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

[rdore.livejournal.com]

What concrete thing could do with the standard real numbers that you couldn't do here?

I want to be able to state whether two objects are equal. Similarly, maybe I can't tell whether a given program will terminate or not, but clearly it *does* terminate or it *doesn't*.

Do you seriously want a logic system where
{(x,y,z)| x,y,z reals and x+y=z}
is not an object that exists?

Like I mentioned in another post, determining if multivariate polynomial equations have integer solutions is an uncomputable problem.

Asking questions about what can be done computably is an interesting and worthwhile problem. But I think its a bit goofy to actually define mathematics as only such problems. It seems like saying physics can only talk about things we can see with the naked eye.

[gustavolacerda.livejournal.com]

dang... I screwed up the definitions. That definition was only for extensionally computable sets.

Obviously {(x,y,z)| x,y,z reals and x+y=z} is an existing set, since you expressed it finitely. So of course it's a computable subset of the reals: but only intensionally.

However, membership in this set is not decidable, since equality is not decidable (neither is it in the regular real numbers).


It seems like saying physics can only talk about things we can see with the naked eye.

I think of atoms as having a more objective existence than uncomputable real numbers, whatever this means.

However, your analogy makes sense in some sense: physics must be grounded in the things we can see with our naked senses. Even the things we see through microscopes are eventually interpreted by our naked eyes.

And this is exactly the point. I'm trying to define mathematics in terms of its interaction with real people. And real people never see uncomputable objects: so their status is necessarily "theoretical".

[rdore.livejournal.com]

But a major point of mathematics is that we can infer the existence of things that we can't tangibly deal with.

[gustavolacerda.livejournal.com]

I don't know what point you're trying to make, but to me everything must return to the source: there must be a reason for us to create abstractions.

The reason we do logic is because what we derive syntactically we can also observe to hold semantically. To me, this is essential to the nature of mathematics. (it never lies, as an old teacher of mine used to say)

Theoretical terms are nice: useful, expedient and even perhaps pointing towards some fundamental truth. But uncomputable objects are not theoretical terms, as in fact they are fundamentally inexpressible. In other words, they don't exist.

Can you express your point better-ly?

[rdore.livejournal.com]

But uncomputable objects are not theoretical terms, as in fact they are fundamentally inexpressible. In other words, they don't exist.

Why do you feel that undefinable means doesn't exist? My main objection is that you claim these two things are the same.

(Also, it is possible to quite explicitly describe real numbers which aren't computable. But that's another can of worms.)

The reason we do logic is because what we derive syntactically we can also observe to hold semantically. To me, this is essential to the nature of mathematics.

You seem to view logic as nothing more than rule pushing. From this definition, what you say is trivially true. I think of logic much more as reasoning about how we reason.

Also, you seem to use 'logic' and 'mathematics' interchangably, which seems to me like calling biology a branch of physics - in a sense is is, but that's not how anyone actually studies it.