>>25
To prove one system, you need another system, which will be subjective.
Yes, but you can get it down to universal computation which can be defined in very simple ways (and in many machines/languages which can be shown to be capable of implementing (or being
translated between) each other).
Mathematics is subjective religion, based on circular reasoning, thay call "infinity".
For a religion, it's the most exact one there can ever be, and it doesn't demand much, if anything.
It doesn't tell you to believe something is absolutely true, it just tells you that if you follow some specific rules, you'll get some specific results. For example, "if PA then ``various arithmetical theorems", "if ZFC then ``more theorems, but some could be like "PA and RA are true"''". Wether you think PA or ZFC is true is another thing, but it doesn't force you to believe in anything, it just tells you to apply the rules you defined and see what follows from them. As a religious statement (provably unprovable), I think PA is likely true (~90%) as the principles that it's based upon seem sound to me, and if for example you were to define a finite variant of it, you'd find it sound, now all that I take on belief is that induction is a valid schema axiom and thus all finite systems derived from it (given some specific axiomatization) are true (of which there are countably infinite). Of course, nobody will force you to believe anything, merely admit that "if PA then <theorems of PA>". Of course mathematical systems are defined in some logics and some of them are trivial (although not for everybody, see constructivism/intuitionistic logic who can at times deny the law of excluded middle). I wouldn't call it 'circular reasoning', merely seeing a principle as true and using it to reason, if you think induction is false, then don't use it, but I'd like to see what you will use instead of it (or if you deny reasoning at all, then I can't help you with it). If you don't assume ANYTHING, reasoning is impossible.
Nope. I propose the reverse - that everything is subjective, which opossite to mathematics, that proposes existence of some eternal and all encopasing truths.
Depends on what you mean by subjective. It is the case that a lot of systems are unprovable, but they are sound and there doesn't seem to be any paradoxes within them, however just because we cannot prove a system's consistency within itself does not mean that there actually exist contradictions within the system, just that any such proof would be infinite and thus impossible for a finite system like ourselves to complete (unless we cheat and use a stronger system, but then we cannot prove the consistency of that system and so on).
I'm a mix between a platonist and a formalist: I think finite computation is absolute and universal, that is I "believe" the Church-Turing thesis, however at the same time you can embed/implement theorem provers and axiomatic systems in computation and thus these systems will be able to arrive at the same truths in any possible universe (which is complex enough to allow computation, which usually just means some very simple rules and some form of time/state transition. Lack of time would also mean lack of computation and lack of any form of consciousness in that system (so no self-aware beings capable of even thinking of math and computation)). I should also mention that PA can embed computation within itself rather easily, but then a lot of systems can. There is a point though: just because ZFC can be embedded in some computation running in PA, does not mean I 'believe' it is true ontologically (it might be, but I'm undecided about it), merely that if you follow its rules, you will arrive at some theorems(truths following from axioms). ZFC's infinities are due its inventors desire of allowing all finite objects (such as numbers) to be taken as a set, and then allowing taking sets of other sets and so on (as long as non-circular, otherwise you get naive set theory which is provably inconsistent by Russell's Paradox). I don't think it's wrong to actually generalize from "finite system up to a natural number k is provably true", then for any k, it's true (in which case you get a countable infinity of such systems).
If you accept computation, but want to make it subjective, does that mean you think that following a system of rules in a specific way will give you DIFFERENT results each time you follow them? That would mean everything is inconsistent (how would we even exist then?). For example by this logic it would mean: on a bad day 1+1=5, and on most days 1+1=2? Is that what you meant?
My other claim is that even if we cannot prove a system's consistency (only inconsistency can be proven), that doesn't mean the system is inconsistent.