The title is the first sentence of Hermann Weyl's 1930 essay, 
"[Levels of Infinity](http://peterpesic.com/levels-of-infinity/)."
He focuses on



> 
> "the distinction between actuality and potentiality, between
>  *Being and Possibility*."
> 
> 
> 


He opines



> 
> "the impossibility of grasping the continuum as a fixed being,"
> 
> 
> 


and concludes his essay with the claim that



> 
> "we can only represent the completed infinite in the symbol."
> 
> 
> 


My question is:



> 
> ***Q***. To what extent is the claim that "Mathematics is the science of the infinite"
>  accepted or justifiable by mathematicians today?
>  Is his essay a relic of the period in which he penned it, 
>  or does it express a generally accepted viewpoint today?
> 
> 
> 


I ask in light of the modern "rise of combinatorics"1 
and of discrete mathematics.2


---


1[Richard Guy: The rise of combinatorics (YouTube)](https://www.youtube.com/watch?v=2RkAvoSC6fs).


2
Ralston, Anthony. "The decline of calculus—The rise of discrete mathematics." In *Mathematics Tomorrow*, pp. 213-220. Springer, New York, NY, 1981.
([Springer link](https://link.springer.com/chapter/10.1007%2F978-1-4613-8127-3_21).)



"Mathematics is the science of the infinite"
I think there is a fairly straightforward answer to this question coming from [reverse mathematics](https://en.m.wikipedia.org/wiki/Reverse_mathematics). According to Wikipedia, "finitistic reductionism" is represented by the system WKL${}\_0$, and reverse math people know a lot about what mathematics can be done in WKL${}\_0$ and what mathematics requires substantially stronger systems. Whether WKL${}\_0$ corresponds to finitism exactly may be debatable, but in any case we know a lot about what can and can't be done around this level.


To answer the question directly, I think it's fair to say that most ordinary mathematics can be done in WKL${}\_0$, and nearly all of it can be done in ACA${}\_0$, which is predicative if not finitistic (although full predicativism [goes substantially further than this](https://arxiv.org/abs/1811.03543)).


