Why I didn’t sign the Fields medallists’ letter
Reflections on the Fields Medallists' Letter: A Personal Perspective
This post has been cross-posted to the blog of Terence Tao.
A Childhood Encounter with Fermat
When I was roughly 11 years old, I first encountered the legend of Fermat's Last Theorem. The simplicity of the problem statement, combined with the historical lore surrounding it, completely fascinated me. I embarked on a genuine attempt to provide a proof.
Naturally, I did not succeed. However, the process of failing was incredibly educational.
At the time, I was unaware that Leonhard Euler had already solved the case for over two centuries prior. I figured that mastering this specific instance would be the perfect stepping stone toward the general proof. Since I was stuck on where to begin, I decided to simplify the challenge further by examining the successive differences of cubes, hoping to demonstrate that such a difference could never be a cube itself.
Because I lacked the formal algebraic vocabulary at that age, I didn't frame this as a Diophantine equation. Instead, I simply calculated differences and analyzed them visually.
(Note: This account is a reconstruction based on fragmented memories from fifty years ago, so it may not be perfectly precise.)
Eventually, I tried taking the difference sequence of the difference sequence and realized it formed an arithmetic progression. This felt like a breakthrough. Through empirical testing, I discovered a general rule: if you start with -th powers and repeatedly take differences, you eventually arrive at a constant sequence.
While this didn't lead me to solve Fermat's Last Theorem—and my mathematical ambitions eventually shifted—this early exploration was invaluable. When I later encountered the formal theory of polynomials and difference sequences in my education, I understood them far more deeply because I had already "played" with them.
The Core of the Debate
I share this story to highlight a central theme of the letter signed by 25 Fields medallists: the intrinsic value of the intellectual struggle. The act of grappling with a problem is where the real learning happens, regardless of whether a solution is reached.
Despite this agreement, I chose not to sign the letter. Much like my approach to the Leiden Declaration, I felt it was more honest to articulate my specific position via a blog post.
To be clear:
I am not opposing the signatories.- I do not wish to see the mathematical community split into hostile camps.
- I agree that we are currently facing a crisis.
I simply believe the nature of that crisis requires a different analysis.
Transparency and Affiliations
To avoid ad hominem dismissals or accusations of being "pro-AI," I want to be transparent about my relationship with OpenAI:
| Relationship Aspect | Detail |
|---|---|
| Financial | I have never been paid by OpenAI. |
| Access | I have had early access to some models (usually a few days prior to release). |
| Subscription | I have been granted free access to Pro models. |
| Network | I maintain contacts within their mathematics group. |
Furthermore, some might assume my stance is biased because I lead a group at Cambridge focused on automatic theorem proving. Ironically, this makes me more "anti-AI" in a sense. My group sought to enable computers to prove theorems by mimicking human-like conceptual insight. Now that LLMs can achieve results without those insights, a primary driver of our research has been eliminated superseded.
We have had to accept what is known as the bitter lesson: the reality that general-purpose computation often outperforms human-engineered heuristics.
The Divergence in Philosophy
While I believe understanding the "how" of human proof is still a worthy intellectual pursuit, I disagree with a specific premise in the medallists' letter.
"But solving problems is only a tool and proxy for achieving the primary goal of conceptual understanding and insight. Forgetting this in the world of AI may turn the tool against the primary goal. Indeed, the mass production at faster and faster pace of “true/false” statements could destroy fertile ground instead of breathing life into new ideas."
My perspective is more nuanced. In my essay The Two Cultures of Mathematics from twenty-five years ago, I argued that there is a spectrum of mathematical attitudes regarding the link between solving problems and gaining conceptual insight.
The Spectrum of Mathematical Motivation:
- End A: Mathematicians driven primarily by the desire to solve specific problems.
- End B: Mathematicians driven primarily by the pursuit of overarching conceptual frameworks.
I do not believe that the "mass production" of solutions necessarily destroys the "fertile ground" of insight, as the relationship between the two is not uniform across the field.
Images and avatars associated with the original discussion:
