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.

[A Toy Windmill]

double post

[A Toy Windmill]

Here is some recent research in finitistic mathematics.

Like all such research, it is metamathematics. The goal is to find methods to translate ordinary classical mathematical proofs to finistic ones, so that mathematicians don't have to make too many changes to their practice. The Hilbert Programme had the boldest ambition, to show that mathematicians needn't make any changes, in effect showing that mathematics was already, at least methodologically, finitistic. With that done, any further arguments could be relegated to the philosophy department. However, various results in proof theory showed that this goal was unachievable.

Goodstein, by the way, came up with a very beautiful computable function that bears his name. It can be written in about half a page of computer code, but you won't get it to terminate with anything but trivial inputs. The proof that it does terminate given enough memory and time requires induction up to ε0, making it one of the first interesting results to be shown outside the scope of Peano Arithmetic.

[LastPoet]

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.

[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.

[Jehanne]

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.

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.

[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.

[LastPoet]

Yeah well, My math analisys is what it is, for someone that needs to explain simple proportionality to my coworkers. I am withness of how hard is to teach math.

Finite groups can be useful on chemistry, but chemistry is a subset of physics, if one gets my drift.

[Jehanne]

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.

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?

[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.