《Peking University, November 15, 2005》
What Are Revolutions in Mathematics?
and How Do They Differ from Those in the Natural Sciences?
Mathematical Truth in the Light of Thomas S. Kuhn's Philosophy of Science
SASAKI Chikara*
**§****1. Toward a New Historiography of Mathematics: Thomas S. Kuhn's Historical Philosophy of Science and the Responses to It by Some Historians of Mathematics**
It must be agreed that the second half of the twentieth century witnessed a kind of revolution in the history and philosophy of science. It may be also admitted that the first edition of a book was playing an overwhelming role for that revolution, Thomas S. Kuhn's *Structure of Scientific Revolutions*, published in 1962. Today, Kuhn's view of science is generally labelled "historical philosophy of science."^{1}
In what follows I will argue whether or not the "historical philosophy of science" can be applied to mathematical truth. For many years since the appearance of aforementioned Kuhn's monograph of 1962, historians and philosophers of mathematics have tended to turn their eyes to the dynamic process by which mathematical knowledge has been acquired rather than the static and formal structure of the products of mathematical research. Many of them, however, have been skeptical towards the applicability of Kuhn's view of science to mathematics. For one, Michael J. Crowe wrote in 1975 as follows:
In the historiography of mathematics, no comparable group of authors seems to have emerged. Moreover, most historians of mathematics acquainted with the new historiography of science have been skeptical as to whether the insights embodied therein can be applied in any direct way to the historiography of mathematics. [...] [T]he major differences between the conceptual structures of mathematics and of science make it questionable whether their histories should exhibit similar patterns of development.^{2}
Further, explaining nine 'laws' concerning patterns of change in the history of mathematics briefly, he concluded his consideration with the tenth 'law', namely:
Revolutions never occur in mathematics.^{3}
My intention in considering the problem whether or not Kuhn's view of scientific revolutions is applicable to mathematical truth in the following lines has been chiefly inspired by my study on the formation of the differential and integral calculus in seventeenth-century Europe, especially the discovery of the fundamental theory of that calculus.
In supporting Kuhn's view of science, several historians of mathematics attempted to refute Crowe's essay of 1975, as exemplified by Joseph W. Dauben's "Conceptual Revolutions and the History of Mathematics: Two Studies in the Growth of Knowledge," published in 1984. By taking examples from Eudoxus's theory of proportion applicable to both commensurable and incommensurable quantities and Georg Cantor's transfinite set theory, Dauben has proposed a new thesis contrary to Crowe's: "Revolutions have occurred in mathematics."^{4}
Crowe himself seems to have changed drastically his view of 1975 in his essay entitled "Ten Misconceptions about Mathematics and Its History." At the beginning of that essay in 1988, he clearly stated: "I have become convinced that ten claims I formerly accepted concerning mathematics and its development are both seriously wrong and a hindrance to the historical study of mathematics."^{5} Despite of Crowe's 'conversion', it is worth considering, I believe, Crowe's view of 1975, since it has been still a firm belief among mathematicians.
Now I state briefly an autobiographical note. As a student of Professor Kuhn at Princeton University, I attended his course on the philosophy of science and proposed an outline of my immature view on the historiography of mathematics to him in May 1977. Before that time, Kuhn had not thought that his theory of science was applicable to mathematics, and this view certainly influenced, it was said, Crowe's essay of 1975. But, in his comment on my report, Kuhn had changed his previous view and wrote clearly that "there must have been revolutions in mathematics." This seemed to me an extremely remarkable statement.
I shall try to support the Dauben camp, by defending the Kuhnian conceptual frame of the "historical philosophy of science" as much as possible *mutatis mutandis*. By doing so, I will try to present the "historical philosophy of mathematics," as it were, an extension of Kuhn's "historical philosophy of science" to mathematical knowledge.
For this purpose, first of all, our understanding of mathematics should be revised. The so-called 'Platonic' or formalist understanding of mathematics must be rejected and a new conception of mathematics should be introduced, for example, mathematics as a "quasi-empirical" science, to use Imre Lakatos's terminology.^{5} Moreover, historians of mathematics so far seem to have discussed the problem, having taken examples from the history of mathematics of modern Europe, at most from that of ancient Greece and modern Europe. In addition to the history of mathematics in Europe, we have to examine further how the mathematical culture in East Asian countries which enjoyed a rather high level was transformed from traditional to the one based on modern Western mathematics. In this sense, our scope of discussions should be extended from the history of Western mathematics to the oecumenical history of mathematics in the world.
