Can the infinite be physically realized? If matter were infinitely divisible, then the infinite would reveal itself here - but matter is not infinitely divisible. If time were infinitely divisible, then the infinite would reveal itself here. Although our simplest models of time are as a continuum, to actually prove (or conclusively demonstrate) such a claim is difficult (impossible?). If space were infinitely divisible, then the infinite would reveal itself here - but already theories of discrete space have emerged (e.g. doubly special relativity ). If some parameter is continuously variable, does this constitute a (counterfactual?) realization of the infinite?
So, shift the question: if the infinite were realized in nature, could we tell? No - in principle, to distinguish between the infinite and the very very large, but finite, is impossible.
Of course, the mathematics of the infinite and the finite are quite different. In the realm of theory and the a priori, we can easily distinguish between finite and infinite systems. In empirical practice, however, distinguishing between the two is in principle impossible.
Nevertheless, the majority of techniques used in the sciences are continuous (this allows for the use of differentiation and integration), and thus infinite.
The question, then is this: if we wish to provide a foundational theory of, say, physics, should the rational reconstruction of physical theory proceed in a finite or an infinite framework? The principle argument for the infinite is that this would provide continuity with the formalisms used by practicing physicists. Important figures here are Truesdell and Noll. The principle argument for the finite is i) that the finite is in principle physically indistinguishable from the infinite, and thus ii) physcial theory should be reconstructable within a finite framework, and iii) most plausible from an a priori standpoint is that the universe itself is finite. A key figure here is Suppes.