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 essay 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.