In Foundations of Set Theory by Fraenkel, Bar-Hillel, and Levy (1973), the authors argue that there have been three distinct periods of crisis in the foundations of mathematics. The first was undergone by the Ancient Greeks:

[...] two discoveries were made that were extremely paradoxical: the first was that not all geometrical entities of the same kind were commensurable with each other, so that, for instance, the diagonal of a given square could not be measured by an aliquot part of its side (in modern terms, that the square root of 2 is not a rational number); the other were the paradoxes of the Eleatic school (Zenon and his circle) developing with many variations the theme of the non-constructibility of finite magnitudes out of infinitely small parts. (pp. 13)

This crisis shocked the Greek mathematicians into obtaining two more brilliant achievements: the theory of proportions, as contained in books 5 and 10 of Euclid’s Elements, and the method of exhaustion, as invented by Archimedes, that was nothing less than a strict, though not sufficiently general, forerunner of modern theories of integration. (pp. 13)

The second involved the foundations of analysis:

[...] in the 17th and 18th centuries, the great power and fruitfulness of the newly invented calculus led most mathematicians of those times into feverish applications of the new ideas without caring much for the solidity of the basis upon which the calculus was founded. However, the shakiness of this basis became clear at the beginning of the 19th century, constituting the second crisis in the foundations of mathematics. (pp. 13)

In order to overcome this crisis, Cauchy, in the eighteen thirties, showed how to replace the irresponsible use of infinitesimals by a careful use of limits, whereas Weierstrass and others, in the sixties and seventies, demonstrated how all of analysis and function theory could be “arithmetized”. (pp. 13-14)

And the third of course was sparked by investigation into set theory and the discovery of the antinomies:

More than the mere appearance of antinomies in the basis of set theory, and thereby of analysis, it is the fact that the various attempts to overcome these antinomies, to be dealt with in the subsequent chapters, revealed a far-going and surprising divergence of opinions and conceptions on the most fundamental mathematical notions, such as set and number themselves, which induces us to speak of the third foundational crisis that mathematics is still undergoing. (pp. 14)

The three crises were motivated by either explicit paradoxes or severe philosophical skepticism about the subject matter and/or validity of the mathematical tools and concepts which were under scrutiny. They also were started by people who are part of Western civilizations (mainly European) and can be seen as part of Western mathematical history (at least in origin, the third crises surely has had global contributions made to it).

My question is:

Are there any examples in non-Western historical cultures, such as classical India or China for example, which had foundational crises in their work on mathematics? Were there any paradoxes or was there severe philosophical skepticism which motivated and lead directly to new tools or conceptions of mathematics?

  • $\begingroup$ I do not think so; "foundational crisis" presupposes the search for foundations abd the deductive structure of mathematics, where all the mathematical facts (at least relative to a discipline; see e.g. Euclid's Elements) are deduced from first principles that are undubitable is typical of the mathematical (and philosophical) tradition deriving from Ancient Greece. $\endgroup$ Jul 12, 2018 at 12:18

5 Answers 5


Let me point out that the Fraenkel-Bar-Hillel "march to rigorization" quote reflects the early 20-th century "rational reconstructions" of history that were worked out by mathematician-historians like Klein, van der Warden and Dieudonné and got enshrined in historical sections of many old and still current textbooks, partly due to their didactic convenience. Nonetheless, they lift patterns and concerns from recent history (formalization, axiomatization, foundations, etc.) and project (impose) them onto prior times. As a result, many of their surmises are largely discredited by contemporary scholarship, including the one about "foundational crises".

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", the 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."

There was no "shock" or "crisis", and Eudoxus was not the savior. Much work on incommensurables was done by Pythagoreans like Archytas, Theodorus and Theaetetus before him, some of it survives in books II and X of Elements. Their work was based on anthyphairesis, a geometric version of what is now called Euclidean algorithm producing what is now called continued fractions, the same process that presumably led to the discovery of incommensurables. There is no evidence of the supposed phase before the discovery, when Pythagoreans thought that all magnitudes were commensurable. Eudoxus's motivations, if we are to speculate about them, likely were extra-mathematical and not internal, they might have had more to do with Platonic push to eliminate motion and process from geometry "corrupted" by them. The same tendency is transparent throughout Elements, and we know that Eudoxus's ingenious mathematical astronomy of nested spheres was a response to Plato's call to "save the phenomena" from their sensible disorder too. Let me also mention in passing that another background assumption of textbook histories, "Euclid the axiomatician", has also been largely discredited, see references in What caused or contributed to Euclid's Elements and Synthetic Geometry falling into disfavor?

