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.

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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

[gustavolacerda.livejournal.com]

I like your question. It's an interesting attempt to get incompleteness to screw up my Platonism.

So, supposing you had a proof that the first counterexample to GC is s, the number of steps until the Goodstein sequence starting with n terminates (or similarly, the first counterexample is BB(n) )...

Q: Does this imply that s doesn't denote a number?
A: No. In fact, I would consider your proof as indication that Goodstein(n) is computable.

(How can an expression not denote a number according to PA?)