What Is Math, Really? Demystifying the Discipline Through Its Greatest Stories

In the winter of 1735, in a dimly lit study in the Winter Palace of St. Petersburg, a 28-year-old mathematician named Leonhard Euler sat hunched over a desk, his eyes milky and unseeing, his fingers flying across a wax tablet as a young assistant read out a stream of astronomical observations. For three days straight, Euler had not slept, working through a problem that had stumped the leading astronomers of Europe: calculating the precise orbit of a newly observed comet that had appeared in the sky that year. By the end of the third day, he had not only solved the orbit problem, but he had also laid the groundwork for a formula that would, 287 years later, be called the most beautiful equation in mathematics: $e^{i\pi} + 1 = 0$.

Euler’s story is the perfect starting point for any conversation about what mathematics really is. He was not a detached, ivory-tower thinker toiling over abstract puzzles for their own sake. He was a working mathematician solving concrete, urgent problems: predicting the movement of celestial bodies, designing better ships for the Russian navy, optimizing tax collection systems for the Prussian state, and even writing a popular science textbook for his teenage students. He produced nearly 900 publications over his 76-year life, covering every branch of mathematics known at the time, and invented much of the standard notation we still use today: $f(x)$ for functions, $i$ for the imaginary unit, $e$ for the base of the natural logarithm, $\Sigma$ for summations, and even the modern notation for trigonometric functions. And he did all of this after going completely blind in his late 50s, relying entirely on his prodigious memory and ability to do complex calculations in his head.

The formula that bears his name is a perfect encapsulation of his life’s work: it weaves together five of the most fundamental constants in all of mathematics, each discovered in a different era for a different purpose, into a single, elegant statement. The 0 and 1 are the building blocks of arithmetic, the $\pi$ is the heart of geometry, the $e$ is the core of calculus, and the $i$ is the foundation of complex numbers. For centuries after Euler’s death, this formula was celebrated as a parlor trick: a neat curiosity that showed how clever mathematicians could be at connecting unrelated ideas. But in the 20th and 21st centuries, it has become a workhorse of science and engineering: it is the backbone of signal processing that powers your phone’s Wi-Fi and Bluetooth, the foundation of quantum mechanics that describes the behavior of subatomic particles, and a core tool of artificial intelligence that powers the algorithms recommending your next Netflix show.

If Euler’s life and work show how mathematics evolves over a single career, a second event, 265 years later, shows how it evolves over centuries. On May 24, 2000, in a ceremony at the Collège de France in Paris, the Clay Mathematics Institute (CMI) announced the seven Millennium Prize Problems: seven unsolved problems in mathematics that the institute’s founders deemed the most important and difficult challenges in the field at the turn of the millennium. Each problem came with a $1 million prize for the first person to provide a rigorous, accepted solution. At the time of the announcement, mathematicians estimated that solving any one of them would require decades of work, and some might remain unsolved for a century or more.

Fast forward to 2024, and only one of the seven problems has been solved: the Poincaré Conjecture, a problem in topology (the study of shape properties that do not change when you stretch or bend an object, but break when you tear or glue it) first posed by French mathematician Henri Poincaré in 1904. The solver was Grigori Perelman, a reclusive Russian mathematician who posted his proof to a public preprint server in 2002 and 2003, refused to attend the International Congress of Mathematicians where he was awarded the Fields Medal (the highest honor in mathematics, often called the “Nobel Prize of math”), and declined the $1 million Millennium Prize, saying he was not interested in money or fame. The other six problems— the Riemann Hypothesis, the P vs NP Problem, the Yang-Mills Existence and Mass Gap, the Navier-Stokes Existence and Smoothness, the Birch and Swinnerton-Dyer Conjecture, and the Hodge Conjecture—remain unsolved, with thousands of mathematicians around the world working on them, publishing partial results, and slowly inching toward a solution.

These two stories, separated by nearly three centuries, are the twin pillars of this book. Euler’s formula shows how a single, elegant insight can weave together centuries of prior work, and become a tool that powers modern technology. The Millennium Problems show how mathematics is still a living, evolving field, full of open questions that will shape science, technology, and our understanding of the world for generations to come. Together, they answer the question that opens this chapter: What is math, really?

The Myth of the Static Rulebook

For most people, the first image that comes to mind when they hear the word “math” is a high school classroom: rows of students hunched over worksheets, memorizing formulas for the area of a triangle or the quadratic formula, drilling for standardized tests where there is only one correct answer and any deviation is marked wrong. This image of math as a static, fixed set of rules invented by dead old men, to be memorized and applied to narrow, artificial problems, is not just wrong—it is one of the greatest barriers to public understanding of one of humanity’s most vibrant and creative endeavors.

To see why, consider the history of the five constants in Euler’s formula. The number $\pi$, the ratio of a circle’s circumference to its diameter, was first approximated by ancient Babylonian and Egyptian scribes around 2000 BCE, who used it to calculate the area of fields and the volume of storage containers. The number $e$, the base of the natural logarithm, was not even discovered until the 17th century, when mathematicians were trying to solve problems related to compound interest and the geometry of curves. The imaginary unit $i$, defined as the square root of -1, was first proposed by Italian mathematician Rafael Bombelli in 1572, and was widely dismissed as a “fictitious” or “imaginary” number with no real-world meaning for nearly 200 years. The constants 0 and 1, which seem so obvious to us now, were not even recognized as distinct numbers in their own right by many ancient cultures, who had no symbol for zero and often did not distinguish between the concept of “one” as a count and “one” as a number.

Each of these constants was discovered, debated, and refined over centuries, as mathematicians solved new problems and built on the work of their predecessors. The rules and formulas we learn in school are not static edicts handed down from on high: they are the accumulated results of thousands of years of human curiosity, trial and error, argument, and collaboration. And the work is far from finished. Every year, mathematicians publish tens of thousands of new papers, proposing new conjectures, proving new theorems, and inventing entirely new branches of mathematics that did not exist 50 years ago. Chaos theory, which studies how tiny changes in initial conditions can lead to wildly different outcomes in complex systems, was not even a formal field until the 1960s, and it now underpins everything from weather forecasting to economic modeling to the design of more efficient airplane wings. Topological data analysis, a field that uses ideas from topology to find patterns in huge datasets, was invented in the 2000s, and is now used by biologists to map protein structures, by sociologists to study social networks, and by computer scientists to improve machine learning algorithms.

Mathematics is not a finished product. It is a process. It is a collective, centuries-long con