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Is it really the case? Was there a proof of sort that shows mathematical facts are more certain than empirical facts? What are the arguments for and against that claim?

  • I think you could ask more specific. I for example don't agree that mathematics > empirics. I think empirics + mathematics > mathematics. The reason is that in order to be useful, it needs empiricism. – mavavilj Feb 23 at 20:34
  • Mathematics is a language, which cannot tell lies. If mathematics says something is true.. it is true. Whether that truth is useful or not, is a different matter. So take gauge algebra which seems to predict particles with 'imaginary' (i, the square root of -1) properties (Tachyons are one such predicted this way). The maths is valid. But do such particles make any sense or is it just Mathematical absurdity? – Richard Feb 23 at 23:17
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Yes, there is a reason why mathematical facts are more certain for us than empirical facts:

One can prove mathematical theorems.

But nobody could prove general empirical statements, e.g., the laws of nature. In the domain of science, which builds on empirism, we can at best confirm our general hypotheses, but all confirmation does not prevent from finding a counter example at a later time.

  • But we can prove certain things with controlled experiments like how light behaves under certain conditions since they are reproducible, we can say under the same conditions something will happen. – repomonster Feb 23 at 20:45
  • @repomonster One can never be sure that by reproducing an experiment all relevant conditions are the same, because one does not know all conditions which are relevant. Every day the Genova-bride was passed by nearly the same nunmber of cars, but one day in 2018 it collapsed. – Jo Wehler Feb 23 at 21:00
  • Wait, what?? Proving a mathematical theorem tells us NOTHING about the world. Have you looked at the axioms of ZFC? Most of them are demonstrably false about the world. Infinity, Choice, Powerset being the three biggest offenders of reality, but not the only ones. Can you clarify what you mean? Certainly we can use math to model our empirical observations. That's the best you can say. – user4894 Feb 24 at 1:23
  • @user4894 I think neither the OP nor my answer deal with a relation between mathematics and the physical world. To whom do you address your objection? – Jo Wehler Feb 24 at 6:07
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    @user4894 - Maths is more certain precisely because it is a formal game like chess. It does not rely on induction. You're right to say it can prove nothing about Reality (as Aristotle observes) since reality might not obey the rules, but what it proves it proves with certainty. . – PeterJ Feb 24 at 12:46

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