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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Unprovable Truths
Date: 2005-02-09 10:40 pm (UTC)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?