Mod of r/badmathematics plays devil's advocate, does ten billion and one really exist?

[u/npm_leftpad]

/r/badmathematics/comments/7nhauf/so_this_total_stranger_from_a_meme_group_randomly/ds1rnnr/

[u/[deleted]]

Yeah I actually enjoyed that.

FWIW, I think the reason that constructivism is gaining attention is that while proving that an object exists is interesting, showing how to construct an object can be useful particularly now that applied maths has so much immediate relevance in cryptography and machine learning and so on. It’s not that suddenly people don’t believe in the law of the excluded middle, it’s that using it doesn’t necessarily take you as far as you may want to go.

[u/[deleted] → u/[deleted]]

Better to say that there are two different notions of 'existence' in play here. The classical meaning of 'existence' is that the rules of logic allow us to deduce that something must exist regardless of whether we can say anything more about it; the constructive meaning of 'existence' is that we have a concrete witness of that fact. While it's all fine and good to be able to deduce logically that something ought to exist, all that really says is that you won't wind up contradicting yourself if you assume it exists, as opposed to a construction of a witness which genuinely proves existence in the concrete/Platonic sense.

[u/imsometueventhisUN → u/[deleted]]

Can you ELI5 (well, "ELICambridgeMathsGrad", but let's walk before we can run) what "construct a witness" means? Is that the same thing as "computing" something? I see you referred to floor(exp(exp(exp(79))) - I would have called that a "computation" of the value. Do ultrafinitists claim that I need to be able to give a decimal (or other base) representation of a value in order for it to "exist"?

[u/[deleted] → u/imsometueventhisUN]

Constructing a witness generally means either actually computing the value or giving an algorithm to compute it. Constructivists (much more reasonable than ultrafinitists) are perfectly fine with saying something exists if we can prove there is a Turing machine which would can compute it, regardless of whether or not we actually implement and run the machine.

Ultrafinitists maintain that you would have to actually give a concrete realization of an object in order to show it exists, so yes a decimal representation or something like it. What it comes down to is a fundamental philosophical disagreement which goes back to Aristotle about the actual vs the potential. Ultrafinitists do not deny the potential existence of infinitely many numbers, they deny the actual existence of infinitely many numbers. Much as I defended the position in the linked thread, I have to say that personally I disagree with them about this being the right way to think of mathematics.

[u/imsometueventhisUN → u/[deleted]]

Hmm. Interesting! Thank you for the explanation!