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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.

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    $\begingroup$ 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 $\endgroup$ – Conifold Jan 12 '18 at 22:23
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    $\begingroup$ "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". $\endgroup$ – Conifold Jan 13 '18 at 7:26
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    $\begingroup$ Possible duplicate of how sophisticated was Egyptian and Babylonian mathematics compared to the Greeks $\endgroup$ – José Carlos Santos Jan 14 '18 at 20:16
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    $\begingroup$ @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. $\endgroup$ – RP_ Jan 15 '18 at 18:17
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    $\begingroup$ @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. $\endgroup$ – RP_ Jan 18 '18 at 20:27
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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.

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  • $\begingroup$ I thought there has been some scholarly debate that the Babylonians had a least a proto form of the proof? $\endgroup$ – user4281 Jan 12 '18 at 5:31
  • $\begingroup$ @user4281: Last sentence of your question is "Would you agree with that claim?" I answered exactly this question: I strongly disagree. One side in this debate is wrong. $\endgroup$ – Alexandre Eremenko Jan 12 '18 at 12:45
  • $\begingroup$ @user4281: If you really wanted to know "why there is some scholarly debate on this subject? ", rephrase your question accordingly, and I will try to answer this too. $\endgroup$ – Alexandre Eremenko Jan 12 '18 at 20:42
  • $\begingroup$ Ok, Alexandre, I reformatted my question? $\endgroup$ – user4281 Jan 12 '18 at 21:28
  • $\begingroup$ It is better if you give a specific reference on the "debate". Who exactly expresses the opposite point of view? $\endgroup$ – Alexandre Eremenko Jan 13 '18 at 1:00
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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 .

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