# The Essential Role of Mathematics in Physics

Mathematics is the language that enables us to quantitatively describe the physical world. Without mathematics, physics would be stuck at a qualitative level, unable to precisely formulate theories or make predictions. Math gives physics its precision and rigor. As physicist Eugene Wigner wrote, “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”

The interplay between math and physics has a long history. Ancient Greek philosophers like Pythagoras believed numbers ruled the universe. In the 17th century, Galileo pioneered the mathematical description of motion and mechanics Isaac Newton invented calculus to provide the mathematics needed to describe dynamics and gravitation Mathematics unlocked revolutionary ideas like electromagnetism, relativity and quantum mechanics in the 19th and 20th centuries.

As physics advanced to tackle ever more complex problems the math got more sophisticated too. Cutting-edge theories like string theory now rely on highly abstract mathematical frameworks. Physics has inspired whole new fields of mathematics like geometry and topology. Physics and math motivate and enrich each other in an endless reciprocal dance.

So how exactly does math enable physics? Here are some of the key roles mathematics plays

## 1. Quantification

Math allows us to go beyond vague descriptions and quantitatively measure and calculate physical properties. Using math, a physicist can precisely calculate numbers like the mass of an electron, the speed of light and the force between two charged particles. Physics is rooted in measurable, quantifiable facts and observations. Math provides the numerical tools to quantify and analyze the physical world.

## 2. Models

Mathematical models are simplified abstractions of physical systems that capture their essential features. Models like the ideal gas law, the simple harmonic oscillator and the two-body problem in gravity allow physicists to calculate system behavior quantitatively. Models may leave out complicating details while retaining accuracy for certain conditions. Good models are essential to theoretical physics.

## 3. Theories

Physical theories like electromagnetism, thermodynamics and general relativity are expressed in concise mathematical formulas and equations. Beautiful theoretical frameworks like Newton’s laws of motion are constructed from mathematical relationships. Theories make predictions by mathematically calculating outcomes for different conditions. Testing those predictions drives new discoveries.

## 4. Problem-Solving

Math is an indispensable problem-solving tool for physicists. Problems ranging from calculating trajectories to analyzing electrical circuits to quantum systems are solved using mathematics. Many problems are too complex to solve purely analytically, requiring numerical methods and computation. Mathematical techniques like calculus, differential equations, vectors and matrices are routinely employed to tackle problems.

## 5. Data Analysis

Experimental results and measurements don’t mean much until mathematical analysis extracts meaningful patterns. Using mathematical methods like statistics, signal processing and data fitting, physicists interpret data and draw conclusions. Data analysis techniques range from simple graphs to multivariate regression and machine learning algorithms. Math extracts insights from raw data.

## 6. Concise Communication

Physics relies on math to express concepts unambiguously. Equations present complex ideas concisely using symbols with precise definitions. By avoiding imprecise language, math strips away ambiguity. Compared to long verbal descriptions, the equation F=ma communicates succinctly. Mathematical notation provides efficient and clear communication of physical relationships.

## 7. Generalizing Results

Mathematics allows results to be derived in a general form, not just for specific cases. For example, Maxwell’s equations describe electromagnetic phenomena for arbitrary charges and currents. Mathematics enables physicists to find universal governing principles. Laws can be abstracted from the particular to the general using mathematics.

## 8. Discover New Physics

Sometimes math itself guides physicists toward new discoveries. By pursuing mathematical extensions of theories, new physics emerges. Quantum mechanics developed when researchers followed where wave equations led. String theory arose from exploring mathematical consistencies. Pure math research occasionally produces concepts physicists can harness, like manifold geometry in general relativity.

Mathematics will continue fuelling physics discoveries. The two share a symbiotic relationship, enabling profound insights into the fundamental workings of reality. Physics needs mathematics to elucidate patterns in nature, just as mathematics looks to physics for motivation and meaning. Their interdependence makes physics and math an incredible tool for comprehending the universe.

