Elements of the History of Mathematics

by Nicolas Bourbaki

This work gathers together, without substantial modification, the major ity of the historical Notes which have appeared to date in my Elements de M atMmatique. Only the flow has been made independent of the Elements to which these Notes were attached; they are therefore, in principle, accessible to every reader who possesses a sound classical mathematical background, of undergraduate standard. Of course, the separate studies which make up this volume could not in any way pretend to sketch, even in a summary manner, a complete and con nected history of the development of Mathematics up to our day. Entire parts of classical mathematics such as differential Geometry, algebraic Geometry, the Calculus of variations, are only mentioned in passing; others, such as the theory of analytic functions, that of differential equations or partial differ ential equations, are hardly touched on; all the more do these gaps become more numerous and more important as the modern era is reached. It goes without saying that this is not a case of intentional omission; it is simply due to the fact that the corresponding chapters of the Elements have not yet been published. Finally the reader will find in these Notes practically no bibliographic or anecdotal information about the mathematicians in question; what has been attempted above all, for each theory, is to bring out as clearly as possible what were the guiding ideas, and how these ideas developed and reacted the ones on the others."

Genres: Mathematics, History, Science, Nonfiction

309 pages, Paperback

First published January 1, 1960

About the author

Nicolas Bourbaki is the collective pseudonym under which a group of (mainly French) 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for rigour and generality. Their work led to the discovery of several concepts and terminologies still discussed.
Bourbaki congress, 1938.

While Nicolas Bourbaki is an invented personage, the Bourbaki group is officially known as the Association des collaborateurs de Nicolas Bourbaki (Association of Collaborators of Nicolas Bourbaki), which has an office at the École Normale Supérieure in Paris.

Reviews

[William Bies · Rating 3 out of 5]

Nicolas Bourbaki is the pen name of a group of French mathematicians who were quite influential in setting the trend of mathematics during the twentieth century. For the convenience of interested readers, the present volume gathers in one place the historical notes from the series of monographs it sponsors. As usual, the authors remain anonymous under the collective Bourbaki moniker. Certainly, the intended audience will be those who know some advanced mathematics at the upper undergraduate or beginning graduate level or higher; these will appreciate the rather technical explanations involving explicit formulae expressed in modern notation. Strong on formal logic, algebraic number theory, commutative algebra, non-commutative algebra, some basic ideas in topology and real analysis and some aspects of functional analysis and the theory of integration. The individual subject synopses are generally good on questions having to do with the genealogy of those concepts that play a major role in modern mathematics as seen by its practitioners; e.g. what was the Lagrange resolvent and what led Lagrange to it? What did Abel do in the theory of equations and how does it compare with Galois’ contributions? How were Kummer’s ideas in number theory expounded and consolidated by Dedekind et al.? Not especially concerned with philosophical issues or with applications to allied fields such as theoretical physics; Bourbaki is purest of the pure!

Let us turn to Bourbaki’s perspective on the history of his subject. All around, he seems to have a lucid, lynx-eyed synthetic grasp of how past developments fit into the overall scheme as it would be pictured by today’s mathematician. A discovery is justified, in his view, first by the immediate results to which it leads; and second, by the further conceptual breakthroughs it renders possible. The first, 44-page chapter on the foundations of mathematics, logic and set theory will perhaps garner the most attention from outsiders (meaning anyone who cares about the subject of mathematics without himself being a professional member of the academic community). Logic, of course, has formed a discipline in its own right ever since antiquity and the middle ages which Bourbaki has neither the scope nor the purpose to cover in all its ramifications; this chapter therefore limits itself to the question of how the evolving understanding of logic impinged on mathematics through the ages. The following quotation illustrates Bourbaki’s own position:

That there was a well-developed prehellenic mathematics is not today in any doubt. Not only are the notions (already very abstract) of whole number and of the measurement of quantities commonly used in the most ancient documents which have reached us from Egypt or Chaldea, but Babylonian algebra, because of the elegance and sureness of its methods, should not be thought of as a simple collection of problems solved by empirical fumbling. And, if nothing is found in the text resembling a “proof” in the formal meaning of the word, it is reasonable to believe that the discovery of such methods of solution, whose generality appears through particular numerical applications, was not possible without a minimum of logical links (perhaps not entirely conscious, but rather like those on which a modern algebraist depends when he undertakes a calculation, before “setting down formally” all the details).

The essential originality of the Greeks consisted precisely of a conscious effort to order mathematical proofs in a sequence such that passing from one link to the next leaves no room for doubt and constrains universal assent. That Greek mathematicians made use, in the course of their research, just like modern mathematicians, of “heuristic”, rather than convincing, arguments, is what was proved for example (if there were any need), by the “treatise of method” of Archimedes; note also in this, allusions to results “found, but not proven” by earlier mathematicians. But from the first detailed texts that are known to us (and which date from the middle of the Vth century), the ideal “canon” of a mathematical text is properly settled. It will find its highest expression in the great classics, Euclid, Archimedes and Apollonius; the notion of proof, in these authors, differs in no way from ours.

