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?