@sed I haven't read it right to the end yet, but it is very interesting. It has been suggested that SmartmanApps is a finitist (not an ultrafinitist - I have seen him give a proof by induction!), but he rejected the label.

Nevertheless, he displays some characteristics of one: he seems to think of infinite decimals as a process rather than actual numbers, and in one of his flawed arguments against the infinitude of intervals of rationals, he goes on a tangent about measurement accuracy and atoms, suggesting he can't, or has difficulty, conceiving of numbers separate from something the number measures.

But his arguments about the finite nature of intervals are just wrong (he states, and does not prove, that there are finitely many rationals in an interval over a given denominator. He states as if it follows, that there are finitely many rationals in the interval.) A proper finitist would have an argument here that isn't completely bogus: she'd say, "we can conceive of a process for generating new rationals in the interval which will never run out, but it is fallacious to conceive of all of them at once as a completed whole because we can never in practice generate them all". (An ultrafinitist would say that there is some actual limit on how fine we can divide the interval, but wouldn't say what the limit is.)

It's kind of annoying that he touches on these very interesting topics but doesn't have the understanding to discuss them properly!