Reading notes on two papers on the subject of the unreasonable effectiveness of mathematics

Eugene Wigner

The Unreasonable Effectiveness of Mathematics in the Natural Sciences

the idea of invariance in science, e.g. in Galileo’s Tower of Pisa experiments the objects fell at the same speed, regardless if they were let go by one person, or by two people, or wether one person was a man and the other a woman.
that often laws have been found to apply outside the situations where the idea originated, and with higher precision than the precision of the initial data

Richard W. Hamming

The Unreasonable Effectiveness of Mathematics

“Most textbooks repeat the Greeks and say that geometry arose from the needs of the Egyptians to survey the land after each flooding by the Nile River, but I attribute much more to aesthetics than do most historians of mathematics and correspondingly less to immediately utility.” [p84]
- Certainly the organisation of geometry as done by Euclid is not motivated by immediate utility. See for example Elements, Book I, Proposition 10, where Euclid’s proof aims to minimise repetition and shorten the proof. Compare this with the approach by Apollonius which is more practical from the number of steps of the construction, but would repeat the proof of Book I Proposition 1.
“Is it not remarkable that 6 sheep plus 7 sheep make 13 sheep; that 6 stones plus 7 stones make 13 stones? Is it not a miracle that the universe is so constructed that such a simple abstraction as a number is possible? To me this is one of the strongest examples of the unreasonable effectiveness of mathematics. Indeed, I find it both strange and unexplainable.” [p84]
- I think that the acceptance of 1 and 0 is a cultural issue so entrenched that it makes it hard for us to reason outside this box, but ultimately it’s an arbitrary choice.
with regards to Euclid’s Elements: “Yet how does it happen that no theorem in all the thirteen books is now false? Not one theorem has been found to be false, though often the proofs given by Euclid seem now to be false.” [p86]
“It is not that the postulation approach is wrong, only that its arbitrariness should be clearly recognized, and we should be prepared to change postulates when the need becomes apparent” [p86]
“We see what we look for.” [p87]
- same as “To a man with a hammer everything looks like a nail”
"”Why should I do all the analysis in terms of Fourier integrals? Why are they the natural tools for the problem?” I soon found out, as many of you already know, that the eigenfunctions of translation are the complex exponentials. If you want time invariance, and certainly physicists and engineers do (so that an experiment done today or tomorrow will give the same results), then you are led to these functions.” [p88]
- Well, I didn’t know. Something to look into.
“And many people believe that the two relativity theories rest more on philosophical grounds than on actual experiments.” [p88]
- I concur.
“Science in fact answers comparatively few problems” [p89]
- An example: the simple pendulum law that the period of oscillation depends on the pendulum length, but not on the mass and initial angle. Yes, it covers a lot of cases, but there are much more cases that are not covered.
“We find, for example, that we can cope with thinking about the world when it is of comparable size to ourselves and our raw unaided senses, but that when we go to the very small or the very large then our thinking has great trouble” [p89]
- e.g. the Euclidian geometry is particularly successful at human scale, but starts to fail at atomic or cosmic scales.
It’s surprising to see what a small world it is. Reference section includes Euclid’s Elements, in particular the Thomas L. Heath translation that I’m currently reading, recommended by Alex Stepanov.

Reference

Eugene Wigner: “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”
Communications in Pure and Applied Mathematics, Feb. 1960, Vol. 13, No. I

Richard W. Hamming: “The Unreasonable Effectiveness of Mathematics”
The American Mathematical Monthly, Feb. 1980, p81