The "feverish applications" without "solidity of the basis" story about calculus conveniently omits its kinematic (late Newtonian) conception, within which Cauchy worked, and most of algebraic analysis of 18th century, that had much more "solidity", to manufacture another "crisis" and leap straight to Weierstrass and Cantor. The boost at the beginning of 19-th century had more to do with institutional changes and Fourier inspired broadening of the repertory of functions than with "the basis".

I think the motivation behind the question is to compare Western mathematics to other cultures to see if it displays unique historical patterns, and this is a valid topic. Unfortunately, the proposed "pattern" in this case is largely fabricated even for the West, so there is no point looking for it elsewhere. An additional difficulty is that our sources for Egyptian, Babylonian, early Chinese and Indian mathematics are too sporadic to form a reliable picture of their development. It would take some work to figure out what a similarly motivated but answerable question might be, perhaps one should look for apparent "big turns". Of course, it will depend on what counts as "big" or "turn", but some promising places to look might be medieval Islamic mathematics and the "Chinese Renaissance". One could perhaps make a case that some traits of "Western" (including Middle East) mathematics were unique, see related Is the Scientific Method uniquely Western?, but we should be mindful that our "sample" for detecting historical patterns just might be too thin to generate anything but artifacts.

  • $\begingroup$ I disagree with your assessment of the question. I used the quoted text just to give context to my question, but I can see how it came come off as setting up my question as being mainly about comparison. That isn't the case. Granting everything you've said in this answer, and taking the word "foundational crisis" out of my question, the question still remains the same: "Were there any major mathematical concepts or techniques that were developed as a response to paradoxes or philosophical skepticism in non Western cultures?" $\endgroup$
    – Not_Here
    Jul 14, 2018 at 0:53
  • $\begingroup$ Everything in your answer besides the last paragraph ignores this, which I think is really the main part of my question. $\endgroup$
    – Not_Here
    Jul 14, 2018 at 0:53
  • $\begingroup$ @Not_Here Sorry, but "paradoxes or philosophical skepticism" is still too vague and modeled too much on recent Europe. Does it refer to "internal" paradoxes? If so the description is dubious, mathematics functioned differently in non-Western societies. There were mathematical developments, like epicyclic geometry in Islamic Middle East, in response to philosophical concerns, the conflict between Ptolemaic astronomy (equant) and Aristotelian physics, but I am not sure if this is a "paradox" or "non-West". I can give you references but you'll have to comb through specific events to see what fits $\endgroup$
    – Conifold
    Jul 15, 2018 at 19:37

Here is a sort of answer. It is merely a memory I have, so no reference. And I cannot say whether it is even true. Maybe someone knows a reference.

In ancient Egypt, if you go back far enough, you find that the taxes for a plot of farmland were calculated based on the perimeter of the plot. Over many years, experience showed that different plots (say, a square and a long thin rectangle) with the same perimeter would yield different amounts of grain. So they had to invent a new notion of "area" to use in computing the taxes.


I'd suggest that there are evidence-based reasons why the present question, though interesting, may be regarded as very difficult or impossible to answer, at least as it stands.

As Mauro Allegranza's comment already pointed out, the question presupposes a search for foundations, clearly a precondition before there can be such a thing as a foundational crisis.

In turn, one may question what the conditions are for a search for foundations. When one looks at historical evidence, one sees that at various periods mathematical knowledge may earn acceptance in (at least) three ways: on the basis of authority (because it was taught by a respected teacher); as a praxis (because it works); or on the basis of justification by an intellectually respectable theory.

The evidence is of several kinds. Acceptance on the basis of authority is with little doubt a phenomenon known worldwide. In Europe it is known at least from the example of acceptance during many centuries of the works of Aristotle and Ptolemy -- and the many mistaken things that they taught (along with other matters of persisting value or validity).

