# Where does the notion of “three crises of mathematics” come from? [duplicate]

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.

### Question

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.