How much of mathematics can ever be truly known? This age-old question is back at the forefront after a recent high-profile lecture at Harvard, where a renowned quantum computing expert explored how the frontiers of computer science, philosophy, and mathematical logic intertwine to define the boundaries of the knowable in mathematics (Harvard Math - Fifth Annual Yip Lecture).
The lecture, titled “How Much Math Is Knowable?”, highlighted that while mathematics is often thought of as the bastion of certainty, the reality is more nuanced and constrained by the very limits of computation itself. The core message: computer science doesn’t just create new tools for mathematicians—it draws the actual lines that define what humanity can, or cannot, ever formally prove or know in mathematics.
This topic holds special relevance for Thai learners and educators, given the growing emphasis on mathematical reasoning in the Thai national education curriculum and the country’s investment in developing more advanced computational thinking skills among students (Ministry of Education, Thailand). Understanding the limitations and the philosophy of mathematics is essential for nurturing a new generation of critical thinkers and innovators.
One of the lecture’s central examples was the famous Goldbach Conjecture, which asserts that every even number greater than two can be expressed as the sum of two prime numbers. While computers have checked this conjecture up to very large numbers, a definitive proof—or disproof—remains elusive. As the analysis pointed out, confirming the conjecture for all cases would require checking an infinite sequence, an impossible task for both humans and machines. Contrast this with the Pythagorean Theorem, where a finite set of logical steps provides a proof that covers an infinite set of triangles. Here, mathematical “regularity” offers a loophole—some infinite problems are tractable, others are not (Wikipedia – Goldbach Conjecture).
To illustrate the challenges around infinity, the lecture presented a whimsical computation: testing each case of the Goldbach conjecture in ever-diminishing time intervals (for instance, the first test in one second, the next in half a second, the next in a quarter, and so on). Mathematically, after two seconds, all checks are done, echoing the ancient Zeno’s Paradox. Yet, real-world physics—including limits set by quantum mechanics like Planck Time—prevents such a fantasy from becoming reality.
The discussion then shifted to the Busy Beaver function, a celebrated problem in theoretical computer science. The Busy Beaver game asks: for a fixed set of rules (an n-state Turing machine), what is the largest number of steps it can take—or symbols it can write—before halting? What makes this function fascinating is that it grows faster than any computable function, rapidly exceeding the limits of both computers and human mathematicians. Even for small values (such as BB(5)), the numbers become astronomical. The implications are staggering: just computing this function for relatively small inputs would solve numerous open mathematical problems, including the Goldbach Conjecture and even the Riemann Hypothesis, by transforming questions of mathematical truth into questions about whether specific machines ever halt (Wikipedia – Busy Beaver function).
The renowned lecturer further tackled the P vs NP problem—one of the most important unsolved problems in computer science. It asks whether every problem whose solution can be quickly checked by computer can also be quickly solved by computer. This has far-reaching consequences not just in abstract mathematics, but in practical fields like cybersecurity, logistics, and even automated medical diagnostics (Wikipedia – P versus NP problem).
Experts at the event stressed that computer science has made it clear: there are strict boundaries to what can be computably proven or decided. This isn’t just a theoretical concern—it shapes the future of mathematics, science, and technology. For instance, Bangkok’s top educators and mathematics outreach specialists note that appreciating the undecidable and the incomputable can inspire students to be both realistic and creative in their approach to mathematics, reminding them that mathematics is rich not just in truths, but also in deep, enduring mysteries.
What does this mean for Thailand? As the Kingdom moves forward with digital transformation and pushes for innovation-driven development—especially through the Thailand 4.0 policy—understanding the intertwining of computability and mathematics becomes increasingly important. Many research and industrial challenges, from AI ethics to cryptography, hinge on recognizing what machines can ultimately deliver and where human intuition, creativity, and philosophical reflection still have the upper hand.
Historically, Thai mathematicians and educators have embraced philosophical thought from both Eastern and Western traditions. The Buddhist concepts of impermanence and the limits of human perception offer a fascinating parallel to the findings of computability theory. Just as not every truth can be grasped in meditation or logic, not every mathematical problem can be definitively resolved—no matter how powerful our machines become.
Looking ahead, expect the debate on the limits of the knowable in mathematics to influence not just university curriculums but also school-level mathematics education. Thai teachers might increasingly challenge students, not just to solve problems, but to consider whether those problems are, in principle, solvable. This critical thinking skill is precisely what will set Thai learners apart as global citizens in a complex, technological world.
For inquisitive minds, the message is clear: embrace both what is known and what may always remain just out of reach. Delving into the boundaries of mathematical knowledge is as much about humility as it is about discovery. Thai educators, policymakers, and students should remain engaged with these global debates to ensure the country’s mathematical progress remains vibrant, balanced, and forward-looking. For students and citizens, curiosity about these issues can lead to a deeper appreciation for the marvel of mathematics—its elegance, its mystery, and its enduring power to puzzle and inspire.
For those interested in exploring these topics further, resources such as Scott Aaronson’s talks, the extensive entry on computability in Wikipedia, and Thai-language educational materials on mathematical theory are recommended as starting points.
Sources: