“The Unreasonable Effectiveness of Mathematics in the Natural Sciences” by Eugene Wigner

It is only relatively recently in human history that the deep connection between math and physics was realized. Early civilizations knew of some simple connections regarding a pretty intuitive level of mathematics. Counting has an obvious connection to our world, as does much of geometry. And yet, that isn’t really what science is any more. It is no longer intuitive kinds of mathematics. Scientific theories today use extremely abstract ideas – including manifolds and tensors, just to name a few – to explain their theories about how our physical world works. Why would something as abstract as a tensor product have anything at all to do with our world?

This is the topic of physicist Eugene Wigner’s famous 1960 essay, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” published in Communications on Pure and Applied Mathematics – an academic journal that primarily publishes papers relating to mathematical science, applied mathematics, and physics. Wigner, who won the Nobel Prize in physics in 1963 for some of his groundbreaking work in elementary particle physics, had a lot to say about the relationship between mathematics and physics. Despite being a convinced atheist himself, he concludes this essay with the following remarkable and surprising line:

“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.” [1]

We ought to be taken aback by an atheistic physicist calling something deep at the heart of modern physics itself a miracle that we don’t understand or deserve. What is going on here? What would lead a Nobel Prize winning physicist to make such a strange remark? For the remainder of this article, I’d like to give a brief summary of Wigner’s deeply interesting article and to expound upon some of his observations.

  • As a note, the square brackets [ ] with numbers inside of them are used for citations, and parentheses ( ) with numbers inside within my summary of Wigner’s article at points that I later want to reference and make commentary on.

Summary of Wigner’s Paper

Introduction to the Idea

“There is a story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. “How can you know that?” was his query. “And what is this symbol here?” “Oh,” said the statistician, “this is pi.” “What is that?” “The ratio of the circumference of the circle to its diameter.” “Well, now you are pushing your joke too far,” said the classmate, “surely the population has nothing to do with the circumference of the circle.”

This is how Wigner’s article opens. Of course, the approach of the classmate it a bit naïve. Yet, there is something eerie about it. Why exactly should the circumference of a circle have anything to do with populations? The classmate goes a bit too far in calling the claim a joke, after all sometimes bizarre things do happen, yet surely there is something unusual here. Wigner points out two things [1]:

  • “The first point is that mathematical concepts turn up in entirely unexpected connections. Moreover, they often permit an unexpectedly close and accurate description of the phenomena in these connections.”
  • “Secondly, just because of this circumstance, and because we do not understand the reasons of their usefulness, we cannot know whether a theory formulated in terms of mathematical concepts is uniquely appropriate.”

Again in Wigner’s words, “We are in a position similar to that of a man who was provided with a bunch of keys and who, having to open several doors in succession, always hit on the right key on the first or second trial. He became skeptical concerning the uniqueness of the coordination between keys and doors.” In other words, is it really true that mathematics is a good way to talk about physics, or are we just getting lucky?

What is Mathematics?

What is it that mathematicians do? Well, to oversimplify, mathematics is very much the invention of concepts that we can exercise our logical skills on in a particularly ‘mathematical’ sort of way (1). Why then would mathematics have anything to do with the real world? Well, from a mathematical perspective, you could argue that the axioms (which are the most basic definitions and assumptions of the mathematician) are themselves grounded in things that we have real-world experience with – like the basics of counting and addition – and so we ought not be surprised that mathematics and the real world line up. This is true, for elementary mathematics. So fair enough, we ought not be surprised that elementary mathematics has anything at all to do with the real world.

But some mathematics isn’t like that. Actually, some isn’t the right word… more accurately, almost all mathematics is not like that. Very, very little interesting mathematics can be done without adding in either additional definitions that are not intuitively grounded in the real world, or additional axioms that are not intuitively grounded in the real world (2). Surely this kind of mathematics can’t have anything to do with the real world?

We will get to that later. For now, the point is that mathematicians, in Wigner’s own words, “ruthlessly exploit the domain of the permissible” with abstract concepts like Borel sets, complex numbers, algebras, and linear operators (particularly on infinite-dimensional spaces). Mathematicians are not even trying to remain within concepts that we see in the physical world – they are basically trying to come up with concepts entirely foreign to the physical world. On this recklessness, Wigner adds the following [1]:

“That his recklessness does not lead him into a morass of contradictions is a mircle in itself: certainly it is hard to believe that our reasoning power was brought, by Darwin’s process of natural selection, to the perfection which it seems to possess.” (3)

So, not only is this process of mathematics entirely unrelated to the physical world, it is not at all clear on a totally physical view of the world how we are even able to do this kind of reasoning without falling into “a morass of contradictions.” But what exactly are we talking about here?

