In 1939, a series of lectures at the University of Cambridge sparked a significant debate between two great minds – Ludwig Wittgenstein, the philosopher, and Alan Turing, the mathematician and computer scientist. The debates took place during Wittgenstein’s lectures on the Foundations of Mathematics, and they are now seen as a significant moment in the development of the philosophy of mathematics.
Wittgenstein, an Austrian-British philosopher, had become increasingly interested in the philosophy of mathematics. He held an unconventional view on mathematics, eschewing the idea that it was a perfect, self-evident truth that must be proven consistent. Instead, he argued that mathematics was more like a game, bound by rules we have agreed upon. He believed that paradoxes and inconsistencies were not fatal flaws but were just a part of the language game of mathematics.
This view was in stark contrast to the conventional mathematical thinking of the time, which was heavily focused on eliminating inconsistencies and paradoxes. Wittgenstein regarded this as a kind of superstition, a baseless fear of inconsistency, as he believed that inconsistency could co-exist with functionality.
On the other side of the debate was Alan Turing, a pioneering mathematician who would later become renowned for his work in computer science. He was fundamentally opposed to Wittgenstein’s views. Turing believed in the primacy of mathematical logic and the absolute necessity for consistency. In his view, without a stable, consistent basis, mathematics could not serve its functional purposes, particularly in fields like engineering.
Turing famously argued that “bridges would fall down” if engineering were based on a “hidden” contradiction. His contention was that an inconsistency, no matter how hidden, could lead to catastrophic failures in practical applications. Thus, in Turing’s view, the pursuit of consistency and the resolution of paradoxes were absolutely critical for mathematics and its applications. So who won this debate? That largely depends on the perspective from which one approaches the question.
From the standpoint of the practical application of mathematics and its role in the sciences, Turing’s argument may seem to hold more weight. After all, we want our bridges to stay up, our computers to compute, and our planes to fly. Mathematical logic and consistency form the bedrock of these applications. In Turing’s world, where mathematics was tied to practical applications and later to digital logic and the design of computers, there was no room for the kinds of inconsistencies that Wittgenstein was willing to entertain.
However, from a philosophical perspective, Wittgenstein’s argument invites us to question our fundamental understanding of what mathematics is and what it means for a statement to be true or false. This doesn’t necessarily affect the process of doing mathematics, or its practical applications, but it’s a fascinating debate in the philosophy of mathematics. Wittgenstein’s approach challenges us to think more deeply about the nature of mathematical proofs, the role of language and agreement in shaping them, and how we can understand the concepts of truth and proof in a broader context.
The Wittgenstein-Turing debate didn’t result in a clear “winner,” but it did highlight the intriguing intersection of philosophy, mathematics, and practicality. Even today, this discussion invites us to explore our understanding of mathematics, its foundation, and its implications. It reminds us that even something as seemingly concrete and straightforward as mathematics can be the subject of deep philosophical debate