unit,
or with any number, or again with any particular thing of any order whatsoever,
quantitative or not; being the absolute and universal All, it contains
Non-Being as well as Being, such that zero itself, whenever it is not regarded
as purely nothing, must also necessarily be considered to be contained within
the Infinite.
In
alluding here to Non-Being, we touch on another meaning of zero quite different
from those we have just considered, the most important from the point of view
of metaphysical symbolism; but in this regard, in order to avoid all confusion
between the symbol and that which it represents, it is necessary to make it
quite clear that the metaphysical Zero, which is Non-Being, is no more the zero
of quantity than the metaphysical Unit, which is Being, is the arithmetical
unit. What is thus designated by these terms is so only by analogical
transposition, since as soon as one places oneself within the Universal one is
obviously beyond every special domain such as that of quantity. Furthermore, it
is not insofar as it represents the indefinitely small that zero by such a
transposition can be taken as a symbol of Non-Being, but, following its most
rigorous mathematical usage, rather insofar as it represents the absence of
quantity, which in its order indeed symbolizes the possibility of non-manifestation,
just as the unit, since it is the point of departure for the indefinite
multiplicity of numbers, symbolizes the possibility of manifestation as Being
is the principle of all manifestation.[107]
This
again leads us to note that zero, however it may be envisaged, can in no case
be taken for pure nothingness, which corresponds metaphysically only to
impossibility, and which in any case cannot logically be represented by
anything. This is all too obvious when it is a question of the indefinitely
small; it is true that this is only a derivative sense, so to speak, due, as we
were just saying, to a sort of approximate assimilation of quantities
negligible for us, to ’ the total absence of quantity; but insofar as it is a
question of this very absence of quantity, what is null in this connection
certainly cannot be so in other respects, as is apparent in an example such as
the point, which, being indivisible, is by that very fact without extension,
that is, spatially null,[108]
but which, as we have explained elsewhere, is nonetheless the very principle of
all extension.[109]
It is quite strange, moreover, that mathematicians are generally inclined to
envisage zero as a pure nothingness, when it is nevertheless
impossible for them not to
regard it at the same time as endowed with an indefinite potentiality, since,
placed to the right of another digit termed ‘significant’, it contributes to
forming the representation of a number that, by the repetition of this same
zero, can increase indefinitely, as is the case with the number ten and its successive
powers for example. If zero were really only pure nothingness, this could not
be so; and indeed, in that case it would only be a useless sign, entirely
deprived of effective value; here we have yet another inconsistency to add to
the list of those that we have already had occasion to point out in the
conceptions of modern mathematicians.
NEGATIVE NUMBERS
If
we now return to the second and more important of
the two mathematical senses of zero, namely that of zero considered as a
representation of the indefinitely small, this is because within the doubly
indefinite sequence of numbers the domain of the latter embraces all that
eludes our means of evaluation in a certain direction, just as within the same
sequence the domain of the indefinitely great embraces all that eludes these
means of evaluation in the other direction. This being so, to speak of numbers
‘less than zero’ is obviously no more appropriate than to speak of numbers
‘greater than the indefinite’, and it is all the more unacceptable—if such is
possible—when zero is taken in its other sense as purely and simply
representing the absence of quantity, for it is totally inconceivable that a
quantity should be less than nothing. In a certain sense, however, this is
precisely what is done when one introduces the consideration of so-called
negative numbers to mathematics, forgetting as a result of modern
‘conventionalism’ that these numbers were originally no more than an
indication of the result of a subtraction that is in fact impossible, in which
a greater number is taken away from a smaller; besides, we have already pointed
out that all generalizations or extensions of the idea of number arise only
from the consideration of operations that are impossible from the point of
view of pure arithmetic; but this conception of negative numbers, and the
consequences it entails, demand some further explanation.
We said earlier that the sequence of whole numbers is formed
starting from the unit, and not from zero; indeed, the unit being fixed, the
entire sequence of numbers is inferred from it in such a
way that one could say
that it is already implied and contained in principle within the initial unit[110]
whereas it is obvious that no number can be derived from zero. Passage from
zero to the unit cannot be made in the same way as passage from the unit to
other numbers, or from any given number to the next, and to suppose the passage
from zero to the unit possible is to have already implicitly posited the unit.[111]
Finally, to place zero at the beginning of the sequence of numbers as if it
were the first in the sequence, can mean only one of two things: either one
admits, contrary to what has already been established, that zero really is a
number, and consequently that its ratios with respect to other numbers are of
the same order as the ratios of these numbers are to each other—which is not
the case, since zero multiplied or divided by a given number is always zero— or
this is a simple device of notation, which can only lead to more or less
inextricable confusions. In fact, the use of this device is never justified
except to permit the introduction of the notation of negative numbers, and if
such notation doubtless offers certain advantages for the convenience of
calculation—an entirely ‘pragmatic’ consideration, which is not in question
here and which is even without any real importance from our point of view—it
is easy to see that it is not without grave logical difficulties. The first of
these is precisely the conception of negative quantities as ‘less than zero’,
an affirmation which Leibnitz ranked among the affirmations that are only toleranter
verae, but which in reality is, as we were just saying, entirely devoid of
meaning. ‘To affirm an isolated negative quantity as less than zero,’ says
Carnot, ‘is to veil the science of mathematics, which should be a science of
the obvious, in an impenetrable cloud, and to thrust oneself into a labyrinth
of paradoxes, each more bizarre than the last.’[112]
On this point we may follow his judgment,
which is above suspicion
and is certainly not exaggerated; moreover, one should never forget in using
this notation of negative numbers that it is a matter of nothing more than a
simple convention.
The
reason for this convention is as follows: when a given subtraction is
arithmetically impossible, its result is nonetheless not devoid of meaning when
this subtraction is linked to magnitudes that can be reckoned in two opposite directions,
as, for example, with distances measured on a line, or angles of rotation
around a fixed point, or again the time elapsed in moving from a certain
instant toward either the past or the future. From this results the geometric
representation habitually accorded negative numbers: taking an entire straight
line, indefinite in both directions, and not in one only, as was the case
earlier, the distances along the line are considered positive or negative
depending on whether they fall one way or the other, and a point is chosen to
serve as the origin, in relation to which the distances are positive on one
side and negative on the other. For each point on the line there is a number
corresponding to the measurement of its distance from the origin, which, in
order to simplify our language, we can call its coefficient; once again, the
origin itself will naturally have zero for its coefficient, and the
coefficients of all the other points on the line will be numbers modified by
the signs + and -, which in reality simply indicate on which side the point
falls in relation to the origin. On a circumference one could likewise
designate positive and negative directions of rotation, and starting from an
initial position of the radius, one would take each angle to be positive or
negative according to the direction in which it lies, and so on analogously.
But to keep to the example of the straight line, two points equidistant from
the origin, one on either side, will have the same number for their
coefficients, but with contrary signs, and in all cases, a point that is
further than another from the origin will naturally have a greater coefficient;
thus it is clear that if a number n is greater than another number m,
it would be absurd to say, as is ordinarily done, that -n is smaller than -m,
since on the contrary it represents a greater distance. Moreover the sign thus
placed in front of a number cannot really modify it in any way with regard to
quantity, since it represents nothing with respect to the measurements of
distances themselves,
but only the direction in
which these distances are traversed, which, properly speaking, is an element of
a qualitative, and not a quantitative, order.[113]
Moreover,
as the line is indefinite in both directions, one is led to envisage both a
positive and a negative indefinite, represented by the signs oo and -oo
respectively, commonly designated by the absurd expressions ‘greater infinity’
and ‘lesser infinity’. One might well ask what a negative infinity would be, or
again what could remain were one to take away an infinite amount from
something, or even from nothing, since mathematicians regard zero as nothing;
one has only to put these matters in clear language in order to see immediately
how devoid of meaning they are. We must further add that particularly when
studying the variation of functions, one is then led to believe that the
negative and the positive indefinite merge in such a way that a moving object
departing from the origin and moving further and further away in the positive
direction would return to the origin from the negative side, or inversely, if
the movement were followed for an indefinite amount of time, whence it would
result that the straight line, or what would then be considered as such, would
in reality be a closed line, albeit an indefinite one. Furthermore, one could
show that the properties of a straight line in a plane would be entirely
analogous to those of a great circle, or diametrical circle on the surface of a
sphere, and that the plane and the straight line could thus be likened
respectively to a sphere and a circle of indefinitely great radius, and
consequently of indefinitely small curvature, ordinary circles in the plane
then being comparable to the smaller circles on the sphere; for this analogy
to be rigorous, one would further have to suppose a ‘passage to the limit’, for
it is obvious that however great a radius might become through indefinite
increase, it always describes a sphere and not a plane, and that the sphere
only tends to be merged with the plane, and its great
circle [or diameter] with
lines, such that plane and line are limits, in the same way that a circle is
the limit of a regular polygon with an indefinitely increasing number of sides.
Without pushing the issue further, we shall only remark that through
considerations of this sort, one can as it were directly grasp the precise
limits of spatial indefinitude; how then, if one wishes to maintain some
appearance of logic, can one still speak of the infinite in all this?
When
considering positive and negative numbers as we have just done, the sequence of
numbers takes the following form: -oo ... - 4, -3, -2, -1, o, 1, 2, 3, 4 ...
+00, the order of these numbers being the same as that of the corresponding
points on the line, that is, the points having these numbers for their
respective coefficients, which, moreover, is the mark of the real origin of the
sequence thus formed. Although the sequence is equally indefinite in both directions,
it is completely different from the one we envisaged earlier, which contained
the whole numbers and their inverses: this one is symmetric not with respect to
the unit, but with respect to zero, which corresponds to the origin of the
distances; and if two numbers equidistant from this central term are to return
to it, it will not be by multiplication, as in the case of inverse numbers, but
by ‘algebraic’ addition, that is, effected while taking account of signs,
which in this case would amount to a subtraction, arithmetically speaking.
Moreover, we can by no means say of the new sequence that it is indefinitely
increasing in one direction and indefinitely decreasing in the other, as we
could of the preceding, or at least, if one claims to consider it thus, this is
only a most incorrect ‘manner of speaking’, as is the case when one envisages
numbers ‘less than zero’. In reality, the sequence increases indefinitely in
both directions equally, since it is the same sequence of whole numbers that is
contained on either side of the central zero; what is called the ‘absolute
value’—another rather singular expression—must only be taken into consideration
in a purely quantitative respect, the positive or negative signs changing
nothing in this regard, since, in reality, they express no more than
differences in ‘situation’, as we have just explained. The negative indefinite
is therefore by no means comparable to the indefinitely small; on the
contrary, it belongs with the indefinitely great as does the positive
indefinite; the only difference, which is not one of
a quantitative order, is
that it proceeds in another direction, which is perfectly conceivable when it
is a question of spatial or temporal magnitudes, but totally devoid of meaning
for arithmetical magnitudes, for which such a progression is necessarily unique
since it cannot be anything other than that of the very sequence of whole
numbers.
Among
the bizarre or illogical consequences of the notation of negative numbers, we
shall further draw attention to the consideration of so-called ‘imaginary’
quantities which were introduced in the solving of algebraic equations and
which, as we have seen, Leibnitz ranked at the same level as infinitesimal
quantities, namely as what he called ‘well-founded fictions’. These quantities,
or what are so called, are presented as the roots of negative numbers, although
in reality this again only corresponds to a pure and simple impossibility,
since, whether a number is positive or negative, its square is necessarily
always positive by virtue of the rules of algebraic multiplication. Even if
one could manage to give these ‘imaginary’ quantities some other meaning,
thereby making them correspond to something real—a possibility we shall not
examine here—it is nonetheless quite certain that their theory and application
to analytic geometry as it is presented by contemporary mathematicians never
appears^ anything but a veritable web of confusions and even absurdities, and
as the product of a need for excessive and entirely artificial generalizations,
which need does not retreat even before manifestly contradictory propositions;
certain theorems concerning the ‘asymptotes of a circle’, for example, amply
suffice to prove that this remark is by no means exaggerated. It is true that
one could say that this is no longer a question of geometry properly speaking,
but, like the consideration of a ‘fourth dimension’ of space,5 only
of algebra translated into geometric language; but precisely because such a
translation, as well as its inverse, is possible and legitimate to a certain
degree, some people would also like to extend it to cases where it can no
longer mean anything, and this is indeed quite serious, for it is the symptom
of an extraordinary confusion of ideas, as well as the extreme result of a
‘conventionalism’ taken so far as to cause some people to lose all sense of
reality.
