The Unreasonable Effectiveness of Mathematics in the Natural Sciences
The opening quote is by Bertrand Russell:
Mathematics, rightly viewed, possesses not only truth, but supreme beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.
This article explores these two main observations:
- Mathematical concepts turn up in entirely unexpected connections, and they often permit an unexpectedly close and accurate description of the phenomena in these connections.
- 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.
It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of laws of nature and of the human mind’s capacity to divine them.
What is Mathematics?
Mathematics is the invention of concepts. It started with inventing concepts to describe the real world (such as fractions, geometry, sequences, even irrational numbers), but the author believes that more advanced mathematical concepts such as complex numbers, Borel sets, etc. were devised only so that “the mathematician can demonstrate his ingenuity and sense of formal beauty”.
What is Physics?
Physics is concerned with discovering the laws of nature.
It is, as Schrodinger has remarked, a miracle that in spite of the baffling complexity of the world, certain regularities in the events could be discovered.
This regularity (such as Galileo’s discovery that two different masses have the same acceleration) is surprising because
- this holds across time and space,
[…] it is true not only in Pisa, and in Galileo’s time, it is true everywhere on Earth, was always true, and will always be true.
- this is independent of so many other factors that could affect it (e.g. who drops it, whether it rains, whether the experiment is carried out indoors or outdoors)
I don’t think I can agree with the second point. A lot of things do matter - air resistance, wind, the force with which it was dropped, etc. Objects only fall at the same rate assuming these are constant, which in reality is a huge assumption.
Wigner emphasizes this point:
the point which is most significant in the present context is that all these laws of nature contain, in even their remotest consequences, only a small part of our knowledge of the inanimate world.
[…] it is not at all natural that “laws of nature” exist, much less that man is able to discover them. [E. Schrodinger, in his What Is Life? (Cambridge: Cambridge University Press, 1945), p. 31, says that this second miracle may well be beyond human understanding.]
Wigner acknowledges that physics chooses mathematical concepts to formulate theories about nature, and the physicist, while doing physics, can develop theories or constructs that have already been conceived before by a mathematician. However (and this is the important part), Wigner believes it is not true that this had to happen, because mathematics concepts are not chosen for their simplicity, but for their amenability to clever manipulations and arguments, and as a result the concepts in mathematics are not conceptually simple concepts that were bound to occur in any formalism.
For example, quantum mechanics we use the complex Hilbert space.
Surely to the unpreoccupied mind, complex numbers are far from natural or simple and they cannot be suggested by physical observations. Furthermore, the use of complex numbers is in this case not a calculational trick of applied mathematics but comes close to being a necessity in the formulation of the laws of quantum mechanics.
In the introduction he gives the example of the example of how pi appears in many places and a classmate thinking, surely the gaussian distribution of a population has got nothing to do with the circumference of a circle.
First rebuttal that comes to mind is that mathematics is the only language we can speak, so of course we think physical phenomena is well-described by mathematics, because we have nothing to compare it to.
In response, Wigner says this:
The mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena. This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language.
Examples:
- planetary motion, started with the law of falling bodies in Italy, turns out it can also describe the motion of the moon
Apparently the law of gravitation was repugnant to Newton’s time and even to himself. He could only verify it with 4% accuracy at that time. It also contains a second derivative, which is not a very intuitive concept (why must laws of nature be intuitive?).
Certainly, the example of Newton’s law, quoted over and over again, must be mentioned first as a monumental example of a law, formulated in terms which appear simple to the mathematician, which has proved accurate beyond all reasonable expectations.
- QM: Born noticed that some rules of computation given by Heisenberg were formally identical with the rules of computation with matrices, established a long time before by mathematicians. They replace position and momentum in classical mechanics by matrices, but didn’t have any rational evidence that this is the correct thing to do, but turns out it is. The initial maths was based on Heisenberg’s theory, but when applied to situations where Heisenberg’s rule cannot be applied, the math is still accurate.
- the “complex spectra”
- QED, the theory of the Lamb shift