To the editors:

In 1972, the Josiah Willard Gibbs Lecture of the American Mathematical Society was delivered by Freeman Dyson. “As a working physicist,” he began, “I am acutely aware of the fact that the marriage between mathematics and physics, which was so enormously fruitful in the past centuries, has recently ended in divorce.”¹ The title of Dyson’s lecture, “Missed Opportunities,” referred to “occasions on which mathematicians and physicists lost chances of making discoveries by neglecting to talk to each other.”

As it turned out, the divorce did not last long. By the time of Dyson’s lecture, James Simons and Chen-Ning Yang, a mathematician and a physicist at Stony Brook University, had realized that fiber bundle connections in mathematics are identical to gauge fields in Yang-Mills theory—a theoretical model of elementary particles that Yang had constructed with Robert Mills two decades earlier by generalizing Maxwell’s theory of electromagnetism.² By the middle of the 1970’s, Yang-Mills theory had been established as a key theoretical ingredient of what is now known as the Standard Model of particle physics, thanks to Gerard ‘t Hooft and Martinius Veltman’s proof of its renormalizability and the discovery of its asymptotic freedom by David Gross, David Politzer, and Frank Wilczek. Because of the central role it plays in mathematics and physics, Yang-Mills theory has allowed mathematicians to dip their toes in the elusive world of quantum field theory (QFT)—even though it has been the basic language in elementary particle physics for almost a century, QFT remains elusive because its mathematical foundation is still lacking.

In her essay, Natalie Paquette reviewed four recent developments in physical mathematics to showcase the power of QFT to inspire new developments in mathematics. Some of the contributors in these examples have been awarded Fields Medals, the highest honor in mathematics. Paquette describes this situation as distinctly odd. “A line of influence,” she writes, “has always run from mathematics to physics.” When attempting to explain “the unreasonable effectiveness of physics within mathematics,” she suggests that

the difference between them may be less a matter of their content than their technique; and that, in the end, they serve to show that there is only one reality to which they both appeal.

Paquette’s claim is a radical departure from the prevailing view of mathematics, that it exists in the Platonic world independently of physical reality, and it can be used to explain not only our own universe, but all logically possible universes.

I would like to propose a different explanation for the unreasonable effectiveness of physics.

Until the seventeenth century, it was widely believed throughout Europe that there was one set of laws governing phenomena on earth, and another set of laws for the heavens. For this reason, it came as a great surprise when Galileo Galilei pointed his newly invented telescope at the night sky and discovered that the moon also has mountains and valleys. Though I am uncertain whether a twenty-three-year-old Isaac Newton did indeed discover his law of gravitation by observing an apple falling from a tree, he must surely have realized by this time that falling apples and the moon’s orbit are due to the same force—a universal law explains gravitational phenomena both on earth and in the heavens. For the purposes of formulating the law and using it to explain planetary orbits in our solar system, seventeenth century mathematics was inadequate. Newton had to invent calculus to work with infinitesimals, define the velocity and acceleration of a trajectory, and to solve his equations of motion. In 1684, Edmond Halley visited Newton in Cambridge. Halley asked Newton what he thought the trajectory might be for a planet’s orbit under the inverse-square law. Newton replied immediately that it would be an ellipse.

Natural languages such as English and Japanese have been invented, developed, and refined over tens of thousands of years in order to describe phenomena in everyday life. During the past few centuries, the realm of human experiences has expanded dramatically. In the early seventeenth century, Galileo’s use of the telescope enabled him to see the moon’s surface a billion meters away. Four centuries later, the LIGO observatories have detected gravitational waves that originated from the collisions of massive black holes located billions of light years away, while the Large Hadron Collider near Geneva, Switzerland is used to observe microscopic phenomena at a billionth of billionth of a meter. With such an expansion of our scientific sphere, it is entirely reasonable that our natural languages are not suitable to explain them. We need to develop a new language—the language of mathematics.

In addition to the lofty goal of discovering the fundamental laws of nature, practical needs in science and engineering have also led to the invention of new mathematical concepts and methods, and they have, in turn, given rise to further scientific discoveries. When Europeans began extensive overseas exploration and colonization during the fifteenth century, the process of determining a ship’s position required precise astronomical measurements and calculations to more than ten decimal places. Astronomers began using trigonometric functions since their addition theorems can be used to transform the multiplication and division of large numbers into the addition and subtraction of angles. In 1614, John Napier invented the logarithm to simplify calculations and published ninety pages of logarithm tables.³ Among its first applications was Johannes Kepler’s discovery of his third law of planetary motion. Kepler announced his first and second laws in 1609, but it took him a further ten years to come up with his third law, which gives the relation between the period of a planetary orbit and its major axis.⁴ When Kepler heard about Napier’s book and the logarithm, he immediately recognized its power and applied it when analyzing data he had inherited from Tycho Brahe. Kepler hit upon the third law when he computed logarithms of the periods and the axes of the six planets known at the time. The logarithm subsequently became an indispensable tool for astronomers. Pierre-Simon Laplace remarked that “by reducing the labor of many months to a few days, it doubles the life of the astronomer.”