REPRESENTATION OF
THE EQUILIBRIUM
OF FORCES
In connection with negative numbers, we
shall now speak of the rather disputable consequences of the use of these
numbers from the point of view of mechanics, even though this is only a
digression with respect to the principal subject of our study; moreover, since
in virtue of its object the field of mechanics itself is in reality a physical
science, the very fact that it is treated as an integral part of mathematics in
consequence of the exclusively quantitative point of view of science today
means that some rather singular distortions have been introduced. Let us only
say that the so-called ‘principles’ upon which modern mathematicians would
build this science, such as they conceive of it, can be referred to as
‘principles’ only in a completely abusive manner, as they are in fact only more
or less well-founded hypotheses, or again, in the most favorable case, only
simple laws that are general to some degree, perhaps more general than others,
if one likes, but still having nothing in common with true universal
principles; in a science constituted according to the traditional point of
view, the laws of mechanics would at most be mere applications of these
principles to an even more specialized domain. Without entering into
excessively lengthy explanations, let us cite as an example of the first case,
the so-called ‘principle of inertia’, which nothing can justify, neither
experience, which on the contrary shows that inertia has no role in nature,
nor in the understanding, which cannot conceive of this so-called inertia that
consists only in a complete absence of properties; one could only legitimately
apply such a word to the pure potentiality of universal
substance,
or to the materia prima of the Scholastics, which is moreover for this very
reason properly ‘unintelligible’; but this materia prima is
assuredly something completely different from the ‘matter’ of the physicists.1
An example of the second case may be seen in what is called the ‘principle of
the equality of action and reaction’, which is so little a principle that it is
immediately deduced from the general law of the equilibrium of natural forces:
whenever this equilibrium is disturbed in any way, it immediately tends to
re-establish itself, whence a reaction of which the intensity is equivalent to
that of the action that provoked it. It is therefore only a simple, particular
case of what the Far-Eastern tradition calls ‘concordant actions and
reactions’, a principle that does not concern the corporeal world alone, as do
the laws of mechanics, but indeed the totality of manifestation in all its
modes and states; and for a moment we propose to dwell precisely on this
question of equilibrium and its mathematical representation, for it is
important enough in itself to merit a momentary pause.
Two
forces in equilibrium are usually represented by two opposed ‘vectors’, that
is, by two line segments of equal length, but aimed in opposite directions: if
two forces applied to the same point have the same intensity and fall along the
same line, but in opposite directions, they are in equilibrium; as they are
then without action at their point of application, it is even commonly said
that they cancel each other out, although this ignores the fact that if one of
the forces is suppressed, the other will immediately act, which proves that
they were never really cancelled in the first place. The forces are characterized
by numerical coefficients proportional to their respective intensities, and two
forces of opposing direction are given coefficients with different signs, the
one positive, the other negative, so that if the one is/, the other will be -f.
In the case we have just considered, in which the two forces are of the same
intensity, the coefficients characterizing them must be equal with respect to
their ‘absolute values’; one then has/=/', from which one can infer as a
condition of their equilibrium that/-/' = o, which is to say that the algebraic
sum of the two forces, or of the two ‘vectors’ representing
them, is null, such that
equilibrium is thus defined by zero. Zero having been incorrectly regarded by
mathematicians as a sort of symbol for nothingness, as we have already said
above—as if nothingness could really be symbolized by anything whatsoever—the
result seems to be that equilibrium is the state of non-existence, which is a
rather strange consequence; it is nevertheless almost certainly for this reason
that, instead of saying that two forces in equilibrium neutralize one another,
which would be more exact, it is said that they cancel one another, which is
contrary to the reality of things, as we have just made clear by a most
elementary observation.
The
true notion of equilibrium is something else altogether. In order to understand
it, it suffices to point out that all natural forces, and not only mechanical
forces (which, let us say again, are no more than a very particular case) but
forces of the subtle order as well as those of the corporeal order, are either
attractive or repulsive; the first can be considered as compressive forces, or
forces of contraction, and the second as expansive forces, or forces of
dilation,[114]
and basically this is no more than an expression in a particular domain of the
fundamental cosmic duality itself. It is easy to understand how, given an
initially homogenous medium, for every point of compression there will
necessarily correspond an equivalent expansion at another point, and inversely,
such that two centers of force must be envisaged correlatively, each of which
could not exist without the other; this is what one can call the law of
polarity, which is, in all its various forms, applicable to all natural phenomena,
since it, too, derives from the duality of the very principles that preside
over all of manifestation; in the specialized domain with which physicists
occupy themselves, this law is above all evident in
electrical and magnetic
phenomena, but it is by no means limited to them. Now if two forces, the one
compressive, the other expansive, act upon the same point, then the condition
requisite for them to be in equilibrium, or to neutralize one another, the
condition, that is, which when fulfilled will produce neither contraction nor
dilation, is that the intensities of the two forces be equivalent; we do not
say equal, since the forces are of different species, and since this is
moreover a question of a truly qualitative, and not simply quantitative,
difference. The forces can be characterized by coefficients proportional to the
contraction or dilation they produce, in such a way that if one considers a
compressive force and an expansive force together, the first will have a
coefficient n>i, and the second a coefficient n' < 1; each of
these coefficients will be the ratio of the density of the space surrounding
the point in consideration, under the action of the corresponding force, to the
original density of the same space, which in this regard is taken to be
homogenous when not subject to any forces in virtue of a simple application of
the principle of sufficient reason.[115]
When neither compression nor dilation is produced, the ratio is necessarily
equal to one, since the density of the space is unchanged; in order for two
forces acting upon a point to be in equilibrium, their resultant must have a
coefficient of one. It is easy to see that the coefficient of this resultant is
the product, and not, as in the ordinary conception, the sum of the
coefficients of the two forces under consideration; these two coefficients, n
and n', must therefore each be the inverse of the other: n' = Vn,
and we will then have (n)(n') = i as the condition for equilibrium;
equilibrium will thus no longer be defined by zero, but by the unit.[116]
It
will be seen that the definition of equilibrium with respect to the unit—its
only real definition—corresponds to the fact that the unit occupies the
mid-point in the doubly indefinite sequence of
whole numbers and their
inverses, while this central position is as it were usurped by zero in the
artificial sequence of positive and negative numbers. Far from being the state
of non-existence, equilibrium is on the contrary existence considered in and
of itself, independent of its secondary, multiple manifestations; moreover, it
is certainly not Non-Being, in the metaphysical sense of the word, for
existence, even in this primordial and undifferentiated state, is still the
point of departure for all differentiated manifestations, just as the unit is
the point of departure for the multiplicity of numbers. As we have just
considered it, this unit in which equilibrium resides is what the Far-Eastern
tradition calls the ‘Invariable Middle’; and according to the same tradition,
this equilibrium or harmony is the reflection of the ‘Activity of Heaven’ at
the center of each state, and of each modality of being.
FIXED QUANTITIES
Let
us now return to the question of the justification
of the rigor of the infinitesimal calculus. We have already seen that Leibnitz
considers quantities to be equal when their difference, while not strictly
null, is nonetheless incomparable with respect to the quantities themselves; in
other words, infinitesimal quantities, though not nihila absoluta
[absolute nothingness], are nevertheless nihila respectiva [nothingness
in some respect], and as such must be negligible with respect to ordinary
quantities. Unfortunately, the notion of ‘incomparability’ is still too
imprecise for an argument based on it alone to be fully sufficient to establish
the rigorous character of the infinitesimal calculus fully; from this point of
view, the calculus appears to be in short but a method of indefinite approximation,
ànd we
cannot say with Leibnitz that ‘once this is affirmed, it follows not only that
the error is infinitely small, but that it is nothing at all’;[117]
but is there no more rigorous means of arriving at this conclusion? We must at
least admit that the error introduced into our calculations can be rendered as
small as desired, which is already saying a great deal; but does not precisely
this infinitesimal character of the error do away with it completely when one
considers, not only the course of the calculation itself, but its final
results?
An infinitesimal difference, that is, one decreasing
indefinitely, can only be the difference between two variable quantities, for
it is obvious that the difference between two fixed quantities can itself only
be a fixed quantity; it would thus be meaningless to speak of an infinitesimal
difference between two fixed quantities. Hence, we
have the right to say that
two fixed quantities are ‘rigorously equal the moment that their would be difference
can be supposed as small as one likes’;[118]
now, ‘the infinitesimal calculus, like ordinary calculation, really has in
view only fixed and determined quantities’;[119]
in short, it introduces variable quantities only as auxiliaries having a purely
transitory character, and these variables must disappear from the results,
which can only express ratios between fixed quantities. Thus, in order to
obtain these results, one must pass from a consideration of variable quantities
to one of fixed quantities; and this passage has precisely as its result the
elimination of infinitesimal quantities, which are essentially variable, and
which can appear only as the differences between variable quantities.
It
will now be easy to understand why, in the definition we cited earlier, Carnot
insisted that infinitesimal quantities as employed in the calculus, are able to
be rendered as small as one likes ‘without one’s being obliged on that account
to vary the quantities to which they are compared.’ It is because these latter
quantities must in reality be fixed quantities; it is true that in the
calculus they are considered to be limits of variable quantities, but these
latter merely play the role of simple auxiliaries, as do the infinitesimal
quantities which they bring with them. In order to justify the rigor of the
infinitesimal calculus, the essential point is that only fixed quantities must
figure in the results; in terms of the calculus, therefore, it is ultimately
necessary to pass from variable quantities to fixed quantities, and this is
indeed a ‘passage to the limit’, but not as conceived by Leibnitz, since there
is no result or ‘final term’ of the variation itself; now—and this is what
really matters—the infinitesimal quantities are eliminated of themselves in
this passage, and this quite simply by reason of the substitution of fixed
quantities for variable quantities.[120]
But must one view their elimination merely as the result of a
simple ‘compensation of errors’, as Carnot would have it? We think not, and it
indeed seems that one really can see more in it as soon as one distinguishes
between variable and fixed quantities, observing that they constitute as it
were two separate domains, between which there doubtless exists a correlation
and analogy—which moreover is necessary in order to be able to pass from one to
the other, however such a passage is effected—but without their real ratios
ever establishing any kind of interpenetration, or even continuity; furthermore,
this implies that an essentially qualitative difference exists between the two
sorts of quantity, in conformity with what was said earlier concerning the
notion of the limit. Leibnitz never made this distinction clearly, and here
again, his conception of a universally applicable continuity no doubt prevented
him from doing so; he was unable to see that ‘passage to the limit’ essentially
implies a discontinuity, because for him no discontinuity existed. However, it
is this distinction alone that allows us to formulate the following
proposition: if the difference between two variable quantities can be rendered
as small as one likes, then the fixed quantities that correspond to these
variables and which are regarded as the respective limits of the latter, are
rigorously equal. Thus, an infinitesimal difference can never become nothing;
but such a difference can exist only between variables, and between the
corresponding fixed quantities, the difference must indeed be nothing; whence
it immediately follows that to an error capable of being rendered as small as
one likes in the dopiain of variable quantities (in which there can in fact be
no question of anything more than indefinite approximation precisely by reason
of the character of these quantities) there necessarily corresponds another
error that is rigorously null in the domain of fixed quantities. The true
justification for the rigor of the infinitesimal calculus essentially resides
in this consideration alone, and not in any others, which, whatever they might
be, are always more or less peripheral to the question.
which one passes, it suffices to keep
in mind the actual destination of the calculations. With each of the ratios,
one must look not at what it seems to express at the moment, but at that which
it will later express, after its limits have been found’.
