“Philosophical Problems in Logic § Ultrafinitism”, Harvey M. Friedman2002-09-25 (computability, math, logic, ontology; similar)⁠:

…We now come to the main tool that Platonists use for defense (attacking the opposition). They ask “where do you draw the line?” The point is that wherever you draw the line, there is a natural slightly higher place, and one has to defend why the stuff on one side of the line is OK whereas the stuff just barely on the other side of the line is not OK.

…I have seen some ultrafinitists go so far as to challenge the existence of 2¹⁰⁰ 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

2¹, 2², 2³, …, 2¹⁰⁰

do we stop having “Platonic 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 Yessenin-Volpin during a lecture of his. He asked me to be more specific. I then proceeded to start with 2¹ and asked him whether this is “real” or something to that effect. He virtually immediately said yes. Then I asked about 2², and he again said yes, but with a perceptible delay. Then 2³, 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 2¹⁰⁰ times as long to answer yes to 2¹⁰⁰ than he would to answering 2¹.

There is no way that I could get very far with this.

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