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Late to the party, but...there& #39;s a big divide between people who think mathematical truths are things we discover ("maths talks about things that exist") or things that we create ("maths talks about things that we define"). Both these views have problems: welcome to my thread. https://twitter.com/aIeturner/status/1298372968838508546">https://twitter.com/aIeturner...
What problems? Well, if theorems are true because they refer to things that exist, where are they? Can you show me "3"? (Not 3 dogs, 3 people, 3 pieces of toast, or even the symbols "3" or "III", but the mathematical object "3"?) How do we have any knowledge about this concept?
OTOH, maybe I think that maths is something we invented. Then we have an answer - we know about "3" because we defined it! But now I& #39;ve got a different problem: how do I know that mathematical statements are true? Sure, they& #39;re consistent within our creation "maths", but so what?
Maybe the nice mathematical structures we& #39;ve defined are just a piece of fiction. If I say "Sherlock Holmes has a website", that might be true within some contexts (ask Benedict Cumberbatch) and false in others (ask Arthur Conan Doyle).
You might even say it doesn& #39;t make sense to talk about truth/falsehood here - there& #39;s no such person as Sherlock Holmes, so it doesn& #39;t make sense to say that he does or doesn& #39;t have a website. It& #39;s just something we kind of agree on and don& #39;t need to think about too much.
This problem - is maths true but possibly unknowable, or knowable but possibly untrue? - is known as Benacerraf& #39;s Dilemma after his article "What numbers could not be" (see https://en.m.wikipedia.org/wiki/Benacerraf%27s_identification_problem">https://en.m.wikipedia.org/wiki/Bena... for some more detail)
How do mathematicians respond? Well, most of us just ignore it on a day to day basis. I have a soft spot for it as a philosophical problem because I once wrote my dissertation about it as a maths/philosophy undergrad...
