Mathematicians who are finitists.

[polymath257]

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.

Everyone agrees (or, should agree) that transfinite arithmetic is a productive area of study, in that there are numerous papers that have been published on the subject, along with textbooks.  You can take classes on the topic.  I think that finitists are like Biblical creationists, in that they deny the obvious.

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