computational philosophy of mathematics
Feb. 5th, 2005 10:53 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
guslI 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.
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.
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(no subject)
Date: 2005-02-06 10:33 am (UTC)I believe it's consistent to assume that nothing has infinite information. For one thing, all of our math books are finite. So it seems strange to postulate the existence of infinite-information objects. Why should we talk about them at all?
(This is especially true if it's consistent to suppose physics is finite.)
I believe we can do math without objects with infinite information, and I'm trying to justify that.
(no subject)
Date: 2005-02-06 10:43 am (UTC)Hmmm... here again, I'm faced with the feeling that this is circular: you can never be "given" (i.e. described, or given an algorithm to compute) a mathematical quantity with an infinite amount of information, but you can infer some of their properties (for example - the fact that the uncomputable real numbers are uncountable, that when they are adjoined to the computable numbers you get a unique complete ordered field, etc...), and these properties constitute real mathematics, whose information seems useful and descriptive of both the real world and of aspects of math that seem reasonable.
We're not "postulating" the existence of any uncomputable, infinite information number any more (or less) than we "postulate" the existence of "1". Why should we talk about the integers at all?
But in fairness, I can see that it's an interesting exercise to figure out how much math you can do with only finite information objects, but I'm willing to bet that a better logician than I could poke holes a few miles wide in this in just a few minutes. Sadly, I'm not up to par on that, at least not tonight... but I'll think about it.