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The title says it all. What exactly are axioms? I mean, I know that is a statement which do not need to be proven. But what are the requirements to be an axiom? For instance, may I state something that seems to be false as an axiom? May I state an axiom that is already a conclusion of another theorem or conclusion?

Are there any books on how to build axiomatic systems and explains this "philosophic" part of mathematics? Is Russell's a nice option for this?

Thanks

EDIT

Thanks for such many responses. I am asking this because recently, I have read an "article" in which the author tries to use the non-aggression principle as an axiom, like mathematicians and scientists do, to prove some things like mathematicians do: Establish the axioms and prove conclusions. But this principle, in my view, seems not be true, since aggression does exist.

I guess this question should be redirected to some exchange on Philosophy...

Mr. N
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  • Usually theorems are derived from axioms, not the other way around – J. W. Tanner Nov 25 '20 at 23:48
  • Yes, I see that. But what if I do the opposite? Would this be a good axiom? Why can't I do that? That's the question about. – Mr. N Nov 25 '20 at 23:50
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    Answer to both questions, yes. ZFC's axiom of infinity is clearly false in real life, we still use it. And given any axiomatic system you can make any provable theorem into an axiom but that would be completely pointless and defies the philosophical purpose of maths which is to make everything trivially provable. –  Nov 25 '20 at 23:55
  • You usually have to be somewhat careful when deciding which statements will be axioms. First of all, they should be consistent so that they don't contradict each other and second, they should be independent. That is, they shouldn't be provable from previous axioms. Someone more knowledgeable can give a better reccomendation but I enjoyed GEB by Hofstadter a lot and it's all about this kind of thing. Russel's intro to mathematical philosophy is also interesting albeit slightly dated. GEB is more relevant to this question, though. – Daniel Nov 26 '20 at 00:01
  • The way I think of axioms in practice is that they specify the domain of study. That is, the purpose of taking a set of axioms as premises is to explore properties of the objects that satisfy the axioms. – Daniel Hast Nov 26 '20 at 01:24
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    @SenZen But I think your claim that the axiom of infinity is false in real life is subjective. It may be hard to assume and regard it to be intuitive, but it is not clear that it is definitely false in real life. – Hermis14 Nov 26 '20 at 01:32
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    @Hermis14 Alright let me put it this way. Axiom of infinity is as false in real life as unicorns existing. –  Nov 26 '20 at 02:33
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    @SenZen Nobody can show that unicorns do not exist. – Hermis14 Nov 26 '20 at 08:44
  • @Hermis14 Exactly so you agree. –  Nov 26 '20 at 12:59
  • NOT exactly; it is a statement that we assume as not in need to be proven in the context of a specific theory. – Mauro ALLEGRANZA Nov 26 '20 at 15:16
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    See D.Hilbert, Axiomatic Thinking as well as Hilbert and the concept of axiom – Mauro ALLEGRANZA Nov 26 '20 at 15:19
  • Thanks for your references! I'll try to read them all. – Mr. N Nov 26 '20 at 17:35
  • Don't we have this question every couple of years? – Asaf Karagila Nov 26 '20 at 17:52
  • @AsafKaragila Well, I tried to search similar questions, but did not find any. – Mr. N Nov 26 '20 at 22:14
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    https://math.stackexchange.com/questions/1801970/meaning-of-the-word-axiom and maybe also stuff in the "Linked" section to the right of the main question. – Asaf Karagila Nov 26 '20 at 22:16

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See Schlimm - Axioms in Mathematical Practice

This paper has a nice philosophical discussion of axioms.

Dominic Petti
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It's fine to use some non-aggression principle as an axiom and prove all kinds of things from that. However, to what extent the proven theorems apply to our reality depends on the extent to which the axioms apply to our world.

This is how it always is with mathematics. Whatever we prove in mathematics will be true for any world to which the axioms apply. That may include our world, but it need not be. We can prove all kinds of things using the axioms of Euclidean geometry, for example, but if our world turns out to be non-Euclidean, then the theorems need not reflect the state of affairs in our world.

Note, though, that even if our axioms don't perfectly describe our world, they may still be a pretty good approximation. Hence, any theorems we derive from them may still be 'close enough' to our reality to learn something from.

J. W. Tanner
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Bram28
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Axioms are nothing more than the premises of your specific problem that are justified by the meanings of their predicates. The axioms should not conflict with the other axioms you may assume. Therefore, if they are false in your axiomatic system, they cannot be added as axioms (premises) to your problem. The propositions that are conclusions of the other theorems are theorems, but not axioms.

Refer to 2011 - Barker-Plummer - Language, Proof, and Logic.

J. W. Tanner
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Hermis14
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