Mathematicians who are finitists.

[Jehanne]

Finitism is stricter than simply assuming there are no infinite sets. The finitist goes so far as to say that one cannot even quantify over the natural numbers! The most cited formal system which tries to capture what limited reasoning is left is "Primitive Recursive Arithmetic."

The finitist will accept that "n + 1 > n", but will regard this as a schema. to be filled in with particular numbers that they can muster.

Well, that does depend on the brand of finitism you subscribe to. Yours is one of the more extreme versions.

Not many mathematicians are finitists; in fact, after my OP, I updated the Wikipedia article:

Wikipedia -- Finitism

to reflect that Reuben Goodstein, a finitist mathematician, had died in the 1980s.