Wittgenstein on Mathematical Certainty
What makes so certain? Wittgenstein spent years wrestling with the foundations of mathematics, arriving at surprising conclusions about the nature of necessity.
The early Wittgenstein, in his Tractatus, saw mathematics as a system of tautologies—statements that say nothing about the world but reveal the logical structure of our language. Mathematical truths are necessary because they are empty.
But the later Wittgenstein grew dissatisfied with this view. In his Remarks on the Foundations of Mathematics, he explored a more radical idea: perhaps mathematical certainty is a matter of how we use mathematical statements, not their correspondence to abstract objects.
Consider what happens when you teach a child to add. At first, they follow rules tentatively, making mistakes. Gradually, certain answers become "natural"—they couldn't imagine answering differently. This is what Wittgenstein calls "agreement in form of life."
Does this mean mathematics is merely conventional? Not exactly. The conventions aren't arbitrary—they're shaped by our practices, our needs, our forms of life. Mathematics works because we are the kind of creatures who can agree on it.
This view remains controversial. Many mathematicians feel it fails to capture the sense that mathematical truths are discovered, not invented. The debate continues.