There is yet another way for physics to influence mathematics: reasoning based on physics has led to the discovery of new mathematical theorems. In the third century BCE, Archimedes of Syracuse sent a manuscript to Eratosthenes, the director of the Great Library of Alexandria. The manuscript, in the form of a letter, became known as The Method because it began as follows:

Since I see that you are a capable scholar and a prominent teacher of philosophy, and also that you understand how to value a mathematical method of investigation when the opportunity is offered, I have thought it well to analyze and lay down a peculiar method by means of which it will be possible for you to derive instruction as to how certain mathematical questions may be investigated by means of mechanics. And I am convinced that this is equally profitable in demonstrating a proposition itself; for much that was made evident to me through the medium of mechanics was later proved by means of geometry because of the treatment by the former method had not yet been established by way of a demonstration.⁵

The use of mechanics by Archimedes in anticipating geometric theorems was in the same spirit as the use of mirror symmetry in physics by Philip Candelas et al. when computing the Gromov-Witten invariants, as recounted in Paquette’s essay.

These examples show that mathematics and physics have been in equipoise over the past millennia. Physical mathematics is not a recent phenomenon. As science progresses, we are constantly expanding the frontiers of human knowledge. The new natural phenomena we experience are not necessarily describable or explainable using existing languages and new mathematical concepts and tools are often needed. Once invented, mathematics acquires its own life and starts to grow according to its own logic. Given a set of axioms, mathematical theorems are true everywhere in our own universe, or any other possible universe—perhaps even without any universe at all. In this sense, mathematics exists independently of physical reality. Physics leads us to fruitful new areas of mathematics by posing challenging questions and predicting their answers by physical reasoning. Physics is particularly effective at influencing mathematics because it is the most quantitative of all the sciences and its observations can often be formulated as precise conjectures.

This still leaves us with the question of why the relationship between mathematics and physics has been so active and productive during the last four decades, whereas at the time of Dyson’s lecture the two were almost divorced. To answer this question, I need to explain QFT and its place at the current interface of physics and mathematics.

The simplest way to define QFT would be to state that it is a theoretical framework to unify special relativity and quantum mechanics.⁶ Special relativity tells us that energy can be converted into the masses of particles and that particles can be spontaneously created from the vacuum by focusing a sufficient amount of energy in a small region of space. Werner Heisenberg’s uncertainty principle, on the other hand, tells us that energy and time cannot be simultaneously measured with arbitrary precision and that the conservation of energy can and must be violated for a brief period of time. Any framework to unify special relativity and quantum mechanics needs to allow for processes where the number of particles changes by the uncertainty in energy. Every particle is characterized by its position, possibly with additional numbers such as the electric charge, called the degrees of freedom of a particle. If each particle possesses a finite number of such degrees of freedom, the theory that describes an arbitrary number of particles should be able to accommodate infinitely many degrees of freedom. QFT is a program to achieve this by assigning independent degrees of freedom at each point in space. The term field is used here in the same sense as in the electric field and the magnetic field, whose value can vary from point to point.

The infinitely many degrees of freedom in QFT posed enormous challenges for physicists when they tried to make sense of the theory. Initial attempts to use the theory to calculate quantum effects often produced infinities for what should have been physically observable quantities. Though the renormalization techniques invented by Richard Feynman, Julian Schwinger, and Shin’ichirō Tomonaga provided a means to extract finite and meaningful answers from the infinities in QFT computations, infinite degrees of freedom continued to be a major obstruction to a precise mathematical formulation of the theory. The lack of mathematical definition made it difficult for physicists using QFT to explain their research results to mathematicians and impeded fruitful collaborations. It also limited the applicability of QFT—the theory was useful only when forces between particles are weak and when approximate calculations can be performed by iterative integrals. It seems plausible to suggest that the pessimistic tone of Dyson’s Gibbs lecture reflected the status of QFT in the late 1960s. He had made a major contribution to QFT during the late 1940s by establishing the relationship between the seemingly different approaches of Feynman, Schwinger, and Tomonaga.⁷ Dyson must surely have been aware of the shortcomings of the theory.

Over the last four decades the situation has improved dramatically. The mathematical interpretation of Yang-Mills theory introduced powerful geometric methods to QFT, allowing us to glimpse the theory beyond the approximate treatment of weak forces, and inspired the invention of new techniques such as solitons, instantons, and the large N expansion. The renormalization method, which was thought to be a tentative resolution of the infinity problem, turned out to be an intrinsic and fundamental property of QFT and was re-invented as the renormalization group by Kenneth Wilson. The atmosphere of the late 1970s was vividly recounted in the preface to a collection of lecture notes by Sidney Coleman, a legendary figure in the development of QFT.

