Wigner’s Puzzle – The Incredible Effectiveness of Mathematics

Why can we describe the world so well with mathematics of all things? In 1960, Nobel laureate in physics EP Wigner published a philosophical essay entitled The unreasonable effectiveness of mathematics in the natural sciences , which addresses the question posed above.Shortly after its publication, the catchphrase Wigners Puzzle was coined to indicate the amazing power of mathematics.

1.      Endless amazement at the world

2.     Applied Mathematics – Wigner’s Puzzle

3.     The magic of math

4.     Some attempts to explain Hamming

5.     We recognize what we are looking for

6.     We choose the math that suits us

7.     All science gives us limited answers

8.     Conclusion – Hamming’s doubts about his attempts

1.      What is the historical development of the explanation of the cosmos?

2.     What exactly does Wigner’s Puzzle mean?

3.     What amazing phenomenon does Wigner describe? What’s amazing about that?

4.     What are Hamming’s attempts to explain the wignant puzzle?

5.     What conclusion does Hamming have to end with?

Endless amazement at the world

As far as we can understand, man has always been amazed at himself, the world and life. The history of mankind knows innumerable creation myths, narratives and stories of the past which try to give us information about how God or several deities created the universe and man.

However, these theological attempts at explanation are unsatisfactory for one simple reason. They all share the characteristic that it doesn’t make any sense to ask why the world is happening, because dogmatically we always arrive at the same point: God created the world this way because he created it this way – after all, it is God.

It is clear that, contrary to the theological explanation, people finally began to investigate the question of the how and why of the universe in a different way. With the philosophy of the pre-Socratics, an alternative explanation of the world came about for the first time, and a few centuries later, thanks to science, we have reached that differentiated point of view that is common today.

Of course, even modern science cannot really explain the why behind the world – nobody knows why we exist, why there is anything at all, and so on. But science can give us so many details about the how of the world that we now get the impression that we have at least a rough feeling for the why of the world. [1]

The next question, of course, is how science knows how to do this. How does science manage to give us an understanding of the how and maybe even the why of things? This is exactly where the wignerious puzzle already mentioned comes into play.

Applied Mathematics – Wigner’s Puzzle

The mathematician and Turing Prize winner RW Hamming gave a lecture in 1979 in the form of a replica of Wigner’s essay. Based on this, he chose the title The unreasonable effectiveness of mathematics , which is more general , to which he tried to give an answer.

We note that our main tool for performing long chains of deduction within strict logic, which science invariably requires, is without a doubt mathematics. Indeed, mathematics could be defined as the mental tool that serves this very purpose.

Wigner pointed precisely to this central position of mathematics in all natural sciences. ›Wigner’s Puzzle‹ therefore deals with the question of the undeniable fact that leads to the following interesting philosophical problem: Why should natural scientists find that they cannot even express their own theories without specifying abstract mathematical theories?

The magic of math

In other words: why can mathematical language, which is developed, so to speak, by aesthetic standards and completely independent of empirical facts, make such incredible references and predictions about the physical world possible?

How does the mathematician – closer to the artist than the explorer – by turning away from nature, arrive at its most appropriate descriptions? [2]

This amazing phenomenon is by no means reserved for physics professors or scientists at the LHC, but we already encounter it in fairly simple everyday situations. An example of what could happen to any curious child is as follows:

Even if the mathematics’ applicability to the physical situation is straightforward, something like Wigner’s puzzle could emerge. A child playing with solid square cards notices that sometimes a larger square (5 × 5) can be rearranged into two smaller square (3 × 3 and 4 × 4).