DIFFERENTIATIONS
The preceding still leaves a difficulty
regarding the consideration of different orders of infinitesimal quantity: how
can one conceive of quantities as infinitesimal not only with respect to
ordinary quantities, but with respect to other quantities that are themselves
infinitesimal? Here again Leibnitz has recourse to the notion of ‘incomparables’, but
this is much too vague to satisfy us, and it does not sufficiently explain the
possibility of successive differentiations. No doubt, this possibility can best
be understood by a comparison or example from mechanics: ‘As for ddx, it
is to dx as the conatus [force] of weight or the centrifugal
tendency is to speed.’[121]
And Leibnitz develops this idea in his response to the objections of the Dutch
mathematician Nieuwentijt, who, while admitting differentials of the first
order, maintained that those of higher orders could only be null quantities:
Ordinary quantity, the first
infinitesimal or differential quantity, and the second infinitesimal or
diffentio-differential quantity, are to each other as movement, speed, and
solicitation,[122]
which is an element of speed. Movement describes a line, speed an element of
the line, and solicitation an element of the element.[123]
But here
we have only a particular example or case, which can in short serve only as a
simple ‘illustration’, not an argument, and it is necessary to furnish
justification of a general order, which this example, moreover, in a certain
sense contains implicitly.
Indeed,
differentials of the first order represent the increases—or, better, the
variations, since depending on the case they could as easily be in the
decreasing as in the increasing direction—that are at each instant received by
ordinary quantities; such is speed with respect to the space covered in a given
movement. In the same way, differentials of a given order represent the
instantaneous variations of differentials of the preceding order, which in turn
are taken as magnitudes existing within a certain interval; such is
acceleration with respect to speed. Thus the distinction between different
orders of infinitesimal quantities in fact rests on the consideration of different
degrees of variation, much more than on that of incomparable magnitudes.
In
order to state precisely the way in which this must be understood, let us
simply make the following remark: one can establish among the variables
themselves distinctions analogous to those established earlier between fixed
and variable quantities; under these conditions, to go back once again to
Carnot’s definition, a quantity is said to be infinitesimal with respect to
others when one can render it ^s small as one likes ‘without one being obliged
thereby to v^y these other quantities.’ Indeed, this is because a quantity th^
is not absolutely fixed, or even one that is essentially variable—as is the
case with infinitesimal quantities, whatever the order in question—can
nevertheless be regarded as fixed and determined, that is, as capable of
playing the role of fixed quantity with respect to certain other variables.
Only under these conditions can a variable quantity be considered the limit of
another variable, which, by the very definition of the term limit, presupposes
that it be regarded as fixed, at least in a certain respect, namely relative to
that which it limits; inversely, a quantity can be variable not only in and of
itself or, what amounts to the same, with respect to absolutely fixed
quantities, but even with respect to other variables, insofar as the latter are
regarded as relatively fixed.
Instead
of speaking in this regard of degrees of variation, as we have just done, one
could equally well speak of degrees of indeterminacy, which ultimately would
be exactly the same thing, only considered from a slightly different point of
view: a quantity, though indeterminate by its nature, can nevertheless be
determined in a relative sense by the introduction of certain hypotheses, which
allow the indeterminacy of other quantities to subsist at the same time; these
latter quantities will therefore be more indeterminate, so to speak, than the
others, or indeterminate to a greater degree, and they will therefore be
related to the others in a manner comparable to that in which the indeterminate
quantities are themselves related to quantities that truly are determined. We
shall confine ourselves to these remarks on the subject, for however summary
they might be, we believe that they are at least sufficient for understanding
the possibility of the existence of differentials of various successive orders;
but, in connection with this same question, it still remains for us to show
more explicitly that there is really no logical difficulty in considering
multiple degrees of indefinitude, and this as much in the order of decreasing
quantities, to which infinitesimals and differentials belong, as in that of
increasing quantities, in which one can likewise envisage integrals of
different orders, which are as it were symmetric with respect to the successive
differentiations; and this is moreover in conformity with the correlation that
exists between the indefinitely increasing and the indefinitely decreasing, as
we have explained. Of course, in all this it is only a question of - degrees of
indefinitude, and not of‘degrees of infinity’, such as Jean Bernoulli
understood them, which notion Leibnitz dared neither adopt nor reject
absolutely in this regard; and here we have yet another case in which the
difficulties can be immediately resolved by substituting the notion of the
indefinite for that of the so-called infinite.
20
OF INDEFINITUDE
The logical difficulties, and
even contradictions which mathematicians run up against when they consider
‘infinitely great’ or ‘infinitely small’ quantities that differ with respect to
one another, and even belong to different orders altogether, arise solely from
the fact that they regard as infinite that which is simply indefinite. It is
true that in general they do not seem very concerned with these difficulties,
but they exist nonetheless, and are no less serious for all that, as they cause
the science of mathematics to appear as if full of illogicalities, or, if one
prefer, of ‘para-logicalities’, and such a science loses all real value and
significance in the eyes of those who do not allow themselves to be deluded by
words. Here are some examples of the contradictions introduced by those who
would allow the existence of infinite magnitudes, when they apply this notion
to géométrie magnitudes:
if a straight line is considered to be infinité, its infinitude must be
less, and even infinitely less, than the infinitude constituted by a surface
such as a plane, in which both that line and an infinite number of others are
also contained, and the infinitude of the plane will in turn be infinitely less
than that of three-dimensional space. The very possibility of the coexistence
of all of these would-be infinities, some of which are supposed to be infinite
to the same degree, others to different degrees, suffices to prove that none of
them can be truly infinite, even apart from any consideration of a more
properly metaphysical order; indeed, as these are truths which we cannot
emphasize enough, let it be said again: it is obvious that if one supposes a
plurality of distinct infinites, each will have to be limited by the others,
which amounts to
saying that they will
exclude one another. Moreover, to tell the truth, the ‘infinitists’, for whom
this purely verbal accumulation of an ‘infinity of infinities’ seems to produce
a kind of ‘mental intoxication’, if such an expression be permissible, do not
retreat in face of such contradictions, since, as has already been said, they
see no difficulty in asserting that various infinite numbers exist, and that
consequently one infinity can be greater or smaller than another; but the
absurdity of such utterances is only too obvious, and the fact that they are
commonly used in contemporary mathematics changes nothing, but only shows to
what extent the sense of the most elementary logic has been lost in our day.
Yet another contradiction, no less blatant than the last, is to be found in
the case of a closed, hence obviously and visibly finite, surface, which
nevertheless contains an infinite number of lines, as, for example, a sphere,
which contains an infinite number of circles; here we have a finite container,
of which the contents would be infinite, which is likewise the case, moreover,
when one maintains, as did Leibnitz, the ‘actual infinity’ of the elements of a
continuous set.
On
the contrary, there is no contradiction in allowing the coexistence of a
multiplicity of indefinite magnitudes of various orders. Thus a line indefinite
in a single dimension can in this regard be considered to constitute a simple
indefinitude of the first order; a surface, indefinite in two dimensions, and
embracing an indefinite number of indefinite lines, will then be an
indefinitude of the second order; and three-dimensional space, which embraces
an indefinite number of indefinite surfaces, will similarly be an indefinitude
of the third order. Here it is essential to point out once again that we said
the surface embraces an indefinite number of lines, not that it is constituted
by an indefinite number of lines, just as a line is not composed of points, but
rather embraces an indefinite multitude of them; and it is again the same in
the case of a volume with respect to its surfaces, three-dimensional space
being itself none other than an indefinite volume. This, moreover, is basically
what we said above on the subject of ‘indivisibles’ and the ‘composition of the
continuous’; it is questions of this kind that, precisely by reason of their
complexity, most make one aware of the necessity of rigorous language. Let us
also add in this regard that if from a certain point
of view one can
legitimately consider a line to be generated by a point, a surface by a line,
and a volume by a surface, this essentially presupposes that the point, the
line, or the surface be displaced through a continuous motion, embracing an
indefinitude of successive positions; and this is altogether different from
considering each of these positions in isolation, that is, regarding the
points, lines, and surfaces as fixed and determined, and as constituting the
parts or elements of the line, the surface, or the volume, respectively.
Likewise, but inversely, when one considers a surface to be the intersection
of two volumes, a line the intersection of two surfaces, and a point the
intersection of two lines, these intersections must not, of course, by any
means be conceived of as parts common to the volumes, surfaces, or lines; they
are only limits or extremities of the latter, as Leibnitz has said.
According
to what we have just said, each dimension introduces as it were a new degree of
indeterminacy to space, that is, to the spatial continuum insofar as it is
subject to indefinite increase of extension and thus yields what could be
called successive powers of the indefinite;[124]
and one can also say that an indefinite quantity of a certain order or power
contains an indefinite multitude of indefinite quantities of a lower order or
lesser power. As long as it is only a question of the indefinite in all of
this, these considerations, as well as others of the same sort, remain
perfectly acceptable, for there is no logical incompatibility between multiple
and distinct indefinite quantities, which, despite theij indefinitude, are
nonetheless of an essentially finite nature, and which, like any other
particular and determined possibility, are therefore perfectly capable of
coexisting within total Possibility, which is alone infinite, since it is
identical to the universal All.[125]
These same considerations take on an impossible and absurd form only when the
indefinite is confused with the infinite; thus, as with the notion of the
‘infinite multitude’, we once again have an instance in which the contradiction
inherent in a so- called determined infinite is concealed, deforming another
idea that, although in itself not at all contradictory, is nonetheless rendered
virtually unrecognizable.
We
have just spoken of various degrees of indeterminacy in relation to quantities
taken in the increasing direction; by applying the same notion to the
decreasing direction we have already justified above the consideration of
various orders of infinitesimal quantity, the possibility of which is all the
more understandable in the light of the correlation we noted earlier between
indefinitely increasing and indefinitely decreasing quantities. Among
indefinite quantities of various orders, those of orders apart from the first
will always be indefinite with respect to those of the preceding order as well
as to ordinary quantities; inversely, among infinitesimal quantities of various
orders, it is just as legitimate to consider those of each order as
infinitesimal not only with respect to ordinary quantities, but also to the
infinitesimal quantities of the preceding orders.[126]
There is no absolute heterogeneity between indefinite quantities and ordinary
quantities, nor again between infinitesimal quantities and ordinary quantities;
in short, it is only a question of a difference of degree, not of kind, since,
in reality, the consideration of indefin- itude, whatever the order or power in
question, never takes us out of the finite; again, it is the false conception
of the infinite that introduces the appearance of a radical heterogeneity between
the different orders of quantity, which at bottom is completely incomprehensible.
In doing away with this heterogeneity, a kind of continuity is established
quite different from that which Leibnitz envisaged between variables and their
limits, and much better grounded in reality, for contrary to what he believed,
the distinction ’ between variable and fixed quantities essentially implies a
difference of nature.
Under
these conditions, ordinary quantities themselves can in a way be regarded as
infinitesimal with respect to indefinitely increasing quantities, at least
when we are dealing with variables, for, if a quantity is capable of being
rendered as great as one likes with respect to another, inversely the latter
will by the same token become as small as one likes with respect to the former.
We say that it must be a question of variables because an infinitesimal
quantity must always be conceived of as essentially variable, and this restriction
is inherent in its very nature; moreover, quantities belonging to two different
orders of indefinitude are inevitably variable with respect to one another, and
this property of relative and reciprocal variability is perfectly symmetric,
for, in accordance with what was just said, to consider one quantity to be
indefinitely increasing with respect to another, or this latter indefinitely
decreasing with respect to the first, amounts to the same thing; without this
relative variability there could be neither indefinite increase nor indefinite
decrease, but only definite and determined ratios between the two quantities.
In
the same way, whenever there is a change in position with respect to two bodies
A and B, to say that body A is in motion with respect to body B,
and, inversely, that body B is in motion with respect to body A, also
amounts to the same thing, at least insofar as the change is only considered in
and of itself; in this regard the concept of relative motion is just as
symmetric as that of relative variability, which we "Were just
considering. This is why, according to Leibnitz, who used it to demonstrate the
inadequacy of Cartesian mechanism as a physical theory claiming to furnish an
explanation for all natural phenomena, one cannot distinguish between a state
of motion and a state of rest when one is limited solely to the consideration
of changes in position; to do so one must bring in something of another order,
namely, the notion of force, which is the proximate cause of such changes, and
which alone can be attributed to one body rather than to another, as it allows
the true cause of change to be located in one body and in that body alone.[127]
IS ANALYTICALLY
INEXHAUSTIBLE
In
the two cases just considered, that of the indefinitely
increasing and that of the indefinitely decreasing, a quantity of a given order
can be regarded as the sum of an indefinitude of elements, each of which is an
infinitesimal quantity with respect to the entire sum. In order to be able to
speak of infinitesimal quantities, it is moreover necessary that it be a
question of elements that are not determined with respect to their sum, and
this is indeed the case whenever the sum is indefinite with respect to the
elements in question; this follows immediately from the essential character of
indefinitude itself, inasmuch as the latter obviously implies the idea of
‘becoming’, as we have said before, and consequently a certain"
indeterminacy. It is of course understood that this indeterminacy can only be
relative, and exists only from a certain point of view or with respect to a
certain thing: such is the case, for example, with a sum that is an ordinary
quantity, and hence not indefinite in and of itself, but only with respect to
its infinitesimal elements; at any rate, if it were otherwise, and if this
notion of indeterminacy were not introduced, one would be reduced to the mere
conception of ‘incomparables’, interpreted in the crude sense of the
grain of sand in comparison to the earth, and the earth in comparison to the
heavens.
