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Why its true that the Egyptians did not see much developments. It has been said that the Babylonians were equal to the Greeks in mathematical achievement in terms of having an axiomatic, deductive system.

Would you agree with that claim? If not, why is there scholarly debate about, or more precisely what is the scholarly debate about in terms of this subject.

user4281
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    Neither Babylonians nor Greeks had axiomatic deductive systems, a common misconception that Euclid's Elements are "deductive" is based on reading Hilbert's work, written over 2000 years later, into it, see Rodin's paper. But Greeks did pay much more attention to the demonstrative side of mathematics. Computational mathematics was far more developed by Babylonians than by Greeks, they had a positional system used for sophisticated financial and astronomical bookkeeping. Greeks imported it during Hellenistic times, and it provided great boost to their astronomy – Conifold Jan 12 '18 at 22:23
  • so Conifold, overall was Babylonian mathematics more sophisticated than the Greeks? – user4281 Jan 13 '18 at 05:41
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    "Sophisticated overall" is too vague and subjective a metric to compare anything to anything, each was more advanced in its own way. One can pick the proof aspect as the "most important" and compare them on that, but that would not be "overall". – Conifold Jan 13 '18 at 07:26
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    Possible duplicate of how sophisticated was Egyptian and Babylonian mathematics compared to the Greeks – José Carlos Santos Jan 14 '18 at 20:16
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    @Conifold: Could you give sources for your claim that Euclid's Elements "is not deductive"? (whatever this may mean, but I'm sure you have something in mind.) It sounds like a statement coming straight out of the post-modern school of mathematical historiography. If so, I would recommend including this information, so that people may take the statement with the appropriate amount of salt. – R.P. Jan 15 '18 at 18:17
  • @René It is fairly non-controversial, postmodernists have little interest in Euclid. One reason is that logic required to do Euclidean geometry deductively was not developed until 19-th century (even taking it informally). One source that goes into details is given in the first comment, another is Friedman's paper, neither is by a postmodernist. – Conifold Jan 18 '18 at 18:41
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    @Conifold: If it is non-controversial, then I guess Rodin wouldn't have had to devote a paper to arguing that "Euclid’s theory of geometry is not axiomatic in the modern sense but is construed differently." The trouble I have with the claim is that Rodin is right in a sense, but not a very interesting sense: Euclid didn't advertise his work by saying it was "deductive", whereas Hilbert did, so it follows that Euclid's and Hilbert's method of deduction are different, the one simply being more self-conscious than the other. I could be wrong, but it never seems to amount to more than this. – R.P. Jan 18 '18 at 20:27
  • @René Friedman uses similar words, Rodin is targeting mathematicians, I think, who are raised on geometry textbooks that are not into historical nuance (Bell's Men of Mathematics is still very popular too). At one point it puzzled me too. The trouble is that most people do not read Euclid or only read bits after reading the retellings to "complete" the picture. Once you spend some time on following Euclid as written it becomes clear that he did things very differently, constructing and reading off of diagrams is not deductive at all, and is converted into it only very artificially. – Conifold Jan 18 '18 at 21:21
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    @Conifold: I think I'll have to read Rodin's paper more carefully before I could say more about this. I couldn't find a free on-line version of Friedman's paper, but I could get my hands on his book Kant and the exact sciences, which also happens to contain the claim that "Euclid's system is not an axiomatic theory in our sense at all." I must say that, in spite of my skepticism, it does arouse my interest. – R.P. Jan 19 '18 at 13:17
  • @René The book chapter is almost a reprinting of the paper. You may also want to look at the two chapters written by Manders in Mancosu edited book (freely available), they are a careful study of Euclid's diagrammatic method. – Conifold Jan 20 '18 at 22:07

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The short answer is certainly no. Mathematics as it is practiced today has its own unique method of telling what is true and what is not. This method is called "mathematical proof". This is what distinguishes mathematics from all other activities. No other culture practiced any form of mathematical proof before Greeks. Moreover, there is absolutely no indication that that any culture invented it independently before or after. In this precise sense the Greeks invented mathematics.

Of course one can use a broader definition of mathematics, to include all activities related to counting and measuring something. Then every culture had some form of this of course. But even with this broad definition no other culture stands any comparison with the Greeks in its contribution to mathematics that we practice and study today.

Alexandre Eremenko
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Mathematics does not need axioms and does not need verbalized proofs.

Applied mathematics answers questions like: if I have 3 apples and get 2 apples, how many apples do I have? 3A + 2A = 5A is a theorem. It can be proved and has been proved by simply doing the experiment. (Mathematics is physics where the experiments are cheap (V.I. Arnold).) No axioms are required.

Same holds for calculating the contents of granaries. Since mathematics is abstracted from reality it can best be checked by reality which is a better computer than all human productions of that kind.

In general axioms only are introduced to show post festum that also useful mathematics can be formalized. But mathematics without axioms is not of less value than the Greek mathematics and it has been pursued for thousands of years by Egypts and Babylonians in a much more sophisticated way than by the Greek.

The Egypts were the first to solve a quadratic equation. That is mathematics, if the solution is correct as can be proved by doing a suitable experiment. Further proof is not required.

For the high level of ancient Egyptian mathematics see https://en.wikipedia.org/wiki/Ancient_Egyptian_mathematics .

Franz Kurz
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