**§****2. An Examination of Crowe's Thesis of 1975**
Before narrating our own story on the differential and integral calculus, let us consider Crowe's explanation in 1975 of the tenth 'law'. Crowe is not a classical advocator of the theory of gradual evolution like Hermann Hankel, nineteenth-century German mathematician, who stated, "In most sciences one generation tears down what another has built and what one has established another undoes. In Mathematics alone each generation builds a new story to the old structure."^{6} Crowe tells us that such a quotation "cannot stand alone and without qualification":
For this law depends upon at least the minimal stipulation that a necessary characteristic of a revolution is that some previously existing entity (be it king, constitution, or theory) must be overthrown and irrevocably discarded.^{7}
According to Crowe, "a number of the most important development in science, though frequently called 'revolutionary', lack this fundamental characteristic." He then distinguishes "transformational" or revolutionary discoveries (e. g. the Copernican revolution) from "formational" discoveries. In the latter case, new areas are "formed" or created without the overthrow of previous doctrines. He exemplifies this case by energy conservation or spectroscopy. Later he states:
It is, I believe, the latter process rather than the former which occurs in the history of mathematics. For example, Euclid was not deposed by, but reigns along with, the various non-Euclidean geometries. Also the stress in law 10 on the preposition "in" is crucial, for, as a number of the earlier laws make clear, revolutions may occur in mathematical nomenclature, symbolism, metamathematics (e. g. the metaphysics of mathematics), methodology (e. g. standards of rigor), and perhaps even in the historiography of mathematics.^{8}
Here Crowe presupposes a fixed belief which appears to occupy most working mathematicians of today. He pays attention to that the preposition "in" is crucial. But, we have to ask with Herbert Mehrtens what "in" mathematics means.^{9} Can we strip the contents or the substance of mathematics from mathematical nomenclature, symbolism, metamathematics, and methodology? Then, let us take an example which Crowe has used, that is, the creation of non-Euclidean geometries. Crowe states, "Euclid was not deposed by, but reigns along with, the various non-Euclidean geometries." In this regard, Stephan Körner's statement on the discovery of non-Euclidean geometries would be suggestive:
Until the discovery of non-Euclidean geometries the doctrines of the uniqueness of mathematical reality or intersubjective intuition, of its accessibility, of the solvability of all classes of mathematical problems and of the possibility of the unambiguous and exhaustive reflection of mathematical reality in conceptual or linguistic formulations were neither called into question nor regarded as needing a more precise analysis.^{10}
The discovery of non-Euclidean geometries changed epistemology by refuting the Kantian apriorism. And, in turn, epistemology reinterpreted the contents of non-Euclidean geometries. Crowe, furthermore, seems to be referring to Book I of Euclid's *Elements* or rather Euclidean geometry in the Kantian sense when he talks about Euclidean geometry. We, however, also recall Book V containing the theory of proportion which is applicable to geometrical magnitudes, and Book X which consists of the complicated theory of irrational magnitudes. Crowe has unconsciously but clearly selected a part of the *Elements* which has been able to survive and apply to his view of the historiography. But, Books V and X of the *Elements* must have been discarded historically and superseded totally by the real number theory of Richard Dedekind, for example. One might insist that the propositions of Books V and X are all still true even now, and that owing to the fact which is valid only in mathematics, Euclid's geometrical mathematics is entirely different from Aristotle's physics. This statement is partly true and partly false. Let us explain this in the next section.