The complex numbers are a helpful example. As a reminder, the foundation of all complex numbers is defined by i = \sqrt{-1}. We must immediately remember that even attempting to define such a number radically defies everything we know about multiplication. This number is supposed to satisfy the equation i^2 = -1, but for real numbers – the only kind of numbers that physical reality suggests to us – this is utterly impossible. If you ask the mathematician, they will tell you that the study of complex numbers was not at all motivated by physics when they were first studied – rather, complex numbers arose in the theory of equations, in efforts of describing solutions to cubic equations (and eventually, equations of higher degree).

What is Physics?

We’ve now discussed what mathematics is about. But what is physics about? In order to understand physics, it is critical to look at the concept of a law of nature.

Wigner’s first observation is summarized by a quote from Schrodinger – that it is “a miracle that in spite of the baffling complexity of the world, certain regularities in the events could be discovered” (emphasis mine). This is quite right. None of us can just look around at the world and predict what is going to happen. For instance, try predicting the weather tomorrow without looking at a weather app. You just can’t do this sort of thing. The world seems in our daily lives to be incredibly unpredictable. And yet, the amazing observation is, we actually can predict some things. Finding this predictability in a sea of unpredictability is surprising.

But what are these regularities? A prototypical example is Galileo’s observation that if you drop two rocks from a tower at the same time and same height, they will always hit the ground at the same time. We find two sorts of regularity in the laws – their constancy across various circumstances, and that only an extremely limited set of factors have any influence at all on the fall. Given the same rock and the same height, it will fall in the same amount of time every time. If you hadn’t grown up being taught this, it wouldn’t be at all clear that this should be the case. But even more surprising is the shockingly few factors that play a role. Things like size, shape, color, location, wind, the time of day and even the weight of the rock have nothing whatsoever to do with the fall. It is not at all obvious why some factors determine how the rock falls and others have nothing at all to do with the fall.

Without both of the criteria just described, there could be no physics – or at least no physics that human beings could actually do or understand. Without the regularity, you cannot predict anything with any certainty. Without the extremely limited number of conditions that are relevant to a given event, problems would be too complicated for a human being to even possibly understand. It is not at all clear that ‘laws of nature’ like these should exist [and Schrodinger himself says that this miracle may be beyond human understanding] [2]. It is not at all obvious why there should be laws of nature that enable us to predict future events given present states of affairs, and why human beings should be able to understand them at all compounds the mystery.

These ‘laws of nature’ are conditional statements and they only relate to a very small slice of the world in which we live. And these laws give surprising information. Why ought we be able to always know the acceleration of any falling body anywhere on earth while being completely ignorant of their position and velocity (4)? And, with respect to actually predicting future events, the laws of nature only give us probabilities – we cannot predict future events with perfect precision even given the laws of nature – although the laws get us very close.

How Math and Physics Interact

When we do physics, we use mathematics to make the calculations. We normally don’t think much of it. But, there is an underlying background assumption here. This whole approach assumes that mathematics is the right language for expressing the laws of physics, as the famous quote of Galileo goes. There are some elementary aspects of the physical world – like counting and a large part of geometry – that were obviously developed for the purpose of looking at the physical world, and so we ought not be too surprised when we find that addition, multiplication, and elementary geometry help us do physics.

But most of physics isn’t like this. Take for example quantum mechanics. There, the two basic concepts (states and observables) are vectors in a Hilbert space and self-adjoint operators on those vectors, respectively. Possible observational values are the characteristic values of the operators. Unless you have some sort of degree in mathematics or physics, these terms from the advanced theory of linear operators will probably be meaningless to you – as well the term linear operator. Which is rather the point we want to make. Even once you learn what these things mean – it is not at all clear that such concepts should make an appearance in physics. You can make a similarly daunting list with general relativity and particle physics.

