4 ms·

I'd pose this as caused by the simplicity of calculating in Z as a ring. This has allowed us to experimentally find many interesting true conjectures. Number t

by rocqua 9y ago

I'd pose this as caused by the simplicity of calculating in Z as a ring. This has allowed us to experimentally find many interesting true conjectures. Number theory is no more complicated than any other system. What differs is the ease by which we can find statements that are probably true. If the field appears more complex, that might be because we've been enticed by these probably true statements.

mrkgnao 9y ago
I'm not sure whether I agree or disagree, but I'm reminded of how it's said that Fermat's Last Theorem wasn't really the goal (or even that the truth or falsity of FLT doesn't imply anything interesting on its own), but it's an extremely vital problem from a development-of-math perspective. Ideals were discovered by Dedekind as a solution to an error made by Gabriel Lamé in his claimed proof of Fermat's Last Theorem: he'd overlooked that not all rings have the same properties as Z. In this situation, they were missing "unique factorization", which Kummer "fixed" by hypothesizing the existence of "ideal numbers", which Dedekind later formalized as "ideals". This was a huge step forward for the development of modern algebra. https://www.quantamagazine.org/the-crazy-number-that-almost-solved-fermats-last-theorem-20170303/ https://www.quantamagazine.org/the-crazy-number-that-almost-... And that's setting aside a lot of algebraic number theory, class field theory, and so on.
pmiller2 9y ago
I'm not so sure about that. Take a look at cellular automata. It's pretty trivial to screw around with them (though not as easy as with integers, I'll grant you). Where are the theorems? There seems to be an awful lot of land, but no maps. To a lesser extent, graph theory has this problem as well. In the ~60 years that it's been a subfield in its own right, we haven't got an awful lot of fundamental new methods. Results being proven today could be easily understood by a graph theorist from 40-50 years ago transported ahead in time, perhaps after at most the equivalent of a semester-long grad course.
- rocqua 9y ago
  Cellular automata are much harder to screw around with. Partly that is because computation by hand is a lot harder, partly because there isn't much that one can combine. What makes the integers interesting is that addition, multiplication and exponentiation are all easy, all operate on the same numbers, and yet they behave very differently. This allows one to pose very easy questions (e.g. Can I express every prime as a sum of ...) that have no obvious solution to them.

by rocqua 9y ago

mrkgnao 9y ago

I'm not sure whether I agree or disagree, but I'm reminded of how it's said that Fermat's Last Theorem wasn't really the goal (or even that the truth or falsity of FLT doesn't imply anything interesting on its own), but it's an extremely vital problem from a development-of-math perspective. Ideals were discovered by Dedekind as a solution to an error made by Gabriel Lamé in his claimed proof of Fermat's Last Theorem: he'd overlooked that not all rings have the same properties as Z. In this situation, they were missing "unique factorization", which Kummer "fixed" by hypothesizing the existence of "ideal numbers", which Dedekind later formalized as "ideals". This was a huge step forward for the development of modern algebra. https://www.quantamagazine.org/the-crazy-number-that-almost-solved-fermats-last-theorem-20170303/ https://www.quantamagazine.org/the-crazy-number-that-almost-... And that's setting aside a lot of algebraic number theory, class field theory, and so on.

pmiller2 9y ago

I'm not so sure about that. Take a look at cellular automata. It's pretty trivial to screw around with them (though not as easy as with integers, I'll grant you). Where are the theorems? There seems to be an awful lot of land, but no maps. To a lesser extent, graph theory has this problem as well. In the ~60 years that it's been a subfield in its own right, we haven't got an awful lot of fundamental new methods. Results being proven today could be easily understood by a graph theorist from 40-50 years ago transported ahead in time, perhaps after at most the equivalent of a semester-long grad course.

rocqua 9y ago
Cellular automata are much harder to screw around with. Partly that is because computation by hand is a lot harder, partly because there isn't much that one can combine. What makes the integers interesting is that addition, multiplication and exponentiation are all easy, all operate on the same numbers, and yet they behave very differently. This allows one to pose very easy questions (e.g. Can I express every prime as a sum of ...) that have no obvious solution to them.