Mathematicians who are finitists.

[A Toy Windmill]

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.

While such work is not to be ignored, a finitist mathematician is similar to a intuitionist logician, where reductio is not allowed. Not my school of thought.

Intuitionism allows proof by negation, but does not generally allow proof by contradiction (proving a positive proposition by refuting its negation). Even then, there are domains were proof by contradiction is admissible, even intuitionistically. In finite domains, it is always permitted.

The fragment of logic available to a finitist allows arbitrary proof by contradiction for exactly this reason.