Here’s an exercise in separating fundamentals from conventions: if we held a technical summit with all of the galaxy’s advanced alien civilizations, what would we find we had in common, and what would we find inscrutable?
-
Show this thread
-
Spoken language would be inscrutable. Different physiologies would lead to an inability to speak or even hear each others’ phonemes. Written language would be inscruible, though we might find analogs of nouns, adjectives, and verbs.
2 replies 0 retweets 35 likesShow this thread -
But mathematics would be shared. After translating syntax and symbols, we’d find we had exactly the same constructive axioms, and agree that other axioms are controversial. We’d have the same theorems and the same proofs. They’d rever a Pythagoras and a Leibniz.
15 replies 1 retweet 43 likesShow this thread -
Replying to @TimSweeneyEpic
We already have several axiomatic systems, most notably ZF vs ZFC (as the axiom of choice is/has been controversial). It would be interesting to discover what axioms other intelligent species came up with.
1 reply 0 retweets 0 likes -
Replying to @Lucas_Trz @TimSweeneyEpic
I could never understand why some people would reject AC. It's equivalent to "Cartesian product of a collection of non-empty sets is non-empty". They must have a very weird intuitive concept of was a set is, if it does not satisfy this axiom.
1 reply 0 retweets 0 likes -
It's like looking looking at that cartesian product of an infinite number of infinite sets with all possible combinations of their elements and saying "There is nothing here." "Wait, don't you see all these?" "Oh, no, they are not sets. No sets here." "Huh?"
1 reply 0 retweets 0 likes -
I can understand finitists who say "We restrain our reasoning to finite collections only. Infinite collections are a pure imagination and we could not be sure of anything if we admitted them". But when someone, after accepting the axiom of infinity and the power set axiom, ...
1 reply 0 retweets 0 likes -
... rejects the axiom of choice, I just cannot understand what do they mean by "set". Like they are partially blind and just cannot see some sets for some reason, or do not recognize them as sets.
2 replies 0 retweets 0 likes -
Replying to @vreshetnikov @Lucas_Trz
The core distinction is between constructive logics (in which proofs of existence are guaranteed to produce an example) and non-constructive logics which can prove something exists without any clue as to what it is.
2 replies 0 retweets 1 like
The Axiom of Choice shouldn't be viewed as true or false, but as a tool to explore the space of non-constructively provable propositions. Anything that can be proven without AC ought to be, as a proof is more powerful as it represents a computation.
Loading seems to be taking a while.
Twitter may be over capacity or experiencing a momentary hiccup. Try again or visit Twitter Status for more information.