Conflicts Between Generalization, Rigor, and Intuition: Number Concepts Underlying the Development of Analysis in 17th-19th Century France and Germany ... History of Mathematics and Physical Sciences)

by Gert Schubring

This volume is, as may be readily apparent, the fruit of many years’ labor in archives and libraries, unearthing rare books, researching Nachlässe, and above all, systematic comparative analysis of fecund sources. The work not only demanded much time in preparation, but was also interrupted by other duties, such as time spent as a guest professor at universities abroad, which of course provided welcome opportunities to present and discuss the work, and in particular, the organizing of the 1994 International Graßmann Conference and the subsequent editing of its proceedings. If it is not possible to be precise about the amount of time spent on this work, it is possible to be precise about the date of its inception. In 1984, during research in the archive of the École polytechnique, my attention was drawn to the way in which the massive rupture that took place in 1811―precipitating the change back to the synthetic method and replacing the limit method by the method of the quantités infiniment petites―significantly altered the teaching of analysis at this first modern institution of higher education, an institution originally founded as a citadel of the analytic method.

Genres: History, Mathematics

692 pages, Hardcover

First published January 1, 2005

Reviews

[William Bies · Rating 4 out of 5]

What makes for the difference, at root, between mathematics and physics? Mathematics concerns number and physics, quantity. Hence, mathematics knows only bare relation while physics knows causes. What makes the philosophy of mathematics interesting is that it invites us to ponder deep-seated conceptual issues that, often enough, remain alien to the everyday practice of mathematicians, at least those schooled in the standard curriculum purveyed to doctoral candidates at present-day universities. A most powerful tool in this connection must be the study of the history of mathematics, for here one will be confronted with the views of outstanding scholars in the past who may have thought about things very differently from the way we have been trained to.

The title of the present work by the German historian at the University of Bielefeld, Gert Schubring strikes one as quite a mouthful, promising to be at once recondite and rewarding if one have the patience: Conflicts between Generalization, Rigor and Intuition: Number Concepts Underlying the Development of Analysis in 17-19th Century France and Germany (Springer Sources and Studies in the History of Mathematics and Physical Sciences, 2005). Two of our previous book reviews cover the development of number concepts: Katherine Neal, From Discrete to Continuous: The Broadening of Number Concepts in Early Modern England (Springer Science & Business Media, 2002, see our review here) and Helena M. Pycior, Symbols, Impossible Numbers, and Geometric Entanglements: British Algebra through the Commentaries on Newton’s Universal Arithmetick (Cambridge University Press, 1987, see our review here). In comparison with these authors, Schubring takes us up to the threshold of what we today would regard as modern analysis, in the early nineteenth century. Also, he enters more deeply than do Neal and Pycior into some conceptual issues pertaining to the foundations of analysis; i.e., apart from the question as to what a number (whether real, complex or what have you) essentially is, what does one do with numbers in the course of carrying out an analysis? Thus, as indicated by the question we pose above in the first paragraph of this review, what one can hope to learn from Schubring about the philosophy of mathematics would be to gain insight into the problem of demarcation, between mathematics and physics or between the ideal realm and the world of our experience.

This is a sprawling work, at 678 pages including bibliography and index. The prerequisites happen not to be that steep, for someone familiar with analysis at the beginning graduate level. Schubring’s own view of what the book is about deserves notice:

The intention is to sketch the conceptual frames, in line with the received literature, sufficiently to prepare for the subsequent in-depth analysis within this concept field. This introduction intends to present only those aspects that are relevant here for concept development. This is why I do not adopt Bohlmann’s subdivision of the concept field into elements. Instead, I start from the general position that concepts are subject to continuous differentiation (Ausdifferenzierung). One original notion evolved, by continuous differentiation, into several separate and independent concepts. This kind of continuous differentiation can be established for the foundational concepts relevant here: the concepts of number and of function do not form entirely separate concepts, but have emerged by way of continuous differentiation from the holistic concept of quantity. If you take quantity as the original fundamental idea, you will see that yet another concept differentiated from this original makes up an element of this concept field: this is the concept of variable. Three foundational concepts thus must be considered to have been successively differentiated from that of quantity: the concepts of number, variable and function. And what is called limit by Bohlmann constitutes but one element of the comprehensive field of limit processes in mathematics, which was eventually differentiated further into the concepts of limit, of continuity and of convergence. The concept of the integral shall also be included in this overview, because it is of special importance for developing the concept of infinitely small quantities. [pp. 15-16]

The historical agents again and again understood, and formulated, the polarity between intuitiveness and generalization as the opposition between synthesis and analysis. The immense variance of the meanings assumed for this contrast in the respective constellations confirms the broad scope that has been elaborated in the volume Analysis and Synthesis in Mathematics (1997). What is surprising in view of this variance, however, is how consistently the historical agents respectively concerned agreed in the opposition between synthesis and analysis as one of geometry and mechanics versus arithmetic and algebra. Just as surprisingly, the concept of quantity finally turned out to be a pervasive structural parameter – both for the concept field of negative numbers and for the concept field of the quantités infiniment petites, as the epistemological support of a holistic character in mathematics against a specialization that appears to place culture in jeopardy. [pp. 616-617]

In light of the diffuseness with which the author outlines his ideas in the passages just quoted, it is noteworthy that he can anchor general statements such as these in a close reading of texts, of which one will get one’s heart’s content in the pages that follow. After an overview of the early history of negative numbers in antiquity and in the Middle Ages, the discussion starts in the sixteenth century with Michael Stifel, Petrus Ramus and Girolamo Cardano: the conceit that a negative number is less than zero and the reasons for adopting the rule of signs (plus times plus is plus, plus times minus is minus, minus times minus is plus). The development of an increasing algebraization is traced through the eighteenth century with the controversy between Antoine Arnauld and Jean Prestet and the epistemological rupture initiated by Bernard de Fontenelle’s separation of quantity from quality, along with Alexis Clairaut and Jean le Rond d’Alembert’s views.

Another path towards algebraization was the development of infinitely small quantities, followed from ancient times down to early modern times and the founders of the infinitesimal calculus, in which the law of continuity and the concept of limit play a role (Colin MacLaurin, George Berkeley). Algebraization culminates with Leonhard Euler and Étienne Bonnot de Condillac. Parts five and six of this volume are dedicated to an in-depth presentation of Lazare Carnot around the turn of the nineteenth century and of Augustin-Louis Cauchy in the early nineteeth century in France, while part seven concerns the development of pure mathematics around the same time in Prussia (Jacob Friedrich Fries and several lesser-known authors). Last, part eight describes the conflict between geometry and algebraization in Siméon Denis Poisson and Jean-Marie-Constant Duhamel. Only in the relatively short part nine does Schubring carry the story forward to the present day.

We wish the devote the remainder of the present review to some thoughts prompted by a reading of Schubring’s patient and meticulous scholarship. The general impression that emerges is that the conceptual language we mathematicians employ today, namely Weierstrassian epsilontics together with Fregean predicate calculus, is after all a very powerful instrument and this explains its universal adoption by practicing mathematicians in the latter half of the twentieth century: yet what has been lost in all the profusion of discoveries in pure mathematics ever since is the vital distinction between quantity and number. If the concept of number is obtained by abstraction from quantity, it is interesting to consider possibility of what would happen if one did not take the process of abstraction all the way to its completion but stopped somewhere short of this limit; then one would obtain a concept of quantity with some operational content from experience of the physical world still built into it.

For this reviewer, quantity somehow connotes beyond number (or measure) a tendency to act: as in √–1 = not just the stationary point (0,1) in an Argand diagram, but as generator of the operation of multiplication by i or if seen more conceptually, the infinitesimal generator of the Lie group SO(2) of counterclockwise rotations of the complex plane. The mooted point matters because the modern mathematization of physics means that most often the numbers with which we deal correspond to physical quantities in the world, hence are endowed with an incipient dynamics. As pure mathematicians, we can if we so want prescind from the real world in order to dwell in a timeless Platonic heaven, but – we being human beings! – thus to take flight from mundane reality incurs the cost of sacrificing our spatio-temporal intuitions with which we are so marvelously equipped to synthesize the manifold of sensibility into ordered experience. Paradoxically, we can be better pure mathematicians if we do not flee into the intelligible realm!

As an aside, let us remark that this observation is really the crux of Ernst Cassirer’s complaint in his perceptive essay on the natural scientific treatises of Johann Wolfgang von Goethe (better known to most everyone as a great poet), entitled ‘Goethe und die mathematische Physik’ (see our review of the eminent German-Jewish scholar’s collection of essays, Idee und Gestalt, here). Here is how Cassirer characterizes the modus operandi of the mathematical physicist:

Maßstäbe als die Grundeinheiten einmal gewonnen, so läßt sich jetzt alles, was die Wissenschaft vom Sein und von der Beschaffenheit eines Naturgegenstandes auszusagen vermag, zuletzt auf einem Komplex von Zahlen zurückführen, deren jede die Eigenschaft oder Zuständlichkeit des betrachteten Gegenstandes, nach irgend einer Hinsicht der Größenvergleichung, charakterisiert und festlegt. Die Gesamtheit dieser Zahlen – sofern wir sie als abgeschlossen denken – repräsentiert für den Physiker das Ding: ja sie ist ihm dieses Ding, sofern nur sie es ist, die in seine Erkenntnisse und Urteile, in seine Gesetze und Funktionsgleichungen eingehen kann. [p. 47]

In contrast, Goethe urges us not to forget the world of experience from which the physicist draws the object of his study:

Somit ist es ein bestimmtes Reihenprinzip, das Goethe, gleich dem mathematischen Physiker, in der Auffassung der Einzelerscheinungen durchführt und zur Geltung bringt; – aber wenn letzterer dieses Reihenprinzip dem Gebiete der Zahl, der abstrakten Grundform der Reihe überhaupt, entnimmt, so verlangt Goethe, die Reihe, auf die er alle übrigen bezieht, selbst noch als Lebensphänomen, als Gesetz und Rhythmus des konkreten Gesamtlebens der Natur, anschauen und sie sich in diesem Anschauen innerlich aneignen zu können. [p. 50]

Die Reduktion auf den Zahlwert aber läßt von der empirischen Einzelanschauung, mit der die physikalische Betrachtung beginnt, im qualitativen Sinne nichts zurück. Sie bestrebt sich, die anfänglichen qualitativen Werte in reine Stellenwerte zu verwandeln, die durch nichts anderes als durch ihre Beziehungen zu anderen Reihengliedern gekennzeichnet sind. [p. 57]

Thus, for Goethe and anyone like-minded, reducing the physical problem to numbers throws out something without remembrance of which we can never hope to grasp the entire phenomenon in which it is implicated:

Das verfahren der exakten Wissenschaft besteht im wesentlichen darin, die sinnlich-empirische Mannigfaltigkeit des Gegebenen auf eine andere, “rationale” Mannigfaltigkeit zu beziehen und sie in ihr vollständig “abzubilden”. Aber um diese Umbildung in der logischen Form zu erreichen: dazu muß die mathematische Physik zuvor die Elemente umgestalten, die in diese Form eingehen sollen. Die Inhalte der empirischen Anschauung müssen erst in reine Größen- und Zahlwerte umgesetzt worden sein, ehe sich von ihnen ein gesetzlicher Zusammenhang aussagen läßt; denn die allgemeine Bedeutung und das Grundschema des Naturgesetzes selbst setzt die Form der Kausalgleichung voraus. Goethe dagegen verlangt eine neue Weise der Verknüpfung des Anschaulichen, die den Gehalt eben dieser Anschauung als solcher unangetastet läßt. Er fordert, daß die Elemente selbst synthetisch zusammengeschaut werden: während in der exakten Wissenschaft die Synthese nicht sowohl sie selber, als vielmehr die begrifflichen und numerischen Repräsentanten betrifft, die wir an ihre Stelle setzen. [p. 75]

Bertrand Russell may be taken as an architect of the trend to which Goethe takes exception. In an influential article from 1913 on causation, or the absence thereof in modern physics, Russell stakes out his claim, perhaps controversial at the time but by now probably the dominant view among cognoscenti [On the notion of cause, Proceedings of the Aristotelian Society, 1912-1913, New Series, Vol. 13 (1912-1913), pp. 1-26]:

All philosophers, of every school, imagine that is one of the fundamental axioms or postulates of science, oddly enough, in advanced sciences such as gravitational astronomy, the word ‘cause’ never occurs. Dr. James Ward, in his Naturalism and Agnosticism, makes this a ground of complaint against physics: the business of science, he apparently thinks, should be the discovery of causes, yet physics never even seeks them. To me it seems that philosophy ought not to assume such legislative functions, and that the reason why physics has ceased to look for causes is that, in fact, there are no such things. The law of causality, I believe, like much that passes muster among philosophers, is a relic of a bygone age, surviving, like the monarchy, only because it is erroneously supposed to do no harm. [p. 1]

What this reviewer grasps of Russell’s argument is roughly this: once the physical problem has been mathematized and reduced, as Cassirer says, to a set of numbers describing the system, there is no longer any place for a traditional notion of cause. Rather, all we find are certain functional relationships among the numbers determined by the applicable equation of motion. Thus, we can calculate what happens without ever having to include the term, ‘cause’, in our vocabulary, and if it is thus dispensable, we ought for reasons of logical economy to get rid of it.

In answer to Russell, dynamics always remains implicit in the formulation of a differential equation (which via an all but dead metaphor we refer to as a dynamical system), through the dependence between initial conditions and final conditions ever present in the theorist’s mind when he applies the equation of motion to a problem in the real world. The implicit judgment that the merely mathematical model holds in the world itself cannot be elided, and contains the heart of the matter (as Kant understands with his construction of experience). Causation may be viewed as a statement about conditional probabilities of the connection between initial and final conditions established by the relevant equation of motion. The simplest case would be of the form: if A then B and if not A then not B. In any system possessing a degree of complexity, the prevailing causal relations may be far more difficult to infer and, in recent years, a whole discipline has been founded in order to elucidate the principles behind causal inference in real-world cases. To pursue this circle of ideas, the interested reader might refer to Stuart Glennan, The New Mechanical Philosophy (Oxford University Press, 2017, reviewed by us a while ago here). See also Judea Pearl and Dana MacKenzie’s popular book, The Book of Why: The New Science of Cause and Effect (Basic Books, 2018, reviewed by us here) – this reviewer owes a review of Pearl’s dense monograph for experts, Causality: Models, Reasoning and Inference (Cambridge University Press, 2000), which – fortune permitting – should be forthcoming before long.

Hence, we propose to resolve the standoff between the ontic and the epistemological interpretations of the wave function amplitude by treating it once again as a quantity, which of course summons up the research problem as to what this quantity may actually be. The infamous shut-up-and-calculate attitude is tantamount to reducing it to just a number, but if so, we deprive ourselves of potential for insight into the physics. This constitutes one of the deepest unsolved problems that have lingered ever since quantum mechanics was formulated, namely, that we can calculate everything we want to know but fail to understand what we are calculating with (as notably Niels Bohr and Richard Feynman are on record as maintaining). Aside: David Bohm’s supposed realist approach does not solve or even so much as adequately address this problem, notwithstanding his strident claims to the contrary, why? Because his pilot wave amounts to nothing but an artificial device with no physics behind it, hence fails to contribute to any improved comprehension of the quantum world. Let us grant one can translate back and forth between the orthodox and Bohmian pictures, but the latter itself has never led to any better apprehension (in contrast with what is demonstrably the case in complex function theory, where one has three pictures, viz., Cauchy’s, Riemann’s and Weierstrass’ – there, the dictionary correspondence among the three proves to be quite fruitful because one can choose the picture adapted to the problem at hand).

4 ½ stars – we are reluctant to award five stars as the author merely broaches the philosophical issue of number versus quantity and gives a detailed exposition of how it has figured in the thought of some historical figures down through the mid-nineteenth century, but does not of course do anything original with it. Nonetheless, the aspiring mathematician with enough pluck to range beyond the scope of the sources conventionally approved these days will find much valuable material here to ponder. Let us hope he may be guided thereby to penetrate to a novel view of the natural world and of how it relates to pure mathematics!

[Snail in Danger (Sid) Nicolaides]

According to The Black Count: Glory, Revolution, Betrayal, and the Real Count of Monte Cristo, this has information on Lazare Carnot, an important figure in the French Revolution.