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nyssos | 1 year ago
Constructivists are only interested in constructive proofs: if you want to claim "forall x in X, P(x) is true" then you need to exhibit a particular element of x for which P holds. As a philosophical stance this isn't super rare but I don't know if I would say it's ever been common. As a field of study it's quite valuable.
Finitists go further and refuse to admit any infinite objects at all. This has always been pretty rare, and it's effectively dead now after the failure of Hilbert's program. It turns out you lose a ton of math this way - even statements that superficially appear to deal only with finite objects - including things as elementary as parts of arithmetic. Nonetheless there are still a few serious finitists.
Ultrafinitists refuse to admit any sufficiently large finite objects. So for instance they deny that exponentiation is always well-defined. This is completely unworkable. It's ultrafringe and always has been.
Wildberger is an ultrafinitist.
kthielen|1 year ago
I don’t mean to be pedantic (although it’s in keeping with constructivism) but in the case you describe, you don’t have to provide a particular x but rather you have to provide a function mapping all x in X to P(x). It may very well be that X is uninhabited but this is still a valid constructive proof (anything follows from nothing, after all).
If instead of “for all” you’d said “there exists”, then yes constructivism requires that you deliver the goods you’ve promised.
nyssos|1 year ago
mbivert|1 year ago
It's likely: I purposefully stayed loose about the "infinite processes" to avoid going awry. I do however remembered him justifying his views as such though: he's not going into details, but he's making that point here[0] (c. 0:40). I assumed — perhaps wrongfully — that he got those historical "facts" correct.
https://youtu.be/I0JozyxM1M0?si=IFdWcEWNeNKDid7t&t=39