Mathematics: The Future Of Physics?

"Doing physics is like talking with the universe. Doing mathematics is like talking with the one who designed the universe."

Let us think about the progress in the field of physics in the early 20th century and the early 21st century. In 1900, physicists were scratching their heads over blackbody radiation and ether [1,2]. By 1925, we had discovered the general theory of relativity and much of quantum mechanics [3,4]. In comparison, there has been no significant progress in the fundamental sciences between 2000 and 2025. Of course, overall we have made remarkable progress; for instance, advances in Artificial Intelligence and the discovery of the Higgs boson [5,6]. However, we have been unable to gain any fundamentally new insight about the fundamental nature of reality.

Working on theoretical physics today is more likely to provide one with new insights in abstract mathematics rather than any insight about the physical world. It can no longer be said that theory and experiment go hand-in-hand, like it was during the early 20th century. This has led many prominent physicists to conclude that physics is currently “lost in math,” [7] and we should take physics back to its empirical roots. However, I believe it is not always a bad thing if exploring physics takes us to the land of mathematics, because mathematics is the most fundamental level of understanding that we know of. Mathematical insights might find their use in some physical theory sooner or later. Because mathematics has it all, and studying mathematics could be a “short-cut” to understanding the fundamental structure of the physical universe, without having to rely on any messy experimental methods or data.

It is surprising how many abstract mathematical concepts, years after they were developed, turned out to provide the perfect theoretical framework needed to describe various aspects of the physical world. Even though many of us believe mathematics is a human invention, it is almost as if the world was created using mathematics. Eugene Wigner called this the “unreasonable effectiveness of mathematics.” [8] When linear algebra and the concepts of vectors spaces, matrices etc. were developed so many years back, nobody at that time could think of any direct applications of these ideas in explaining the physical world. (Solving a set of linear equations for a real-life problem - like the ones we used to do back in high-school – is not really an application in understanding the physical world). But now we know that the entire formulation of quantum mechanics is based on a certain type of vector space, and eigenvalues and matrices offer us a well-defined mathematical way of predicting the possibilities and probabilities corresponding to measurements we make on a system in quantum mechanics [9]. Beyond that, quantum mechanics has led us to quantum field theory and the Standard Model of particle physics [10], where to explain interactions between particles, we make extensive use of the mathematics of symmetry groups, which was, again, developed many years back without any application in mind. Although Max Tegmark’s “mathematical universe” hypothesis [11,12] sounds crazy to many, all of this could be suggestive of the fact that probably everything in the physical world has a mathematical description. Tegmark takes the argument further: the universe is not just described by mathematics; the universe IS mathematics!

According to Platonism [13], mathematics is not invented by humans; we are only discovering the mathematical truths, but they have always been there and will always be there, regardless of whether the physical universe exists or not. The physical universe could be a manifestation of some subspace of the most general mathematical space possible, and everything in the physical universe probably has a unique counterpart in this mathematical subspace. Even better, this correspondence could be one-to-one, meaning to every object in the physical universe, there exists a unique corresponding object in the subspace. A mathematical space simply refers to a set of mathematical objects that follow certain mathematical rules among themselves [14]. These objects are not physical objects; they could be abstract mathematical structures which cannot be perceived by our senses. In other words, we can only see them in the equations we write. So, every physical phenomenon could have a mathematical description, and if that is true, then once we find the mathematics behind all physical phenomena, we probably will be able to better appreciate the fundamental structure of the universe and formulate a true “theory of everything.” [15]

Now, let us discuss a specific point where I think mathematics can directly help physics: explaining the emergence of spacetime [16,17]. Since general relativity and quantum field theory seem to work with fundamentally incompatible descriptions of spacetime [18], a lot of physicists speculate that spacetime is not fundamental, rather it is emergent from a deeper level of understanding. It is possible that this “deeper level” has no physical description at all, because it cannot contain within it the notions of “space” and “time” (since spacetime is emergent from this level). It is likely that underneath the spacetime description we are familiar with, there exists a deeper and entirely mathematical description of the universe, with no physical constraints. However, revealing this underlying mathematical structure, assuming it exists, would be no easy feat. We must keep in mind that it must not contain any concept of space or time within it; we must explain how the physical world around us emerges from this structure. In particular, I believe explaining the emergence of time would be challenging. Everything in the physical universe changes with time, but a mathematical space does not change in its entirety. In other words, its constituents and the relations or mathematical rules they follow don’t change. We need to explain how this “illusion” of change, and hence the illusion of time, emerges when we focus on a part of the entire mathematical structure, the part which is the physical world.

So when we move beyond spacetime and to the most fundamental level of understanding, we probably will not find “time” or “space” there. Whatever we will find there can probably be best described in the framework of information [19,20] rather than in the framework of physical concepts like distance and time. A big question here is whether it is possible for the human mind, with all its limitations, to ever achieve such a fundamental level of understanding. But the best we can do is see how far mathematics can take us.

