# Achilles and the Tortoise

An amusing account (at least to a mathematician) by Harvey Friedman of an encounter with eccentric Russian mathematician and dissident Alexander Yessenin-Volpin. Friedman supervised the Stanford PhD of John E. Hutchinson, who taught me calculus.  (Hat tip: AB)

Let me give an example. I have seen some ultrafinitists go so far as to challenge the existence of 2^100 as a natural number, in the sense of there being a series of ‘points’ of that length. There is the obvious ‘draw the line’ objection, asking where in

2^1, 2^2, 2^3, . . . , 2^100

do we stop having ‘Platonistic reality’? Here this . . . is totally innocent, in that it can be easily be replaced by 100 items (names) separated by commas.

I raised just this objection with the (extreme) ultrafinitist [mathematician Alexander] Yessenin Volpin during a lecture of his.  He asked me to be more specific.  I then proceeded to start with 2^1 and asked him whether this is ‘real’ or something to that effect.  He virtually immediately said yes.  Then I asked about 2^2, and he again said yes, but with perceptible delay. Then 2^3, and yes, but with more delay.  This continued for a couple of more times, till it was obvious how he was handling this objection.  Sure, he was prepared to always answer yes, but he was going to take 2^100 times as long to answer yes to 2^100 then he would to answering 2^1.  There is no way that I could get very far with this. (pp. 4-5).

Note: Of course, Friedman is wrong about the . . . being replaced by 100 items. We would expect it to be replaced with just 96 items, since 4 items in the list of 100 are already listed explicitly.
Reference:
Harvey M. Friedman [2002]: Lecture Notes on Philosophical Problems in Logic. Princeton University.