A short essay discussing the role of mathematics in physics.

Role of Mathematics In Physics 

Mathematics is the logical study of patterns.  It typically consists of two things: a set of things, and a set of operations on the things.  For example, think of arithmetic.  This consists of a set of numbers and a set of operations that do things to numbers i.e. additions, subtraction multiplication and division.  The things can be anything.  Irrational numbers, prime numbers, two dimensional shapes, n-dimensional shapes, tensors, groups, knots, diseases, economies, galaxies … anything.

Physics is the logical systematic study of patterns in Nature.  The ‘things’ are physical entities, ranging in size from subatomic particles to clusters of enormous galaxies.  

It follows that not all of the patterns discovered or invented in mathematics are physics.  Even if the objects being studied are real, not all the solutions to physical equations are real.  For example, consider a disc with area equal to Pi square meters.  Its radius can be calculated as √(π/π) = 1 meter.  But there is another answer as well.  Minus one.  This does not mean that there is an alternate universe in which discs of matter have negative radii.

The more complex the equations the more likely it is that there will be a multitude of solutions.  Einstein’s equations connecting the stress-energy tensor to curved four-dimensional space-time consists of ten differential equations.  These are hard to solve and they can have multiple solutions in different circumstances.  Some of the solutions exist in nature, e.g. black holes, and some probably do not, e.g. worm holes into alternate universes.

Having said that there are examples where the patterns in the mathematical models have guided the discovery of new physics.  The standard model of particle physics is a good example.  Symmetries in the group theory used to find patterns amongst known particles formed a kind of jigsaw puzzle with some pieces missing.  One by one the remaining puzzle pieces were discovered, mainly in the form of short lived fragments from high energy collisions.

It is sometimes said that mathematics is the language of physics.  This is an unfortunate phrase because it is too limiting.  A more accurate sentence would be “some parts of mathematics are incredibly useful for defining, exploring, understanding, explaining and discussing some parts of physics.”

Furthermore it can be argued that there are three languages of physics.  Mathematics is just one of the three languages.  The second is normal language, such as English or Chinese.  If the contents of a scientific paper cannot be explained using plain language there is a risk that the contents are mathematics but not physics.  It does not mean that the work is not impressive, and it may be satisfying to many people intellectually.  And it may find application in the real world at some time in the future.

As to the third language, Nature is speaking to us all the time, even if we cannot understand what is being said.  “An experiment is a question asked of Nature.  A measurement is part of Nature’s answer” (Max Planck).  Measurements and observations are words from Nature to us, and it is up to us to put the words together and understand their meaning and implications. 

Even if the mathematics is impressive, and the storyline plausible, the third language is still critical.  For if the contents of a scientific paper or a theory more generally cannot be tested experimentally or make predictions which can be tested experimentally, then the work has not yet met the three criteria for a good scientific theory.  It may have potential, but it has not yet arrived.  And if decades of efforts have passed without progress or promise of progress on this, it may be worth diverting scarce resources to other lines of enquiry.

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Van de Vusse, Sjoerd B.A., 2024,  Some ideas and experiments for issues affecting modern physics,   https://hereticalphysics.com.au
Author contact:  SBAvan@utas.edu.au Author’s location:  Hobart, Australia