**§****3. Mathematics as Quasi-Empirical and Time-Dependent Knowledge**
Mathematical truth is always true and is not time-dependent, therefore, is considered to be perennial in the strict sense of the word. Why does this statement appear to one to be true? One reason for this is that mathematics acquired the firm paradigm from prehistory or at least ancient Greece.^{11} Mathematics was one of the first sciences to employ deductive inference. Thus, mathematics is always related to the ideal world, as distinct from the real world through the procedure of abstraction. Mathematics, after the establishment of the firm paradigm in ancient Greece, is also characterized by the axiomatic method. Mathematics in the Islam civilization, in the Latin Middle Ages, and in modern Europe were all more or less influenced by such aspects (or paradigms) although the extents were various. But, these characteristics of mathematics are not a priori, but historical acquisitions. We can contrast the paradigm of Greek mathematics with that of Chinese mathematics most effectively. In this regard Joseph Needham's observation is very illuminating:
"On account of its more abstract and systematic character"―so came the words of themselves to the keys of the machine. Systematic, yes, there no doubt is possible, but abstract―was that wholly an advantage? Historians of science are beginning to question whether the predirection of Greek science and mathematics for "the abstract, the deductive and the pure, over the concrete, the empirical and the applied" was a wholly a gain. [...] In the flight from practice into the realms of the pure intellect, Chinese mathematics did not participate. [...] For the 1st century B. C., the time of Lohsia Hung and Liu Hsin, the *Chiu Chang Suan Shu* (Nine Chapters of the Mathematical Art) was a splendid body of knowledge. It dominated the practice of Chinese reckoning-clerks for more than a millennium. Yet in its social origins it was closely bound up with the bureaucratic government system, and devoted to the problems which the ruling officials had to solve (or persuade others to solve). Land mensuration and survey, granary dimensions, the making of dykes and canals, taxation, rates of exchanges―these were the practical matters which seemed all-important. Of mathematics 'for the sake of mathematics' there was extremely little. This does not mean that Chinese calculators were not interested in truth, but it was not abstract systematised academic truth after which sought the Greeks.^{12}
Moreover, such paradigms which were acquired in ancient Greece, whose main characteristic was geometrical, had to be reinterpreted by the algebraic language particularly by François Viète and René Descartes it was introduced into the European world and began to flourish there.^{13} The rigor of the modern European mathematics itself is based on the manner of algebraic thinking.
Generally mathematicians have had the belief that mathematics is a "pure deductive science" in the strict sense. László Kalmár summarizes the reasons why this mathematicians' belief is sustained:
(i) The rise of a philosophy of the *a priori*, according to which it was unnecessary to test in practice intuitively evident axioms and rules of inference; (ii) mathematics' becoming proverbial as an "infallible" science, due to its appropriate methods; (iii) mathematics' becoming, because of its precision, an ideal for other fields, beginning perhaps with Spinoza's *Ethica more geometrico demonstrata*, down to the mathematical economics or mathematical linguistics of our own day; (iv) the success of mathematics itself, and the extension of its method of abstraction to more and more general regularities holding for empirical facts, which was reflected in more and more abstract mathematical theorems and algorithms.^{14}
These circumstances, however, do not deny that mathematics has empirical origins and that its foundation has empirical aspects. Rather it would be more suitable to call this characteristic "quasi-empirical", for the mathematically given is closely related to, but not identical with, the empirically given. It was Imre Lakatos who characterized "the present empiricist mood" of the philosophy of mathematics, relying on his historical studies of theorems of topological geometry and analysis, and penetrating the intellectual situation of the post-Gödelian mathematics. Lakatos explains very concisely the impact of Gödel's incompleteness theorem of 1930-31:
The infinite regress in proofs and definitions in mathematics cannot be stopped by a *Euclidean* logic. Logic may *explain* mathematics but cannot *prove* it.^{15}
Thus, his comment on Kalmár's above-quoted presentation has the title: "A Renaissance of Empiricism in the Recent Philosophy of Mathematics?" And he states his thesis:
The present empiricist mood, so strongly represented by Professor Kalmár, originates in the recent recognition that mathematics is not Euclidean or quasi-Euclidean―as had been expected in the first heroic period of foundational studies―but quasi-empirical, and in the increasing attention drawn to the part of mathematics that falls *outside* its Euclidean kernel.^{16}
In a word, mathematics is normatively "pure deductive science," but it is, in fact, "quasi-empirical."
Then, we should return to our question of the infallibility of propositions of mathematics. We have had a question that the propositions of Books V and X of Euclid are infallible and that they are totally different from, for example, the natural philosophical statements of Aristotle's *Physics*. This problem seems to be solved by explaining that mathematics has the particularity which we expounded slightly in the earlier part of this section. Here we should introduce a distinction among mathematical theories. One is the group of mathematical theories which give mathematicians for a time active problems. We may call these theories active ones, which often correspond to "normal science". Also, we shall sometimes use the concept of "normal mathematics," following H. Mehrtens.^{17} The other is for theories which are not active any longer.