The sum in question can by no means be effected in the manner
of an arithmetical sum, since for that it would be necessary for an
indefinite series of
successive additions to be achieved, which is contradictory; in the case in
which the sum is an ordinary and determined quantity as such, it is obviously
necessary, as we already said when we set forth the definition of the integral
calculus, that the number, or rather the multitude, of elements increase indefinitely
while at the same time the magnitude of each decreases indefinitely, and in
this sense the indefinitude of its elements is truly inexhaustible. But if the
sum cannot be effected in this way, as the final result of a multitude of
distinct and successive operations, it can on the other hand be comprehended at
one stroke, by a single operation, namely, integration;[128]
here we have the inverse operation of differentiation, since it reconstitutes
the sum starting from its infinitesimal elements, while differentiation on the
contrary moves from the sum to the elements, furnishing the means of
formulating the law for the instantaneous variations of the quantity of which
the expression is given.
Thus,
whenever it is a question of indefinitude, the notion of an arithmetical sum is
no longer applicable, and one must resort to the notion of integration in order
to compensate for the impossibility of ‘numbering’ the infinitesimal elements,
an impossibility which, of course, results from the very nature of these
elements, and not from any imperfection on our part. In passing we may observe
that as regards the application of this to geometric magnitudes (which,
moreover, is ultimately-the true raison d’être of the infinitesimal
calculus), this is a method of measurement completely different from the usual
method founded on the division of a magnitude into definite portions, of which
we spoke previously in connection with ‘units of measurement’. The latter
always amounts in short to a substitution of the discontinuous for the
continuous by ‘cutting up’ the sum into various portions equal to a magnitude
of the same species
taken as the unit,[129]
in order to be able to apply the resulting number directly to the measurement
of continuous magnitudes, which cannot actually be done except by altering the
nature of the magnitudes in order to make it assimilable, so to speak, to that
of number. The other method, on the contrary, respects the true character of
continuity as much as possible, regarding it as a sum of elements that are
fixed and determined, but that are essentially variable and by virtue of their
variability capable of becoming smaller than any assignable magnitude; this
method thereby allows the spatial quantity between the limits of these elements
to be reduced as much as one likes, and it is therefore the least imperfect
representation of continuous variation one can give, in that it takes account
of the nature of number, which in spite of everything cannot be changed.
These
observations will allow us to understand more precisely in what sense one can
say, as we did at the beginning, that the limits of the indefinite can never be
reached through any analytical procedure, or, in other words, that the
indefinite, while not absolutely and in every way inexhaustible, is at least
analytically inexhaustible. In this regard, we must naturally consider those
procedures analytical which, in order to reconstitute a whole, consist in
taking its elements distinctly and successively; such is the procedure for the
formation of an arithmetical sum, and it is precisely in this regard that it
differs essentially from integration. This is particularly interesting from
our point of view, for one can see in it, as a very clear example, the true
relationship between analysis and synthesis: contrary to current opinion,
according to which analysis is as it were a preparation for synthesis, or again
something leading to it, so much so that one must always begin with analysis,
even when one does not intend to stop there, the truth is that one can never
actually arrive at synthesis through analysis. All synthesis, in the true sense
of the word, is something
immediate, so to speak, something that is not preceded by any analysis and is
entirely independent of it, just as integration is an operation carried out in
a single stroke, by no means presupposing the consideration of elements
comparable to those of an arithmetical sum; and as this arithmetical sum can
yield no means of attaining and exhausting the indefinite, this latter must, in
every domain, be one of those things that by their very nature resist analysis
and can be known only through synthesis.[130]
22
THE SYNTHETIC
CHARACTER OF
INTEGRATION
Contrary to the formation of
an arithmetical sum, which, as we have just said, is strictly analytic in
character, integration must be regarded as an essentially synthetic operation
in that it simultaneously embraces each element of the sum to be calculated,
preserving the ‘indistinction’ appropriate to the parts of a continuum, since,
by the very nature of continuity, these parts cannot be fixed and determined
things. Moreover, whenever one wishes to calculate the sum of the discontinuous
elements of an indefinite sequence, this ‘indistinction’ must likewise be
maintained, although for a slightly different reason, for even if the magnitude
of each may be conceived of as determined, the total number of elements may
not, and we can even say more exactly that their multitude surpasses all
number; nevertheless, there are some cases in which the sum of the elements of
such a sequence tends toward a certain definite limit, even when their
multitude increases indefinitely. Although such a manner of speaking might at
first seem a little strange, one could also say that such a discontinuous
sequence is indefinite by ‘extrapolation’, while a continuous set is so by
‘interpolation’; what is meant by this is that if one takes a given portion of
a discontinuous sequence, bounded by any two of its terms, such a portion will
in no way be indefinite, as it is determined both as a whole and with respect
to its elements; the indefinitude of the sequence lies in the fact that it
extends beyond this portion, without ever arriving at a final term; on the
contrary, the indefinitude of a continuous set,
determined as such, is to
be found precisely in its interior, since its elements are not determined, and
since it has no final terms, the continuous being always divisible; in this
respect each case is thus as it were the inverse of the other. The summation of
an indefinite numerical sequence will never be completed if each term must be
taken one by one, since there is no final term whereby the sequence could come
to an end; such a summation is possible only in the case where a synthetic
procedure lets us seize in a single stroke, as it were, the indefinitude
considered in its entirety, without this at all presupposing the distinct
consideration of its elements, which, moreover, is impossible, by the very fact
that they constitute an indefinite multitude. And similarly, when an indefinite
sequence is given to us implicitly by its law of formation, as in the case of
the sequence of whole numbers, we can say that it is thus given to us
completely in a synthetic manner, and that it cannot be given otherwise;
indeed, to do so analytically would be to lay out each term distinctly, which
is an impossibility.
Therefore,
whenever we have a given example of indefinitude to consider, whether it be a
continuous set or a discontinuous sequence, it will be necessary in every case
to have recourse to a synthetic operation in order to reach its limits;
progression by degrees would be useless here and could never bring us to our
goal, for such a progression can arrive at a final term only on the twofold
condition that both this term and the number of degrees to be covered in order
to reach it, be determined. That is why we did not say that the limits of the
indefinite could not be reached at all, which would be unjustifiable when its
limits do exist, but only that they cannot be reached analytically: the
indefinite cannot be exhausted by degrees, but it can be embraced in its
totality by certain transcendent operations, of which integration is the
classic example in the mathematical order. One could point out that
progression by degrees here corresponds precisely to the variation of quantity,
directly in the case of discontinuous sequences and, in cases of continuous
variation, following therefrom, so to speak, to the extent permitted by the
discontinuous nature of number; on the other hand, synthetic operations
immediately place one outside of and beyond the domain of variation, as must
necessarily be the case according with
what we said above, in
order for a ‘passage to the limit’ actually to be realized; in other words,
analysis pertains only to variables, taken in the very course of their
variation, while synthesis alone attains their limits, which is the only
definitive and really valuable result, since, to be able to speak of results,
one must clearly arrive at something relating exclusively to fixed and
determined quantities.
Furthermore,
one can of course find analogous synthetic operations in domains apart from
quantity, for the idea of an indefinite development of possibilities is clearly
applicable to other things than quantity, as, for example, to a given state of
manifested existence and the conditions, whatever they might be, to which the
state is subject, whether considered with respect to the whole of the cosmos,
or to one being in particular; that is, one can take either a ‘macrocosmic’ or
a ‘microcosmic’ point of view.[131]
One could say that in this case ‘passage to the limit’ corresponds to the
definitive fixation of the results of manifestation in the principial order;
indeed, by this alone does the being finally escape from the change and
‘becoming’ that is necessarily inherent to all manifestation as such; and one
can thus see that this fixation is in no way a ‘final term’ of the development
of manifestation, but rather that it is essentially situated outside of and
beyond that development, since it belongs to another order of reality,
transcendent in relation to manifestation and ‘becoming’; in this regard, the
distinction between the manifested order and the principial order thus
corresponds analogically to that which we established between the domains of
variable and fixed quantities. What is more, when it is a question of fixed
quantities, it is obvious that no modification can be introduced by any operation
whatsoever, and that, consequently, ‘passage to the limit’ cannot produce
anything in this domain, but can only give us knowledge of it; likewise, the
principial order being immutable, arriving at it is not a question of
‘effectuating’ something that did not exist before, but rather of effectively
taking cognizance, in a permanent and absolute manner, of that which is. Given
the subject of this study, we must naturally consider more particularly and
above
all, what properly
concerns the quantitative domain, in which, as we have seen, the idea of the
development of possibilities is translated by the notion of variation, whether
in the direction of indefinite increase or of indefinite decrease; but these
few will suffice to show that by an appropriate analogical transposition all of
this is capable of receiving an incomparably greater significance than that
which it appears to have in and of itself, since integration and other
operations of the same kind will thereby veritably appear as symbols of metaphysical
‘realization’ itself.
By
this one sees the extent of the difference between traditional science, which
allows such considerations, and the profane science of the moderns; and, in
this connection, we shall add yet another remark directly relating to the
distinction between analytic and synthetic knowledge. Profane science, indeed,
is essentially and exclusively analytical; it never considers principles,
losing itself instead in the details of phenomena, of which the indefinite and
indefinitely changing multiplicity are for it truly inexhaustible, such that it
can never arrive at any real or definitive result as far as knowledge is
concerned; it keeps solely to phenomena themselves, that is, to exterior
appearances, and is incapable of reaching the heart of things, for which
Leibnitz had already reproached Cartesian mechanism. This is moreover one of
the reasons by which modern ‘agnosticism’ is explained, for, since there are
things that can be known only synthetically, whoever; proceeds by analysis
alone is thereby led to declare such things ‘unknowable’, since in this respect
they really are so, just as those who kep to the analytic view of the
indefinite believe its indefinitude to be absolutely inexhaustible, whereas in
reality it is so only analytically. It is true that synthetic knowledge is
essentially what one might call ‘global’ knowledge, as is the knowledge of a
continuous set or an indefinite sequence the elements of which are not and
cannot be set out distinctly; but, apart from the fact that this knowledge is
ultimately all that really matters, one can always—since everything is
contained in it in principle—descend from it to the consideration of such
particular things as one might wish, just as, if an indefinite sequence, for
example, is given synthetically through the knowledge of its law of formation,
one can as occasion arises always calculate any of its particular terms, while
on
the contrary when one
takes as one’s starting-point these same particular things considered in and
of themselves, and in all their indefinite detail, one can never rise to the
level of principles; and, as we said at the beginning, it is in this regard
that the method and point of view of traditional science is as it were inverse
to that of profane science, as synthesis itself is to analysis. Moreover, we
have here only an application of the obvious truth that, although the ‘lesser’
can be drawn from the ‘greater’, one can never cause the ‘greater’ to come from
the ‘lesser’; nevertheless, this is precisely what modern science claims to do,
with its mechanistic and materialistic conceptions and its exclusively
quantitative point of view; but it is precisely because this is impossible that
such science is, in reality, incapable of giving the true explanation of
anything whatever.[132]
The
preceding considerations implicitly contain the solution to
all problems of the sort raised by Zeno of Elea in his famous arguments against
the possibility of motion, or at least in what appear to be such when one takes
the arguments only as they are usually presented; in fact, one might well doubt
whether this was really their true significance. Indeed, it is rather unlikely
that Zeno really intended to deny motion; what is more probable is that he
merely wished to prove the incompatibility of the latter with the supposition,
accepted notably by the atomists, of a real, irreducible multiplicity existing
in the nature of things. It was therefore originally against this very
multiplicity so conceived that these arguments origiiially must have been
directed; we do not say against all multiplicity, for it goes without saying
that multiplicity also exists within its order, as does motion, which,
moreover, like every kind of change, necessarily supposes multiplicity. But
just as motion, by reason of its character of transitory and momentary
modification, is not self-sufficient and would be purely illusory were it not
linked to a higher principle transcendent with respect to it, such as the
‘unmoved mover’ of Aristotle, so multiplicity would truly be nonexistent were
it to be reduced to itself alone, and did it not proceed from unity, as is
reflected mathematically in the formation of the sequence of numbers, as we
have seen. What is more, the supposition of an irreducible multiplicity
inevitably excludes all real connections between the elements of things, and
consequently all continuity as well, for the latter is only a particular case
or special
form of such connections.
