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I have seen a claim in some literature that there are three crises in the history of mathematics. The first is the discovery of $\sqrt2$ being irrational in Greek time which shook the belief that every point on a line can be measured by a rational number at that time. The second is the infinitesimal used in the 17th century in Calculus by Newton and Leibniz. The third one is Russell's paradox and other paradoxes discovered from the turn of the 20th century which has shaken the foundation of mathematics.

However, after I googled it online (in English), I can hardly find any discussion or support of such a claim. So I wonder if this is widely accepted in the study of the history of mathematics, or simply a statement without much merit.

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    $\begingroup$ Goedel's name would be a better candidate for a third crisis, but this is just my opinion; what's more, I do not think Russell's result had much importance in mathematics, it just showed the 'naivete' of early set theory . $\endgroup$ – sand1 May 8 at 20:57
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    $\begingroup$ Russell's paradox has a significant impact on the future development of mathematics, i.e. the creation of axiomatic set theories. So it is not just a disproof of naive set theory earlier and definitively has an important impact in mathematics. $\endgroup$ – Math Wizard May 8 at 22:37
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    $\begingroup$ Calculus was essentially discovered back in ancient Greece by Archimedes ; then lost for a while. en.wikipedia.org/wiki/The_Method_of_Mechanical_Theorems $\endgroup$ – Carl Witthoft May 9 at 12:54
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    $\begingroup$ Although Archimedes may have some ideas of limit and derivative, Calculus was invented by Newton and Leibniz in the 17th century, a widely accepted fact. $\endgroup$ – Math Wizard May 9 at 17:58
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No, it is not widely accepted. The language of "crises" is rather obsolete and mostly reflects the attitudes of the early 20th century projected backwards. At that time, the contemporaries did indeed characterize the situation in mathematics (and physics) as a crisis. For example, Weyl's 1920 address was titled “The new foundational crisis in mathematics”, to which Hilbert saw fit to respond with what is now called the Hilbert program (see van Dalen's paper). But the language fits well with the language of (Kuhnian) revolutions, which are precipitated by "crises" (accumulations of anomalies), and provides a convenient shorthand for undergraduate texts to get through the history expediently. After all, the retellings of history in textbooks largely serve the practical purpose of bringing students up to speed on the current mathematics, and this is easier to do by translating all of mathematics into modern terminology, and marking the places where major curriculum pieces were introduced as "crises".

The first "crisis" is almost completely made up based on discredited accounts of the early Pythagorean history. The rational numbers or the real line are sheer modernizations with little relation to the ancient Greece, and even the story of ratios and incommensurables is a dramatized fable. On early Greek mathematics a good source is Fowler's Mathematics of Plato's Academy. Concerning the traditional story about the "discovery of irrationals" and the subsequent "crisis", a reviewer writes:

"It may be somewhat surprising, then, to discover that many, if not most, historians of ancient mathematics disbelieve some or all of this story. In The Mathematics of Plato's Academy, David Fowler gives a convincing account of the reasons for rejecting the standard story, and offers a very interesting alternative reconstruction of the history of early Greek mathematics... One of the important themes in Fowler's book is that one must carefully assess the available sources on early Greek mathematics. He demonstrates that most of the elements of the "standard story" either come from very late Greek sources (500 years or more after the time of Plato) or are reconstructions by modern historians who are really reflecting nineteenth and twentieth century concerns about foundational issues... The resulting picture is quite different from the traditional one. In particular, incommensurability is seen not as a foundational crisis but rather as an interesting discovery that led to significant mathematics."

The second "crisis" is usually associated not with the creation of calculus but with its subsequent problems, such as the inconsistencies with infinitesimals. It was a pretty slow moving "crisis" though, and the kinematic (late Newtonian) conception of calculus, within which Cauchy worked, and most of algebraic analysis of 18th century, were not seen as having some cardinal problems at the time. Only injecting later foundational attitudes and concerns about rigor lets one detect them. Another potential crisis, with a better claim to the name (in the Kuhnian sense at least), were the struggles with the notion of function that followed the discovery of Fourier series, see the introductory chapters to Bressoud's Radical Approach to Real Analysis, who names his opening chapter Crisis in Mathematics: Fourier's Series.