Historical acceptance by practical justification may sometimes be impossible to identify or exclude because of a shortage of evidence. The ancient evidence is often scanty, preferentially showing what was done or used, but not why: it sometimes lacks direct evidence (or even any evidence at all) about reasons for the acceptance and use of the praxis, and whether it was associated with any theoretical justification -- or with any felt need for any such thing. Examples can be seen in parts of the early history of trigonometry, e.g. in D E Smith's History of Mathematics, vol.2 especially chapter 8. This example shows how motivation for acceptance of a mathematical praxis can for lack of evidence remain a doubtful and conjectural matter, much less secure than the historical evidence for the praxis itself.

When there is acceptance in either of these two ways, there is arguably little room for a search for theoretical foundations or for a foundational crisis. (The western example of the long-accepted authority of Aristotle and Ptolemy also arguably casts doubt on the validity of the present framing of the question in terms of possible disjunction between western and non-western traditions.)

So the question perhaps might be usefully be reframed to include a preliminary enquiry for the preconditions of a foundational crisis. It may also be asked whether the example of 'three distinct periods of crisis' that underlies the question is heavily overdrawn, even magnifying by rhetoric alone the difference between 'crisis' and non-crisis.

Thus, the 19th-century work of Cauchy and then the arithmetists 'overcame' the calculus 'crisis' by clarifying contemporary scruples in a directly applicable way. It is arguable that this 'crisis' had involved a wider canvas anyway than problems of the new calculus and its infinitesimals or limits. There were contributory 17th-century scruples about the mathematical legitimacy of other innovative mathematics that had in common that they were ultimately of a character involving transcendentals and/or the infinite, e.g. the so-called 'tentative' solutions of Kepler's problem to evaluate in effect angle from area, and the impossibility of a direct numerical evaluation without something equivalent to infinite series solution or its approximation.

As to the first period, it seems difficult to claim that the problem or 'crisis' of the Pythagorean discovery of irrational numbers was in any analogous way 'overcome' by the theory of proportion or by the method of exhaustion. It arguably raised a particular problem involving the infinite that took a very long time to overcome or answer.

The 'periods of crisis' identified in the book cited in the question are arguably similar only by stretched analogy. If the analogy is allowed to be stretched like that, then the different crises also begin to look more like particular instances of a common and deeper and longer-lasting problem about how mathematically to handle aspects of the infinite. The 'periods' of 'crisis' may then be accountable more simply as periods when the problems gained more conspicuous attention, while at other times attentions were turned elsewhere.

What might be (respectfully) suggested here is a reframing to help towards an answer to an at least related question. It might first be asked whether, when and where there were historical periods and places where the first two mentioned forms of mathematical acceptance were missing, that is, acceptance as praxis or from a teaching authority. For those periods and places, one may perhaps then hope with greater likelihood to find signs of deeper mathematical introspection about foundations -- if indeed, as historians may hope, the relevant evidence was ever recorded and has been preserved.