As we have already said above, atomism necessarily implies the discontinuity of
all things; ultimately, motion really is incompatible with this discontinuity,
and we shall see that this is indeed what the arguments of Zeno show.
Take,
for example, the following argument: an object in motion can never pass from
one position to another, since between the two there is always an infinity of
other positions, however close, that must be successively traversed in the
course of the motion, and, however much time is employed to traverse them, this
infinity can never be exhausted. Assuredly, this is not a question of an
infinity, as is usually said, for such would have no real meaning; but it is no
less the case that in every interval one may take into account an indefinite
number of positions for the moving object, and these cannot be exhausted in
analytic fashion, which would involve each position being occupied one by one,
as the terms of a discontinuous sequence are taken one by one. But it is this
very conception of motion that is in error, for it amounts in short to
regarding the continuous as if it were composed of points, or of final,
indivisible elements, like the notion according to which bodies are composed
of atoms; and this would amount to saying that in reality there is no continuity,
for whether it is a question of points or atoms, these final elements can only
be discontinuous; furthermore, it is true that without continuity there would
be no possible motion, and this is all that the argument actually proves. The
same goes for the argument of the arrow that flies and is nonetheless
immobile, since at each instant one sees only a single position, which amounts
to supposing that each position can in itself be regarded as fixed and
determined, and that the successive positions thus form a sort of discontinuous
series. It is further necessary to observe that it is not in fact true that a
moving object is ever viewed as if it occupied a fixed position, and that quite
to the contrary, when the motion is fast enough, one will no longer see the
moving object distinctly, but only the path of its continuous displacement;
thus for example, if a flaming ember is whirled about rapidly, one will no
longer see the form of the ember, but only a circle of fire; moreover, whether
one explains this by the persistence of retinal impressions, as physiologists
do, or in any other way, it matters little, for it is no less obvious
that in such cases one
grasps the continuity of motion directly, as it were, and in a perceptible
manner. What is more, when one uses the expression ‘at each instant’ in
formulating such arguments, one is implying that time is formed from a sequence
of indivisible instants, to each of which there corresponds a determined
position of the object; but in reality, temporal continuity is no more composed
of instants than spatial continuity is of points, and as we have already
pointed out, the possibility of motion presupposes the union, or rather the
combination, of both temporal and spatial continuity.
It is
also argued that in order to traverse a given distance, it is first necessary
to traverse half this distance, then half of the remaining half, then half of
the rest, and so on indefinitely,[133]
such that one would always be faced with an indefinitude that, envisaged in
this way, is indeed inexhaustible. Another almost equivalent argument is as
follows: if one supposes two moving objects to be separated by a certain
distance, then one of them, even if traveling faster than the other, will never
be able to overtake the other, for, when it arrives at the point where it would
have met the one in the lead, the latter will be in a second position,
separated from the first by a smaller distance than the initial one; when it
arrives at this new position, the other will be in yet a third position,
separated from the second by a still smaller distance, and so on indefinitely,
in such a way that, despite the fact that the distance between the two objects
is always decreasing, it will never disappear altogether. The essential problem
with these two arguments, as well as with the preceding, consists in the fact
that they all suppose that in order to reach a certain endpoint, all the
intermediate degrees must be traversed distinctly and successively. Now, we are
led to one of two conclusions: either the motion in question is indeed
continuous, and therefore cannot be broken down in this way, since the
continuous has no irreducible elements; or the motion is composed, or at least
may be considered to be composed, of a discontinuous succession of intervals,
each with a determined magnitude, as with the steps taken by a man
walking,[134]
in which case the consideration of these intervals would obviously rule out
that of all the various intermediate positions possible, which would not
actually have to be traversed as so many distinct steps. Besides, in the first
case, which is really that of a continuous variation, the end-point, assumed
by definition to be fixed, cannot be reached within the variation itself, and
the fact that it actually is reached demands the introduction of a qualitative
heterogeneity, which this time does constitute a true discontinuity, and which
is represented here by the passage from the state of motion to that of rest;
this brings us to the question of passage to the limit’, the true meaning of
which still remains to be explained.
THE TRUE
CONCEPTION
OF ‘PASSAGE
TO THE LIMIT’
The consideration of‘passage to the limit’,
we said above, is necessary, if not to the practical applications of the
infinitesimal method, then at least to its theoretical justification, and this
justification is precisely the only thing that concerns us here, for simple
practical rules of calculation that succeed in an as it were empirical’ manner
and without our knowing exactly why, are obviously of no interest from our
point of view. Undoubtedly, in order to perform the calculations, and even to
follow them through to the end, there is in fact no need to raise the question
as to whether the variable reaches its limit, or how it can do so;
nevertheless, if it does not r^ach its limit, such a calculus will only have
value as a simple calculus of approximation. It is true that here we are
dealing with an iridefinite approximation, since the very nature of
infinitesimal quantities allows the error to be rendered as small as one might
wish, without it being possible to eliminate it entirely, since despite the
indefinite decrease, these same infinitesimal quantities never become nothing.
Perhaps one might say that, practically speaking, this is the equivalent of a
perfectly rigorous calculation; but, besides the fact that this is not what
matters to us, such is not in question, can the indefinite approximation itself
retain meaning if, with respect to the desired results, one no longer envisages
variables, but rather fixed and determined quantities? Under these conditions,
one cannot escape the following alternative as far as the results are
concerned: either the
limit is not reached, in which case the infinitesimal calculus is then only
the least crude of various methods of approximation; or the limit is reached,
in which case one is dealing with a method that is truly rigorous. But we have
seen that limits, by their very definition, can never exactly be reached by
variables; how, then, do we have the right to say that they are nonetheless
reached? This can be precisely accomplished, not in the course of the
calculation, but in the results, since only fixed and determined quantities,
like the limit itself, must figure therein, while variables no longer do so;
consequently the distinction between variable and fixed quantities, which is a
strictly qualitative distinction, moreover, is the only true justification for
the rigor of the infinitesimal calculus, as we have already said.
Thus,
let us repeat it again, a limit cannot be reached within a variation, and as a
term of the latter; it is not the final value the variable takes on, and the
idea of a continuous variation arriving at any ‘final value’, or ‘final state’,
would be as incomprehensible and contradictory as that of an indefinite
sequence arriving at a ‘final term’, or of the division of a continuum arriving
at ‘final elements’. Therefore a limit does not belong to the sequence of
successive values of the variable, but it falls outside of this series, and
that is why we said that ‘passage to the limit’ essentially implies a
discontinuity. Were it otherwise, we would be faced with an indefinitude that
could be exhausted analytically, and this can never happen. Here the distinction
we previously established in this regard takes on its full signifia cance,
for we find ourselves in one of those cases in which it is a question of
reaching the limits of a given indefinite quantity, according to an expression
we have already used; it is therefore not without reason that the same word
‘limit’ comes up again, but with another, more specialized meaning, in the
particular case we shall now consider. The limit of a variable must truly
limit, in the general sense of the word, the indefinitude of the states or
possible modifications comprised within the definition of this variable; and
it is precisely for this reason that it must necessarily be located outside of
that which it limits. There can be no question of exhausting this indefinitude
through the very course of the variation by which it is constituted; in
reality, it is a question of passing beyond the domain
of this variation, in
which the limit is not contained, and this is the result that is obtained, not
analytically and by degrees, but synthetically and in a single stroke, in a
manner that is as it were ‘sudden’ and corresponds to the discontinuity
produced in passing from variable to fixed quantities.[135]
Limits
pertain essentially to the domain of fixed quantities; this is why ‘passage to
the limit’ logically demands the simultaneous consideration of two different
and as it were superimposed modalities existing within quantity; it is nothing
other than passage to the higher modality, in which what exists only as the
state of a simple tendency in the lower modality, is fully realized; to use the
Aristotelian terminology, it is a passage from potentiality to actuality,
which assuredly has nothing in common with the simple ‘compensation of errors’
that Carnot had in mind. The mathematical notion of the limit implies by its
very definition a character of stability and equilibrium, which applies to
permanent and definite things, and which obviously cannot be realized by
quantities insofar as one considers them in the lower of the two modalities, as
essentially variable; the limit can therefore never be reached gradually, but
only immediately by the passage from one modality to the other, which alone
allows the omission of all intermediate stages, since it includes and embraces
synthetically all of their indefinitude; in this way, what was and could only
be but a tendency within the variable, is affirme^ and fixed in a real and
definite result. Otherwise, ‘passage to the liinit’ would always be an
illogicality pure and simple, for it is obvious that, insofar as one keeps to
the domain of variables, one cannot obtain the fixity appropriate to limits,
since the quantity previously considered to be variable would precisely have to
lose its transitory and contingent character. The state of variable quantities
is indeed an eminently transitory and as it were imperfect state, since it is
only the expression of a ‘becoming’, as we have likewise found to be the case
with the idea at the root of indefinitude itself,
which, moreover, is
closely linked to the state of variation. The calculation will thus only be
perfect, or truly completed, when it arrives at results in which there is no
longer anything variable or indefinite, but only fixed and determined
quantities; and we have already seen how this can be applied through analogical
transposition beyond the quantitative order—which latter will then have no
more than a symbolic value—and will extend even to that which directly concerns
the metaphysical ‘realization’ of being.
There
is no need to stress the importance that the
issues examined in the course of this study present from the strictly
mathematical point of view, as they contain the solution to all the problems
that have been raised concerning the infinitesimal method, whether regarding
its true significance or its rigor. The necessary and sufficient condition for
arriving at this solution is nothing other than the strict application of true
principles, but these are precisely the principles of which modern
mathematicians, along with all other profane scholars, are completely ignorant.
Ultimately this ignorance is the sole reason for so many of the discussions
that, under these conditions, can be pursued indefinitely without ever reaching
any valid conclusion, but on the contrary only further confuse the question and
multiply the confusions, as the quarrel between the ‘finitists’ and
‘infinitists’ shows only too well. Nevertheless all such discussions would have
been cut short quite easily had the true notion of the metaphysical Infinite
and the fundamental distinction between the Infinite and the indefinite been
set forth clearly and before all else. On this subject Leibnitz himself, who
unlike those who have come after him at least had the merit of frankly facing
certain questions, too often says things that are hardly metaphysical, and are
sometimes even as clearly anti-metaphysical, as are the ordinary speculations
of most modern philosophers; thus it is again this same lack of principles that
prevented him from responding to his adversaries in a satisfying and as it were
definitive way, and which consequently opened the door to all subsequent
discussions. No doubt one can say with Carnot that, ‘if Leibnitz was mistaken,
it was solely in raising doubts as to the exactitude of his own analysis,
so far as he really had
these doubts’;[136]
but even if ultimately he did not, he was nonetheless unable to demonstrate its
exactitude rigorously since his conception of continuity, which is most
certainly neither metaphysical nor logical, prevented him from making the
necessary distinctions and consequently from formulating a precise notion of
the limit, which is as we have shown of chief importance for the foundation of
the infinitesimal method.
