Assuming no new catagories, might further mathematical interrelatedness eventually decline? Perhaps another universal language might help - music. Is there an end to new music? For n components, one would have 2^n combinations. And for mathematics, for N components, one would have 2^N combinations. Since N>>>n, then 2^N>>2^n; thus one would have to assume the end to new music, before consideration of the end to new mathematics.
humanino
Jul23-08, 08:13 AM
Mathematics can not end, even in principle. There will always be true statements, the proof of which requiring new axioms (Godel).
Why do you assume n (music) <<< N (math) ? Are you strongly restricting what you call music ?
thomasxc
Jul23-08, 08:35 AM
im not seeing how math can end.
humanino
Jul23-08, 08:41 AM
im not seeing how math can end.Well, think about a system which would be rudimentary enough that you can list and prove everything that is true about it. Such systems exist. But as soon as you have structures able to contain natural numbers, this becomes impossible : there are true statements that can not be proven within your system. You need to enlarge your system with new axioms to prove those true statements.
thomasxc
Jul23-08, 08:49 AM
oh. okay. so, theoretically, mathematical progress could end if we found a set of basic...laws that could explain everything?i'll say its possible but not probable in the least.
humanino
Jul23-08, 08:55 AM
mathematical progress could end if we found a set of basic...laws that could explain everything?That's surely not what I meant to say.
Mathematics already have built in your set of laws : math are build out of axioms. Within those axioms, some things are true and some things are provable. There are true statements that are unprovable, in any given system which can contain natural numbers. My claim is, from this point which is Godel's theorem, that you will be able to prove those statements in a more elaborate system, by adding one or more axioms. This procedure never ends, so mathematics can not end.
thomasxc
Jul23-08, 09:25 AM
ah. i see. iguess i was accidentally referring to progress in general. but i see what youre saying.