A Philosophy of Mathematics Between Two Camps

Abstract

The history of philosophy can partially be characterized by what Hilary Putnam has called the recoil phenomenon: an oscillation between two extreme positions, with each camp reacting to the untenable part of the other, resulting, finally, in two untenable positions. The current recoil ricochets across both analytic and Continental philosophy. On one side are those who deny objectivity in all fields in all ways; there are only incommensurable narratives. On the other side are those who attempt to secure objectivity, but do so at the cost of clothing it in metaphysical mystery. The first side (justifiably) points out the illusions in the second's metaphysics, and then recoils to anarchy. The second (justifiably) shows the inherent contradictions in the anarchist position, and then recoils to building more epicycles in its metaphysical castle. Wittgenstein argued against both sides. His ultimate achievement in the philosophy of mathematics was to stake out a defensible intermediate position between two untenable warring factions. This chapter will explicate Wittgenstein's position by stressing his opposition to each side, emphasizing, as well, the unity of Wittgenstein's later philosophy of mathematics with the Philosophical Investigations . Much of Wittgenstein's post- Tractatus work in the philosophy of mathematics endeavors to expose the delusions and misconceptions behind the second, metaphysical, camp. William James, a philosopher Wittgenstein admired, wrote that “the trail of the human serpent is … over everything.” Wittgenstein attempted to find the proper place for that trail in his post- Tractatus explorations of both mathematical and non-mathematical language. Much of the dispute among commentators concerns not the importance of such a trail, but its precise location. Wittgenstein perceived his chief foe, in mathematics, as altogether denying the human serpent. His adversaries picture mathematics as transcendent: a mathematical proposition has truth and meaning regardless of human rules or use. There is an under- (or over-) lying mathematical reality which is independent of our mathematical practice and language and which adjudicates the correctness of that practice and language.