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> Formal mathematics has no concept of absolute truth ... Well, it does. If you drag the propositional calculus into the fray, "true" is arbitrarily but probab
by gizi 11y ago
> Formal mathematics has no concept of absolute truth ...
Well, it does. If you drag the propositional calculus into the fray, "true" is arbitrarily but probably most elegantly defined as following:
true(x) = x or not x
False is defined like this:
false(x) = x and not x
These definitions are pretty much arbitrary. The lambda calculus does this:
true(a,b)=a
false(a,b)=b
It also works absolutely fine.
> It's just concerned with axioms and theorems (and their proofs)
In the general case, theorems are indeed "provable" or "unprovable" (not "true" or "false").
However, if a theorem has exclusively been derived from propositional calculus, it can also be "true".
Gödel's incompleteness is exactly about statements that are "true but unprovable". In terms of propositional calculus, the theorem is "true" but since you can show that the theorem can never be derived from the axioms, it is also "unprovable".
- zvrba 11y agoWell, it does not. Mathematics is a symbolic system invented by humans, but that system tells you nothing about the state of the world. Used correctly, it can be used to describe the world, but it can also be used to describe phenomena that have not been physically observed. E.g., currently accepted theories allow for existence of wormholes, but we do not know whether the actual physical laws of the universe ("absolute truth") do. So, no, there is no "absolute truth" in mathematics. There are only valid derivations, but taking them as the "absolute truth" is like mistaking the pointing finger for the moon.
- Houshalter 11y agoYes there are absolute truths. 2+2 will not equal 5 in any universe. The laws of physics could be radically different, but I have 2 apples, and you give me two more, I will not suddenly have 5 apples. Math can tell you an awful lot about the state of the world. That's why it's useful.