From
all of this one can see what significance the consideration of principles can
have even for a specialized science considered in and of itself, and without
any intention of going further in support of this science than the relative and
contingent domain to which the principles are immediately applicable. Of
course, this is what the moderns totally misunderstand, readily boasting as
they do that with their profane conception of science they have rendered the
latter independent of metaphysics, and likewise of theology,[137]
while the truth of the matter is that they have thereby only deprived it of all
real value as far as knowledge is concerned. In addition, once one understands
the need to link science back to principles, it goes without saying that there
should no longer be any reason to stop there, and one will quite naturally be
led back to the traditional conception according to which a particular
science, whatever it might be, is less valuable for what it is in itself than
for the possibility of using it as a ‘support’ for elevating oneself to
knowledge of a higher order.[138]
Our intention here has been to present by way of a characteristic example
an idea of precisely what it would be possible to do, at least in certain
cases, to restore to science, mutilated and distorted by profane conceptions,
its real value and scope, both from the point of view of the relative knowledge
it represents directly, and from that of the higher knowledge to which it can
lead through analogical transposition. In this last respect we have been able
to see, notably,
what may be drawn from
notions such as those of integration and ‘passage to the limit’. Moreover, it
should be said that, more than any other science, mathematics thus furnishes a
particularly apt symbolism for the expression of metaphysical truths to the
extent that the latter are expressible, as those familiar with some of our
other works are aware. This is why mathematical symbolism is used so
frequently, whether from the traditional point of view in general, or from the
initiatic point of view in particular.[139]
But it is of course understood that in order for this to be so it is above all
necessary that these sciences be rid of the various errors and confusions that
have been introduced by the false views of the moderns, and we should be happy
if the present work is at least able to contribute in some way to this end.
INDEX
algebra 64 ni, 94
algorithm 69
Anima Mundi 24 09
Archimedes 33
Aristotelian
sense 76 n/
terminology 126
Aristotle
120 arithmetic 26-27, 64 ni, 89 atomism 51, 53, 63,121
Being 804,20-21, 86
Bernoulli Jean, 16 03, 24 09,34, 41-45, 47-48, 52,
60,105,112 ni
Cantor 16
Carnot 35 ni, 39, 80, 82,90, 101-102,104,109 n3,126
Cartesian(ism)
concept 6, 50
mechanism 110,118
rationalism 66
Cauchy 16 n 2
Cavalieri 50
Chinese language 58
Compagnonnage 129
n2
Coomaraswamy, A. K. 14 n 14 cosmos 24 n9,117
Couturat,
L. 10 n6,39 n7,52 09, 64 ni, 69 ni, 70 n3, 84 n2
denary 58
Descartes 11-12,19 ni, 62 equilibrium 57,75,86, 96-99
Euclid 79
Europe 305
evolution
45
Far-Eastern
cosmology 98 04
tradition 96, 99
fraction (s) 26,29,31 ni, 48-49,
54,56 113 U2
Freycinet,
Ch. de 38 n6, 69 n2, 71 n4, 81 n8,101, ns3—4
Galileo 16 n2
geometry 1,36, 45, 61,72, 94,
129 n2
Grandi, Guido 16 n3
‘Great Man’, (Kabbalistic) 45
Greek
3 n5, 58
Harvey 44
Hebrew 305
Hegelian affirmation 77
helix 13 n 12
Hermetic
figure 6 n 6
‘coagulation’ and
‘solution’ 97
n2
Huygens
103 ni
Indian 305
individualism 5
Infinity
degrees of 41-46
symbol of 86
Kabbalah 2,3 n 5
Lagrange 81
Leeuwenhoeck 44
Leibnitz 4-6, 9,11-12,15-24, 31-53,
60-85, 90, 94, 98 n3, 100-110,112 ni, 118,122 ni, 128
Leibnitzian
‘virtuality’ 85 luz, Judaic concept of 45
Mallebranche, R.P. 67 n 5 Marquis de l’Hôspital 36
n4, 79 n3
mathematician(s) 37, 71-72,103 mathematics 1,3-4,36,39,
66-67, 89-90,95,106-107,130
metaphysics 10, 20 n3,
42,129 Middle Ages 3 n5, 5,129 ns 2-3
Newton 5
Nieuwentijt 103
Non-Being 86-87, 99 numbers
incommensurable 28-29, 57 03
whole 16,25-28,54-57,86,89, 93-94,
99,116
ontology 20 n3, 40
Pascal 39 n 9 physics 36,
61, 67 n5 physiologists 121 Platonism 1,14 ni4 principiaLnumerations’ 3 n5
Pythagorean(s) 1-2, 45, 59
Quinta Essentia 6
n6
Renouvier 10 n6,18 n5, 63
Rosicrucian 5,32 n4
Rota Mundi 6
n 6
Saint Thomas Aquinas 20-21
Scholasticism 11
Scholastic distinction 11 doctrines 5
secundum quid 12 n 8 sense 76 n7
Scholastics 8,19-20, 42, 62, 96
Schulenburg 68 n7
Sephiroth
305
Spinoza 8 n2,18 n6
stars 33, 44
Stoics 24 n 9
Tao Te Ching 58
Tetraktys 59
theology 129
Trinity 129 n2
Universal All 7-9,18 n6, 22, 24 n9, 86
Varignon 12 n 11,16 n3,34 ns6-8, 36 n3, 41 n2, 48 n5,
49 n6, 6 6 n3, 67 n5, 74 n2,75 n4, 80 n4
Wallis 16 n 3
Wolf, V. Cl. Christian 74 n 1
yang 98
n4
yin 98
n4
Zeno of Elea 120-121
Zero 83-88
The
Metaphysical Principles
of the Infinitesimal Calculus
René Guenon
|
R |
ené Guénon (1886-1951) was
one of the great luminaries of the twentieth century, whose critique of the modern
world has stood fast against the shifting sands of intellectual fashion. * His
extensive writings, now finally available in English, are a providential
treasure-trove for the modern seeker: while pointing ceaselessly to the
perennial wisdom found in past cultures ranging from the Shamanistic to the
Indian and Chinese, the Hellenic and Judaic, the Christian and Islamic, and
including also Alchemy, Hermeticism, and other esoteric currents, they direct
the reader also to the deepest level of religious praxis, emphasizing the need
for affiliation with a revealed tradition even while acknowledging the final
identity of all spiritual paths as they approach the summit of spiritual
realization-
Guenon’s early and abiding
interest in mathematics, like that of Plato, Pascal, Leibnitz, and many other
metaphysicians of note, runs like a scarier thread throughout his doctrinal
studies. In this late text published just five years before his death, Guénon devotes
an entire volume to questions regarding the nature of limits and the infinite
with respect to the calculus both as a mathematical discipline and
as symbolism for the initiatic path. This book therefore extends and
complements die geometrical symbolism he employs in other works, especially
The Symbolism of the Cross, The Multiple Stales of the Being, and Symbols
of Sacred Science.
According
to Guénon, the
concept 'infinite number' is a contradiction in terms. Infinity is a metaphysical
concept ar a higher level of reality than that of quantity, where all
that can be expressed is the indefinite, not the infinite. But although
quantity is the only level recognized by modern science, the numbers that
express it also possess qualities, their quantitative aspect being
merely their outer husk Our reliance today on a mathematics of approximation
and probability only further conceals die qualitative mathematics’ of the
ancient world, which comes to us most directly through the Pythagorean-Platonic
tradition.
The
Collected Works of René Guenon
brings together the writings of one of the greatest prophets of our time, whose
voice is even more important today than when he was alive.
Huston
Smith. The World’s Religion
SP
SOPHIA PERBNNIS
[1] See The Reign of Quantity and the Signs of
the Times.
[2] See Miscellanea, pt.'j, chap. i. Ed.
[3] The French calculhas the double meaning
of‘calculus’ and ‘calculation’. Ed.
[4] It is the same with certain ‘pseudo-esoterists’,
who know so little of what they wish to speak about that they likewise never
fail to confuse the two in the fanciful ravings they presume to substitute for
the traditional science of numbers!
[5] Hebrew and Greek are two examples, and Arabic
was equally so before the introduction of the use of numerals of Indian origin,
which then, being more or less modified, passed from there to Europe in the
Middle Ages; in this connection one can note that the word cipher’ [French chiffre, ‘numeral’]
is itself none other than the Arabic sifr, though this word is in
reality only the designation for zero. On the other hand, it is true that in
Hebrew saphar means ‘to count’ or ‘to number’, and at the same time ‘to
write’, whence sepher, ‘scripture’ or ‘book’ (in Arabic sifr,
which designates in particular a sacred book), and sephar, ‘numeration’
or ‘calculation’; from this last word also comes the designation of the Sephiroth
of the Kabbalah, which are the principial ‘numerations’ assimilated to the
divine attributes.
[6] The undeniable mark of this origin is to be
found in the Hermetic figure placed by Leibnitz at the head of his treatise De
Arte Combinatoria: it is a representation of the Rota Mundi, in
which, at the center of the double cross of the elements (fire and water, air
and earth) and qualities (hot and cold, dry and moist), the quinta essentia
is symbolized by a rose with five petals (corresponding to ether considered in
itself and as principle of the four other elements); naturally, this
‘signature’ has been passed over completely by all academic commentators.
[7] It is in a rather similar sense that Spinoza
later used the expression ‘infinite in its kind’, which naturally gives rise to
the same objections.
[8] One could say further that it leaves outside
itself only the impossible, which, being a pure nothing, could not limit it in
anyway.
[9] This is equally true for determinations of a
universal and no longer simply general order, including even Being itself,
which is the first of all determinations; but it goes without saying that this
consideration does not enter into the uniquely cosmological applications we are
dealing with in the present study.
[10] In response to any astonishment that might arise
on account of our use of the expression ‘semi-profane’, we will say that it is
justified, in a very precise manner, by the distinction between effective
initiation and merely virtual initiation, which we shall have to explain on
another occasion. [See Perspectives on Initiation, chap. 30. Ed.]
[11] As a characteristic example, let us here cite
the conclusion of L, Couturat’s thesis De l’infini
mathématique, in
which he tried to prove the existence of an infinity of number and of
magnitude by stating that his intention had been to show thereby that, ‘in
spite of neo-criticism [that is, the theories of Renouvier and his school], an
infinitist metaphysics is plausible’!
[12] One should, in all logical rigor, distinguish
between a ‘false notion’ (or, if one prefer, ‘pseudo-notion’) and an ‘incorrect
notion’; an ‘incorrect notion’ is one that does not correspond adequately to
reality, though it does, however, correspond in a certain measure; on the
contrary, a ‘false notion’ is one that implies contradiction— as is the case
here—and is therefore not really a notion, not even an incorrect one, though it
appears as such to those who do not perceive the contradiction; for, expressing
only the impossible, which is the same as nothingness, it corresponds to
absolutely nothing; an ‘incorrect notion’ can be rectified, but a ‘false
notion’ can only be rejected altogether.
[13] These words seem to refer to the Scholastic secundum
quid, and thus it could be that the primary intention of the sentence cited
had been to criticize indirectly the expression infinitum secundum quid.
[14] Principes de la Philosophie, i,
26.
[15] Ibid., i, 27.
[16] Thus in his correspondence with Leibnitz on the
subject of the infinitesimal calculus, Varignon uses the terms ‘infinite’ and
‘indefinite’ indifferently, as if they were virtually synonymous, or at the
very least as if it were unimportant, so to speak, that the one be taken for
the other, even though it is on the contrary the difference in their meanings
that should have been regarded as the essential point in all these discussions.
[17] Again, we should note that, as we have explained
elsewhere, such a cycle is never truly closed, and it seems so only so long as
one places oneself in a perspective that does not allow one to perceive the
distance really existing between its extremities, just as a helix situated
along a vertical axis appears as a circle when it is projected on a horizontal
plane.
[18] It is thus of no use to say that space, for
example, could be limited only by something still spatial, such that space in
general could no longer be limited by anything; it is on the contrary limited
by the very determination that constitutes its own nature as space and that
leaves room, outside of it, to all the non-spatial possibilities.
[19] Cf. the remark of A.K. Coomaraswamy on the Platonic
concept of‘measure’, which we have cited elsewhere (The Reign of Quantity
and the Signs of the Times, chap. 3): the ‘non-measured’ is that which has
not yet been defined, which is to say, in short, the indefinite, and it is at
the same time and by the same token that which is only incompletely realized
within manifestation.
[20] ‘In spite of my infinitesimal calculus,’ he
wrote, ‘I do not admit a true infinite number, though I do confess that the
multitude of things surpasses all finite numbers, or rather all number.’
