In pure math there are lots of worlds, each with its own rules. What follows in one world doesn’t in another because the rules are different. In Boolean algebra 1 + 1 = 1, which works fine in computing but not in accounting.
you’re not using “world” in the way i was using it, but no matter: my point is simply that it is possible to have real knowledge of mathematical “worlds” (in your sense) independently of experience.
which means that experience is
not the only source of knowledge.
(indeed, as i pointed out before, the proposition “only empirical experience can yield knowledge” is itself incapable of empirical demonstration, and is so far forth self-refuting.)
inocente:
And how do we know which math world applies to a real world problem? By experience.
well
that’s certainly true, and tautologically so: “the only way to know if theorem X applies to the empirical world is to examine the empirical world empirically”
my point has only been that there is more to the world than its material (read: empirical) aspect, and thus sources of knowledge other than empirical experience.
inocente:
Philosophy is founded on the notion that we are rational computers. Turns out we’re not, we don’t have a logic unit and arithmetic unit in our head. All our reasoning has to be learned, which means the way your parse “if a and b then a” is down to your unique set of experiences. This means that since we don’t reason uniformly, we must have some way of testing our reasoning.
In math the reasoning rules are tightly defined and mathematicians don’t disagree whether a statement is a theorem. That’s a good test.
Whereas in philosophy the rules are so loose that philosophers rarely agree and it’s just a matter of opinion.
this - what you’re doing right here - is philosophy. and you yourself think that it yields knowledge (i.e. you think your reasoning here is valid and sound and therefore that its conclusions are true and that you know them.)
and i’m afraid it’s self-refuting: let P be the proposition “disagreement about A entails that there is no truth of the matter concerning A” (i.e. any belief about A is simply and irreducibly “opinion”). then the fact that you and i disagree about the truth of P entails that P is
not true (or false).
inocente:
This sounds like special pleading: there’s a mystic Truth™ out there but it’s unknowable. But if it’s unknowable, it’s neither use nor ornament.
nope, it’s perfectly knowable.
inocente said:
“if a and b then a” is only true in worlds where the physics keeps a in existence long enough to parse the statement (and where it keeps us in existence long enough). We have to test it and the test, as Tunare points out, is the proof of the pudding is in the eating.
you don’t have to test “if a and be, then a”. it is deductively true, and cannot fail to be true. if you
did need to test it, then you’d never be able to know it was true. the same goes for the pythagorean theorem, and cantorian set theory and transfinite math and…
i mean, there’s a reason that you can’t prove goldbach’s conjecture, for example, simply by going through every even number and seeing if it is the sum of two primes.
of course, if what you mean (again) is just that there is no necessary link between math and the world (that we know of), then sure, but so what? i’m not disagreeing that experiment is required to determine fit between math and the empirical world: i’m simply saying that empirical experiment is (demonstrably) not the only source of knowledge.