R.A. Aronov - The Pythagorean syndrom, Historia i Filozofia Matematyki

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50 RUSSIAN STUDIES IN PHILOSOPHY
Russian Studies in Philosophy
, vol. 41, no. 2 (Fall 2002), pp. 50–69.
© 2003 M.E. Sharpe, Inc. All rights reserved.
1061–1967/2003 $9.50 + 0.00.
R.A. A
RONOV
The Pythagorean Syndrome
in Science and Philosophy
From numbers they make what has heaviness and lightness.
—Aristotle
The problem of the relationship between mathematics and objective re-
ality, which arose in early antiquity, is still a subject of heated discus-
sion. The discussions are mainly about the question that probably was
posed most clearly by Immanuel Kant in his
Critique of Pure Reason
:
“How do subjective conditions of thought have objective validity, that
is, how do they become conditions of the possibility of all knowledge of
objects?”
1
Is it because they are themselves elements of objective real-
ity, or because they are contained in thought a priori, before and inde-
pendently of experience, or, finally, because they are subjective images
of corresponding facets and aspects of objective reality?
Underlying the first two responses to the question is the Pythagorean
syndrome, which was first described in the famous thesis of Pythagorean
philosophy: “All things are numbers.” What this meant was that num-
bers, existing only in human consciousness, were identified with exist-
ing things that are external and independent: “The ancient Pythagoreans
English text © 2003 by M.E. Sharpe, Inc., from the Russian text © 1996 by the
Presidium of the Russian Academy of Sciences. “Pifagoreiskii sindrom v nauke i
filosofii,”
Voprosy filosofii,
1996, no. 4, pp. 134–46. A publication of the Institute of
Philosophy, RAS.
Rafail Aronovich Aronov is a candidate of philosophical sciences, a correspond-
ing member of the Russian Academy of Sciences and the prorector of Maimonides
State Hebrew Academy
50
FALL 2002 51
taught that things consist of numbers in the same sense in which, ac-
cording to the teachings of their predecessors and contemporaries, things
consist of water, air, fire, and so forth, that is, numbers are the very
substance and the prime matter of all things. Nor did the Pythagoreans
separate number from matter; they did not accept a special world of
numbers, accessible to reason and independent of the sensuous world.”
2
The present article discusses the subsequent fate of the Pythagorean
syndrome; how it influenced the various domains of the world of cul-
ture, and above all science and philosophy; what effects this had; what
is the reason for the endurance of this syndrome; what is its case history;
is there any effective remedy for it, and if so, what? Of course, we can-
not examine all the known manifestations of the Pythagorean syndrome
in science and philosophy from Parmenides’s doctrine that “an idea and
what it is about are one and the same” to the rejection of Einstein’s
general theory of relativity by A.A. Logunov’s relativistic theory of gravi-
tation in which abstract space, existing only as an element of the theory,
is identified with real space and time existing external to and indepen-
dent of the theory. I shall limit myself here to an examination of only
those manifestations of the Pythagorean syndrome that in my view have
been the most influential in our times.
Let me say, to begin with, that the Pythagorean syndrome from its
inception in Pythagorean philosophy is not merely the identification of
things with numbers, as it may seem at first glance. It includes also
(although in implicit form) the thesis that things are identical with geo-
metric figures. This is connected with the fact that for the Pythagoreans
numbers themselves possess geometric form. As Brunschvicg wrote quite
aptly: “Before saying that things are numbers the Pythagoreans began
by understanding numbers as things. The expressions ‘square number’
or ‘triangular number’ were not metaphors. These numbers were liter-
ally square and triangular to the eye and the mind.”
3
The transition
from the identification of things with “figural numbers” to their iden-
tification with the corresponding geometric ideas occurred explicitly
later in connection with the discovery of the incommensurability of
magnitudes. This occurred first with the discovery of the incommen-
surability of the diagonal with the side of a square. As I.M. Iaglom
writes, “The demonstration that the length of the diagonal of a square
with a side of one unit cannot be expressed by any number was a shock
to the Pythagoreans (to their credit, they promptly appreciated the sig-
nificance of this discovery).”
4
52 RUSSIAN STUDIES IN PHILOSOPHY
Although I agree that the Pythagorean thesis that “all things are num-
bers” contradicted the thesis of the incommensurability of the diagonal
with the sides of a square, I do not share Iaglom’s opinion about the
Pythagoreans’ reaction to this discovery (rather, I see in it an element of
black humor). The Pythagoreans, indeed, promptly appreciated the sig-
nificance of this discovery: according to tradition, Hippasus, who told
them of the incommensurability of the diagonal and the sides of a square,
was immediately thrown overboard by the angry Pythagoreans. The tran-
sition from an old paradigm to a new one did not occur “quickly” in
those days (nor does it today) . . .
From its inception the Pythagorean syndrome has appeared in one
way or another as a form of logical-epistemological pathology and has
had a corresponding influence on the further development of philoso-
phy and science. It enters the history of philosophy and science as one
of the epistemological roots of rationalism, which identifies what is logi-
cally proved with what really exists; it is the line of least resistance in
dealing with the question of why real phenomena come under theories
that describe directly only their idealized analogies; and it promotes the
development of various philosophical systems and scientific theories in
which physical objects and relationships between them are interpreted
as what the main character of I.A. Goncharov’s novel
An Ordinary Story
[Obyknovennaia istoriia], Aleksandr Aduev, describes as “the physical
signs of nonphysical relationships.”
