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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Uncomputable Real Number
Date: 2005-02-06 11:03 am (UTC)Well for any real number that has a single algorithm that can approximate it to any desired accuracy only contains a finite amount of information.
My point for![[livejournal.com profile]](https://www.dreamwidth.org/img/external/lj-userinfo.gif)
pbrane was that uncomputable numbers cannot be approximated to any desired accuracy. I felt I should give an explicit example.
Re: Uncomputable Real Number
Date: 2005-02-06 03:13 pm (UTC)I was thinking about this, and for numbers like pi (which we can approximate with a finite algorithm), it would still take an infinite-length input in order to get the algorithm to output pi to infinite precision.
So when we think of pi, we must think of it in an interactive way: not as a fixed number, but as a an algorithm that returns an approximation to (the perimeter of a circle of radius 1 / 2) to whatever precision you desire. But you yourself are limited in how much precision you can ask.
My point is that, when defining such numbers, we need to discount the information in the input. I'm not sure if the exact value of pi is computable... and I'm not sure this makes sense.