Mathematics, if we regard this discipline as a science which develops historically, has empirical origins and cannot get absolute rigor. Stephan Körner analyzed the philosophical significance of the post-Gödelian mathematics insightfully, comparing it with the existence of competitive non-Euclidean geometries:
Disregarding always possible appeals to some dogmatic metaphysics, one must conclude that the metamathematical discoveries of the present century imply the falsehood of the common doctrines shared by the classical philosophies of non-competitive mathematical theories, but not that they imply the positivist doctrine that if mathematics is not true-or-false in the actual or every possible world, it is therefore meaningless.^{18}
Körner then refutes the dogmatic formalism by stating that competitive mathematical theories have their relation to experience and that their historical validity and survival is sustained by their applicability to the real world:
They support on the other hand the view that the various unequivalent, mutually compatible theories are meaningful, i. e. true in different possible worlds, even though none of them is actual, and that their agreement or conflict with each other lies in their relation to experience, namely the coidentifiability or otherwise, of at least some of their axioms or theorems with empirical propositions.^{19}
Therefore, supporting that the reasoning of syllogism is always true, we may state that a mathematical proposition is necessarily true if its postulates are true. But, the postulates and even the standard of rigor are time-dependent, as will be exemplified below. The world of mathematics is *essentially* open to the real world though it is idealized. Moreover, such idealization cannot transcend history. Mathematical terms are fastened to the historical world in which they flourish. Let us quote the first definition of Euclid's *Elements*: "A point is that which has no part." As lots of commentaries on this definition show, the concept of point has been reinterpreted again and again. Exactly speaking, the mathematical terms of Hilbert's *Grundlagen der Geometrie* (1st ed., 1899) are not identical to Euclid's. If Euclid's theory is eternal, it would be true that the theory is concerned only with elementary statements and with the world which is common both to the Greeks and the modern men. But from the seventeenth to the beginning of the twentieth century Euclideanism has been on a great retreat as Lakatos states:
The fallible sophistication of the empiricist programme has won, the infallible triviality of Euclideans has lost. Euclideans could only survive in those underdeveloped subjects where knowledge is still trivial like ethics, economics etc.^{20}
We have already been accustomed to explain historically discarded physical theories―for example, Aristotle's natural philosophy―by examining the related *Weltanschauung* and terms such as motion, the void, space, and place. We sense here a discontinuity between the ancient Aristotelian physics and the modern Newtonian mechanics. Kuhn has called this discontinuity "incommensurability" and has admitted a scientific revolution.^{21} Although we might admit that the statements of natural sciences are much more easily refuted by the new statements than mathematics, we cannot dream the nineteenth-century mathematicians' dream: "In mathematics alone each generation builds a new story on the old structure."
An example which we should mention here is Judith V. Grabiner's exciting essay "Is Mathematical Truth Time-Dependent?" in *American Mathematical Monthly*, Vol 81, 1974. Grabiner here pursues the reason why standards of mathematical rigor change, for nineteenth-century analysts, beginning with Augustin-Louis Cauchy and Bernard Bolzano, demanded rigorous proofs of the propositions in infinitesimal algebraic analysis which had not been treated rigorously in eighteenth-century analysts. Stating as an internal reason that we must point out that the need to avoid errors became more important near the end of the eighteenth century, when there was increasing interest among mathematicians in complex functions, in functions of several variables, and in trigonometric series, she adds a major social change:
Before the last decade of the century, mathematicians were often attached to royal courts; their job was to do mathematics and thus add to the glory, or edification, of their patron. But almost all mathematicians since the French Revolution have made their living by teaching.^{22}
Grabiner asks, "Why might the new economic circumstances of mathematicians―the need to teach―have helped promote rigor?" Then, she answers, "Teaching always makes the teacher think carefully about the basis for the subject. A mathematician could understand enough about a concept to use it, and could rely on the insight he had gained through his experience."^{23}
Let us put our problem in order. Certainly, mathematical propositions appear to be infallible besides the standard of rigor which is time-dependent but transferable relatively easily. And they are understandable and justifiable however complicated and dependent upon the *Weltanschauung* in which they were created. This aspect of mathematics is, no doubt, important and quite different from the natural sciences. This is because mathematical knowledge is conditional and concerned with the ideal world in the Platonic sense.
But, we have to ask about active mathematics. Books V and X of the *Elements* are not active any longer in modern Europe except as a subject of special historical interests. Even Book I is not so active among mathematical practitioners, because the way of mathematical thinking has changed. There was a certain change, as will be argued below.
With such preliminary considerations, we turn to our main concern on the differential and integral calculus.
