Mathematicians who are finitists.

[A Toy Windmill]

I see it more as a discipline to see just how much can be done without the assumptions concerning infinite sets.

Not as much, no doubt; how many papers have been published in the peer-reviewed mathematical journals by finitists over the last 50 years?  How  many textbooks are there devoted to the subject?  Of the 3K+ 4-year colleges in North America, where can I, as a student, take a class on finitism?  Are there any journals devoted to finitism?

It's not clear to me what anxieties are soothed by finitism that are not adequately soothed by intuitionism, which agrees that mathematical constructions are finite and which enjoys far more active research. It also has practical advantages to computer scientists.