### A Tale of Two Disciplines

Math and physics are two closely connected fields. For physicists, math is a tool used to answer questions. For example, Newton invented calculus to help describe motion. For mathematicians, physics can be a source of inspiration, with theoretical concepts such as general relativity and quantum theory providing an impetus for mathematicians to develop new tools.

But despite their close connections, physics and math research rely on distinct methods. As the systematic study of how matter behaves, physics encompasses the study of both the great and the small, from galaxies and planets to atoms and particles. Questions are addressed using combinations of theories, experiments, models, and observations to either support or refute new ideas about the nature of the universe.

In contrast, math is focused on abstract topics such as quantity (number theory), structure (algebra), and space (geometry). Mathematicians look for patterns and develop new ideas and theories using pure logic and mathematical reasoning. Instead of experiments or observations, mathematicians use proofs to support their ideas.

While physicists rely heavily on math for calculations in their work, they don’t work toward a fundamental understanding of abstract mathematical ideas in the way that mathematicians do. Physicists “want answers, and the way they get answers is by doing computations,” says Tony Pantev, Class of 1939 Professor of Mathematics. “But in mathematics, the computations are just a decoration on top of the cake. You have to understand everything completely, then you do a computation.”

This fundamental difference leads researchers in both fields to use the analogy of language, highlighting a need to “translate” ideas in order to make progress and understand one another. “We are dealing with how to formulate physics questions so it can be seen as a mathematics problem” says Mirjam Cvetič, Fay R. and Eugene L. Langberg Professor of Physics. “That’s typically the hardest part.”

“A physicist comes to us, asks, ‘How do you prove that this is true?’ and we immediately show them it’s false,” says Ron Donagi, Professor of Mathematics. “But we keep talking, and the trick is not to do what they say to do but what they mean, a translation of the problem.”

In addition to differences in methodology and language, math and physics also have different research cultures. In physics, papers might involve dozens of co-authors and institutions, with researchers publishing work several times per year. In contrast, mathematicians might work on a single problem that takes years to complete with a small number of collaborators. “Sometimes, physics papers are essentially, ‘We discovered this thing, isn’t that cool?’” says Randy Kamien, Vicki and William Abrams Professor in the Natural Sciences. “But math is never like that. Everything is about understanding things for the sake of understanding them. Culturally, it’s very different.”

Randall Kamien, Vicki and William Abrams Professor in the Natural Sciences, works on physics problems that have a strong connection to geometry and topology.

When asked how mathematicians and physicists can bridge these fundamental gaps and successfully work together, researchers refer to a commonly cited example that also has a connection to Penn. In the 1950s, Eugenio Calabi, now Thomas A. Scott Professor of Mathematics Emeritus, conjectured the existence of a six-dimensional manifold, a topological space arranged in a way that allows complex structures to be described and understood more simply. After the manifold’s existence was proven in 1978 by Shing-Tung Yau, currently the William Caspar Graustein Professor of Mathematics at Harvard University, this new finding was poised to become a fundamental component of a new idea in particle physics: string theory.

Proposed in the 1970s as a candidate framework for a “theory of everything,” string theory describes matter as being made of one-dimensional vibrating strings that form elementary particles, like electrons and neutrinos, as well as forces, like gravity and electromagnetism. The challenge, however, is that string theory requires a 10-dimensional universe, so physicists turned to Calabi-Yau manifolds as a place to house the “extra” dimensions.

Calabi-Yau manifolds, conjectured in the 1950s by Eugenio Calabi, now Thomas A. Scott Professor of Mathematics Emeritus, are a fundamental component of research in both particle physics and cutting-edge mathematics.

Because the structure is so complex and only recently proven by mathematicians, it wasn’t simple to directly implement into a physics framework. Physicists “use differential geometry, but that’s been known for a long time,” says Burt Ovrut, Professor of Physics and Astronomy. “When all of a sudden string theory launches, who the heck knows what a Calabi-Yau manifold is?”

