## Revolution and Realism?

Abstract:

In the seventeenth century the practice of mathematics was fundamentally transformed; and this transformed mathematics led in turn to a transformed practice of physics. It was at the time very natural to think that although we had in the past gotten things wildly wrong—fire does not want to go up, the sun does not revolve around the earth, things are not really colored, or flavored, or sounding—now we had things right. Such thinking cannot survive a second revolutionary transformation such as occurred in mathematical practice in the nineteenth century and in the practice of physics in the twentieth. The fact that, in principle, this second revolution could be followed by a third, then a fourth, and so on, provides grounds for skepticism about the truth of our scientific theories; it suggests that we have no good grounds for scientific realism. But if one considers these two revolutions more closely, a very different picture begins to emerge. As is especially clear in the case of the mathematical revolutions that underwrite those in physics, the two revolutions are essentially different. Instead of “one damn thing after another”, we see in these two revolutions an organic growth of knowledge that provides grounds for a very robust form of scientific realism.
The first revolution in mathematics was inaugurated by Descartes. Although Viète had already developed a symbolic notation suitable for algebraic manipulations, the notation had for him only instrumental value. It was Descartes who learned to read the symbolism of arithmetic and algebra as a fully meaningful language; mathematics, from being about objects of various sorts, was now to be conceived as a science of relations among arbitrary quantities as expressed in equations. This first revolution has the form of a metamorphosis in mathematics and mathematical practice. The second, nineteenth century revolution is different; it is a rebirth of the science of mathematics as a whole. All of mathematics is to be reborn as a purely conceptual enterprise and reborn in a way that fully realizes the aspirations of early modernity: mathematics is revealed to be, has become, the work of pure reason. And this new, purely rational and conceptual mathematics enables in turn a new form of fundamental physics that does not merely use mathematics as, say, Newton’s physics does, but instead simply is mathematics. There is no physical correlate. And because this mathematics is purely rational, because it has been purged of all sensory content, it is correct to say that the aspects of reality it reveals, in special and general relativity and in quantum mechanics, is maximally objective, the same for all rational beings. This, then, is a new form of scientific realism, one that is structuralist without being quite what is generally meant by structural realism. What it shows is that far from being incompatible with scientific realism, revolutions in the practice of science can be constitutive of scientific realism.