The Quine/Putnam indispensability argument is regarded by many as the chief argument for the existence of platonic objects. We argue that this argument cannot establish what its proponents intend. The form of our argument is simple. Suppose indispensability to science is the only good reason for believing in the existence of platonic objects. Either the dispensability of mathematical objects to science can be demonstrated and, hence, there is no good reason for believing in the existence of platonic objects, or their (...) dispensability cannot be demonstrated and, hence, there is no good reason for believing in the existence of mathematical objects which are genuinely platonic. Therefore, indispensability, whether true or false, does not support platonism. (shrink)

ABSTRACT Historical structuralist views have been ontological. They either deny that there are any mathematical objects or they maintain that mathematical objects are structures or positions in them. Non-ontological structuralism offers no account of the nature of mathematical objects. My own structuralism has evolved from an early sui generis version to a non-ontological version that embraces Quine’s doctrine of ontological relativity. In this paper I further develop and explain this view.

An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world and (...) are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge. (shrink)

Discussions of the applicability of mathematics in the natural sciences have been flawed by failure to realize that there are multiple senses in which mathematics can be ‘applied’ and, correspondingly, multiple problems that stem from the applicability of mathematics. I discuss semantic, metaphysical, descriptive, and and epistemological problems of mathematical applicability, dwelling on Frege's contribution to the solution of the first two types. As for the remaining problems, I discuss the contributions of Hartry Field and Eugene Wigner. Finally, I argue (...) that there are epistemological problems concerning the applicability of mathematics that nobody in the philosophical community has yet confronted, though the problems are well known to physicists. (shrink)

Everyone appreciates a clever mathematical picture, but the prevailing attitude is one of scepticism: diagrams, illustrations, and pictures prove nothing; they are psychologically important and heuristically useful, but only a traditional verbal/symbolic proof provides genuine evidence for a purported theorem. Like some other recent writers (Barwise and Etchemendy [1991]; Shin [1994]; and Giaquinto [1994]) I take a different view and argue, from historical considerations and some striking examples, for a positive evidential role for pictures in mathematics.

Contemporary empiricism is closely allied with naturalism. Not only do empiricists hold that all our knowledge is based upon sensory experience, but they also tend to offer some sort of causal account of how this experience comes about. The causal ingredient in knowledge seems very plausible — after all, my knowing that there is a tea cup on my desk is based on sense impressions which are caused by the cup itself. Photons come from the cup to my eye; a (...) signal is then sent down the optic nerve into the visual part of the brain, and so on. And without that causal process, I likely wouldn't have the knowledge that I do have. (shrink)