Ultrafinitism

Ultrafinitism is the philosophical view that only comparatively small or accessible numbers exist.

Ultrafinitism is the philosophical view that only comparatively small or accessible numbers exist. According to ultrafinitism, the various extremely large numbers that mathematicians conventionally take themselves to describe, such as 2¹⁰⁰ or 10^100!, do not actually exist, and it is a kind of illusion to speak of them. Indeed, often we find ourselves unable to answer basic questions about such numbers. For example, consider Skewe’s number—the exponential tower:

\(\Large e^{e^{e^{79}}}\)

Is it an integer? We don’t know.

Merely denying the existence of all extremely large numbers, however, does not seem to make one an ultrafinitist. After all, other positions in the philosophy of mathematical ontology—I am thinking of certain forms of formalism, fictionalism, nominalism, and so forth—deny in a sense the existence of numbers altogether, but one would not ordinarily classify these views as automatically ultrafinitist. Ultrafinitism, rather, is specifically about a difference in the nature of existence of small versus large numbers. An ultrafinitist accepts the existence of the small or feasible numbers as unproblematic, yet denies the existence of very large numbers. Nevertheless, to be sure, there does seem to be a friendly affinity or overlap between those positions I mentioned and the ultrafinitist attitude toward very large numbers.

Welcome to this first in a series of three critical essays on ultrafinitism, which will appear here on Infinitely More in the coming weeks in the ultrafinitism tag. These essays grow out of a talk I gave at the conference Ultrafinitism: Physics, Mathematics, and Philosophy held at Columbia University last April, 2025.

Harvey Friedman (2002, p. 4-5) raised the “draw the line” objection with ultrafinitist Alexander Yessenin-Volpin, concerning existence of 2¹⁰⁰.

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

\(2^1, 2^2, 2^3,...,2^{100}\)

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 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 then he would to answering 2¹. There is no way that I could get very far with this.

Let’s discuss it.