Update: It can be traced back to Fraenkel-Bar-Hillel's Foundations of Set Theory, originally published in 1958. Further discussions can be seen at the linked question.

The notion of "three crises of mathematics" is extremely popular in China. Open any book, textbook or encyclopedia about the history of mathematics, one will find a version or variant of this standard description.

Throughout the history of Western mathematics, there have been three major crises on its foundation due to new discoveries, and led to major advancements in mathematics. Namely,

  1. The first crisis is the discovery of $\sqrt 2$ (by Hippasus according to the legend) and other incommensurables that challenged the foundation of Pythagoreanism, not resolved until Eudoxus of Cnidus and Euclid offered geometric treatments.

  2. The second crisis is the inconsistent uses of infinity and infinitesimal by Newton and Leibniz that raised doubts on the theoretical soundness of calculus, with vocal critics such as George Berkeley, eventually fully resolved by Cauchy who came up with rigorous definitions that are widely accepted, e.g. the epsilon–delta method.

  3. The third crisis is the discovery of various paradoxes in Cantor's naive set theory, such as Russell's paradox, it threatened the set-theoretic foundation of mathematical logic. The attempts of solving the problem included Russell's type theory and the Zermelo set theory, and eventually led to the creation of ZF and ZFC set theory.

It was persuasive to me. Until I realized the notion of "three crises of mathematics" does not exist in English at all.

In English literature, although almost everyone recognizes the treatment of incommensurables by Greeks or infinitesimal by Newton as major theoretical problems in history, some authors even used the word "crises", but different authors presented these issues from different perspectives, for example, In The third great crisis in mathematics by C.K. Gordon, published in 1968 in IEEE Spectrum (doi:10.1109/MSPEC.1968.5214636), focused on later technical problems in set theory, such as the well-ordering theorem and the Banach-Tarski paradox. Another paper by Ernst Snapper, The Three Crises in Mathematics: Logicism, Intuitionism and Formalism, discusses modern issues as the title indicated, not historical ones. In conclusion, a single standard version of "three crises of mathematics" doesn't exist.

Furthermore, a critical read can raise even more questions.

  • According to Ten Misconceptions from the History of Analysis and Their Debunking, Berkeley’s logical criticism was not effective (and his metaphysical criticism was largely ideological), "Fermat planted the seeds of the answer to the logical criticism of the infinitesimal, a century before George Berkeley ever lifted up his pen", and Cauchy did not replace infinitesimals by modern, meaning epsilon-delta, definition of continuity.

  • Only the so-called "third crisis", known as the foundational crisis of mathematics in the 20th century, is recognized in history, the "first" and the "second" crises were simply the subjective assessments by finding similar examples in history, and well before the establishment of modern mathematical logic.

  • The description of these crises is incomplete. The debates in the 20th century were important, but this popular account only mentioned Russell's paradox in naive set theory. Many other issues, such as Gödel's incompleteness theorems, which had much more impact on the foundation of mathematics, were not even mentioned at all. The halting problem was ignored as well.

  • And arguably, the historical examples are incomplete. Personally, I think that, if finding persuasive examples in history regardless of context is the primary goal, I would definitely add the discovery of non-Euclidean geometry into the list, as it raised doubts on the notion of absolute rationality and authority, which triggered some great debates among philosophers. Also, more recent issues, such as Vladimir Voevodsky's claim of his suspicion that an inconsistency in first-order Peano arithmetic might be discovered in the future, summarized in the essay The Consistency of Arithmetic, deserves an inclusion.


My question is, where did the standard notion of "three crises of mathematics" come from?

I'm not referring to three independent issues, but specifically asking for the origin of the "three issues as a complete package" version in the beginning of my question. I suspect it must come from somewhere authoritative, I suspect it came from either German literature or Soviet Union literature, or possibly less known English-speaking historians, but I'm not sure.

Note: I don't think this question is a duplicate of What is the status of the three crises in the history of mathematics? or Foundational crises in non-Western historical mathematical communities. Although the same concept was mentioned, neither were an explicit requests of sources or origins of this concept.

  • $\begingroup$ The second link gives one source in the question, Fraenkel-Bar-Hillel's Foundations of Set Theory, originally published in 1958. But the idea was commonplace in the Whiggish "rational reconstructions" (Carnap's term) of history by positivist philosophers in 1930-50s after the "scientific" view of mathematics was presented in Russell's Principia. It was then enshrined in historical notes to Bourbaki volumes (1947+) and spread into popular accounts, like Kline's (1953+), and multiple textbooks. There are echos of it even in current Western texts, and China is having more of a lag time. $\endgroup$
    – Conifold
    Sep 19, 2019 at 1:59
  • $\begingroup$ @Conifold Thanks for the comment. I know Morris Kline's works, and indeed he mentioned all of these issues, but I don't think he specifically picked three questions and call them the crises. I have Kline's Mathematics in Western Culture on my bookshelf, and checked the book again, and surely there is no such mention, perhaps he mentioned it in another book? BTW, having a list of references to Bourbaki's volumes would be nice. $\endgroup$ Sep 19, 2019 at 2:51
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    $\begingroup$ Kline does not call them crises, but tells us how Pythagoreans discovered irrationals "to great dismay" (p.36) and how Greeks "failed to develop the number system" and other modern goods (p.38), etc. Ditto with Bourbaki. Weyl talked of a "new crisis" back in 1920. Casting this sort of narrative into the handy shorthand of "three crises" on the path to modernity might have been done by various authors, Fraenkel-Bar-Hillel being among the first. $\endgroup$
    – Conifold
    Sep 19, 2019 at 7:57
  • $\begingroup$ @CarlWitthoft Okay, so the same notion dates back to at least 1958 in Fraenkel-Bar-Hillel's Foundations of Set Theory (although this date was not given in that question). I'll close the question as a duplicate. $\endgroup$ Sep 19, 2019 at 13:29
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    $\begingroup$ A possible source is the divulgative successfull book of Howard Eves (1911-2004) on Foundations and Fundamental Concepts of Mathematics (1st ed 1958, Dover reprint 1997) that speaks of a crisis with the Greek discovery of the irrationality of the ration between the side and the diagonal of the square (page 16), of a second crisis with the discovery of the calculus (page 263) and of a third one with Cantor (page 263). $\endgroup$ Sep 19, 2019 at 15:23