This was a great time to be a high-energy theorist, the period of the famous triumph of quantum field theory. And what a triumph it was, in the old sense of the word: a glorious victory parade, full of wonderful things brought back from far places to make the spectator gasp with awe and laugh with joy.⁸

The superstring revolution that began in 1984 brought a new perspective to the study of QFT. While the theory contains both special relativity and quantum mechanics, gravity cannot be included in its current framework because this would generate infinitely many new types of infinities that cannot be removed even by renormalization. For this purpose, a unification of general relativity and quantum mechanics would be required, and the only credible candidate we have found thus far is superstring theory. In 1984, a series of discoveries initiated by Michael Green and John Schwarz’s anomaly cancellation mechanism showed that superstring theory also contains all the theoretical ingredients for the Standard Model of particle physics. QFT is used in many aspects of superstring theory, and it also arises as various limits of superstring theory. By way of superstring theory, seemingly different kinds of QFT’s became related to each other. Many technically difficult problems have also been transformed into simpler problems, deepening our insight into QFT.

Just as physical reasoning based on mechanics led Archimedes to discover geometric theorems, QFT calculations have produced many interesting conjectures in mathematics. Even though QFT itself is still lacking a solid foundation, conjectures it has produced can be precisely formulated and many have been proven mathematically. These conjectures often connect different areas of mathematics in surprising ways. As noted by Paquette, mirror symmetry relates enumerative problems in geometry to period integrals and monstrous moonshine relates the representation theory of finite groups to the theory of modular forms. QFT’s unreasonable effectiveness in producing powerful conjectures in mathematics is a reflection of the fact that it was invented and developed to understand the microscopic frontier of physics, and that we are struggling to invent a new language of mathematics to describe it. QFT is a work in progress; its treatment of infinities is ad hoc, and it is not clear whether various limits one takes to compute physically observable quantities are well-defined. Experimental evidence overwhelmingly supports its correctness. QFT calculations of the magnetic moment of the electron have been experimentally verified up to twelve decimal places, which is perhaps the most precise confirmation of any scientific theory. The unreasonable effectiveness of QFT suggests that there must be a place for the theory in the Platonic world of mathematics.

Eleven years ago, I worked with my friends in Japan on a proposal for a new research institute in Tokyo—a place where mathematicians could work with physicists and astronomers to solve some of the most fundamental questions about the universe, such as how it began, what it is made of, and what its future might be. The decision to include mathematics as one of the core areas of the proposed institute was inspired, in part, by the following passage from Galileo’s Il Saggiatore:

Philosophy is written in this grand book—I mean the Universe—which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics…

We also wanted to include the word mathematics in the name of the institute itself. The name we came up with was the Institute for the Physics and Mathematics of the Universe.⁹ As non-native speakers, we were unsure if “Mathematics of the Universe” is correct English. I then found a book review by Roger Penrose with exactly the same title.¹⁰ Published in Nature, it was a review of “The Large Scale Structure of Space-Time” by Stephen Hawking and George Ellis, concerning geometric approaches to general relativity developed by Hawking, Penrose and others. General relativity had, in fact, played a central role in the development of cosmology and astrophysics during the twentieth century. For this reason, it seemed justifiable to refer to it as the mathematics of the universe.

For the ancient Egyptians and Greeks, the mathematics of the universe must have been plane geometry. The word geometry means “measure of land,” suggesting that it was developed to understand the space around us. Eratosthenes used geometry to measure the radius of the earth and by combining this result with Aristarco’s observations of a lunar eclipse he was able to determine the radius of the moon. To measure astronomical distances Hipparchus developed the parallax method. Many ancient Greeks did not believe in the heliocentric view of the solar system, but this was not solely due to religious reasons. If the earth was orbiting around the sun, they thought it should have been possible to observe stellar parallax. They were unable to do so. Their logic was, in fact, perfectly sound, but the stars are much further away than they imagined. The annual parallax of Alpha Centauri, the nearest star to earth, is mere 0.0002 degrees.

Even in the early seventeenth century, plane geometry was still the mathematics of the universe. Galileo’s Il Saggiatore continues:

[The Universe] is written in the language of mathematics and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth.

Soon, however, plane geometry would cede preeminence to calculus, without which Newton’s equation of motion and his law of gravitation could not have been formulated.

If the mathematics of the universe in the seventeenth century was calculus, and in the twentieth it was general relativity, what will be the mathematics of the universe in the twenty-first century? Whatever it may turn out to be, it should be able to deal with infinite degrees of freedom in a more powerful way. Just as calculus, originally invented to make sense of velocity and acceleration, has become an essential tool in all areas of sciences and technology, a new framework to deal with infinite degrees of freedoms will find many important applications beyond fundamental physics. I would like to think that the four recent developments in physical mathematics mentioned in Paquette’s essay are simply previews of the rich pastures of infinite analysis, which are waiting to be discovered in mathematics.