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What you describe has to do with mathematical constructivism[0], and also with the Axiom of Choice, and it's really complicated (to me, I'm not a mathematician,

What you describe has to do with mathematical constructivism[0], and also with the Axiom of Choice, and it's really complicated (to me, I'm not a mathematician, and I only sort-of get it). You may have heard about the Banach-Tarski paradox[1], which tells you that if you assume "Real numbers" are actually reality, and the Axiom of Choice, you can divide a sphere into five pieces (one of which is just a point) and rearrange them into two spheres of the same volume. Obviously you can't do that in reality. This is the point where it gets really complex and I won't fill in the details as I understand them, because they are probably wrong. R. Buckminster Fuller also wrote some ideas on this constructivism (or something a lot like it), basically advocating against the set of "Real Numbers" as an accurate depiction of reality. For instance if you make a 1m x 1m square out of a real material. The sides of that square will have N atoms, and the number of atoms (and how far they are apart, which is a property of the material used) is really what describes the length. This is a whole number. But according to geometry math, the diagonal of this square has a length of sqrt(2) metres, or sqrt(2) * N atoms. That can't be because you can't have an irrational number of atoms. Then what is really going on suddenly depends on how these atoms are packed inside the square, and again, it gets really complex and that's where my understanding (of both atomic physics and mathematical constructivism) stops. [0] http://en.wikipedia.org/wiki/Constructivism_(mathematics) http://en.wikipedia.org/wiki/Constructivism_(mathematics) [1] http://en.wikipedia.org/wiki/Banach-Tarski_paradox http://en.wikipedia.org/wiki/Banach-Tarski_paradox

graycat 13y ago
> Banach-Tarski Yes, but in that construction, not all the sets are Lebesgue measurable. It's a curious fact, already true just in one dimension, i.e., the reals, that we can't have a nice way to assign a length, area, or measure to all the subsets. Instead there have to be some subsets to which we can't assign a length, etc. Here the usual proof on the reals does use the axiom of choice! With measurable sets, which already can be wildly bizarre, can't do anything like Banach-Tarski. So, suddenly in this thread, instead of hackers, we are all turning into undergraduate pure math majors!
tel 13y ago
Constructivism is significantly more fundamental than AC. AC is notable for being famously perplexing, less foundationally perplexing. For a more graspable investigation of the perplexities and arbitrariness of our choice of foundational set theory, take a look at Vicious Cycles where the author investigates what happens when set theory does not assume a terminating world (and thus introduces a lot of things that we use in CS often, try to model on sets, and are left dissatisfied because we really wanted costa, not data). From there please take a deep look into topos theory and you'll discover again that AC is just one choice of properties that make for a meaningful "foundation of mathematics (of a kind)" and a not particularly remarkable one at that.
- tripzilch 13y ago
  "Costa, not data", what does that mean? And Vicious Cycles is a book, I presume? Who is the author?
deleted 13y ago
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graycat 13y ago

> Banach-Tarski Yes, but in that construction, not all the sets are Lebesgue measurable. It's a curious fact, already true just in one dimension, i.e., the reals, that we can't have a nice way to assign a length, area, or measure to all the subsets. Instead there have to be some subsets to which we can't assign a length, etc. Here the usual proof on the reals does use the axiom of choice! With measurable sets, which already can be wildly bizarre, can't do anything like Banach-Tarski. So, suddenly in this thread, instead of hackers, we are all turning into undergraduate pure math majors!

tel 13y ago

Constructivism is significantly more fundamental than AC. AC is notable for being famously perplexing, less foundationally perplexing. For a more graspable investigation of the perplexities and arbitrariness of our choice of foundational set theory, take a look at Vicious Cycles where the author investigates what happens when set theory does not assume a terminating world (and thus introduces a lot of things that we use in CS often, try to model on sets, and are left dissatisfied because we really wanted costa, not data). From there please take a deep look into topos theory and you'll discover again that AC is just one choice of properties that make for a meaningful "foundation of mathematics (of a kind)" and a not particularly remarkable one at that.

tripzilch 13y ago
"Costa, not data", what does that mean? And Vicious Cycles is a book, I presume? Who is the author?

deleted 13y ago

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