The unreasonable effectiveness of mathematics

«The Unreasonable Effectiveness of Mathematics in the Natural Sciences, » by E. Wigner, in Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960) ;
http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

« The Unreasonable Effectiveness of Mathematics, » by R.P. Hamming, in The American Mathematical Monthly, vol 87, No II (February 1980) ;
http://www-lmmb.ncifcrf.gov/~toms/Hamming.unreasonable.html

An idea that is present in both essays is that mathematics work because they are to a large extent inspired by our perception of the world we live in. I had the following thought about natural numbers.

Why on Earth do natural numbers seem so natural? It may be the simple fact that physical objects can be distinguished from one another. If they could not be distinguished, the very concept of counting would not be relevant at all. I don’t mean if the objects were identical, I mean if there was really no way to tell “this is one object and this is another one”. In these conditions, the idea of natural numbers would probably not have come to anybody’s mind. To some extent, the obviousness of natural numbers –and presumably their properties- has something to do about physics.

In quantum mechanics, identical particles are indistinguishable, even in principle. I don’t know, however, if this means they cannot be counted. For instance, we might be able to say “there are for sure 20 photons between these two mirrors, but there is no way we can label them”. If you are an able physicist, I would definitely be interested to discuss this.

It has often been said that Euclid’s geometry would not have been discovered if our everyday environment had been the curved space of general relativity. I am now wondering whether natural numbers would have been discovered by anybody but the most insane of us if the everyday physics had been quantum mechanics.

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