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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Date: 2005-02-06 10:33 am (UTC)I'm just not sure how to demonstrate exactly what you lose, and how. Maybe it's based on the intuition that we never state whether something we're dealing with is computable or not (sometimes it's obvious that we've got something that's associated with an algorithm, sometimes not), and when we aren't sure, the results that pop out are of indeterminite computability, and your system will be very fragile to insertion of any noncomputable elements - whenever they enter, the whole system can come crashing down if they're not dealt with properly.
It seems that Godel's theorem will stick you with undecidables somehow too, but that's a whole 'nother kettle of fish, and one you seem to be willing to ignore because it deals with self-reference (although even this I'm not sure how consistent that view is either...)