Why a little bit goes a long way: Logical foundations of scientifically applicable mathematics

Source abstract

Abstract Here we consider answers of an underlying character to these questions, that is, from the point of view of the foundations of mathematics.1 In this respect, both Quine and Putnam were led to accept set-theoretical notions and principles to some significant extent or other. However, neither one relied on any detailed examination of just what is needed for scientifically applicable mathematics in arriving at their positions. Nor did they seem to consider whether any of the alternative foundational schemes actively developed during this century-namely, those of predicativism, constructivism, and finitism-ought to be preferred on philosophical grounds, particularly when natural science is given such primacy. On the face of it, scientific realism is at odds with the strong form of platonic realism required to justify set theory through its assumption of the independent existence of abstract entities (such as sets of sets of sets ... of unbounded infinite cardinality). The failure to consider other foundational approaches no doubt stems from the common impression that-whatever their philosophical merits these schemes are simply inadequate to meet the needs of everyday mathematics by being too restrictive and too foreign to practice. This impression needs to be corrected: there has been considerable logical work in recent years which has established in some detail the unexpected mathematical reach of each of these programs. Moreover, one result of the work in question is that surprisingly meager (in the proof-theoretical sense) predicatively justified systems suffice for the direct formalization of almost all, if not all, scientifically applicable mathematics.