Are the traditional philosophies of mathematics really incompatible?
The paper argues that a view entitled `constructive nominalism' shows that (at least ``moderate'' versions of) the traditional foundationalist schools in the philosophy of mathematics (intuitionism, logicism, platonism and nominalism) are compatible. Constructive nominalism identifies truth with what holds in a certain topos, essentially, the finite-type category generated by certain linguistic terms. (For details, see the author and \textit{P. J. Scott}: Introduction to higher order categorical logic (1986; Zbl 0596.03002).) The view is an interesting one in its own right, and certainly includes some of the aspects of each traditional view; but it would seem to sacrifice central planks of each (as the paper -- mostly -- points out), and so cannot demonstrate their compatibility. (i) Though the mathematics delivered is intuitionist, a good deal more is required for non-elementary intuitionist mathematics. (ii) Against logicism, the natural numbers cannot be defined, but numerals have to be taken as primitive. (iii) The central claim of platonism, that mathematical structures are mind-independent, hardly seems to be respected: such mind-dependence would seem to entail both the law of excluded middle and the omega completeness of arithmetic, both of which fail in the construction. (iv) Finally, though the ontology of the theory is linguistic, the items would appear to be types, not tokens, and so traditional nominalism is not respected.
scientific article; zbMATH DE number 566369
scientific article