This is unlike the previous situation, because now the concepts are not ‘so obvious that they were sure to arise in a physical theory.’ Take as an example the complex numbers, which we mentioned earlier. One would at first guess that \sqrt{-1} can’t possibly have anything to do with the physical world – after all, it defies the laws of multiplication that are familiar to us. And yet, they are absolutely crucial in essentially all of modern physics. Complex numbers are indispensable. The application of so-called analytic functions to quantum mechanics is similar – there is no reason whatsoever to expect the mathematical theory of analytic functions to play the huge role in physics that it in fact does play.

The so-called “empirical law of epistemology” – the assumption that the laws of nature shall play out as mathematical constructions that human beings are capable of understanding – is “an article of faith of the theoretical physicist.”

Why the Miracle?

In light of all this, Wigner says the following: “It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that a human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of the existence of laws of nature and of the human mind’s capacity to divine them.” The closest explanation we have is Einstein’s statement that we will only ever accept beautiful physical theories – but that is a statement of what kinds of theories we want to look at, not about what theories might be true about the world.

One possible explanation would be to accuse the physicists of irresponsibly assuming there is a deep mathematical connection going on every time he sees something he recognizes as looking like some other thing he learned in math class. Perhaps this is so. But then how in the world does such crude experience lead to extremely precise predictions? And it is even more surprising, because the predictions made upon a certain evidence base are able to make discoveries that have no relation to the dataset used to construct the theory. Take for example Newton’s gravitational laws. There was very little empirical evidence at the time to back up his theory, and the fact that the second derivative played such a massively important role was repugnant to some, or perhaps many, at his time. Why the second derivative and not the first derivative, or just the underlying function itself? And yet this crudely developed theory has now been shown to have an accuracy of less than one ten thousandth of one percent, and Newton’s law of gravity was used to discover a planet never before seen.

Elementary quantum mechanics has similar features. After it was initially developed, some physicists noticed a parallel between the mathematics of matrix algebra and of this rudimentary quantum mechanics. So, at the time there was no evidence whatsoever that their matrix mechanics would actually work in real-world scenarios. The move to use matrices was purely based on a cursory mathematical similarity with little to no empirical evidence, and yet the matrix mechanics worked, and solved extremely difficult problems about hydrogen and helium atoms to a degree of accuracy of less than one part in one million. In Wigner’s words: “Surely in this case we “got something out” of the equations that we did not put in.

We reemphasize our point with the example of Newtonian gravity again. Why should we expect that, even though there is no simple equation that tells us the velocity location of an object at any moment in time, there is a simple equation that tells us the acceleration of any object whatsoever at any moment. Why should there be an equation like that? It is patently absurd to think that this is self-evidently true (i.e. obviously true). Some may argue that things will be much clearer once we have a unified view of all of physics – but it is not even clear that physics can be unified. The two most successful scientific theories ever – general relativity and quantum mechanics – make logically contradictory assumptions about the way the universe is structured (5). One operates in an infinite dimensional Hilbert space, and the other in a four-dimensional Riemann space. These are very, very different sorts of objects. This ought to at least give us serious doubts about whether unifying physics is even possible. And even if we did unify physics, it is very likely that the non-obviousness of these laws of nature will be equally present in a unified theory, if not more so. Even though essentially all physicists believe a unifying theory is possible, this is by no stretch a foregone conclusion.

There is yet more surprise here. Because of the contradictory nature of general relativity and quantum mechanics, we know that at least one of them has a foundational assumption that is false. But then why does a theory with such a deep flaw at its very center make such incredibly accurate predictions about the world? This is not the dream scenario of the “article of faith” mentioned before – we are now entering the nightmare of a totally false theory producing amazingly accurate results. And this nightmares, for all we know, may actually be real.

In conclusion, Wigner writes the following:

“Let me end on a more cheerful note. 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.”

Additional Discussion

I’ll start of by saying that I find Wigner’s actual article quite readable and not all that long. I’d highly recommend it to anyone interested in the discussion here. I have provided a citation at the end.

As a mathematician myself (or a mathematician-in-training, depending on your definitions), I agree with what Wigner says about mathematics. It is very similar to the opinion defended by most – or more likely all – professional mathematicians that I know, summarized best in G.H. Hardy’s A Mathematician’s Apology. The point here is that mathematics – as practiced by the mathematician – is comparable to poetry, painting, or theatre. It is an art form, seeking to shed light on beautiful patterns and to weave logical ideas into interesting stories. Although we use a different medium from the painter and poet, mathematics is a form of art. I think that probably the closest direct comparisons would be poetry and music.

