Alan Baker Mathematics , Indispensability and Scientific Progress

Are there good reasons for including mathematical objects such as numbers, sets, and functions as part of our ultimate catalogue of the furniture of the universe? Recent debates within the philosophy of mathematics over this sort of general ontological question have centered on the pros and cons of the so-called Indispensability Argument. The basic idea behind this argument is quite straightforward. When faced with a general existence question such as ‘Do mathematical objects exist?’, we should look to our best available theories of the world for guidance. Our current best theories of the world – by general consensus – are the theories of empirical science. And current science (especially physics) quantifies over mathematical objects. Hence we have good reason to believe in the existence of mathematical objects, unless and until we can do science without postulating them. In short, mathematics is indispensable for science. One way of formulating the Indispensability Argument is as follows;