my computationalist Platonism

[gusl]

In a philosophical conversation with spoonless today, I expressed my view succinctly:

"To me, the purpose of axioms should be to discover truths about mathematics, not to make truths."

If the Goldbach conjecture is false, all good axiom systems will agree about this. (You can replace "Goldbach conjecture" with any statement about the integers)

The situation is not the same for infinitary questions. The truth of statements like the Continuum Hypothesis, the determinacy of infinite games, etc is axiom-dependent. Therefore, these statements don't have an objective truth-value. Their meaning is thus relative to a model.

---

UPDATE:
I found someone talking about the phenomenon of axioms making truths:

"Their existence is axiomatic" -Robert Alain, about the non-standard integers

Such fictional objects are useful to the extent that they can give us information about the real objects.

Of course, even standard real analysis deals with non-real objects (the real real numbers are the computable real numbers, which form a countable set). Theorems in real analysis are useful because they tell us something about the computable real numbers.

Comments

[easwaran.livejournal.com]

But it seems quite plausible that there is indeed a fact of the matter about the consistency of any theory T - the fact that PA doesn't settle Con(T) just illustrates the weakness of PA. I suppose you could also take the position that Con(T) doesn't properly express the consistency of the theory T - after all, it depends on the arithmetization of the notion of proof, just as Turing's results depend on the formalization of the notion of computation. I'm working on a paper on this stuff right now, trying to show that people really have to accept that there's a fact of the matter at least about these sorts of statements.

Also, you might want to restrict your original claim to Π⁰₁ statements about the integers, rather than statements with arbitrary quantifier complexity. Though I do suspect that many further ones will have determinate facts of the matter.

[jcreed.livejournal.com]

I have the same qualms about there being a unique "fact of the matter" about consistency claims as I do in the comment I made below.

I'm very probably comfortable with whatever arithmetization you might make of Con(T) as "properly expressing the consistency of the theory", though, for what it's worth.

It's just that Con(T) says that there's no finite proof of bottom from T, and I still have to wonder, "ok, what do you mean by "finite"?" and this is tantamount to asking what the structure of the natural numbers "really" is.

[rdore.livejournal.com]

It's just that Con(T) says that there's no finite proof of bottom from T, and I still have to wonder, "ok, what do you mean by "finite"? and this is tantamount to asking what the structure of the natural numbers "really" is.

Are you talking about nonstandard integers? That is an interesting question. But it seems like you always might have a metatheory infected with nonstandard integers, so I'm not sure how you'd deal with that.

non-standard integers

[gustavolacerda.livejournal.com]

I'd never heard of them until now.

Looking it up, I found a book talking about the phenomenon that axioms make truths, which this post is about. See red highlight

[jcreed.livejournal.com]

yes, exactly --- how do we determine which numbers are "standard" and which are "nonstandard"? Your metatheory is always at risk for being "infected" by the "nonstandard" ones, because you can't know ahead of time (in a uniform, computable way that would manifest itself as a decision procedure for which axioms to include or not) whether you are surprised to the point of rejecting some number (some axiom) as nonstandard.

[easwaran.livejournal.com]

One way to determine which are standard and which are non-standard is by saying that the standard ones are the ones that are in the unique (up to isomorphism) model of PA that injects into all others. Or do you question the existence and uniqueness of such a model?

[rdore.livejournal.com]

the fact that PA doesn't settle Con(T) just illustrates the weakness of PA

Maybe I don't understand what you're saying, but it seems like Godel's second incompleteness theorem means you always have this problem.

[gustavolacerda.livejournal.com]

That's correct (though "problem" seems like a loaded word). Thence the Feferman hierarchy.