The Hidden Math That Shapes Reality

In October 2024, independent researcher Luke Durant of San Jose announced he had identified the largest known prime number — a value so vast it would take years to write out in full. While such discoveries may seem the province of specialists, prime numbers are widely regarded as fundamental building blocks of mathematics, and mathematics itself underpins everything from quantum theory and smartphone algorithms to the structural safety of bridges and the odds at a casino.

People have an intuitive relationship with numbers from an early age: experiments show infants possess a rudimentary sense of quantity. For some, that early aptitude grows into a lifetime of inquiry. In 2023, talented high school students published a trigonometric proof of the Pythagorean theorem, and the 2025 Abel Prize was awarded to Masaki Kashiwara in recognition of nearly seven decades of deep contributions to the field.

The hunt for ever-larger primes continues, and researchers are devising new techniques — including approaches that use integer partitions — to expand the frontier. Many mathematical problems carry long histories: the Langlands conjectures have guided decades of research, and some questions remain unresolved after more than half a century. The discipline is also rich in paradoxes and subtle thought experiments. For example, the infinite alternating series 1 − 1 + 1 − 1 + ... resists a single naive interpretation, and the famous barber paradox — a barber shaves those, and only those, who do not shave themselves — vividly illustrates self-reference and logical tension. These puzzles connect to deeper debates about the nature and interpretation of probability.

Mathematics is tightly woven into physics and cosmology. Complex numbers are central to quantum theory, and in 2025 a group of mathematicians announced a claimed breakthrough on “Hilbert’s sixth,” presenting a proof that links several physical descriptions of fluid motion. Topology — the study of shapes that can be stretched or deformed without tearing — is reshaping methods in data analysis, and researchers continue to search for an "einstein" tile: a single shape that tiles the plane without producing a repeating pattern, a challenge that has intrigued mathematicians for at least a century.

Mathematical ideas appear in everyday life in surprising ways. Equations determine the stability of structures such as bridges and airplanes; statistical methods can, in particular contexts, improve odds in games of chance; and algorithms influence strategic decision-making in activities like poker. New computational models can design citizens' assemblies so they more fairly reflect the populations they represent, potentially strengthening democratic processes. Even the everyday practical problem of moving a sofa around a corner has inspired mathematical analysis.

Over thousands of years, mathematics has proven itself a powerful tool for explaining form, predicting behavior, and optimizing systems — from physical phenomena to social mechanisms and perhaps even aspects of human cognition. Seen this way, mathematics is not an arcane or alien pursuit but a reflective lens on human thought and the world we inhabit.