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

[pbrane.livejournal.com]

The only part which uses Occam's razor is the "preferring fewer moving parts" part of it - the rest is just, "we describe what we observe", and so far, we've reached most of the conclusions of modern science with the premise of continuity. It's not *logically necessary* for there to be arbitrarily accurate continuity, but so far, that's all we have evidence for. But in some sense, Occam's razor in science is just for expediency: we don't claim that any model "is" the real world - it's just a model of it, and the more accurate the model, for the least amount of work, the better.

But who knows, maybe it will be easier to understand things / make new discoveries if physicists start thinking this way.

Possible, but so far, in the history of science, we typically have discretized continous things for computational purposes (recently) or to avoid singular problems that were artifacts of the construction (which we could remove and then take the discretization length to zero at the end of the day).

It still doesn't seem like *meta*physical assumption, but an actual physical one, and an unjustified one at that. Show me some evidence for discreteness, or else show me some nice theoretical problems solved by the proposal, or at least show me how it simplifies current descriptions, or else I'm not sure why it's being done: just because we're scared of the infinite? Seems pretty arbitrarily limiting.