computational philosophy of mathematics

[gusl]

I only believe in mathematical objects with finite information (i.e. with a finite expression). Things can expressed in any way you wish, intensionally, extensionally, whatever. So, while the natural numbers exist, not all subsets of it do.

While the set of real numbers exists (R can be expressed as the power set of N), not all of the traditional real numbers exist (only the computable ones do). So, in my definition, the cardinality of R is aleph_0 (i.e. the same as N). In fact, no set can have greater cardinality, for that would imply it had non-existing elements (since only computable things exist).

Can we design a set theory this way? How much of traditional set theory can be translated? Does we lose any good mathematics this way?

Whereas people seem to view uncomputability as a fundamental property of a problem, I tend to view it as another form of self-reference paradoxes. All "well-defined" problems are computable. Again, this is not a theorem or mathematical insight, but a re-definition, just like my "R only has countably many existing numbers" is a definition. But the point isn't just to change names and keep everything the same... it's to change the intuition that goes with the names.

Comments

[easwaran.livejournal.com]

That is, if you're only going to admit the existence of things for which a computation can be given, then I think certain existential statements will not be provable, even though a contradiction can be reached from its negation. Thus, neither the statement nor its negation will be "true" in your sense, and this is basically the intuitionist position.

[gustavolacerda.livejournal.com]

even though a contradiction can be reached from its negation

hey, that's a proof to me!

Does this seem somehow inconsistent with my other beliefs here?

[easwaran.livejournal.com]

I think I have to think about this a bit more. There might be some statements that end up with different truth values than their classical counterparts, but maybe you're right that you can do this without ending up denying excluded middle.

Though intuitionists also don't deny any particular instance of excluded middle - they just don't assert every instance. Their idea is that assertion of a negation is just denial of the asserted statement, and there are many statements for which we have no grounds to either assert or deny them.

I think it's possible for your system that there be an existential statement that is not satisfied by any computable real number, but for which this non-existence has no computable proof. In fact, Gödel's theorem might require there to be such statements. On your computational methodology, it seems to make no sense to either assert or deny this statement. Would you then deny that it is a meaningful statement?

[gustavolacerda.livejournal.com]

I'm familiar with intuitionism, although I think it's a terrible way to do math.


I think it's possible for your system that there be an existential statement that is not satisfied by any computable real number, but for which this non-existence has no computable proof. In fact, Gödel's theorem might require there to be such statements. On your computational methodology, it seems to make no sense to either assert or deny this statement. Would you then deny that it is a meaningful statement?

I would consider such a statement as true but not provable. But this can be fixed for any individual instance by adding a new axiom. Of course, you'll always have new undecidable statements... even with infinitely many axioms, I think.

We humans tend to consider ourselves to be outside of any axiomatic system... but I think this is wrong.
For example, when we talk something being true but not provable, we claim to know the theorem in a somehow transcendent way. But we are just a meta-level higher than the system.

Unprovable Truths

[henriknordmark.livejournal.com]

If you are a devoted Platonist, there is no problem with having mathematical statements that are true but not provable. It simply means that in the Platonic Realm the mathematical statement is true, but we will never be able to know that truth by simple deductive reasoning in an axiomatic system like Peano Arithmetic (PA).

However, for me personally, I find this problematic because if we are not using logical deduction in an axiom system to determine an unprovable truth Phi(x), then on what grounds can we really claim that Phi(x) holds? Do you start believing in some mysterious mystical ability to apprehend Phi(x) to be true as the Platonists believe? Woodin has a whole research program around determining whether the Continuum Hypothesis (CH) is true despite that we know that CH is independent of ZF. It seems that Woodin wants to show that (CH) is false because it is mathematically intuitive to have a "healthy dosis of large cardinals", which would contradict CH. But why should we believe we need this healthy dosis of large cardinals in the first place?