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This might be more of a philosophical question, but why in mathematics is the tendency to only accept formal proof as a means of finding out what's true? In the physical sciences there's no such thing, and yet we feel we know a lot about the world around us through experimentation and "softer" inferences. For example, if we search exhaustively for counterexamples to some proposition, why is a failure to find them never considered good enough? Could one idea be that when it's at least possible to prove a statement, that a lack of proof is a kind of evidence that it's false?

dsaxton
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  • "...as a means of finding out what's true?" What exactly is the meaning of the word true in this context? – drhab Dec 10 '15 at 15:48
  • I'm confused by your question. It means the same thing it would in any other context. – dsaxton Dec 10 '15 at 17:24
  • In mathematics we're often busy with finding out what's true within the context of some collection of axioms. The question: "is this true?" can for instance be answered with: "yes if you accept the axiom of choice, but it cannot be shown to be true if you don't." In that sense it differs from how we look at it in real life. – drhab Dec 10 '15 at 17:53
  • Oh, I see what you mean. Still it seems one might be able to make approximate inferences even within a system of axioms. Like when one goes searching for counterexamples, presumably you're searching in some space where a set of axioms either holds or doesn't. Also, is there anything at all to be said about "almost" proofs, or are these as useless as no proof at all? – dsaxton Dec 10 '15 at 18:11
  • I never encountered "almost" proofs in maths. There are so-called conjectures, though. If something is proved based on such conjecture then of course that must be mentioned explicitly. I would not give them the label "useless", but "interesting". – drhab Dec 10 '15 at 18:52
  • By almost proof I meant an argument with a small leap of logic somewhere. – dsaxton Dec 10 '15 at 21:20
  • You might be interested in reading this. In particular you might like the answer by Qiaochu Yuan. – Anguepa Dec 11 '15 at 10:59
  • @Anguepa Yes, this thread is exactly what I'm thinking about. Thanks. – dsaxton Dec 11 '15 at 17:27

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