Gödel, Wittgenstein, & the Limits of Knowledge

Michael D. McGranahan takes us to the edge of language, mathematics and science.

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There are things we can never know. There are questions that have no answer. There will always be uncertainty.

People typically reject these notions. We like to think we can delve into any problem and eventually find the solution, examine any mystery and uncover the truth. But this just isn't so. There are limits to what we can know about the world. Here I want to explore the convergence of Gödel and Wittgenstein on the limits of knowledge, against a background of Russell, Hilbert, and Heisenberg. We will find that mathematics, logic, and science all lead to the same truth: that not everything is knowable.

What Can We Know?

Reality is like an image in a cloud. From a distance, things seem clear: surely that's a rabbit. But move closer and the shape distorts: it looks less and less like a rabbit. Closer still, and the image becomes fuzzier and fuzzier, until eventually it's a blur. Finally, you're swallowed up in fog and nothing makes sense. Why is that?

Why does a close examination of reality only yield more questions?

Let’s turn back the clock about a hundred years to Kurt Gödel (1906-78) and Ludwig Wittgenstein (1889-1951). It is striking that over a century later we’re still grappling with their insights.

Let me say at the outset that these thinkers were dealing with difficult concepts, but we won’t be delving into the gritty details. Instead I'll summarize their ideas, distilling down to the essential results, and leave it to the curious reader to investigate the details.

We start with Wittgenstein. He fought in WWI, was captured, and while in a prisoner of war camp wrote up the notes that he had scribbled in the trenches. This became his