"From an era of proof scarcity to an era of proof abundance"
Programming Note: As a reminder, every Sunday, I write pieces that are predominantly based on personal experiences which may or may not be loosely connected with investing. If you are reading MBI Deep Dives everyday, I think it would be rather useful for my readers to understand my personal lens a bit better since that presumably affects (at least in some capacity) the way I analyze businesses as well.
Back in 2012, I visited Harvard for the first time to attend a conference. During that conference, I attended a social get together where I met a bunch of undergrads at Harvard (I myself was a Junior in college at that time back in Bangladesh). It was still socially acceptable to ask for each other’s Facebook account and stay connected through Facebook. So, after the event, I became “friends” with a bunch of Harvard undergrads.
A couple of weeks ago, one such friend (who once won Gold at International Mathematical Olympiad) posted the below status update on Facebook:
“As International Mathematical Olympiad is being held in Shanghai right now, I’m having strange feelings while experimenting with AI model performance at the most well known global academic competition in the world. Our conclusion is that AI models are now practically better than almost all humans, even at IMO. Day 1 perfect score is confirmed, and likely to be the same on day 2 as well. This surpasses our previous expectation which was merely a gold medal level performance.
If I was 15 year old right now, I know I wouldn’t be able to motivate myself to get better at math as I did back then, this makes me sad. At that age, I felt real romantic love towards mathematics, and AI is like a much stronger bully stealing my love. It may sound absurd, but that’s how I feel!”
Then a couple days later, he confirmed the AI models’ supremacy in another status update on Facebook:
“GPT 5.6 Pro solved all 6 problems from IMO 2026 on the first attempt without any human help or steering…The problems are considered incredibly hard, usually a performance at this level is only accomplished by < 5 contestants from the whole world.”
While reading these status updates, I was reminded of another interaction I had with a girl when I was visiting Bangladesh a couple of months ago. She was just in her 9th grade, and it was pretty evident to me that she was perhaps one of the smartest 9th graders I have ever interacted with. Her father used to teach my wife Mathematics when my wife was in High School, and he wanted to visit us with his daughter when he learned that we are in town.
The daughter clearly got the Mathematics bug from her father and she was telling me how she spends much of her day solving Math problems and rummaging through different problem sets and discussions in different internet forums. it was a good reminder that despite being born in a second-tier city of a third world country, a gifted kid can still keep pace with the world through the blessings of the internet. She wanted to represent Bangladesh at IMO and was curious to learn more about college admission process in the US. The reality is most US colleges would be lucky to have her in their class, so I encouraged her to not fret over such mundane things. However, I did have one suggestion for her: “keep your identity small…don’t make mathematical proficiency your primary identity, rather just internalize that it is one of your gifts that the world may or may not value in a decade or two. Nurture the inherent joy of Mathematics that got you hooked in the first place so that even if the world becomes less appreciative of your gifts, you still have your reservoir of curiosity and joy to remain infatuated with Mathematics.”
AI’s rapid progress may make these questions more urgent than I may have internalized even a couple of months ago, but while this discomforting reality may be a relatively new phenomenon for today’s mathematically gifted kids, many other gifted kids indeed grappled with such reality for a long time. Imagine a kid in India who is the national Champion in some random Olympic sport but fail to make the Olympic podium. Can you imagine how challenging it is to become national champion in anything in a country of a billion and half population? And yet, there are perhaps numerous such gifted kids who do not find much of an economic value for their gifts.
Of course, Mathematics is no equivalent of a random Olympic sport as it is far more instrumental and fundamental in our understanding of the world. But I do often wonder Ilya Sutskever’s October 2023 tweet: “if you value intelligence above all other human qualities, you’re gonna have a bad time”
Indeed, if intelligence becomes too cheap to meter in not-so-distant future, that may fundamentally reshape how we approach mathematics (and mathematicians) too. Terence Tao, one of the greatest living Mathematicians, actually gave a profound (and very accessible) presentation titled “Mathematics in the age of AI” just two days ago at “International Congress of Mathematicians” (ICM). He didn’t mince the soul searching that may be required in the field of mathematics sooner rather than later (emphasis mine):
“…in the early twentieth century, discoveries such as Russell’s paradox (1901) or the Gödel incompleteness theorems (1931) forced practicing mathematicians to critically re-examine their implicit assumptions about the foundations of mathematics.
This crisis in foundations (∼ 1900–1930) was a turbulent period for mathematics. But the end product was extremely valuable: an explicit, rigorous, and standardized foundational framework. There is scope for further improvement. But our current foundations have survived strenuous testing and are a trusted environment for mathematics.
I believe we are entering a similarly turbulent period —a crisis in the foundations of mathematical values and practices.”
Since there are plenty of people out there who remain deeply skeptical of AI’s actual capability (especially given how jagged it can be), Tao was careful in explaining his “AI Capability Conjecture”:
At some point in the near future, some AI tools will, at some expense, and with some level of human supervision, be able to correctly accomplish some research-level mathematical tasks in some fields of mathematics, with some non-trivial success rate, and at some level of correctness and quality.
He then cited some independent assessment that led him to formulate his “working hypothesis” for AI’s capability:
AI tools will, reasonably soon, become capable of performing a reasonable fraction of research-level mathematical tasks, with reasonable levels of success, quality, supervision, and cost.
In such a world, what should be the “Goals” of today’s and tomorrow’s Mathematicians? Tao goes through different iterations of the goals in his presentation and decided on the following in his fifth attempt:
Solve unsolved problems, verify them to be correct, ensure they are clearly communicated, and have them digested, accepted, and incorporated into the definitive theory of the field.
The last bit may not seem super prestigious today, but Tao reminded a wonderful quote by William Thurston in ““On proof and progress in mathematics”:
"We are not trying to meet some abstract production quota of definitions, theorems and proofs. The measure of our success is whether what we do enables people to understand and think more clearly and effectively about math."
Of course, digestion and acceptance may be the actual bottleneck relative to the generation of novel proofs, but that’s exactly where humans can leave their mark. From Tao:
Community acceptance of a result, by its nature, is slow and human. It can be encouraged with good exposition and careful writing. But it is ultimately an external process that cannot be optimized purely by the authors and their AI tools
Our current publication infrastructure relies on human editors and referees to voluntarily provide this community acceptance as a service. This work is often regarded as less prestigious than that of generating proofs in the first place. But it is an essential component of our profession. It is also how we convert the individual achievements of mathematicians into collective progress and understanding.
Near the end of the presentation, Tao acknowledged as we are likely moving from an “era of proof scarcity to an era of proof abundance”, the mathematics community needs to evolve with it:
In short, we will transition from an era of proof scarcity to an era of proof abundance
we need to decrease the emphasis on proof generation, and of being the “first” to solve a problem; and increase the emphasis on “proof digestion”: exposition, publication, and canonicalization.
my suggested rule of thumb: if the authors cannot convincingly demonstrate that they can give a clear, expert-level talk on their results, that is correct and properly attributed, then the result should not be published.
The whole presentation is worth going through; you can sense the internal struggle due to the rapidly progressing AI models, and the wisdom that came out of it from one of the greatest living Mathematicians. As the foundations of mathematical values shift from generation to digestion, the best path forward might just be the one I suggested to that 9th grader in Bangladesh. When the novelty of simply finding the answer fades, it is the inherent joy of the process and our ability to share that clarity with others that will truly endure. Ultimately, our greatest enduring moat may simply be our curiosity and our shared human experience of understanding.
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