The point Wigner makes that I agree so firmly with is that, by and large, all but the most absolutely basic of mathematical concepts are motivated by aesthetic reasons rather than practical ones. This is why Wigner sees such surprise in advanced mathematics applying to physics – it would be like discovering an actual poem found uniformly across nature. It is not obvious why such abstract mathematics should be relevant in the world of physics.

I am no physicist, but it seems that again Wigner is perfectly correct in his analysis of physics and the relationship between mathematics and physics. We find that framing laws of nature as mathematical formalisms has been extremely useful and accurate, and yet there really is not any a priori reason to think that laws of nature ought to be mathematical. For if such a thing were a priori obvious, why didn’t more ancient cultures have this idea? Ancient Greeks had engineering and some pretty advanced mathematics, but never applied the same degree of mathematical precision to study nature itself. It took monumental figures like Galileo and Newton to get the “physics is written in the language of mathematics” theme to actually catch on.

The mystery here is quite real. On the one hand, the study of, say, Riemann surfaces or Hilbert spaces, was entirely motivated by a desire to study a rich mathematical beauty and had nothing to do with physics. And yet, these beauty-motivated studies pop up in physics all over the place. And it isn’t just a few concepts. You find the same pattern all over physics. What is going on here? Is this actually true? And if so, how do we explain this incredibly non-obvious fact about the world?

I know of only one way. If the universe were created by an intelligent being capable of recognizing mathematical beauty and structuring the physical world according to beautiful ideas, then this apparently miraculous fact would make a lot of sense. This same hypothesis can also explain why human beings are capable of observing this structure – because the intelligent being that created us purposefully gave humanity the ability to see both beauty and logical structure in the world around us. This sounds quite a lot like theism – the view that God exists (polytheism would be shaved away by Ockham’s razor unless some evidence were be provided that more than one intelligent being is required).

Wigner himself was an atheist, but he never provided any explanation for how or why the world has this mathematical structure. So you can maintain that there is no God and yet recognize the facts that Wigner presents. But I’m not aware of any account other than theism that makes sense of why the world has this surprising feature.

My purpose here is not to lay out the full argument for the existence of God, so I won’t do so. But I have provided resource links to helpful videos in which the points Wigner brings up are synthesized into a more rigorous presentation of the idea I have just put forward. I also have in the video resources a rebuttal by a popular online atheist and a response to that rebuttal in order to give both perspectives.

I hope anyone who has read this far finds Wigner’s observations in this famous essay as interesting and thought-provoking as I have.

Commentary

(1) This includes operations with ideas like quantity, size, shape, and most importantly, studying pattern, symmetry, and structure.

(2) The axiom of choice would be a good example of an axiom that cannot be grounded in our physical experience of the world.

(3) The problem of arriving at a system capable of abstract logic via natural selection aimed only at survival is quite a serious problem apart from the difficulties it brings here. See Christian philosopher Alvin Plantinga’s book Where the Conflict Really Lies or atheist philosopher Thomas Nagel’s book Mind and Cosmos: Why the Materialist Neo-Darwinian Conception of Nature is Almost Certainly False for in-depth discussion of this conflict.

(4) To know the position and velocity of a body in the future, you must first know its position and velocity in the present. This is not so with acceleration. Acceleration (due to gravity) is constant, and so you do not need present acceleration to know future acceleration. But, since acceleration is defined as the first derivative of velocity and the second derivative of position, it is surprising that such closely related concepts do not have similar levels of predictability.

(5) The way general relativity and quantum mechanics define space is contradictory. In quantum mechanics, space is quantized (built out of discrete, indivisible bits) and in general relativity, space is a manifold (completely smooth, not made out of discrete, indivisible bits). Both of these conditions cannot be simultaneously true.

Video Resources

Citations

[1] Wigner, E.P. (1960). “The unreasonable effectiveness of mathematics in the natural sciences. Richard Courant lecture in mathematical sciences delivered at New York University, May 11, 1959”. Communications on Pure and Applied Mathematics. 13: 1-14. (Link)

[2] Schrodinger, E. What is Life? (Cambridge: Cambridge University Press, 1945), p. 31.

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