[blather snipped] > We can think of mathematics as a science which studies the phenomena > observed in the world of computation. All of the mathematics that has > the potential to be applied can be thought of that way. As a conceptual > aid, we can think of the (abstract) computer as both a microscope and a > test tube: it helps us peer deeply into the world of computation, and > it gives us a way to perform experiments within the world of > computation. Mathematics studies what we observe when we look through > that microscope. Then, roughly speaking, a statement may be said to > have testable consequences if it makes predictions about the results of > computational experiments.
Mathematical concepts involving uncountable infinities such as infinite-dimensional vector spaces can be used to model real-life phenomena. That such concepts do not directly adhere to the physical reality we experience is irrelevant. Things like optimization algorithms rely on fundamental properties that exist outside the narrow confines of R^n (or worse, pseudo-physical R^n as perpetrated by cranks who think mathematics is a subset of physics).
That David Petry has a philosophical problem against uncountable infinities does not remove the utility of such models, nor will it persuade mathematicians to recast all their work in terms of simplistic finite computations which amount to nothing more than worshipping a particular tool.