Kobi Kremnitzer, a pure mathematician from the University of Oxford. He has been meditating for years and is particularly interested in how meditation relates to consciousness. His research explores the mathematical structure of consciousness and its connection to quantum mechanics.
René Descartes did “meditate” in a philosophical sense. His famous work, Meditations on First Philosophy, is structured as a series of six meditations, where he systematically doubts everything he previously believed to be true in order to establish a foundation of absolute certainty. He wrote as if he were engaging in deep reflection over six days, each meditation building upon the last.
Of course, this wasn’t meditation in the modern sense of mindfulness or relaxation—it was an intellectual exercise aimed at uncovering fundamental truths about existence, knowledge, and God. His approach led to his famous conclusion: Cogito, ergo sum (“I think, therefore I am”).
Srinivasa Ramanujan – He often credited his mathematical insights to divine inspiration, which suggests a meditative or spiritual approach
Mathematician is a machine which turns coffee into theorems. - Paul Erdős
Pythagoras – His teachings incorporated elements of mysticism and self-discipline, which could align with meditative practices.
Pythagoras (c. 570–495 BCE) was a Greek philosopher and mathematician best known for the Pythagorean theorem, which relates the sides of a right triangle. He founded the Pythagorean school, a group that combined mathematics, mysticism, and philosophy. His followers believed that numbers held deep spiritual significance and that reality could be understood through mathematical relationships. Pythagoras also contributed to music theory, discovering numerical ratios that define musical harmony. His teachings influenced later philosophers like Plato and Aristotle
(Meditating and Mathematicians may not always apply)
Marijn Heule operates at the intersection of computer science and mathematics. Some might call him a “mathematical problem solver,” but he is not a mathematician in the classical sense like Terence Tao or Andrew Wiles.
Mathematics has seen some incredibly long proofs, but one of the longest ever recorded is the proof for the Boolean Pythagorean triples problem. This proof, generated by a supercomputer, is a staggering 200 terabytes in size—so vast that it would take a human 10 billion years to read it!
The Boolean Pythagorean triples problem is a fascinating question from Ramsey theory that asks whether the positive integers can be colored red and blue in such a way that no Pythagorean triple (numbers satisfying a2+b2=c2a^2 + b^2 = c^2) consists entirely of one color.
Marijn Heule, Oliver Kullmann, and Victor W. Marek solved this problem in May 2016 using a computer-assisted proof. They found that such a coloring is only possible up to 7824, but not beyond 7825. The proof was generated using a SAT solver and required 4 CPU-years of computation on the Stampede supercomputer at the Texas Advanced Computing Center. The final proof was 200 terabytes in size but was later compressed to 68 gigabytes.
This proof is one of the longest ever produced and won the best paper award at the SAT 2016 conference. Interestingly, mathematician Ronald Graham had offered a $100 prize for solving this problem, which was awarded to Heule.
Marijn Heule is a Dutch computer scientist specializing in SAT solvers—powerful tools used to solve complex mathematical problems. He is currently an Associate Professor at Carnegie Mellon University. Heule has made significant contributions to formal verification, number theory, and extremal combinatorics.
Some of his most notable achievements include:
Heule has also worked on trusted computing, ensuring that automated reasoning tools produce correct and verifiable results. His research aims to make computer-generated proofs more understandable for humans.
Do we truly “understand” a proof if we can’t read it ourselves?
The Boolean Pythagorean triples problem can certainly be understood conceptually, but fully comprehending its proof is a different challenge altogether.
At its core, the problem asks whether we can color the natural numbers red and blue without creating a monochromatic Pythagorean triple. That idea is simple. However, the proof—spanning 200 terabytes—is beyond human comprehension in a practical sense. It was computer-generated, meaning the reasoning involved is not neatly formatted like a traditional mathematical proof. Instead, it consists of a massive set of logical steps verified by a computer, making it inaccessible for manual verification.
So, while the problem itself is well within the grasp of mathematicians, the proof is a prime example of how modern mathematics is increasingly reliant on computational power. Some researchers aim to develop ways to simplify or verify these enormous proofs so humans can engage with them meaningfully. But as of now, no single person can fully comprehend the proof in its entirety.