We have no texts allowing us to follow the first steps of this “deductive method”, which seems to us already near perfection at the exact moment when we become aware of its existence. One can only think that it fits fairly naturally into the perpetual search for “explanations” of the world which characterizes Greek thought and which is so discernible already amongst the Ionian philosophers of the VIIth century; moreover tradition is unanimous in ascribing the development and refinement of the method to the Pythagorean School, in a period which fits between the end of the VIth century and the middle of the Vth century.

It is on this “deductive” mathematics, fully conscious of its goals and methods, that the philosophical and mathematical thought of subsequent times will be concentrated. We will see on the one hand the establishment little by little of “formal” Logic modeled on mathematics, to conclude with the creation of formalized languages; on the other hand, mainly starting at the beginning of the XIXth century, the basic concepts of mathematics will be queried more and more and a great effort will be made to clarify their nature, especially after the birth of the Theory of sets. (pp. 1-2)

There follow fairly detailed subsections on Leibniz and his characteristica universalis, Frege’s foundational work on the logicization of arithmetic, Peano’s goal of rendering mathematics into a formal language and Russell and Whitehead’s Principia Mathematica. The bulk of the chapter is given over to a discussion of the notion of truth in mathematics, the rise of the theory of sets during the nineteenth century, the paradoxes and crisis in foundations encountered around the turn of the twentieth century and lastly metamathematics, a novel discipline which gained currency around then because it figures centrally in Hilbert’s logical-formalist program of the first three decades of the twentieth century.

Another advantage from which the reader who may not have the experience of having been a practicing mathematician will profit can be sought in occasional stray remarks by the authors, all of whom are established mathematicians who know their craft well. The following passage would be a prime instance in point:

It is not with impunity besides, that one can, in mathematics, allow a ditch to be dug between discovery and proof. In favorable times, the mathematician, without abandoning rigor, only has to write down his ideas almost as he conceives them; sometimes even he can hope to do it so that it would be thus, at the cost of a felicitous change in the language and notations assumed. But often he must resign himself to choosing between incorrect and possibly fruitful methods of exposition, and correct methods that no longer allow him to express his thought except by distorting it and at the cost of a tiring effort. Neither one nor the other way is exempt from dangers. The Greeks followed the second, and it is perhaps there, even more than in the sterilizing effect of the Roman conquest, that the reason for the surprising halt must be sought. It has been suggested, without it being unlikely, that the oral teaching of the successors of Archimedes and Apollonius could have contained many a new result without their having thought it necessary to inflict on themselves the extraordinary effort required for a publication which conformed to the received canons. It is no longer such scruples in any case that stopped the mathematicians of the XVIIth century, when, faced with the new problems which were being stated in droves, they sought in assiduous study of the writings of Archimedes the means to bypass it. (p. 170)

Before closing, it will be well briefly to mention some of the drawbacks that beset Bourbaki’s historical writing. First off, the English translation from the French can often resort to awkward turns of phrase, as if it were composed by a non-native speaker. Second, the table of contents skips over complex function theory altogether, is weak on differential geometry, manifold theory, ordinary and partial differential equations and operator and spectral theory. The apparent reason for such wide gaps in coverage is that the chapters reprinted here are limited to the corresponding volumes of Bourbaki’s ambitious series of ground-breaking textbooks, the Eléments de Mathématique published to date as of 1984 (which aims eventually to embrace all of mathematics – a hopeless task, one might fear, as the rate at which new mathematics is being invented probably exceeds that at which it can be captured in textbooks, and moreover is ever accelerating!).

A more serious fault (in some quarters) would be that Bourbaki’s style is unapologetically Whiggish. This will present no particular problem to the aspiring professional who merely wishes to bring himself up to date on the field as it currently stands, but does entail a certain obliviousness on the part of the author concerning the motivations behind the researches commented upon, whose inclusion here is justified solely in terms of relevance to the later tradition in the West rather than in terms of contemporary importance [Belang]. Although the original accomplishments of other great civilizations (Babylonian, Egyptian, Greek-Hellenistic, Arabic, Hindu, Chinese etc.) are duly cited when germane, scant detail is provided until they enter the mainstream of western mathematics beginning around the fifteenth century. Thus, for instance, Bourbaki entirely misses the spirit of Egyptian arithmetic and geometry, concerning which Otto Neugebauer’s tireless reconstructions have furnished so much insight (vide our review elsewhere).

The reader who knows what he is getting and what not, then, will derive no little benefit from ruminating on Bourbaki’s historical commentaries and so familiarizing himself the more with the interconnection and genealogical dependence among mathematical ideas. Nevertheless, none of the authors intends his historical notes as anything more than an ancillary to other original research into mathematics itself not reprinted here. Hence, one will miss the perspective of a true historian of ideas. For this reason, too, the historical notes in the present collection lack verve and philosophical depth – in light of which this reviewer judges it fitting to assign but three stars for work that may be competent in its line, though hardly genial.

[Renard · Rating 3 out of 5]

Cest vraiment un livre pour ceux qui s'y connaissent bien en mathématiques. Dans ce cas il doit être super intéressant. Pour moi (prépa + école d'ingé), c'était trop techniques.
Néamoins il vous montre comment les mathématiques se sont formées, entre pratique et théorie, entre côté brouillon et rigueur.