Abstract
Bourbaki saw his project of the Éléments as the culmination and synthesis of two trends leading to the state of mathematics in the thirties of last century: axiomatics and set theory. We argue that, in Bourbaki’s rendering, it was an unholy marriage. To stay together, set theory had to be reduced to a language and logic to the grammar of that language, and Hilbert to the caricature of a formalist. Both set theory and mathematical logic have suffered from this misrepresentation; they couldn’t force their entrance in the house, as other disciplines did, capitalizing on their necessity, since they were already, nominally and distorted, inside.
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Notes
- 1.
Nicolas Bourbaki was born in 1935 and the first publication appeared in 1939, Fascicule de Résultats de la Théorie des Ensembles, but there was the war: “Bourbaki survived during the war with only Henri Cartan and Jean Dieudonné—according to Cartier. But all the work that had been done in the thirties blossomed in the fifties”.
- 2.
Cartier (1998). Cartier himself was a bourbakist. We shall refer grammatically to Bourbaki as a single individual, male of course, although Cartier describes four generations of members of the family.
- 3.
From the minutes of the Bourbaki’s meetings, see later n. 32.
- 4.
See Hermann (1986). Cartier recalls how André Weil (1906–1998), staying in Göttingen in 1926 and studying with David Hilbert (1862–1943), was unaware of the inception of quantum mechanics.
- 5.
Only Radon measure is presented; it is not by chance, because the primary interest was oriented to the theory of integration (we are not suggesting that Bourbaki was ignorant on this topic). The grand project was born from the idea of writing an up to date analysis textbook.
- 6.
Only crystal groups are mentioned, thanks to Cartier’s insistence; it was him who pointed to their relationship with Lie groups and introduced Coxeter’s work to the others bourbakists.
- 7.
“There were various attempts within the group to focus on new projects. For instance, for awhile the idea was that you should develop the theory of several complex variables, and many drafts were written. But it never matured, I think partly because it was too late. There were already many good textbooks on several complex variables in the seventies, by Grauert and other people. [...] There was a whole generation of textbooks, and books, which were under his influence”, Cartier (1998).
- 8.
Cartier (1998).
- 9.
Dieudonné (1939), reprint p. 544. The translations from Dieudonné’s essay are of the author.
- 10.
Bourbaki (1949), p. 1.
- 11.
ivi, pp. 2–3.
- 12.
ivi, p. 7.
- 13.
ivi, p. 8.
- 14.
Hilbert (1900).
- 15.
Dieudonné (1939), reprint p. 544.
- 16.
And probably wrote this essay; “Dieudonné was the scribe of Bourbaki”, according to Cartier.
- 17.
Bourbaki (1948), English transl. p. 223.
- 18.
ivi, p. 222.
- 19.
ivi, p. 230.
- 20.
ivi, p. 223. In the last two quotations it is noticeable Poincaré’s influence.
- 21.
ivi, p. 228. We shall not dwell on the concept of structure, which is dealt with in other contributions to this volume. See also Corry (2003).
- 22.
Dieudonné (1939), reprint p. 544.
- 23.
Dieudonné does not mention Giuseppe Peano (1858–1932) and the work of his school, even on new theories such as vector spaces. Could be for disdain of Peano’s artificial language, or national rivalry.
- 24.
Dieudonné says “nombre fini expérimental”, meaning a number upon which “one can perform the operation of counting”.
- 25.
Dieudonné (1939), reprint p. 556.
- 26.
- 27.
Dieudonné (1939), reprint p. 553.
- 28.
The lack of reference to Gödel is forcefully denounced and lamented by A. R. D. Mathias (1944) in Mathias (1992).
- 29.
See Rosser’s comment in n. 34.
- 30.
In Hilbert (1922) for example, he describes how through formalisation “real mathematics o mathematics in a strict sense, becomes an array of provable formulae”; by applying the contentual [inhaltlich] finitary methods one obtains “copies of the transfinite propositions of usual mathematics”.
- 31.
Bourbaki (1960), English transl. p. 40. The historical notes were the best place to treat polemical issues (des subjets polémiques) as it is said in La Tribu, n. 25, p. 9, see next foootnote.
- 32.
