The Future of Mathematical Discovery: A Conversation with Terence Tao

Kepler as a High-Temperature LLM

Kepler's Scientific Process

To understand the current state of AI in mathematics, it is helpful to look back at the history of astronomy. Johannes Kepler’s discovery of the laws of planetary motion provides a perfect case study. Kepler built upon the work of Copernicus, who had refined the earlier heliocentric models by proposing that planets moved in perfect circles. While Copernicus’s model fit existing data from Greek, Arab, and Indian sources, Kepler noticed that the ratios of the orbital sizes seemed to possess a geometric meaning. He famously theorized that the five Platonic solids—the cube, tetrahedron, icosahedron, octahedron, and dodecahedron—could be inscribed between the spheres of the six then-known planets. To him, this was a beautiful manifestation of divine mathematical perfection.

However, Kepler needed data to confirm his theory, leading him to the work of Tycho Brahe, a wealthy Danish astronomer who had spent decades collecting remarkably precise naked-eye observations of planets like Mars and Jupiter. After eventually securing this data, Kepler discovered to his great disappointment that his Platonic solid theory was off by about 10%. He spent years attempting to "fudge" his models to fit the data, but he eventually abandoned them to derive the actual orbits. Through this painstaking analysis, he discovered that planetary orbits are ellipses and that a planet sweeps out equal areas in equal times. A decade later, he derived his third law: the time it takes for a planet to orbit the Sun is proportional to a power of its distance from the Sun.

The Analogy to Modern AI

There is a compelling argument that Kepler functioned like a "high-temperature" Large Language Model (LLM). Throughout his career, he tested a vast array of random relationships, many of which were purely astrological or musical—such as his theory that Earth's "note" was mi-fa-mi, explaining earthly misery. Within this "slop" of speculative ideas lay the empirical regularities that would later be validated by Newton.

Modern AI models operate in a similar fashion. They can experiment with thousands of variables and random hypotheses for years, provided they have a verifiable dataset like Brahe's to check against. As long as a model can verify its results, these empirical regularities can drive deep scientific progress. We often celebrate the "eureka" moments of idea generation, but the reality of science—then and now—is a cycle of trying many things, failing, and only occasionally hitting upon a truth that can be verified. The challenge today is that AI has driven the cost of idea generation to near zero, making verification and validation the new bottlenecks of scientific progress.

The Evolution of Scientific Paradigms

From Hypothesis to Data-Driven Discovery

Traditionally, the scientific method involved making an observation, forming a hypothesis, and then collecting data to test it. In the 20th century, this was supplemented by numerical simulation and, eventually, the era of "big data." Modern science is increasingly reversing the classic method: researchers now collect massive datasets first and use statistical analysis to extract hypotheses.

While Kepler is often seen as a traditional theorist, his process mirrors this modern shift. Without Brahe’s massive dataset, Kepler would have remained a theorist writing books about harmonics and Platonic objects. His reliance on regression—fitting a curve to six planetary data points—was an early form of data science. However, even Kepler knew to be tentative. Johann Bode later used similar curve-fitting to predict a missing planet between Mars and Jupiter, which appeared to succeed with the discovery of Uranus and Ceres, only to fail spectacularly with Neptune. Data points alone do not guarantee a law of nature, and we must be wary of "numerical flukes."

Managing the Flood of AI Output

The scientific community is currently struggling to adapt to the scale of AI-generated content. We built peer-review systems to filter human-generated research, but those systems are now being overwhelmed by the volume of AI submissions. We lack a scalable way to determine which AI-generated findings represent real progress versus which are "red herrings."

History shows us that scientific progress is rarely straightforward. Copernicus’s heliocentric model was initially less accurate than the established, highly-tweaked Ptolemaic system. It took centuries of refining assumptions—such as the Aristotelian notion that objects naturally remain at rest—for the truth to become clear. Identifying a "unifying concept" within the noise of millions of AI papers will likely require a "test of time" that we cannot currently simulate. We need systems that can distinguish between a breakthrough that reshapes a field (like the "bit" in information theory) and noise that only resembles progress.

The Bottleneck: Breadth vs. Depth

The Complementary Nature of Human and AI Researchers

AI currently excels at breadth, while humans excel at depth. This is a complementary relationship that we haven't yet mastered. In mathematics, we have traditionally focused on deep, singular problems because that is what human expertise allows. However, we could be using AI to map out entire fields of science, identifying "islands of difficulty" where human experts can later focus their efforts.

Regarding the recent surge in AI-solved Erdős problems, it is important to maintain perspective. While the 50 solved problems receive significant publicity, they represent a small fraction of the total challenges. In systematic studies, the success rate of current AI tools on hard problems is quite low—around 1% or 2%. The AI appears successful because it can operate at scale; it can "buy" success by attempting thousands of variations until one hits the mark. This is not necessarily an advancement in the model's internal understanding of mathematics, but rather a powerful, brute-force search capability.

Developing a New Mathematical Language

Mathematics has been unique among the sciences for its near-total focus on theory. We haven't historically treated problem-solving as an experimental process. As we integrate AI, we need to develop a "semi-formal" language for mathematical strategies—not just the formal proofs that tools like Lean provide. We need a way to quantify plausibility and communicate the "narrative of discovery" that helps scientists decide which paths are worth pursuing.

We have a clear framework for what constitutes a valid proof, but we have no formal framework for why a conjecture is plausible. This is the missing link. We have faith in the Riemann hypothesis or the security of cryptography not just because of brute-force checking, but because of a collective, heuristic, and probabilistic model we have built over centuries. If we could formalize how these beliefs are constructed, we might be able to create a system where AI and humans collaborate on the strategy of science, not just the verification of its outputs.

The Future of Scientific Work

Changing the Daily Routine

AI has already changed the nature of mathematical research. Papers are becoming richer and broader, filled with plots, code, and deep literature reviews that would have been too time-consuming to generate previously. However, the core of mathematical creativity—the actual "solving" of the most difficult portions of a problem—remains a human task. AI currently serves as a highly efficient research assistant, cleaning up formatting, managing parentheses, and accelerating secondary tasks, but it is not yet a replacement for the intuition required to navigate the frontier of a field.

Embracing Serendipity

There is a danger that in our quest to optimize the scientific process, we are destroying the serendipity that leads to breakthrough moments. The transition to remote work and the ability to instantly search for the exact paper you need has replaced the accidental discovery of interesting ideas while browsing library shelves. We need a certain level of "distraction" and randomness to maintain inspiration.

For those entering a career in mathematics, the path is more unpredictable than ever. Traditional credentials will remain important for some time, but students should prepare for an era where they can contribute to research far earlier than was previously possible. The goal is to develop an adaptable mindset, leveraging AI to handle the "boilterplate" of science while keeping the human capacity for curiosity, serendipity, and deep, conceptual thought at the center of the work. We are in a period of unprecedented change, and while the future of mathematics may be unrecognizable, it is also filled with new opportunities for discovery.