Mathematics has long been one of the most traditional academic disciplines: blackboards, small collaborations, and individual problem-solving have dominated for centuries. Unlike other sciences that embraced large-scale collaboration and computation, mathematics has historically evolved more slowly. Still, computational tools have always played a role, from the abacus and tables of logarithms to early calculators. These uses have evolved gradually, paving the way for today’s computational mathematics.
Mathematicians once relied on tables of values (like trigonometric, logarithmic, and prime tables) to facilitate calculations—now replaced by computational tools and databases. The Online Encyclopedia of Integer Sequences (OEIS) is a modern, essential resource for identifying and connecting mathematical objects by their sequences.
Empirical, computer-driven investigations have sometimes preceded theory, as seen with the prime number theorem or the Birch and Swinnerton-Dyer conjecture. Databases of mathematical objects (such as elliptic curves) have also revealed unexpected relationships.
Machines excel at number crunching and can even solve logical problems such as Boolean satisfiability (SAT). Notable example: The Boolean Pythagorean triples problem, solved by exhaustive computer search, producing a proof certificate hundreds of gigabytes in size.
Terence Tao highlights three emerging tools and approaches that, while not yet a "killer app," are already reshaping mathematical practice:
These are used to detect patterns in large mathematical datasets that are beyond human recognition. For example, neural networks have helped discover new links between geometric and combinatorial knot invariants, leading to fresh conjectures and, eventually, new proofs.
LLMs like ChatGPT and Claude are beginning to assist mathematicians with literature searches, code and LaTeX generation, and brainstorming. They sometimes even solve Olympiad-level problems (though still unreliable for detailed calculations). Their best use is to generate code or formal statements, which can then be checked rigorously.
Proof assistants (such as Lean and Coq) rigorously check the correctness of formalized proofs, reducing human error. Major theorems (like the Four Color Theorem and Kepler Conjecture) have been formalized this way. Formalization also enables collaborative “crowdsourcing” of proofs, where large groups can independently verify small lemmas, making group proofs and inclusivity feasible at unprecedented scales.
The Kepler Conjecture's formalization required over a decade and vast collaborative effort, setting a new bar for certainty in computer-assisted proofs.
Recent projects (such as the Polynomial Freiman–Ruzsa conjecture) have shown that 20+ mathematicians—many of whom were previously strangers—can collaboratively formalize complex results with proof assistants.
Automating the exploration of millions of "laws of algebra" has created new datasets and networks of mathematical implications, offering insights unimaginable without modern computation.
Many mathematical fields (especially outside combinatorics and algebra) are still “data-poor.” There is a growing need for better databases—including records of negative results and partial attempts, not just published theorems.
The way mathematical data is represented (e.g., graphs as adjacency matrices vs. edge lists) can make a big difference for machine learning. Some hope higher-level frameworks (like category theory or homotopy type theory) will facilitate robust formalization and cross-domain work, but real adoption is still limited.
AI-powered textbooks and interactive platforms could revolutionize math education—enabling adaptive, on-demand explanations and personalized tutoring at scale.
Mathematical research might shift toward more modular, data-driven, and failure-tolerant workflows—where proof assistants and computational checks ensure rigor, and experimentation plays a bigger role in discovery.
Listening to Terence Tao’s perspective, I’m struck by how much the landscape of mathematical practice is evolving. For me, several key takeaways stand out—practical habits and attitudes I want to embrace going forward:
Tao’s examples reinforce the value of not just abstract reasoning, but also “experimenting” with math using code, databases, and even neural networks. I want to get more comfortable with running computational checks, searching for patterns in data, and leveraging platforms like OEIS as part of my workflow.
The rise of proof assistants (Lean, Coq, etc.) is inspiring. While the initial learning curve can be steep, I’m motivated to start learning at least one formal verification tool, so I can write and check small proofs—and maybe even contribute to bigger collaborative projects in the future.
Mathematics is becoming more collaborative and modular. I want to become comfortable sharing partial ideas or results, contributing to joint projects, and even formalizing “work in progress”—knowing that the collective effort (with help from computers) can often go further than working alone.
I’m encouraged to treat failed attempts, negative results, or partial findings not as dead ends, but as useful data. Recording and sharing these may help others—and myself—build toward eventual progress, echoing Tao’s point about the importance of data-rich mathematical communities.
Tools like ChatGPT are not a replacement for understanding, but they’re great for brainstorming, coding, LaTeX formatting, and checking intuition. I want to use them not just for convenience, but as genuine partners in my mathematical process, while still verifying everything rigorously.
Above all, Tao’s talk gives me confidence that blending traditional rigor with the new computational and collaborative possibilities will make me not just a better problem-solver, but a better mathematician—open to change, resilient, and eager to contribute to a rapidly evolving field.
Mathematics stands at the threshold of transformation. While we haven’t yet seen a single “killer app” for mathematics, the synergy of experimental computation, machine learning, LLMs, and formal verification is already reshaping how math is explored, proven, communicated, and taught. The next decade will likely see these tools become even more integrated—enabling new discoveries and welcoming new generations of mathematicians.