Continuing an old discussion under a more apropos name, quantum physicist Nicolas Gisin writes a summary of his position on free will in NewScientist (21 May 2016, paywall):
But are the mathematical real numbers physically real? Certainly not! Most real numbers are never-ending strings of digits. They can be thought of as containing an infinite amount of information – they could, for example, encode the answers to all possible questions that can be formulated in any human language. Yet a finite volume of space-time can only hold a finite amount of information. So the position of a particle, or the value of any field or quantum state in a finite volume, cannot be a real number. Real numbers are non-physical monsters.
That’s a bit puzzling. Sure, they can be thought of containing an infinite amount of information, but that’s just one interpretation, one amongst many. He’s posted a longer paper to the academic pre-print server arxiv:
The use of real numbers in physics, and other sciences, is an extremely efficient and useful idealization, e.g. to allow for differential equations. But one should not make the confusion of believing that this idealization implies that nature is deterministic. A deterministic theoretical model of physics doesn’t imply that nature is deterministic. Again, real numbers are extremely useful to do theoretical physics and calculations, but they are not physically real.
The fact that so-called real numbers have in fact random digits, after the few first ones, has especially important consequences in chaotic dynamical systems. After a pretty short time, the future evolution would depend on the thousandth digit of the initial condition. But that
digit doesn’t really exist. Consequently, the future of classical chaotic systems is open and Newtonian dynam-ics is not deterministic. Actually most classical systems are chaotic, at least the interesting ones, i.e. all those that are not equivalent to a bunch of harmonic oscillators. Hence, classical mechanics is not deterministic, contrary to standard claims and widely held beliefs.
Discerning the difference between reality and modeling is interesting, but I can’t help but notice the argument is merely symmetrically applicable, i.e., you also can’t use the argument to disprove the suggestion the Universe is deterministic. The rest of his argument is either way beyond me, or gibberish – I can’t tell.
On an unrelated note, he also gives the reason for the name real numbers, which I thought was interesting. From the NewScientist article:
It took me a long time to identify what I believe is the key to the problem: a crucial detail of the mathematics we use to describe the world. Fittingly, it goes back again to Descartes. He gave the name “real” to the numbers commonly used in science: 1, 2, ¾, 1.797546… His point was to distinguish them from the imaginary numbers based on the square root of -1, numbers that intuitively cannot exist in the real world.