Let us look at a second example of how mathematics might accelerate the progress in theoretical physics. As anyone with even the slightest idea of the incompatibility of general relativity and quantum field theory knows only too well, it is possible that we perceive the existence of some phenomenon in the physical world, or some aspect of a theory describing the physical world, which seems to violate some other theory describing some other aspects of the physical world. Now, I think it could be the case that this apparent “problem” would vanish if we shift to a more fundamental level of understanding, which can probably be achieved by mathematics. Mathematics is much more general than physics in the sense that it deals with the space of all objects and structures (which are following certain mathematical relations between them), and it does not exclude any possibility. Physics, on the other hand, is only concerned about those possibilities that occur in the physical world. This means by studying physics we can only see a part of the whole, and the “problems” that arise when studying just this part might be automatically resolved if we look at the whole (mathematics). A simple illustration could be the fact that open systems like living organisms decrease entropy locally, and if you do not take the surroundings into account (the whole) and focus just on the system (a part of the whole), it may seem that entropy is decreasing and the second law of thermodynamics is violated [21]. However, the overall entropy of the universe, as we know, always increases. Although it is not yet clear how to use this reasoning to solve the problems that arise when we attempt to reconcile gravity with quantum theory, the same approach (coupled with some good mathematics) might do the trick.

In conclusion, the key to unlocking the secrets of the universe might not lie in devising better experiments and formulating theories based on the results of these experiments. This approach has been remarkably successful in past, but I think the future of physics belongs to mathematics - the only place where truth and beauty mean the same!

References

[1] Three Failures of Classical Physics. Weber State University.

https://physics.weber.edu/carroll/honors/failures.htm

[2] L. Motta. Ether. Eric Weisstein's World of Science.

https://scienceworld.wolfram.com/physics/Ether.html

[3] J. Norton. Einstein's Pathway to General Relativity. University of Pittsburgh.

https://sites.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_pathway/index.html

[4] V. F. Weisskopf. Physics in the 20th Century. CERN.

https://cds.cern.ch/record/1728977/files/vol9-issue10-p295-e.pdf

[5] W. Henshall. 4 Charts That Show Why AI Progress Is Unlikely to Slow Down. Time

Magazine. https://time.com/6300942/ai-progress-charts/

[6] How did we discover the Higgs boson? CERN. https://home.cern/science/physics/higgs-boson/how

[7] S. Hossenfelder. Lost In Math. Lost in Math: How Beauty Leads Physics Astray. Basic

Books. https://g.co/kgs/8KYXcw9

[8] E. Wigner. The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

University of Edinburgh. https://www.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf

[9] D. Tong. The Formalism of Quantum Mechanics. University of Cambridge.

https://www.damtp.cam.ac.uk/user/tong/qm/qmhtml/S3.html

[10] W. Hollik. Quantum field theory and the Standard Model. https://cds.cern.ch/record/1281946/files/p1.pdf

[11] M. Tegmark. Our Mathematical Universe: My Quest for the Ultimate Nature of Reality.

Knopf. https://g.co/kgs/Pkr2scR

[12] M. Tegmark. The Mathematical Universe. Foundations of Physics.

https://arxiv.org/pdf/0704.0646

[13] O. Linnebo. Platonism in the Philosophy of Mathematics. Stanford Encyclopedia of

Philosophy. https://plato.stanford.edu/entries/platonism-mathematics/

[14] M. Sklar. What is a mathematical space? The Local Maximum.

https://www.localmaxradio.com/questions/what-is-a-mathematical-space

[15] J. Barrow. Theories of Everything: The Quest for Ultimate Explanation. Oxford

University Press. https://g.co/kgs/rR4rBgi

[16] M. Belan, C. Wood. The Thought Experiments That Fray the Fabric of Space-Time.

Quanta Magazine. https://www.quantamagazine.org/the-thought-experiments-that-fray-the-fabric-of-space-time-20240925/

[17] A. Gefter. The Logic That Must Lie Behind a New Physics. Quanta Magazine.

https://www.quantamagazine.org/the-logic-that-must-lie-behind-a-new-physics-20240925/

[18] C. Powell. Relativity versus quantum mechanics: the battle for the universe. The

Guardian. https://www.theguardian.com/news/2015/nov/04/relativity-quantum-mechanics-universe-physicists

[19] R. Kuhn. Forget Space-Time: Information May Create the Cosmos. Space.com.

https://www.space.com/29477-did-information-create-the-cosmos.html

[20] J. Wheeler. Information, Physics, Quantum: The Search for Links.

https://philpapers.org/archive/WHEIPQ.pdf

[21] R. Oerter. Does Life On Earth Violate the Second Law of Thermodynamics? Geroge

Mason University. http://physics.gmu.edu/~roerter/EvolutionEntropy.htm