Through the combined efforts of Ed Witten, a theoretical physicist and professor of mathematical physics at the Institute for Advanced Study in Princeton, and the late Sir Michael Atiyah, a British-Lebanese mathematician specializing in geometry, researchers found a way to apply Calabi-Yau manifolds in string theory. It was the ability of Witten to help translate ideas between the two fields that many researchers say was instrumental in successfully applying brand-new ideas from mathematics into up-and-coming theories from physics.

Donagi, Pantev, and Antonella Grassi, Professor of Mathematics Emeritae, as well as physicists Cvetič, Kamien, Ovrut, and Jonathan Heckman, Assistant Professor of Physics and Astronomy, have also recognized the importance of speaking a common language as they work across the two fields. They credit Penn as being a place that’s particularly adept at fostering connections and bridging gaps in cultural, linguistic, and methodological differences, and they credit their success to time spent listening to new ideas and developing ways to “translate” between languages.

“[The physicists] speak our language, and they can explain the questions they are struggling with in a way that we can understand and approach them,” says Tony Pantev (R), Class of 1939 Professor of Mathematics, pictured with Mirjam Cvetič, Fay R. and Eugene L. Langberg Professor of Physics, and Ron Donagi, Professor of Mathematics.

For Donagi, it was a chance encounter with Witten in the mid-1990s that led the mathematician to his first collaboration with a researcher outside of pure math. He enjoyed working with Witten so much that he reached out to Penn physicists Cvetič and Ovrut to start a “local” crossover collaboration. “I’ve been hooked since then, and I’ve been talking as much to physicists as to other mathematicians,” Donagi says.

During the mid-2000s, Donagi and Ovrut co-led an interedisciplinary research group supported by the National Science Foundation and the Department of Energy with Pantev and Grassi that was supported by the U.S. Department of Energy. The collaboration marked a successful first official math and physics crossover collaboration at Penn. As Ovrut explains, the work was focused on a specific kind of string theory and required extremely close interactions between physics and math researchers. “It was at the very edge of mathematics and algebraic geometry, so I couldn’t do this myself, and the mathematicians were very interested in these things,” says Ovrut.

Cvetič, a longtime collaborator with Donagi and Grassi, says that Penn’s mathematicians have the expertise they need to help answer important questions in physics and that their collaborations at the interface of string theory and algebraic geometry are “extremely fruitful and productive.”

“I think it’s been incredibly productive and helpful for both our groups,” Donagi says. “We’ve been doing this for longer than anyone else, and we have a really good, strong connection between the groups. They’ve almost become one group.”

And in terms of embracing cultural differences, physicists like Kamien, who works on problems with a strong connection to geometry and topology, encourages his group members to try to understand math the way mathematicians do instead of only seeing it as a tool for their work. “We’ve tried to absorb not just their language but their culture, how they understand things, how sometimes understanding a problem more deeply is better,” he says.

Burt Ovrut (L), Professor of Physics and Astronomy, was one of the co-leads of the successful joint math and physics program.

Craig Lawrie and Ling Lin, current and former postdocs working with Cvetič and Heckman, know firsthand about both the challenges and opportunities of working on a problem that combines cutting-edge math and physics. Physicists like Lawrie and Lin, who work on problems in two different branches of string theory, are trying to figure out what types of particles different geometric structures can create while also removing the “extra” six dimensions.

Adding extra symmetries, a physical or mathematical feature that remains constant when undergoing a transformation (think of a ball rotating in front of a mirror), makes string theory problems easier to work with and allows researchers to ask questions about the properties of geometric structures and how they correspond to real-world physics. Building off previous work by Heckman, Lawrie, and Lin were able to extract physical features from known geometries in five-dimensional systems to see if those particles overlapped with standard model particles. Using their knowledge of both physics and math, the researchers showed that geometries in different dimensions are all related mathematically, which means they can study particles in different dimensions more easily.

