# How are mathematical objects defined from an ultrafinitist perspective?

I remember attending a lecture given by an ultrafinitist who denied that curves are a set of points, he would only say that any particular point may or not be on the curve. Similarly for algebraic or analytic objects, the only way I know how to define them is as a set with operations on the elements of that set. Since ultrafinitists cannot use definitions with infinite sets, what sort of definitions do they use?

• I'm not an ultrafinitist, but I would imagine they would say that the whole idea of considering an infinite set is misguided according to their philosophy; "potentially-infinite" (as in, unbounded/unlimited) is OK, but a completed infinite set is not. – Zen Harper Jul 18 '11 at 3:05
• @Zen Harper: I am not an ultrafinitist either, but that sounds more like a finitist viewpoint. As I understand ultrafinitism, it wouldn't even allow "potential infinity". Troelstra and van Dalen (Constructivism in Mathematics) succinctly summarise ultrafinitism as rejecting the idea that we can view 2^1024 as a sequence of units, even in principle. I have argued with a few people calling themselves ultrafinistists, I think they generally accept that, and those (one?) that don't will still agree that "potential infinity" is not kosher. – Daniel Mehkeri Jul 18 '11 at 4:52
• @Daniel Mehkeri, I remember reading somewhere that one of the famous mathematicians and logicians argued with Alexander Esenin-Volpin about it. At the end he decided to try to find what is the largest natural number that Alexander Esenin-Volpin was ready to admit that it exists. After sometime the famous mathematicians noticed that this is not going to work, because as the numbers increased Alexander Esenin-Volpin waited proportionally more time before replying with yes, he accepts that number exists. – Kaveh Jul 18 '11 at 6:44
• @Kaveh: I told the story here: mathoverflow.net/questions/44208/… – Todd Trimble Jul 18 '11 at 6:49
• @Todd, :) also here: scottaaronson.com/blog/?p=103 or even maybe on FOM mailing list. – Kaveh Jul 18 '11 at 7:57