I hope this question is not too soft for MO.


The Wikipedia says about *finitism* that it is an extreme form of *constructivism*. See <http://en.wikipedia.org/wiki/Finitism>. I doubt that this is correct.


As I understand, there were different approaches to solve the crisis of the foundation of mathematics. One was constructivism, where there must be a witness of an object. Finitism was another approach were one (Hilbert) tried to give existing mathematics a foundation in finitism. But it was not a rejection of certain mathematical methods, just finding a foundation. Finally, the third approach was just adapting the logics, which led to ZFC and type-theory.


Hilbert opposed the intuitionism of Brouwer. So, it is a little bit strange to count them to the same family.


In more modern finitism, related to reverse mathematics, one tries to prove that a finitism result obtained by infinitism methods, has a finitism proof. This has many successes and it has been shown that this is at least true for large parts of mathematics. Again, this is not a rejection of infinitism methods, but just an attempt to find a better foundation. While, constructivism is really a rejection of certain arguments.


So, based on above arguments, I believe the statement in the Wikipedia is totally wrong.


Maybe someone has more historical knowledge than me? If I am right, I can try to correct the article.


Regards,


Lucas


**Edit:** Thanks for the answers. I agree with Mike Schulman that both can mean a variety of things. I do think that the article in the Wikipedia needs some rewriting. It might be the case that finitism is more strict, however, I think it is not a subset of constructivism by definition (after lots of reasoning, one might conclude that).



Is finitism an extreme form of constructivism?

> 
> this is not a rejection of infinitism methods, but just an attempt to find a better foundation. While, constructivism is really a rejection of certain arguments.
> 
> 
> 


I think that's arguable. There do exist (or, at least, there have existed) people who are *philosophically* finitist, in that they really do reject infinitistic arguments as wrong or meaningless. On the other hand, many people working on "constructive mathematics" nowadays are not philosophically constructivist like Brouwer, but would phrase their study as preferring to have constructive proofs than nonconstructive ones. (For instance, this is useful even if one "believes" in classical mathematics, since in the internal logic of a topos, only constructive mathematics is valid.)


I think that "finitist" and "constructivist" can both mean a variety of different things -- they can refer to a philosophical viewpoint, or merely to a mathematical enterprise. In the latter sense, I think finitism is indeed a "more restrictive" framework than constructivism. In the former sense, I think finitism is one extreme type of constructivism, although as you point out it is incompatible with some other types of constructivism.


