woah hm what's constructive mathematics sounds interesting
Conversation
several possible ways of phrasing it :)
one possible way: you only allow objects that are computable (need not be _efficiently_ computable, simply computable)
this excludes pretty much all of the reals (except for a few cases we can compute) etc.
things get weird from there
1
4
(I'm always a little worried stating somewhat hand wavy definitions around bc he's a Real Mathematicianโข and I'm decidedly not and he definitely has much better thoughts on all of this than I do)
1
4
but the point is, you can't say the object exists unless, essentially, you can *construct it* by, roughly speaking, writing a program that "generates it" in some sense or another
(or, equivalently, giving rules to a mathematician who can then follow them to get the object "out")
1
3
hmmmm this is cool
i was snooping around some foundations of mathematics stuff a while ago and came across a particular (proof/ving? axiomatic?) system that
outlawed all non direct proofs (like ones that relied on negation or contradiction)
sounds lik this may be different ?
1
3
nope this is exactly the same, that's just another formulation :)
(proofs by contradiction and nonconstructive proofs are prohibited de-facto from here; essentially this is just the refusal to have the law of excluded middle)
1
3
of course, there are many "flavors" of constructive math, some more simple imo, than others
but overall I like the general idea/thinking much more than I like its execution
1
3
(for example, a silly thing that isn't constructive is that, given two real numbers (a, b), you cannot prove that that exactly one of a > b, a = b, or a < b is true)
1
3
on the other hand, a fun thing is that, under most constructive formalisms, all functions are continuous :)
1
4
This Tweet was deleted by the Tweet author. Learn more
not a direct example but phase transitions
This Tweet was deleted by the Tweet author. Learn more
Show replies