The first to attempt to unravel the riddle of the Pythagorean syndrome
was, apparently, Aristotle. Concluding that the Pythagorean syndrome
“is an impossible thing,”
5
he countered it with the thesis that things ac-
tually contain not numbers, not mathematical concepts, but their proto-
types; that mathematical concepts are abstracted from the real world
and for that reason are applicable to it. (Later, Engels reproduces this
thesis of Aristotle’s almost word for word in his
Anti-Dühring
). Admit-
tedly, Aristotle understood the class of objects that are the prototypes in
the real world of mathematical concepts in a narrow sense: it did not
include all the properties of and interrelations among physical objects.
The Sophists also took exception to the Pythagorean syndrome, and
refuted it with the thesis that mathematical concepts are not and cannot
be in things, that the existence of numbers, nonextended points, lines
without width, and so forth, in things contradicts experience and the
evidence of the senses. For, as Protagoras put it, the measure of things is
not the things themselves, “the measure of all things is man, of existing
FALL 2002 53
things that they exist, and of nonexisting things that they do not exist.”
The fact that neither Pythagoras himself nor his disciplines understood
this (and much less) was one of the reasons for Heraclitus’s conclusion:
“Learning of many things does not teach intelligence; otherwise it would
have taught . . . Pythagoras.
6
The subsequent development of science and philosophy introduced
certain refinements to Heraclitus’s statement. The knowledge of many
things that mankind had acquired by the end of the twentieth century
(that the Pythagoreans and their numerous disciples as well as their less
numerous opponents had lacked), did teach us something (including how
to diagnose and avoid the Pythagorean syndrome in science and phi-
losophy, as will become clear below).
Yet this syndrome continued to spread in one form or another in sci-
ence and philosophy down to our own day. Perhaps the influence of the
Pythagorean syndrome was manifested most clearly in Campanella’s
and Galileo’s doctrine of “the two books,” Kant’s a priorism, Poincaré’s
conventionalism, and Logunov’s relativistic theory of gravitation men-
tioned above.
The Pythagorean syndrome is responsible for the fact that the math-
ematization (geometrization) of science was explained in the course of
the intellectual revolution of the sixteenth and seventeenth centuries in
Europe as a consequence of the mathematization (geometrization) of
nature. Ultimately, this is the sense of A. Koyré’s well-known comment
that this revolution was based on the “mathematization (geometriza-
tion) of nature and, consequently, the mathematization (geometrization)
of science.”
7
Thus was born Campanella’s and Galileo’s doctrine of “the
two books.” In his
Mathematical Discourses and Demonstrations Con-
cerning Two New Sciences,
Galileo explained this doctrine as follows: it
is based on the theory of double truth, according to which there are two
classes of truth—the truths of theology and the truths of philosophy.
The first are represented in the Bible, the Book of Divine Revelation,
written in ordinary language, and the second in the Book of Nature, in
the “greatest book that is always before our eyes (I am speaking of the
universe), but cannot be understood without first learning to understand
the language and to distinguish the signs in which it is written. It is, in
fact, written in mathematical language.”
8
The Pythagorean syndrome had a somewhat different influence in
Kant’s philosophy, according to which mathematical concepts are con-
tained in the human understanding a priori and appear in reality as a
54 RUSSIAN STUDIES IN PHILOSOPHY
consequence of the fact that in experience man introduces them into
reality. According to Kant, the fact that mathematical concepts are con-
firmed by empirical facts does not mean that they are derived from them.
In Kant’s opinion this is confirmed by the universality of mathematical
concepts, by the circumstance that they are clearly thinkable indepen-
dently of any connection with concrete empirical content. This holds
above all for Euclidian space: it is contained in the human understand-
ing as an a priori form of sensibility and appears in reality as a conse-
quence of the fact that man introduces it there. For this reason, according
to Kant, all things as phenomena appear in Euclidian space: abstract
space contained in the human understanding a priori is the space in which
all things as phenomena are located.
The secret of the Kantian a priori (the fact that it once had been
a posteriori) was cleared up finally only in the twentieth century, when
it was demonstrated that mathematical concepts are not universal and
that there are limits to their applicability. Neither Kant nor the science
of his time were aware of this.
Some historians of science believe that “had Kant paid more atten-
tion to the developments in the mathematics of his time, perhaps, he
would not have insisted that the ordering of spatial sensations in the
image and likeness of Euclidian geometry is the only ordering reason
can admit.”
9
One can hardly accept this opinion. First, the mathematics
of Kant’s time was not clear on this issue: the works of N.I. Lobachevskii
and J. Bolyai on non-Euclidian geometry were published more than a
quarter of a century after Kant’s death, and K.F. Gauss, who had real-
ized as early as 1792 that non-Euclidian geometry was possible, com-
municated this only in 1831 and, then, only in a letter to his friend H.C.
Schumacher.
Second, in this comment M. Kline clearly confuses two different ques-
tions: (1) is “Euclidian geometry . . . the only one reason can admit?”
and (2) is the “ordering of spatial sensations in the image and likeness of
Euclidian geometry . . . the only ordering reason can admit?” As Kline
assumes, had Kant paid greater attention to developments in the math-
ematics of his time, then possibly he would have answered the first of
these two questions negatively. However, this would not have predeter-
mined the answer to the second question, for its answer, in the final
analysis, depends on what geometry is objectively realized in the spatio-
temporal realm within the reach of our sense organs and, as a result, is
necessarily manifested in our sensations.
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