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In this wikipedia article on the foundations of mathematics, it says:

In practice, most mathematicians ... do not work from axiomatic systems

Is this correct? If so, what is an example of this?

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closed as off-topic by Ryan Budney, Daniel Moskovich, Qiaochu Yuan, Steven Landsburg, Nik Weaver Feb 12 at 3:35

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I believe this just means that most mathematicians, most of the time, do not go out of their way to make explicit exactly what axioms (e.g., choice) they are using; and also do not write complete (e.g. two-column or semantic tableaux) proofs in practice. I'm not sure this question is appropriate for MO. –  Noah S Feb 12 at 2:50
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An example would be almost any mathematics paper in the literature. –  Ryan Budney Feb 12 at 2:51
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Yes, I think the sentence is unclearly written. Surely the intent is to say that mathematicians commonly do mathematics without paying much if any attention to foundational issues. (Not that group theorists don't use the axioms of group theory, or that topologists don't use the axioms of topological spaces.) –  Tom Goodwillie Feb 12 at 3:01
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A key point - drawing out what Tom Goodwhillie said - is that most results in group theory are actually proved in some stronger theory (for example, the axioms of group theory do not include any axioms that would allow for the construction of a product group from two given groups). The stronger theory in which day-to-day mathematics is carried out is not formalized, although of course it can be. But I suspect few mathematians can state all the axioms of ZFC, for example, which casts doubt on the claim that the are really working in ZFC in their informal mathematics. –  Carl Mummert Feb 12 at 13:44

1 Answer 1

up vote 3 down vote accepted

It's correct in a sense, but it makes me want to rewrite that Wikipedia section.

If you replace the preposition "from" with "in", you have a foundational question.

Most mathematics is not done in axiomatic systems. As a trivial example, consider a typical MathOverflow post: it is in English, is not in a formal language, and does not specify an axiom system. The more important claim is different: $\ $ Most mathematics can be readily formalized in axiomatic systems, and it's sometimes useful to do so.

If you insist on the preposition "from", you have a more psychological question.

I would say mathematicians work more from intuition than from axiomatic systems. Most mathematicians would have difficulty writing out the axiomatic system for ZFC -- it would require writing out rules for first-order logic, being precise about bound and free variables in quantifications. Perhaps they could do it, but they would find it a strange request and one irrelevant to their ordinary mathematical work -- which shows that they are probably not working from the formal system in the first place.

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