This is a simple physical characteristic of the tiles. The purely mathematical fact that 3² + 4² = 5² together with the facts about the plane geometry are relevant to provide an explanation for this. The child might wonder whether something similar happens with solid cubes: Can a large cube made up of smaller cubes be rearranged into two smaller cubes? [3]

Some attempts to explain Hamming

Before we can consider some of Hamming’s attempts to explain the incredible effectiveness of mathematics, we should know his own “definition” of mathematics. To this end, he names four central facets of the mathematical discipline, which he summarizes as follows [cf. 1, p.83]:

  1. The ability to perform long chains of deduction
  2. The concept of geometry that evolved into topology
  3. The concept of number that gave rise to modern algebra
  4. The sense of aesthetics that promoted the elegance of mathematics

Hamming’s most important notion, however, remains that mathematics is a man-made tool, at most an apt representation of his thought process. This view is based on a formalism that regards mathematics as the pointless shifting around of signs according to set rules. It is therefore also clear why Hamming cannot see anything absolute in mathematics:

Mathematics was created solely by humans and is subject to constant change and development. [see. 1, p.86]

Against this background, Hamming is now trying to provide an explanation for Wigner’s puzzle.

  1. We recognize what we are looking for

The first explanation Hamming proposed for the incredible effectiveness of mathematics resembles a constructivist perspective. Normally, according to Hamming, in the natural sciences we think of the sequence: first the experiment, then the mathematical description of the observed phenomena.

However, this is fundamentally wrong. Countless times, purely mathematical considerations have led to predictions about the world that often only came true decades later. An example of this would be the prediction of neutrinos by Wolfgang Pauli in 1930 (the existence of neutrinos was confirmed experimentally in 1956). [4]

According to Hamming’s thesis, Pythagoras, Galileo and Newton also derived from mathematics what they are world-famous for today. It is usually the case that the physical fact can only be derived from the mathematical description. Hamming uses an appropriate metaphor for this:

If you put on blue eyeglasses, you will suddenly see the whole world blue. [see. 1, p.87]

In this sense, Hamming believes that the original natural phenomenon emerges from the mathematical tools we use and not from the real world. One must therefore be ready to accept that much of what we see comes solely from the glasses that are unshakably placed on our noses. [see. 1, p.88]

  1. We choose the math that suits us

From the history of mathematics it is readily apparent that this repeatedly comes up against its limits and must be expanded. The natural numbers were supplemented by rational numbers, then by irrational ones and so on. Vectors were added to the scalars because another tool was needed to describe physical forces. And so on.

According to Hamming, this fact only confirms once again that mathematics is a human invention that can be expanded and changed at will. The effectiveness of mathematics would then be rooted in the fact that, in the course of applying our mathematical models, we have come to a fairly good version of the description of empirical facts that have resulted from the interaction with physics.

So my second explanation is that we always choose the math to match the situation. It’s just not true that the same math works in every place! [see. 1, p.89]

  1. All science gives us limited answers 

According to Hamming, the effectiveness of the natural sciences and especially mathematics is, to a certain extent, a lie. An enormous number of aspects of reality cannot be described, let alone explained, by physics, chemistry, biology et cetera. These include, for example, central philosophical terms such as beauty, justice and truth.

What can we understand by these three terms? Hamming comments on this with these words:

So long as we use a mathematics in which the whole is the sum of the parts we are not likely to have mathematics as a major tool in examining these famous three terms of philosophy. Indeed, to generalize, almost all of our experiences in this world do not fall under the domain of science or mathematics. [1, p.89]

Conclusion – Hamming’s doubts about his attempts

Even without its modern title, Wigner’s puzzle amazed countless researchers from the past centuries and, unfortunately, Hamming doesn’t have much to counter it. In the conclusion of his speech he states, defeated:

From all of the foregoing, I must conclude that math is unreasonably effective, and that all of the suggestions I have tried to make are simply insufficient to explain why it is in the first place. [see. 1, p.90]

Perhaps Wigner’s puzzle is one of those secrets that we humans will never be able to find out. All that remains for us is to search with all our enthusiasm for a possible explanation for the incredible effectiveness of mathematics, for our approaches today are not much more convincing than those of the ancient Greeks. We are still blind to the origins of the power of those tools that we use too much.


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