The distinction between mathematical and physical objects has probably played a greater role shaping the philosophy of mathematics than the distinction between observable and theoretical entities has had in defining the philosophy of science. All the major movements in the philosophy of mathematics may be seen as attempts to free mathematics of an abstract ontology or to come to terms with it. The reasons are epistemic. Most philosophers of mathematics believe that the abstractaess of mathematical objects introduces special difficulties in (...) accounting for our ability to know them, to refer to them and even to entertain beliefs about them. These difficulties—supposedly absent even in the case of the most theoretical physical objects—make mathematical objects especially problematic and philosophically unattractive.Few have questioned this epistemic thesis or the ontic distinction it presupposes. LaVerne Shelton (1980) challenged the abstract-concrete distinction some years ago in an unpublished APA address. (shrink)

Much of the current thought concerning mathematical ontology and epistemology follows Quine and Putnam in looking to the indispensable application of mathematics in science. A standard assumption of the indispensability approach is some version of confirmational holism, i.e., that only "sufficiently large" sets of beliefs "face the tribunal of experience." In this paper I develop and defend a distinction between a pure mathematical theory and a mathematized scientific theory in which it is applied. This distinction allows for the possibility that (...) pure mathematical theories are systematically insulated from such confirmation in virtue of their being distinct from the "sufficiently large" blocks of scientific theory that are empirically confirmed. (shrink)