Mirror Mirror on the Wall, Why Does Math Work at All?

Why does mathematics work?

Eugene Wigner, in his much celebrated paper, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", asked this very question, re-igniting a catharsis among the scientific community, which has lasted until the present hour.

Why does math work? Why is it that a person, sitting in his pyjamas under a dim lamp can conjure up complex theories related to the smallest of particles and to the heaviest of heavenly objects? Why is it that the mathematical patterns found in cartesian spaces are replicated in the chaotic and random natural world? Why are mathematical truths absolute and timeless to the effect that they possess the quality of everlasting applicability and utility in all of the human eras? These extremely dense questions bordering on philosophical exercises, have no clear answers.

First, an ode to the sheer utility of mathematics for the human experiment touched upon by Wigner himself. The famous Erwin Schrodinger considered it a miracle that in spite of the baffling complexity of the world, certain regularities of the real world could be discovered. One such regularity, discovered by Galileo, is that two rocks, dropped at the same time from the same height, reach the ground at the same time. Commenting on the same, Wigner said,

"The first reason that it is surprising is that it is true not only in Pisa, and in Galileo's time, it is true everywhere on the Earth, was always true, and will always be true. The second surprising feature is that the regularity which we are discussing is independent of so many conditions which could have an effect on it. It is valid no matter whether it rains or not, whether the experiment is carried out in a room or from the Leaning Tower, no matter whether the person who drops the rocks is a man or a woman. It is valid even if the two rocks are dropped, simultaneously and from the same height, by two different people"

This very persistence of mathematical truths, expressing these regularities, is what makes our modern life, as we know it, possible. We can predict with surprising accuracy, how two objects will react when they come in contact, based on Newton's third law of motion: For every action, there is an equal and opposite reaction. We can predict how stars with masses incomprehensible to the human mind, will end their lives based on the Chandrasekhar's limit. We can calculate how close we can get to any object in space before we get sucked into it, by calculating its Schwarzschild radius. Lastly, at the expense of technical obscurity to drive home the point, we can predict with an almost annoying accuracy through the Dirac equation the magnetic strength and orientation of an electron (The Magnetic Moment) of an atom. Such is the beauty of this equation that, Richard Feynman said, "it is as if calculating the distance between New York and Los Angeles to an accuracy of less than the thickness of the human hair!"

Marvelled by the unreasonable effectiveness of mathematics in what it does and what it helps us do, it becomes even more surprising that this powerful tool of mathematics is actually a luxury we have. This line of thought, of asking why math works, naturally leads us to ask, what is the origin of mathematics as a language and a tool. A moment of thought and it becomes apparent that asking ‘where does math come from?’ is intertwined with the question ‘why does math work?’. Thus, it is by unraveling its origin, that we can hope to understand the unreasonable effectiveness of mathematics.

Sadly, yet unsurprisingly the origin of mathematics is as divisive a question as a question can be. There are some like Roger Penrose and Wigner himself, who believe math is discovered and has an existence outside the human mind and in the physical realm in the form of regularities that were mentioned above. While on the other hand there are some, who believe that math is an invention, a tool developed by the human mind to understand the coherence and the regularities of the physical world. An invented logical exercise, to create useful but ultimately artificial order in a chaotic world. 19th century German mathematician, Leopold Kroneker, would be a proponent of this idea. His ideas summed up by his famous statement,

"God created the natural numbers, all else is the work of man"

Taking both of these arguments to their conclusion one by one, two reasons become apparent as to why mathematics works. If math is indeed discovered, and has an existence outside of the human mind, we could conclude that there could not exist a world where math wouldn't work; as math itself is the source code of all natural occurrences. As Euclid would put it, "the natural world is a manifestation of mathematical formulae". While if we took the argument of math as an invention, one may conclude that the human brain wouldn't invent a framework for understanding the real world, if in fact it wasn't logical and was on the other hand, contradictory. A need for a coherent foolproof system to comprehend our chaotic world is what renders mathematics its unreasonable effectiveness. Whether this effectiveness is real in the absolute sense or just a result of the human brain's trickery, is a question we cannot answer as all of us are limited to the use of a human brain itself to understand this phenomenon of effectiveness of mathematics.

Furthermore, there are some experts who give midway answers to this question. Like Grant Sanderson from 3blue1brown, the highly popular Youtube channel exploring the marvels of mathematics. He said this on the lex fridman podcast, “I think there’s a cycle at play, where you discover things about the universe, that tell you what math would be useful and that math itself is invented”. He further adds, “But of all the possible maths that you could have invented, it’s discoveries about the world that tell you which one’s are”. What Mr. Sanderson is saying is that our observations of the physical universe leads us to genuine discoveries such as concepts of distance, trigonometric relations and axiomatic truths and, while exploring those concepts (possibly in abstraction) we invent the math that would be useful to do said explorations. These explorations then lead to further discoveries and the cycle persists.

In this framework, mathematics is invented and it works because the invention is underpinned by a genuine discovery of the physical world which has an absolute existence outside the human brain.

Hoping to have informed the reader that mathematics is a hodgepodge of ideas, perspectives and frameworks, we arrive at the latter end of the blog and my repository of ideas with regards to this topic. At this juncture I would urge the reader to pause and ponder; think about what makes mathematics work and why we are even asking this question. Any and all answers will be welcome in the comments section below.

To conclude it would be ideal to quote Wigner himself on this whole affair,

"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. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning".

Discovered or invented or anything in the middle, we need mathematics to retain its unreasonable effectiveness if we are to solve the critical questions of our time and progress as a species and as a better civilisation.