The Mathematical Experience
In precisely what sense do universals (such as blueness, goodness, cowness, squareness, and threeness) exist? For Plato they are transcendent things, independent of the universe. Aristotle agreed that they are outside human minds, but he pulled them down from Plato’s heaven to make them inseparable from the world. During the Middle Ages the nominalists and conceptualists shifted universals sideways from the outside world to the inside of human heads.
In the philosophy of mathematics, with which The Mathematical Experience is primarily concerned, this ancient controversy over universals takes the form of speculating on what it means to say that such abstractions as the number three, a triangle, or an infinite set “exist,” and the companion problem of what it means to say that a theorem about these ideal objects has been “proved.” Let us not get bogged down in the technical and ambiguous differences between such schools as the logicism of Bertrand Russell, the formalism of David Hilbert, and the constructivism (or intuitionism) of L.E.J. Brouwer. All of these are briskly discussed, along with many other central mathematical issues, by the book’s two distinguished authors, mathematicians Philip Davis and Reuben Hersh. Let us consider instead the more fundamental question which cuts across all the schools. Do mathematical structures have a reality independent of human minds?
It is easy to caricature what mathematicians mean when they call themselves realists. They certainly do not suppose (I doubt if Plato did) that were we transported to some far-off realm we would see luminous objects floating about which we would recognize as pi, the square root of minus one, transfinite sets, pure circles, and so on; not symbols or models, but the undefiled universals themselves. Realists mean something less exotic. They mean that if all intelligent minds in the universe disappeared, the universe would still have a mathematical structure, and that in some sense even the theorems of pure mathematics would continue to be “true.” On its ultimate microlevel (if it has one) the universe may be nothing but mathematical structure. “Matter” has a way of vanishing on the microlevel, leaving only patterns. To say that these patterns have no reality outside minds is to take a giant step toward solipsism, for if you refuse to put the patterns outside human experience, why must you put them outside your experience?
For a mathematical realist a tree not only exists when nobody looks at it, but its branches have a “tree” pattern even when no graph theorist looks at them. Not only that, but when two dinosaurs met two dinosaurs there were four dinosaurs. In this prehistoric tableau “2 + 2 = 4” was accurately modeled by the beasts, even though they were too stupid to know it and even though no humans were there to observe it. The symbols for this equality are, obviously, human creations, and our mental concepts of two, four, plus, and equals are by definition mind-dependent. If mathematical structure is taken to mean only what is inside the …
Facing Up to Realism January 21, 1982