**§****4. The Measure of Areas and the Tangent Method Before the Establishment of the Differential and Integral Calculus: From Archimedes to Pierre de Fermat**
Now we should take a look at how the differential and integral calculus was formed in seventeenth-century Europe.^{24} Before examining this historical event closely, we have to trace briefly back its forerunners from Archimedes to Pierre de Fermat.
We begin with the theory of integration. The theory of integration can be said to have been derived from the art of measuring or calculating areas in antiquity. It was undoubtedly among the most fundamental knowledge which human beings acquired. They knew that the area of a rectangle with one side *a* and the other side *x* is equal to ∫*a*d*x *= *ax* (Figure 1.a). And the area of a triangle with its basis *x* and its height *y *= *ax *was also known to be equal to ∫*ax*d*x *= *ax*^{2}/2 (Figure 1.b).
Then, what Archimedes attempted at in his the *Quadrature of the Parabola* was to get the area of a segment of the parabola. Through both sophisticated geometrical and mechanical methods, he obtained the area of a segment of the parabola to be ∫*ax*^{2}d*x *= *ax*^{3}/3 (Figure 1.c) in the modern algebraic language. It is known today, thanks mainly to Roshdi Rashed, that Ibn al-Haytham (Latinized as Alhazen from his first name) of the late tenth and early eleventh centuries became a mighty mathematician in no way inferior to Archimedes himself on the infinitesimal geometry in the Archimedean tradition.^{25}
With the almost total recovery of the works of Archimedes during the Renaissance, humanists and mathematicians started to learn the Archimedean method seriously and tried to go beyond it. In his monumental *Geometria indivisibilibus continuorum* published in Bologna in 1635, Bonaventura Cavalieri (*c*. 1598-1647) certainly introduced a new method of integration with his name and derived the Archimedean formula ∫*ax*^{2}d*x *= *ax*^{3}/3 for the parabola, but he failed to get a new result. In his later work with the title *Exercitationes geometriae sex* published in Bologna in 1647, however, he succeeded in obtaining the following formulae for some generalized or higher parabolas for *n = *3, 4, ..., 9:
∫*ax*^{n}d*x *= *ax*^{n}^{＋１}/ (*n*+1).
Figure 1
As may be imagined, the method which Cavalieri used for those results was a natural extension of the Archimedean geometrical method for the quadrature of the parabola. It would not be so difficult for talented contemporary mathematicians to conjecture the following formulae inductively and generally (Figure 1.d):
∫*ax*^{n}d*x *= *ax*^{n}^{＋１}/ (*n*+1) for *ax*^{n }(*n*≧3).
Pierre de Fermat (1601-1665) further generalized the above formulae for *n* being positive rational numbers and negative integers other than －1.
A modern method of drawing tangents can be said to have been born with Fermat. In his manuscripts with the title *Methodus ad disquirendum maximam et minimam et de tangentibus linearum curvarum*, which may be ascribed to the period of 1629-36, Fermat obtained algebraic methods concerning how to get values of maxima and minima and how to draw the tangent to the parabola.^{26}
^{ } We see his treatment of the latter problem of finding the tangent *PT* to a curve at a point *P* on the curve. He did this by finding the length of the subtangent *TN*. We may think that he further extended his method of tangent for the generalized parabola of which equation is *y *= *x*^{n }for *n* being a positive integer.^{27} In Figure 2, let *P* be (*x*, *y*) and *P*' is a neighbouring point (*x* + *e*, *y* + *d*) on the curve. Now we have
*y* + *d* = (*x* + *e*)^{n} = *x*^{n }+ *nx*^{n}^{－}^{1 }*e *+ {terms in *e*^{2} and higher powers of *e*}.
Figure 2
Then, *d*/*e =* *nx*^{n}^{－}^{1 }+ {terms in *e*}. Consequently, the slope of the curve at *P* is obtained as the formula which is familiar to us,
*dy*/*dx* = *dx*^{n }/*dx* = *nx*^{n}^{－}^{1 }.
For the algorithmic method of tangent, it may be said that Fermat thus attained the result *dy*/*dx *= *nax*^{n}^{－}^{1 }for the generalized formula *y *= *ax*^{n}^{ }with some anachronistic modifications But, unfortunately, he failed to see the mutual inverse relationship between the algorithm of quadrature and that of tangent. Such is a general situation of infinitesimal mathematics in the middle of the seventeenth century.
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