The Relationship Between Physics and Mathematics
“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.” – Eugene Wigner
One of the most puzzling mysteries of modern day Physics is why mathematics is such a useful tool for uncovering the workings of the universe. Although mathematics has always been the only acceptable language in which to phrase serious physics, letting mathematics guide the direction of physics is a late 20th/21st century revolution in scientific thinking.
The works of Newton, Maxwell and Einstein were all written in the language of mathematics, but their work was inspired by the world they saw around them, not the mathematics they were writing down.
But in modern theoretical physics, the search for new laws of nature is guided by the analysis of symmetries in the mathematical structures of our current physics. Instead of finding particles and trying to fit them into our theories, our mathematics is predicting the existence of fundamental particles and telling us what to search for in the world.
Mathematical symmetries have seemingly overtaken nature itself as being the guide to the development of physics. A large part of what the LHC at CERN does now is searching for the particles predicted by mathematical models.
The failure of the LHC to find any new particles beyond the current Standard Model however has cast doubt over our enthusiasm to entrust the progress of physics to mathematics.
The Unreasonable Effectiveness of Mathematics
As our physical theories have gotten more and more advanced, physicists have had to learn more and more mathematics in order to formulate the laws of their field. Maxwell’s equations of electromagnetism are formulated in the language of vector calculus. Einstein’s Special Relativity is formulated in the language of 4-vectors. Quantum Mechanics is formulated in the language of an abstract vector space known as a Hilbert Space. General Relativity is formulated in the language of Tensor Calculus.
Looking further afield into esoteric areas of 21st century physics, a physicist needs to know about ideas from a plethora of mathematical fields – group theory, algebraic topology, algebraic geometry, representation theory, differential geometry, … the list goes on.
It’s even got to the point now where certain areas of physics research actually help to advance areas of mathematical research! Over time, physics and mathematics have drawn closer and closer to each other.
Why is it that mathematics is so crucial to physics? Without a doubt, mathematics has proven itself to be the only language in which we can properly express our physics, but what does this mean about the physical status of mathematics?
Does mathematics form the very logical structure of the universe? Or is mathematics an invention of ours, that has applications to physics, but doesn’t represent anything real itself?
The Radical Mathematical View
Certain physicists, the most well known being Max Tegmark, support the radical hypothesis that the universe is just a mathematical structure. Furthermore, this view claims that every mathematical structure exists in a multiverse of mathematical structures.
This would explain why mathematics is so useful in our physical theories, but also why there seems to be so much more mathematics than we can apply to our universe. Some of the mathematics we discover is simply not realised in this universe, but applies in others.
Our mathematics then is a representation of fundamental logical structures that have an independent existence from our minds.
The Selective Realism of Mathematics
On the other side of the spectrum, physicists such as Lee Smolin have claimed that mathematics doesn’t really represent anything fundamental in the universe at all. Mathematics is just a convenient way of describing causal connections in the present day universe.
In Smolin’s philosophy, there isn’t any such thing as a law of nature. The fundamental entity in the universe is time, and everything both exists in time and is affected by it. This includes the laws of physics.
This entails that as time marches on, the causal relations between objects in the universe change, and thus, so do the laws of nature that we use to codify these relations. Mathematics is a useful way of writing these laws of nature because mathematics is an abstraction – a timeless one. In the present epoch of our universe the laws of physics do not change noticeably, and so it is most convenient to use a timeless language such as that of mathematics to describe the laws.
This worldview has a good go at explaining why mathematics is so useful to us in describing the world, but it doesn’t seem to address the puzzling issue of why there is so much more mathematics than we can apply to the universe, and why it takes on such a life of its own seemingly distinct from the minds of humans.
It is the prevailing view in theoretical physics that the mathematical nature of the universe is the far more appealing and sensible philosophy. The recent shortcomings however of letting mathematics guide our physics has made some physicists wary that we’re blindly going down the alley of mathematics without properly considering why we’re doing it.
The Platonic Realm 
Smolin is a bit off. Everything that is, exists in the context of time and in fact all relationships are defined by time.
The measurement of time is however unnatural and time is divided into units that are used to symbolize time. Symbology is not the thing just like dividing space into inches. There are no Cartesian increments in space.
Mathematics is one of the two academically acceptable means of making an argument; 1. Language 2. Mathematics
Both are defined by linear rules such as grammar and algebra. And so are far more restricted than the phenomena that they are attempting to describe. A tree is not dependent on inches and the sky is not reliant on sentences. The real world is ineffable by these systems of symbology and are not constrained by them.
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In my viewpoint, yes, everything is mathematics, in so much as mathematics is the art of logic and reasoning. Without the ability to provide arguments and determine their validity there would be no way of forming thoughts on, models of, or descriptions of any phenomena. However, mathematics explains why things occur with given assumptions. It is therefore the responsibility of the applied researcher to show that the situation they are looking at in fact satisfies the assumptions they are making and to determine if the assumptions will hold in general.
In response to the previous reply, the limitations of notation and language to naturally describe things does not imply a limitation of mathematics to describe situations. Rather, it describes the current limitations of the human race in being able to come up with such a way to adequately describe what it observes. As humanity wishes to extend its own understanding of the universe, it must provide new ways to describe things.
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Wow ! This post is just what I needed today. I really appreciate the way that the inseparable connection between Mathematics and Physics(and indeed all other fields of science) has been described here. Thanks!
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Well thank you! I’m glad you liked it so much, I hope you enjoy the rest of my writing too!
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I enjoyed your article. I’m terrible at mathematics, but I can’t deny its importance to the human race. Good luck with future articles.
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