[21] This was done by Cauchy, who attributed the
argument, moreover, to Galileo (Sept leçons de Physique générale, third lesson).
[22] Already at the time of Leibnitz, Wallis was
envisaging spatiaplus quam infinita [more than infinite space]; this
opinion, denounced by Varignon as implying contradiction, was equally held by
Guido Grandi in his book De Infinitis infinitorum [Concerning the
Infinite of infinites]. On the other hand, Jean Bernoulli, in the course of his
discussions with Leibnitz, wrote, Si dantur termini infiniti, dabitur etiam
terminus infinitesimus (non dico ultimus) et qui eum equuntur [If the limits of
[23] Thus Renouvier thought that number is applicable
to everything, at least ideally, that is, that everything is ‘numerable’ in itself,
even if we are in fact incapable of ‘numbering’ it; he therefore completely
misunderstood the meaning Leibnitz gives to the notion of ‘multitude’, and he
was never able to understand how the distinction between the latter and number
allows one to escape the contradiction of an ‘infinite number’.
[24] We have said, however, that every particular or
determined thing, whatever it might be, is limited by its very nature, but
there is absolutely no contradiction here: indeed, it is limited by the
negative side of this nature (for, as Spinoza has said, omnis determinatio
negatio est [all determination is a negation]), that is, its nature considered
insofar as it excludes other things and leaves them outside of itself, so that
finally it is really the coexistence of these other things that limits the
thing in consideration; this is moreover why the universal All, and it alone,
cannot be limited by anything.
[25] Descartes spoke solely of‘clear and distinct’
ideas; Leibnitz specified that an idea can be clear without being distinct, in
that it only allows one to recognize it and to distinguish it from all other
things, whereas a distinct idea is that which is not only ‘distinguishing’ in
this sense, but ‘distinguished’ in its elements; moreover, an idea can be more
or less distinct, and the adequate idea is that which is so completely and in
all its elements; but, while Descartes was of the opinion that one could have
‘clear and distinct’ ideas of all things, Leibnitz on the contrary believed
that mathematical ideas alone can be adequate, their elements being as it were
of a definite number, whereas all other ideas enclose a multitude of elements,
of which the analysis can never be completed, so that they will always remain
partially confused.
[26] We will cite only one text among others, which
is particularly clear in this regard: Qui diceret aliquam multitudinem esse infinitum, non diceret earn esse
numerum, vel numerum habere; addit etiam numerus super multitudinem rationem
mensurationis. Est enim numerus multitudo mensurata per unum ... et propter hoc numerus ponitur species quantitatis
discretae, non autem multitudo, sed est de tran- scendentibus [If one were to
say that some multitude is infinite one would not be saying that it is a number
or has a number, for number adds to multitude the idea of measure. For a number
is multitude measured by one ... and for this reason number is categorized as
a species of discrete quantity but multitude is not, but rather is one of the
transcendentals (Saint Thomas Aquinas, in Physics, in, 1.8).
[27] We know that the Scholastics, even in the
properly metaphysical part of their doctrines, never went beyond the
consideration of Being, so that for them metaphysics is in fact reduced solely
to ontology.
[28] Système nouveau de la nature et de la communication des substances.
[29] The French word unité means both ‘unit’ and
‘unity’, as Guénon himself explains. Ed.
[30] Observatio quod rationes sive proportiones
non habeant locum circa quantitates nihilo minores, et de vero sensu Methodi infinitesimalis [An Observation that Calculations and Proportions Do Not Apply to
Diminishing Quantities, and About the True Understanding of the Infinitesimal
Method], in the Acta Eruditorum of Leipzig, 1712.
[31] Cf. ibid., Infinitum continuum vel discretum proprie nec unum, nec totum, nec quantum est [The continuous or
discrete infinite is properly speaking neither one nor a whole nor a quantity],
where the expression nec quantum seems to imply that for him, as we
indicated above, the ‘indefinite multitude’ must not be conceived of
quantitatively, unless by quantum he had meant solely a definite
quantity, as the so- called‘infinite number’would have been, the contradiction
of which he had already demonstrated.
[32] On this point, see further The Multiple States
of the Being, chap. 1.
[33] Letter to Jean Bernoulli.—Leibnitz here rather
gratuitously attributes to the ancients in general an opinion that in reality
was held by only some of them; he obviously had in mind the theory of the
Stoics, who conceived of God as uniquely immanent, identifying him with the Anima
Mundi. It goes without saying, moreover, that it is here a question only
of the manifested Universe, that is, the cosmos, and not of the universal All,
which comprehends all possibilities, the non-mani- fested as well as the
manifested.
[34] Letter to Jean Bernoulli, June 7,1698.
[35] ‘Whole numbers’ (nombres entiers) is
simply what is nowadays termed ‘integers’ which is to say that the term ‘whole
number’ (even though everyone will immediately understand what is meant) is not
currently idiomatic. It appears, moreover, that when Guénon speaks of nombres entiers, he means the positive integers, or so-called
natural numbers. Ed.
[36] It will be seen in what follows, concerning the
geometric representation of negative numbers, why we must take into
consideration here only half a straight line; besides, the fact that the series
of numbers develops only in a single direction, as we said earlier, should
already suffice to indicate the reason.
[37] This will be made still clearer when we come to
speak of negative numbers.
[38] Note that we did not say the points composing or
constituting the line, which would betray a false understanding of continuity,
as considerations we shall later explain will show.
[39] Nova Methodus pro maximis et minimis, itemque tangentibus, quae necfractas
nec irrationales quantitates moratur, et singulare pro Ulis calculi genus [A New
Method for Greatest and Smallest Quantities as Well as Tangents, Which Does Not
Involve Either Fractional or Irrational Quantities, and a Unique Kind of
Calculus For Them], in the Acta Eruditorum of Leipzig, 1684.
[40] De Geometria recondita et Analysi indivisibilium atque infinitorum
[On the Hidden Geometry and the Analysis of Indivisible and Infinite Quantities],
1686. Subsequent works all relate to the solving of particular problems.
[41] First in his correspondence, and then in Historia
et origo Calculi
differntialis [The History and Origin of
Differential Calculus], 1714.
[42] In Rosicrucian language one would say that this,
as much as and even more than the failure of his projects of characteristica
universalis, proves that even if he did
[43] Letter to Varignon, February 2,1702.
[44] Imaginary roots are roots of negative numbers;
later we shall speak more of the question of negative numbers and the logical
difficulties to which they give rise.
[45] Letter to Varignon, April 14,1702.
[46] Mémoire already cited above, in the Acta Eruditorum of Leipzig, 1712.
[47] It is in this consideration of practical utility
that Carnot believed he had found a sufficient justification; it is obvious
that from the time of Leibnitz to him, the ‘pragmatist’ tendency of modern
science had already become much more pronounced.
[48] Previously cited Mémoire, in the Acta Eruditorum of Leipzig,
1712.
[49] Previously cited letter to Varignon, February
2,1702.
[50] Letter to Marquis de 1’Hospital, 1693.
[51] ‘Considerations sur la différence qu’il y a entre l’Analyse ordinaire et le nouveau Calcul
des transcendantes’, in the Journal des Sçavans, 1694.
[52] Ch. de Freycinet, De
VAnalyse infinitésimale, pp 21-22. The author adds: ‘But the first
expression [that of infinitely small] having prevailed in the language, we
believe it should be retained.’ This is assuredly quite an excessive scruple,
for usage does not suffice to justify the mistakes and improprieties of
language, and, if one never dared to raise oneself above abuses of this kind,
one could never even try to introduce more exactitude and precision to terms
than that which they carry in current usage.
[53] See especially L. Couturat, De l'infini mathématique, p 265, note: ‘One can logically constitute the infinitesimal
calculus on the sole notion of the indefinite....’ It is true that the use of
the word ‘logically’ here implies a reservation, for it is opposed, for the
author, to ‘rationally’, which is moreover a rather strange terminology; the
admission is nonetheless interesting to keep in mind.
[54] Réflexions sur la
Métaphysique du Calcul infinitésimal, py, note; cf. ibid., p2o. The title of this work is scarcely justified, for in reality there
is not to be found in it the least idea of a metaphysical order.
[55] Pascal’s overly celebrated conception of ‘two
infinities’ is metaphysically absurd, and it is again only the result of a
confusion of the infinite with the indefinite, the latter being taken in the
two opposite directions of increasing and decreasing magnitude.
[56] Letter to Jean Bernoulli, November 18,1698.
[57] Previously cited letter to Varignon, February
2,1702.
[58] On this subject, see The Multiple States of
the Being.
[59] Letter of July 23,1698.
[60] See The King of the World, chap. 7.
[61] Previously cited letter to Jean Bernoulli,
November 18,1698.
[62] The Symbolism of the Cross, chap. 6. On
the distinction between ‘possibles’ and ‘compossibles’, on which the notion of
the ‘best of worlds’ further depends, cf. The Multiple States of the Being,
chap. 2.
[63] Monadologie, 67; cf. ibid., 74.
[64] Monadologie, 65.
[65] Letter to Jean Bernoulli, July 12-22,1698.
[66] Previously cited letter of July 23,1698.
[67] Letter of July 29,1698.
[68] On this subject, see The Reign of Quantity
and the Signs of the Times.
[69] The Symbolism of the Cross, chap. 16.
[70] Cf. L. Couturat, De l'infini mathématique, P467: ‘The sequence of natural numbers is given
entirely by its law of formation, as moreover is the case with all other
infinite sequences and series: in general a formula of recurrence suffices to
define them entirely, such that their limit or sum (when it exists) is on that
account completely determined.... It is thanks to this law of formation of the
sequence of natural numbers that we have an idea of every whole number, and in
this sense they are altogether given by this law.’—One can indeed say that the
general formula expressing the nth term of a sequence contains,
potentially and implicitly, though not actually and distinctly, all the terms
of the sequence, since any of them can be derived from it by giving to n
the value corresponding to the position the term occupies in the sequence; but,
contrary to what Couturat thought, this is certainly not what Leibnitz meant to
say ‘when he maintained the actual infinity of the sequence of natural
numbers.’
[71] According to the definition of inverse numbers,
the unit appears first in the form 1 and then again in the form h, such that
(i) (Vi) = 1; but, as on the other hand Vi = 1, it is the same unit that is
thus represented in two different forms, and it is consequently, as we said
above, its own inverse.
[72] We say indivisible because whenever one of the
two numbers forming such a pair exists, the other also necessarily exists by
that very fact.
[73] It goes without saying that the incommensurable
numbers, in relation to magnitude, are necessarily interspersed among the
ordinary numbers, which are whole or fractional according to whether they are
greater or smaller than the unit; this demonstrates, moreover, the geometrical
correspondence we pointed out earlier, as well as the possibility of defining
such a number by two convergent sets of commensurable numbers, of which it is
the common limit.
[74] Meditatio nova de natura anguli contactas et osculi, horumque usu in practica
Mathesi ad figuras faciliores
succedaneas difficilioribus substituendas [A New
Reflection on the Nature of Angles of Contact and Tangency and on the Use of
These in Practical Mathematics for Substituting Easier Figures for the More Difficult],
in the Acta Eruditorum of Leipzig, 1686.
[75] Cf. L. Couturat, De l’infini mathématique, P140: ‘In general, the principle of continuity has
no place in algebra, and cannot be invoked to justify the algebraic
generalization of number. Not only is continuity by no means necessary to
speculations concerning general arithmetic, it is repugnant to the spirit of
the science, and to the very nature of number. Number, indeed, is essentially
discontinuous, as are nearly all its arithmetical properties.... One therefore
cannot impose continuity on algebraic functions, however complicated they might
be, since the whole numbers, which furnish their elements, are discontinuous,
“jumping”, as it were, from one value to the next without any possible
transition.’
[76] Previously cited letter to Varignon, February
2,1702.
[77] See The Multiple States of the Being,
chap. 11.
