[drdeca]

Doing math without it is pretty ugly.

(Granted, there are other objects that may seem ugly which can only be constructed by reference to infinite things.)

You can do a lot of things without assuming that the natural numbers keep going, but it is plain awful to work with.

[chithanh]

It turns out rather ok without actual infinity, by limiting oneself to potential infinity. Think a Turing Machine where every time it reaches the end of its tape, an operator ("tape ape") will come and put in another reel.