By combining their physics intuition with their knowledge of math, Lawrie and Lin were able to make new discoveries that wouldn’t have been possible if approaches from the two fields were used in isolation. “What we found seems to suggest that theories in five dimensions come from theories in six dimensions,” explains Lin. “That is something that mathematicians, if they didn’t know about string theory or physics, would not think about.”

Lawrie adds that being able to work directly with mathematicians is also helpful in their field since understanding new math research can be a challenge, even for theoretical physics researchers. “As physicists, we can have a long discussion where we use a lot of intuition, but if you talk to a mathematician they will say, ‘Wait, precisely what do you mean by that?’ and then you have to pull out your important assumptions,” says Lawrie. “It’s also good for clarifying our own thought process.”

Rodrigo Barbosa, GR’19, also knows what it’s like to work across fields, in his case coming from math to physics. While studying a seven-dimensional manifold as part of his Ph.D. program, Barbosa connected at a conference with Lawrie over their shared research interests. The two researchers were then able to combine their experiences through a successful interdisciplinary collaboration. The work was motivated by Barbosa’s Ph.D. research in math that included both junior and senior faculty, as well as postdocs and graduate students, from physics.

While Barbosa says that the work was challenging, especially being the only mathematician in the group, he also found it rewarding. He enjoyed being able to provide mathematical explanations for certain difficult concepts and relished the rare opportunity to work so closely with researchers outside of his field while still in graduate school. “I’m very grateful that I did my Ph.D. at Penn because it’s really one of a handful of places where this could have happened,” he says.

Jonathan Heckman (L), Assistant Professor of Physics and Astronomy, joined the physics faculty in 2017 and is already active in a number of collaborations with the math department.

Faculty in both departments see the next generation of students and postdocs as “ambidextrous,” having fundamental skills, knowledge, and intuition from both math and physics. “Young people are extremely sophisticated and open-minded,” says Pantev. “In the old days, it was very hard to get into physics-related research if you were a mathematician because the thinking is completely different. Now, young people are equally versed in both modes of thinking, so it’s easy for them to make progress.”

Heckman is also a member of this new ambidextrous generation of researchers, and in his two years at Penn he has co-authored several papers and started new projects with mathematicians. He says that researchers who want to be successful in the future need to be able to balance the needs of both fields. “Some students act more like mathematicians, and I have to guide them to act more like physicists, and others have more physical intuition but they have to pick up the math,” he says.

It’s a balance that requires a blend of flexibility and precision, and is one that will be a continuing challenge as topics become increasingly complex and new observations are made from physics experiments. “Mathematicians want to make everything well-defined and rigorous. From a physics perspective, sometimes you want to get an answer that doesn’t need to be well-defined, so you need to make a compromise,” says Lin.

This compromise is something that’s attracted Barbosa to working more with physicists, adding that the two fields are complementary. “Problems have become so difficult that you need input from all possible directions. Physics works by finding examples and describing solutions, while in math you try to see how general these equations are and how things fit together,” Barbosa says. He also enjoys that physics provides him with a way to make progress on answering questions more quickly than in pure math, where problems can take years to solve.

## Feynman-“what differs physics from mathematics”

Why is math important in physics?

Math plays a critical role in scientific calculation in the field of physics. Physics carries significant crossover with several mathematical fields, including algebra for basic physics and calculus for advanced physics.

What is the relationship between mathematics and physics?

Generally considered a relationship of great intimacy, mathematics has been described as “an essential tool for physics” and physics has been described as “a rich source of inspiration and insight in mathematics”.

What fields of mathematics do physics professionals use?

The most common fields of mathematics that physics professionals use include: Algebra is a foundational study for most advanced forms of mathematics. It teaches students how to solve problems that include variables, allowing a user to find the value of one variable given conditional values for other variables in an equation.

What role does mathematisation play in physics education?

Accordingly, physical–mathematical modelling is the core of physics and its methodology. Therefore, mathematics and mathematisation in connection with physics play a significant role in physics education on all school levels, at university and also in the in-service education of teachers.