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  • $\begingroup$ "Crises are obsolete". Call them what you like but mathematicians did (and do) get shocked by surprising results: nowhere differentiable continuous functions, Peano curves, Godel's result, etc. Similarly, there may well have been no "foundational crisis" for the Greeks but that does not mean that irrationals didn't shock their understanding of mathematics. $\endgroup$ – Chrystomath May 9 at 9:40
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    $\begingroup$ @Chrystomath The language of "crises" comes from "foundational crisis" model of 19th-20th century, more specific than just unexpected result in general . Peano curve and Gödel's theorem fit into this pattern. Peano curve upset the naive idea of what a curve might be (it goes with the reconsideration of the notion of function after Fourier), Gödel's result upset the Hilbert program. It is harder to assimilate ancient Greece or calculus results into this pattern, because concerns with foundations were not current then. The issue with the language is just that it encourages too much modernization $\endgroup$ – Conifold May 9 at 20:39
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These notions were popular in the middle of 20th century, in the aftermath of the third crisis. More precisely they were called the "crises of foundations of mathematics". Of course the first two were essentially the reconstructions of historians of mathematics.

Mathematics was conceived as a rigorous method of obtaining true statements by proofs. The starting point was a set of premises which were considered self-evident without any doubt. This basic believe was seriously shaken several times in the course of history, and the principal such episodes are called "crises of foundations". You can read about this in the middle 20th century histories of mathematics.

About the "first crisis" we can only speculate. The earliest complete mathematical writings which we have originate at the time after the crisis.

The second one is well documented. It is related to invention of Calculus which it was difficult to justify to the same level of rigor which was considered standard since the Greeks. It took about 200 years to obtain full clarity about foundations of calculus.

The third crisis was triggered by Cantor's discoveries, then the discovery of paradoxes, attempts to create new rigorous foundations, and spectacular failure of these attempts. In some sense this third crisis continues, since it was never resolved. But my impression is that most mathematicians just lost interest to these questions, so it is not perceived as a crisis anymore.

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  • $\begingroup$ I find it strange to claim that we've reached full clarity about the foundations of calculus, when no modern calculus book formalises concepts like infinitesimals, differentials, higher order differentials, variables, constants or the idea that something is a function of something else. All these were central in the original calculus up to ~1900. Physicists still use them, but modern mainstream mathematics just avoids them (and any attempts to make them rigorous). $\endgroup$ – Michael Bächtold May 9 at 16:29
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    $\begingroup$ @MichaelBächtold, Keisler's calculus uses infinitesimals, and has an accompanying "foundations" book. "Nonstandard methods in Stochastic Analysis and Math Phys" by S. Albeverio, J. Fenstad, R. Hugh-Kruhn, and T. Lindstrom. Also, "Nonstandard Analysis in Practice", ed. Diener and Diener. (And A. Robert's very nice little introduction.) $\endgroup$ – paul garrett May 9 at 16:38
  • $\begingroup$ @paulgarrett thanks for the references. I might be wrong, but my perception is that NSA is ignored by 99.9% of mathematicians (as is synthetic infinitesimal analysis). I'm aware that these approaches formalize infinitesimals, but I believe they don't formalize the other things I mentioned. $\endgroup$ – Michael Bächtold May 9 at 17:16
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    $\begingroup$ @alexandre-eremenko Do you not consider ZFC to be a resolution? Why not? Just curious... $\endgroup$ – sfmiller940 May 9 at 17:29
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    $\begingroup$ @sfmiller: essentially because we do not know (and probably will never know for sure ) that ZFC is free of contradiction. And we don't know whtether new axioms will be necessary in the future. $\endgroup$ – Alexandre Eremenko May 9 at 17:34

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