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Edited question: In response to Qiaochu Yuan's comment asking me to narrow down the question,the edited question is as follows: When did mathematicians first think of axiomatically building set theory with the level of rigour seen nowadays?(I am asking this because I learnt of the Peano's axioms and was amazed at how little needs to be assumed to build the number system)

Original question: I am interested to know when and under what circumstances mathematicians first thought of axiomatically building the number system and defining binary operations like addition and subtraction or for that matter the theory of sets ?

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    This is a big question. My impression is that the person who did the most to really push the axiomatic method in modern mathematics is Hilbert, but there are a lot of other names and directions to mention. Could you possibly make your question a little narrower?2012-05-07
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    The modern notion of abstract group is apparently due to Cayley, so I think the answer lies a little bit before Hilbert.2012-05-07
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    @Zhen: yes, but the realization and popularization of the notion that the axiomatic method could be applied to all kinds of mathematical objects besides groups certainly comes later (e.g. Fréchet did not introduce the metric space axioms until 1906).2012-05-07
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    The axiomatisation of arithmetic happened pre-1900 though: Dedekind (1888) and Peano (1889). Analysis earlier still...2012-05-07
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    The theory of sets is quite a different matter -from binary operations and arithmetic. It was invented, in the sense of developing its foundations and yoga, by Cantor, but there must clearly have been people (philosophers?) thinking about it before.2012-05-07
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    I think I remember something about van der Waerden axiomatizing vector spaces, in a textbook... From wiki: "Walther von Dyck (1882) gave the first statement of the modern definition of an abstract group." And regarding vdW and his textbook "Algebra" wiki says: "This work systematized an ample body of research by Emmy Noether, David Hilbert, Richard Dedekind, and Emil Artin."2012-05-07
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    I would comment that Gauss, Cauchy, Riemann, and big minds around there, perhaps Euler, probably thought about defining the natural numbers and their operations via successor. Dedekind defined primitive recursive functions. I would say that Gauss probably thought about primitive recursive functions, and perhaps about groups, axiomatically. I don't think he came to the idea of sets, he seems to have been biased against such philisophical topics, and he probably did not have enough examples to feel the need for/use of category theory.2012-05-07
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    Well, on the other hand, I remember Gauss commented about Fermat's last theorem being "unprovable" (in his lifetime) or Goldbach's conjecture, and many similarly simple statements in number theory, so he must clearly have thought very generally about the process of doing research in mathematics. Then it is natural to assume that he also understood how we create definitions, auxiliary/novel structures in mathematics, with operations and relations (which is why we call them "structures"), and look at their properties as wholes, therefore that he had a basic understanding of category theory.2012-05-07
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    @plm: I can't tell if you're satirising bad historiography or being serious. As for vector spaces (and exterior algebra!): it has to be Grassmann – though his work was forgotten for a while.2012-05-07
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    The "British" school of algebraists (Hamilton, Boole, Cayley, others) was an early user of an axiomatic approach, but had little direct influence on the Continent.2012-05-07
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    @ZhenLin, At least I did it in comments, not in a response. :) I was serious (though consciously speculative), I even quoted wikipedia. I also really liked the insight that people well before Dedekind certainly had probably (or not) thought about axiomatizing the natural numbers. And the comments on Gauss too. Of course I may be wrong but if I am right the fact that it's puzzling to you would show that was real insight. :)2012-05-07
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    @plm: Not really. It's easier to confuse people who already know something than people who know nothing. You should read [this](http://www.dpmms.cam.ac.uk/~piers/maths%20and%20whig%20historiography%202010.pdf).2012-05-07

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