Penelope Maddy's original solution to the dilemma posed by Benacerraf in his 'Mathematical Truth' was to reconcile mathematical empiricism with mathematical realism by arguing that we can perceive realistically construed sets. Though her hypothesis has attracted considerable critical attention, much of it, in my view, misses the point. In this paper I vigorously defend Maddy's account against published criticisms, not because I think it is true, but because these criticisms have functioned to obscure a more fundamental issue that is well (...) worth addressing: in general -- and not only in the mathematical domain -- empiricism and realism simply cannot be reconciled by means of an account of perception anything like Maddy's. But because Maddy's account of perception is so plausible, this conclusion raises the specter of the broader incompatibility of realism and empiricism, which contemporary philosophers are frequently at pains to forget. (shrink)

In many diagrams one seems to perceive necessity – one sees not only that something is so, but that it must be so. That conflicts with a certain empiricism largely taken for granted in contemporary philosophy, which believes perception is not capable of such feats. The reason for this belief is often thought well-summarized in Hume's maxim: ‘there are no necessary connections between distinct existences’. It is also thought that even if there were such necessities, perception is too passive or (...) localized a faculty to register them. We defend the perception of necessity against such Humeanism, drawing on examples from mathematics. (shrink)

A response is given here to Benacerraf's 1973 argument that mathematical platonism is incompatible with a naturalistic epistemology. Unlike almost all previous platonist responses to Benacerraf, the response given here is positive rather than negative; that is, rather than trying to find a problem with Benacerraf's argument, I accept his challenge and meet it head on by constructing an epistemology of abstract (i.e., aspatial and atemporal) mathematical objects. Thus, I show that spatio-temporal creatures like ourselves can attain knowledge about mathematical (...) objects by simply explaininghow they can do this. My argument is based upon the adoption of a particular version of platonism — full-blooded platonism — which asserts that any mathematical object which possiblycould exist actuallydoes exist. (shrink)

The main task of this paper is to defend anti-platonism by providing an anti-platonist (in particular, a fictionalist) account of the indispensable applications of mathematics to empirical science.

The aim of the present paper is to offer a new analysis of the multifarious relations between mathematics and reality. We believe that the relation of mathematics to reality is, just like in the case of the natural sciences, mediated by instruments . Therefore the kind of realism we aim to develop for mathematics can be called instrumental realism. It is a kind of realism, because it is based on the thesis, that mathematics describes certain patterns of reality. And it (...) is instrumental realism, because it pays atten-tion to the role of instruments by means of which mathematics identifies these patterns. The article concludes by offering solutions to some famous semantic paradoxes based on the diagonal construction as corroboration for this claim. (shrink)