  • $\begingroup$ See my comment to Conifold's answer, but basically I find the same issue with your answer. I grant that my question was set up possibly in a confusing way with too much focus on the cited text, but that wasn't my intended question, it was merely to give an idea of what I meant by "foundational crisis". The real question I meant to (did) ask was, in regards to non Western cultures, "Were there any paradoxes or was there severe philosophical skepticism which motivated and lead directly to new tools or conceptions of mathematics?" $\endgroup$
    – Not_Here
    Jul 14, 2018 at 0:58
  • $\begingroup$ In regards to your next to last paragraph, again I disagree, the analogy between the three given events (whether or not they are historically well represented in the text) is that they were all three motivated by either explicit paradoxes or severe philosophical skepticism about the subject matter and techniques used. Other ancient cultures besides just Western cultures wrote about and contended with both of those things, so I do not see how asking for a comparison is futile. Again, throw out the word "foundation", because that was not my focus. $\endgroup$
    – Not_Here
    Jul 14, 2018 at 1:01
  • $\begingroup$ And I would say as well that Mauro's comment is not correct. It points to philosophical reflect on the subject matter, not a search for foundations. For example, Panini's work on Sanskrit grammar was clearly a case of reflection on the subject matter and a desire to work towards formalization, not necessarily towards a "foundation". It is not the exact same as the Greek conception of the axiomatic method, but it is comparable and I am very sure he did it for similar, philosophically motivated reasons. $\endgroup$
    – Not_Here
    Jul 14, 2018 at 1:06
  • $\begingroup$ @Not_Here : well thank you for your comments, I find them interesting too, but I do think the question looks very different now after the edits. You've taken out inter alia any reference to foundations, so I'm now not sure what you intended by this expression before. I'll have to consider again whether there is anything that I could usefully offer. $\endgroup$
    – terry-s
    Jul 14, 2018 at 1:29
  • $\begingroup$ Before, by 'foundational' what I meant was something that was motivated by explicit paradoxes or severe philosophical skepticism. That term derailed my question, as evidenced by your and Conifold's answer which both focused on how either the text I quoted was incorrect in its analysis or that other cultures didn't have the same idea of foundation that stemmed from the Greek axiomatic approach. The text argued that in the Greek case, Zeno's paradoxes partially motivated the method of exhaustion; philosophical skepticism about infinitesimals motivated the epsilon-delta definition; $\endgroup$
    – Not_Here
    Jul 14, 2018 at 3:09

Don't have the reputation to comment on Gerald Edgar's answer, but it seems not-quite-accurate from a few minutes of googling, at least all the sources I could find on ancient Egyptian land taxes mentioned "area" but not "perimeter". An weaker but still interesting claim does seem to be true, according to the article "Measuring the Accuracy of an Ancient Area Formula":

In assessing the area of a plot of land, they used a faulty formula for computing the area of a general quadrilateral. Given sides of lengths $a,b,c,d$, they said $A = .5(a+c) \cdot .5(b+d)$, that is, they averaged the lengths of opposing sides and multiplied the two results. The formula is wrong but not as outrageously wrong as using the perimeter as a proxy, so it's slightly less embarrassing that they were still using it "200 years after Euclid had lived and taught in Egypt" (the paper spends a few pages analyzing how far off the formula is for different quadrilaterals).

And of course, there is no formula for the area of a quadrilateral that uses only its sides. Brahmagupta's formula works for cyclic quadrilaterals, but that was not until 7th century AD India. And Heron's formula does the trick for triangles, but I'm not sure if that was discovered after the relevant time period. (Full disclosure: I'm not an expert on ancient Egypt or its mathematics).


In his recent book A New History of Greek Mathematics (2022) Reviel Netz writes concerning the history of reception of Pythagoras and other early writers:

The argument that Aristotle is an unreliable narrator of the earlier history of philosophy was made forcefully already by H. F. Cherniss, Aristotle’s Criticism of Pre-Socratic Philosophy (Baltimore, MD: John Hopkins Press, 1935).

Netz details some of Aristotle's misconceptions about Pythagoras and others. So the fairy tales about Pythagoras certainly don't originate with early 20th century mathematicians, contrary to what user Conifold's answer seems to suggest.

Conifold does not contest the legitimacy of describing the early 20th century "foundation crisis" as such. This leaves us with the remaining item on the list by Fraenkel, Bar-Hillel, and Levy: the alleged foundation crisis in the first half of the 19th century having to do with Fourier series and other developments that were pushing the envelope of what kind of functions mathematical analysis dealt with. Here Conifold is similarly sceptical concerning the existence of a "crisis", and I would like to disagree with Conifold. Consider, for example, Abel's reaction to Cauchy's work. On the one hand, while forcefully disagreeing with Cauchy's politics, Abel clearly stated that Cauchy is the only one who knows how to do analysis. On the other hand, Abel is the one who claimed in 1826 to find a purported mistake in Cauchy's work on the convergence of series. It emerges that according to Abel, the only one who could do analysis was publishing mistaken theorems in analysis. My appreciation of the situation in the first half of the 19th century is that it can indeed be described as a crisis, even if we have challenged the Cauchy-Weierstrass tale that sometimes passes for the history of 19th century analysis; see this list of publications.

So my conclusion, at variance with Conifold, is that two out of three crises that Fraenkel, Bar-Hillel, and Levy mentioned are legitimate.


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