I am indebted to A. R. D. Mathias for directing me to the minutes of Bourbaki’s meetings (Congrès), written in the form of a journal, La Tribu. Bulletin œcoumémique, apériodique et bourbachique. The relevant issues for our concerns are n. 25, 5/26-7/8 1951 and n. 28, 5/25-7/8 1952 (Congrès de la motorisation de âne qui trotte). The conclusions one can draw have been already signalled in Mathias (2014). Mathias also notes that Claude Chevalley (1909–1984) had read parts of the 1940 monography of Gödel on constructible sets and urged for a stronger system modelled on Gödel’s. Page numbers in the rest of this section refer to the typewritten La Tribu minutes of these two meetings.
- 33.
The exception was the discussion for the acceptance of Hilbert’s \(\epsilon \)-formalism, endorsed by Weil and Chevalley.
- 34.
Rosser (1950). His remarks: the axioms are a weak version of the original Zermelo system; the union of two sets exists only if they are subsets of a same set; individuals without elements are not sets, as in Zermelo, but only one can be proved to exist, the null class specifically taken as non-set; there is only one reference to the relevant literature; the notion of synonymity seems to formalise some personal feeling, but should probably better be left out.
- 35.
With possibly some variants; for instance the well-ordering property instead of induction.
- 36.
An Italian figure of speech easily understood.
- 37.
Cardano (2009).
- 38.
To this end it is sufficient in fact a weak theory, of strength comparable to that of Z, in which to justify a sparing construction of sets and operations on sets inside a given set. See e. g. Mathias (2001).
- 39.
Actually Tarski did not use the term “structure” but that of “model”, defining it as a pair K, R of a set and relation such that the axioms of a simple toy theory with a single relational symbol could be interpreted and become true. See Tarski (1941).
- 40.
van der Waerden (1931). “What van der Waerden had done for algebra would have to be done for the rest of mathematics”, according to Cartier.
- 41.
Enriques (1922), p. 140.
- 42.
A term coined gratuitously, given the by then universally acceptance of “categorical”. It is true that in the historical notes Bourbaki defines categoricity as completeness, as we have seen; but then we have three terms, “univalent, categorical, complete” to account for. If univalent means complete, the two words “univalent and categorical” name the same concept; if univalent means categorical it cannot mean complete. We’ll propose below a reading coherent with the accepted meanings of these logical terms, but any reading turns out ambiguous.
- 43.
Bourbaki (1948), p. 230.
- 44.
The sequence of events which brought to the solution of the apparent paradox of two contradictory theorems for a while cohabitant has been tortuous but in the end the puzzle has been worked out, and is now easily explained, of course with some knowledge of modern logic. All the ingredients were ripe and well known when Bourbaki was born. But again, the puzzle requires the acknowledgment of the presence of logic in the mathematical thought.
- 45.
Without the replacement axiom of ZF the theory of ordinals and transfinite induction could not be justified. Bourbaki could rebut that the latter is useless, or does not belong to mainstream mathematics, but it seems bizarre to exclude from mathematics the theory of cardinality.
- 46.
Bourbaki (1948), English transl. p. 230.
- 47.
Cartier (1998). As for analysis, even in the engineering schools reached by Bourbaki’s influence the calculus textbooks began with a chapter in set theory, and often developed analysis not on the base of real or complex numbers but on abstract structures such as metric or topological spaces.
- 48.
Roubaud (1997), pp. 158–9. Jacques Roubaud (1932) is a bourbakist and a poet. To him the outline of the first pages of the topology volume of the Éléments conjured up the inescapable structure of Wittgenstein’s Tractatus.
- 49.
See Cipra and MacKenzie (2015). It is a series of booklets began by Barry Cipra in 1993 (Dana Mackenzie joined as co-author from vol. 6 on) offering reports on recent results and researches.
- 50.
Devlin and Lorden (2007).
- 51.
- 52.
MacLane (1972).
- 53.
In the fifties in the USA the panic caused by the success of the soviet space program, with the urge to bridge the gap with URSS in scientific and technological knowledge, encouraged to accept all the projects for a better teaching which were ready at hand, even if conceived with other concerns; this was the case with the Illinois project from which the new maths originated: see the remarks by W. W. Sawyer (1911–2008) in http://www.marco-learningsystems.com/pages/sawyer/sawyer.htm.
- 54.
A famous general criticism with detailed denunciations is due to Kline (1961).
- 55.
Roubaud (1997), p. 96.
- 56.
ivi, p. 21.
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Lolli, G. (2020). Bourbaki and Foundations. In: Peruzzi, A., Zipoli Caiani, S. (eds) Structures Mères: Semantics, Mathematics, and Cognitive Science. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 57. Springer, Cham. https://doi.org/10.1007/978-3-030-51821-9_2
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