[78] From the same letter to Varignon.—The first
explanation of the ‘law of continuity’ had appeared in the Nouvelles de la République des Lettres in July of 1687, under this rather significant title, and from the
same point of view: Principium quoddam generale non in Mathematicis tantum sed et Physicis utile, cujus ope ex con- sideratione Sapientiae Divinae
examinantur Naturae Leges, qua occasione nata cum R.P. Mallebranchio
controversia explicatur, et quidam Cartesianorum errores notan- tur [A
Certain General Principle, Useful Not Only In Mathematics But In Physics Also,
By Which the Laws of Nature are Examined In Reference to Divine Wisdom, and By Which
the Controversy Started By R. P. Malebranche Is Explained and Some Errors of the Cartesians Are
Pointed Out],
[79] Specimen Dynamicum pro admirandis Naturae
Legibus circa corporum vires et mutuas actiones detegendis et ad suas causas revocandis [A Dynamic Specimen for
Studying the Laws of Nature Regarding the Forces of Bodies and Discovering
Their Interactions, and For Tracing Their Causes], Part II.
[80] Letter to Schulenburg, March 29,1698.
[81] L. Couturat, De l’infini
mathématique, introduction,
p23.
[82] Ch. de Freycinet, De
l’Analyse infinitésimale, preface, p 8.
[83] L. Couturat, De l’infini
mathématique, p268,
note.—This is the point of view expressed, notably, in the Justification du Calcul des infinitésimales par celui de l’Algèbre ordinaire.
[84] Ch. De Freycinet, De
l’Analyse infinitésimale, pi8.
[85] It would be more exact to say that one of them
can come closer and closer to the other since only one of the objects is
variable, while the other is essentially fixed; thus, precisely by reason of
the definition of the limit, their coming together can in no way be considered
to constitute a reciprocal relation, in which the two terms would be as it were
interchangeable; moreover, this irreciprocity implies that their difference is
of a properly qualitative order.
[86] Ibid., p 19.
[87] See The Reign of Quantity and the Signs of
the Times, chap. 4.
[88] Epistola ad V. Cl. Christianum Wolfium, Professorem
Matheseos Halensem, circa Scientiam Infiniti
[Letter to V.C1. Christian Wolf, Mathematics Professor Halensem, concerning the
Science of the Infinite], in the Acta Eruditorum of Leipzig, 1713.
[89] Previously cited letter to Varignon, February
2,1702.
[90] Specimen Dynamicum, previously cited
above.
[91] Justification du Calcul des infinitésimales par celui de l’Algèbre ordinaire, note added to the letter from Varignon to Leibnitz of May 23,1702,
in which it is mentioned as having been sent by Leibnitz to be inserted in the
Journal de Trévoux. Leibnitz takes the word ‘continual’ in the sense
of‘continuous’.
[92] ‘Epistola ad V. Cl. Christianum Wolfium’, previously cited above.
[93] Initia Rerum Mathematicarum Metaphysica [The
Metaphysical Principles of Mathematicals]. Leibnitz’s exact words are genus
in quasi-speciem oppositam desinit, and use of the singular expression
‘quasi-species’ seems at the very least to indicate a certain difficulty in
giving a more plausible appearance to such a statement.
[94] The words ‘actuality’ and ‘potentiality’ are of
course taken here in their Aristotelian and Scholastic sense.
[95] For Leibnitz, °/o = i, since, as he says, ‘one
nothing is the same as another’; but as (o)(n) is also equal to o, for any
value of n, it is obvious that one could just as well write °/o = n,
and this is why the expression °/o is generally thought of as representing
what is called an‘indeterminate form’.
[96] The difference between this and the comparison
of the grain of sand is that as soon as one speaks of ‘vanishing quantities’,
it is necessarily a question of variable quantities, and no longer of fixed and
determined quantities, however small one might suppose them to be.
[97] Letter to Marquis de 1’Hospital, June
14-24,1695.
[98] Previously cited letter to Varignon, February
2,1702.
[99] Réflexions sur la
Métaphysique du Calcul infinitésimal, P36.
[100] On this subject, see the preceding note.
[101] With this difference, namely that for him the
ratio °/o is not indeterminate, but always equal to i, as we pointed out
earlier, whereas in fact the value in question differs in each case.
[102] Cf. Ch. de Freycinet, De l’Analyse infinitésimale, PP45-46: ‘If the increases are reduced to the state of pure zeros,
they will no longer have any meaning. Their property is to be, not rigorously
null, but indefinitely decreasing, without ever being confounded with zero, in
virtue of the general principle that a variable can never coincide with its
limit.’
[103] In ordinary notation this would be represented
by the formula (o)(oo) =1; but in fact the form %o is again, like °/o, an
‘indeterminate form’, and one could write (o)(oo) = n, where n
stands for any number, which moreover shows that, in reality, o and oo cannot
be regarded as representing determined numbers; we shall return to this point
later. In another respect, one could remark that (o)(oo) is for the ‘limits of
sums’ of the integral calculus what °/o is for the ‘limits of ratios’ of the
differential calculus.
[104] Indeed, the arguments of L. Couturat in his
thesis De l’infini mathématique rest, in large part, on this postulate.
[105] From this it further results that zero cannot be
considered a limit in the mathematical sense of the word, for by definition a
true limit is always a quantity; moreover, it is evident that a quantity that
decreases indefinitely has no more of a limit than does a quantity that
increases indefinitely, or at least that neither can have any other limits than
those that necessarily result from the very nature of quantity as such, which
is a rather different use of the word ‘limit’ although there is a certain
connection between the two meanings, as will be shown later; mathematically,
one can speak only of the limit of the ratio of two indefinitely increasing or
indefinitely decreasing quantities, and not of the limit of these quantities
themselves.
[106] On this subject, see the preceding note.
[107] On this subject, see The Multiple States of
the Being, chap. 3.
[108] This is why the point can in no way be
considered as constituting an element or part of length, as we said earlier.
[109] See The Symbolism of the Cross, chap. 16.
[110] Similarly, by analogous transposition, all the
indefinite multiplicity of the possibilities of manifestation is contained,
‘eminently’ and in principle, within pure Being, or the metaphysical Unit.
[111] This will appear completely obvious if, in
conformity with the general law of formation for the sequence of numbers, one
represents this passage by the formula 0 + 1 = 1.
[112] ‘Note sur les quantités
négatives’, placed at the end of the Réflexions sur la Métaphysique du Calcul infinitésimal, pi/3.
[113] See The Reign of Quantity and the Signs of
the Times, chap. 4. One might wonder whether there is not to be found some
sort of unconscious memory, as it were, of this qualitative character in the
fact that mathematicians still sometimes designate numbers taken ‘with their
sign’ that is, considered to be positive or negative, by the name of‘qualified
numbers’, although they otherwise do not seem to attach any very clear meaning
to this expression.
[114] If one considers the ordinary notion of
centripetal and centrifugal forces, one will easily see that the first fall
under the category of compressive forces, the second under that of expansive
forces; likewise, frictional force can be assimilated to the expansive forces,
since it is exerted away from its point of application, and an impulse or
impact can be assimilated to the compressive forces, since it is on the
contrary exerted toward its point of application; but if one envisages things
with respect to the point of emission, the inverse will be true, and this is
moreover demanded by the law of polarity. In another domain, Hermetic
‘coagulation’ and ‘solution’ also correspond to compression and expansion,
respectively.
[115] When we speak thus of the principle of
sufficient reason, we of course have in mind only the principle in itself,
apart from any of the specialized and more or less contestable forms that
Leibnitz or others may have wished to give it.
[116] This formula corresponds exactly to the
conception in Far-Eastem cosmology of the equilibrium of the two complementary
principles of yang and yin.
[117] Fragment dated from March 26,1676.
[118] Carnot, Réflexions sur la
Métaphysique du Calcul infinitésimal, pi9.
[119] Ch. de Freycinet, De l’Analyse infinitésimale, preface, pviii.
[120] Cf. Ch. de Freycinet, ibid., p2.2.0: ‘The equations Carnot called “imperfect” are properly speaking
unfulfilled equations, or equations of transition, which are rigorous insofar
as they are made to serve only for the calculation of limits; they would be
absolutely inaccurate, on the contrary, if their limits did not actually have
to be found. In order for there to be no doubt as to the value of the ratios
through which
[121] Letter to Huygens, October 1-11,1693.
[122] By ‘solicitation’ is meant that which is
commonly designated by the term ‘acceleration’.
[123] Responsio ad nonnullas difftcultates a Dn.
Bernardo Nieuwentijt circa Metho- dum differentialem seu infinitesimalem notas [The Answer to Several
Difficulties Raised by Mr Bernard Nieuwentijt About the Differential or
Infinitesimal Method], in (he Acta Eruditorum of Leipzig, 1695.
[124] Cf. The Symbolism of the Cross, chap. 12.
[125] Cf. The Multiple States of the Being, chap. 1.
[126] In accordance with common usage, we reserve the
denomination ‘infinitesimal’ for indefinitely decreasing quantities, to the
exclusion of indefinitely increasing quantities, which, for the sake of
convenience, we can call simply ‘indefinite’; it is rather strange that Carnot
brought both together under the name of‘infinitesimal’, contrary not only to
common usage but even to the obvious origin of the term. While we shall
continue to use the word ‘infinitesimal’ in the sense just given, we cannot
refrain from pointing out that the term has one serious shortcoming, namely
that it is clearly derived from the word ‘infinite’, which renders it scarcely
adequate to the idea it really expresses; to be able to use it without any
drawbacks, its origin must be forgotten, so to speak, or at least accorded a
solely‘historical’ character, as arising from Leibnitz’s conception
of‘well-founded fictions’.
[127] See Leibnitz, Discours de
Métaphysique, chap.
18; cf. The Reign of Quantity and the Signs of the Times, chap. 14.
[128] The terms ‘integral’ and ‘integration’, which
have prevailed in usage, are not Leibnitz’s, but Jean Bernoulli’s; in their
place Leibnitz used only the words ‘sum’ and ‘summation’, with the drawback
that these terms seem to indicate an analogy between the operation in question
and the formation of an arithmetical sum; we say only that they seem to do so,
for it is quite certain that the essential difference between the two
operations could not have escaped Leibnitz.
[129] Or by a fraction of this magnitude, which
matters little, since the fraction would then constitute a secondary, smaller
unit that is substituted for the first in the case in which division by the
original magnitude cannot be carried out exactly; and, in order to obtain an
exact, or least a more exact, result, one instead uses this fraction.
[130] Here, and in what follows, it should be
understood that we take the terms ‘analysis’ and ‘synthesis’ in their true and
original sense, and one must indeed take care to distinguish this sense from
the completely different and quite improper sense in which one currently speaks
of ‘mathematical analysis’, according to which integration itself, despite its
essentially synthetic character, is regarded as playing a part in what one
calls ‘infinitesimal analysis’; it is for this reason, moreover, that we prefer
to avoid using this last expression, availing ourselves only of those of‘the
infinitesimal calculus’ and ‘the infinitesimal method’, which lead to no such
equivocation.
[131] On this analogical application of the notion of
integration, cf. The Symbolism of the Cross, chaps. 18 and 20.
[132] On this last point, one can again refer to the
considerations set forth in The Reign of Quantity and the Signs of the
Times.
[133] This corresponds to the successive terms of the
indefinite series Vi + Vz + V4 + V8 + ... = 2, used by Leibnitz as an example
in a passage already cited above.
[134] In reality, the motions comprising his walking
are indeed continuous, like any other motion, but the points where he touches
the ground form a discontinuous sequence, such that each step marks a
determined interval, and the distance traversed can thus be broken down into
such intervals, the ground not being touched at any intermediate points.
[135] This ‘sudden’ or ‘instantaneous’ character could
be compared, by way of an analogy from the order of natural phenomena, to' the
example we gave above concerning the breaking of the rope: the rupture itself
is also a limit, namely of the tension, but it is by no means comparable to
tension, whatever the degree.
[136] Réflexions sur la
Métaphysique du Calcul infinitésimal, P33.
[137] We recall somewhere having seen a contemporary
‘scientist’ who was indignant at the fact that in the Middle Ages, for
example, the Trinity had been spoken of in connection with the geometry of the
triangle; he probably did not suspect that this is still the case today in the
symbolism of the ‘Compagnonnage’.
[138] For an example on this subject, see The
Esoterism of Dante, chap. 2, on the esoteric or initiatic aspect of the
‘liberal arts’ of the Middle Ages.
[139] On the reasons for the very special value of
mathematical symbolism, numerical as well as geometric, one may refer
particularly to the explanations given in The Reign of Quantity and the
Signs of the Times.