Occasionally I see the claim, that mathematics was constructive before the rise of formal logic and set theory. I'd like to understand the history better.
- When did proofs by contradiction or by excluded middle become accepted/standard? Can one find them for instance in classical works (Archimedes, Euclid, Euler, Gauss, etc.)?
- Was there ever a debate about their validity before Brouwer?
My interest was sparked after reading the following "proof" from Newton's Principia that seems to use contradiction:
LEMMA I.
Quantities, and the ratios of quantities, which in any finite
time converge continually to equality, and before the end of that time
approach nearer the one to the other than by any given difference,
become ultimately equal.If you deny it, suppose them to be ultimately unequal, and let $D$ be
their ultimate difference. Therefore they cannot approach nearer to
equality than by that given difference $D$; which is against the
supposition.
Best Answer
I want to echo the other answer, that Brouwer presents the first robust challenge to the Law of Excluded Middle (LEM), but I do want to add some history and background, and maybe recommend a related paper.
The main paper by Brouwer is De onbetrouwbaarheid der logische principes or in English The unreliability of the logical principles published in 1908. In some sense, this is a continuation of his dissertation in 1907, but I think it's better to just speak of 1908 as the starting point for Brouwer's views on LEM.
In a paper [1] offering a new translation of Brouwer's paper, the authors have a section devoted to "precursors" (this is only four pages long, too long to quote in full, but worth reading). At first glance, the references given do not mention LEM, and the relationship to Brouwer seems slightly strained, but I think the authors adequately defend the relevance of the section. It starts with a quote from Kronecker in 1882 about constructability issues. It seems this was extended by Kronecker's student Molk, who said in 1885
The authors then devote an entire page of quotes from Molk (in 1904), noting the similarities "even down to some of the finer detail" to Brouwer's line of thought, though Molk doesn't continue as far as Brouwer. For instance, here's one sentence from the page of quotes:
I think it should be pointed out that the authors tried but failed to establish any record that Brouwer was aware of Molk's writings. A footnote points out that Brouwer himself seems to have become aware of the similarities some years later.
Following the discussion on Molk, the authors give a correspondence by Lebesgue in 1905 which contains an afterthought very similar to LEM:
but this seems to be more a curious coincidence than serious mathematical reasoning.
I think the final sentence summarizes the section well:
Edit: Adding a little bit more history, just some more sources that say the same as above.
In [4], Kleene devotes the very first sentence to a brief background on intuitionism (he goes on to talk about Brouwer in the next paragraph, not included here):
Above, Kleene references his Introduction to Meta-Mathematics [5], which says the following at the opening of section 13
I extracted the references for Poincaré and Borel, they are listed at the end of the references section below.
I could not find a copy of Brouwer’s Intuitionism. Studies in the History and Philosophy of Science, vol. 2 by W. van Stigt (1990) to read, so that may or may not have other useful information. However, I did find the following paragraph in a book review [2], I wanted to include this just to give some context for LEM and when it gained prominence.
Finally, there is a related paper [3] that might be relevant to the original question (though I think the "crisis" talk is a bit sensationalized from a modern perspective, and could have been moderated a bit more in the paper). It looks into the history of science and math about Intuitionism in particular, discusses Kuhnian paradigm shifts, and some difficulties in why Intuitionism never "caught on."
There's a lot of background, probably the only section of interest is section 4 which begins
[1] Mark van Atten, Göran Sundholm. "L.E.J. Brouwer's `Unreliability of the logical principles'. A new translation, with an introduction" https://arxiv.org/abs/1511.01113
[2] Wim Ruitenburg. "Review of W. P. van Stigt, Brouwer's Intuitionism". Modern Logic 2 (1992), no. 4, 424--430. https://projecteuclid.org/euclid.rml/1204834908
[3] Bruce Pourciau. "Intuitionism as a (failed) Kuhnian revolution in mathematics" Studies in History and Philosophy of Science Part A. Volume 31, Issue 2, June 2000, Pages 297-329. https://doi.org/10.1016/S0039-3681(00)00010-8
[4] S. C. Kleene, R. E. Vesley. "The Foundations of Intuitionistic Mathematics" North-Holland, 1965, page 1.
[5] S. C. Kleene. "Introduction to Meta-Mathematics" Ishi Press, New York 2009. Page 46.
Kleene's references (apologies for bad French)
Borel 1898 Leçons sur la théorie des fonctions. Paris (Gauthier-Villars), 8+136 pp. 4th ed. (Leçons sur la théorie des fonctions; principes de la théorie des ensembles en vue des applications à la théorie des fonctions). Paris (Gauthier-Villars), 1950, xii+295 pp. Available on archive.org, e.g. https://archive.org/details/leconstheoriefon00borerich
Poincaré 1902. La Science et l'hypothèse. Paris, 284 pp. Translated by G. Bruce Halstead as pp. 27-197 of The foundations of science by H. Poincaré, New York (The Science Press) 1913; reprinted 1929.
Poincaré 1905-6 Les mathématiques et la logique. Revue de métaphysique et de morale, vol. 13 (1905), pp 815-835, vol. 14 (1906), pp. 17-34, 294-317. Reprinted in 1908 with substantial alterations and additions.