Maths in Europe: John Conway

Sometimes I wish I hadn’t invented that game.
– John Horton Conway about the Game of life ⁽³⁾

The most famous mathematical invention by John Conway was undoubtedly the Game of life, that was popolirized by Martin Gardner on his column published on Scientific American ⁽¹⁾, but today I want to tell you something about the free will theorem.

The theorem was proposed by Conway with Simon Kochen, inspired by the question about the interpretation of quantum mechanics. The statement is:

If the choice of directions in which to perform spin 1 experiments is not a function of the information accessible to the experimenters, then the responses of the particles are equally not functions of the information accessible to them. ⁽²⁾

That is, the theorem states that the outcome of the experiment is independent of the choices made by the experimenters. Or if you prefer that the eyes of the experimenters does not influence the outcome of the experiment (and therefore does not change the universe).

In the following the two mathematicians are concerned with showing the consistency of the theorem with quantum mechanics, first of all showing that this is only one of the last arguments against the hidden variables of David Bohm, proposed to restore the determinism lost by quantum mechanics.

The other interesting effort of the two mathematicians is to reconcile everything, at least from a logical-philosophical point of view, with relativity. In their judgment, after introducing a series of concepts, such as actual randomness (the universe would appear random from any reference system), Conway and Kochen show a series of errors of interpretation related to the EPR paradox that would push physicists to believe the existence of violations of the principle of randomness (and therefore of relativity). The point is that if in the system of the two connected states A, B the spin A is measured and it is down, it is much more correct to say that if the measurement is performed on B its spin will be up and not the spin of B is instantly up.

The situation is undoubtedly much more complex and detailed than this, but overall the feeling is that the two mathematicians wrote an explicit invitation to physicists to take the question more relaxed. Which seems confirmed by the phrase that I think perfectly summarizes the work of Conway and Kochen:

God lets the world run free.

Card Colm Mulcahy, an irish mathematician and Conway’s friend, on 11 april 2020 published on twitter the news of the death of Conway. His source was a close associate of his and confirmed by the family.

I written this little post in his honour: good bye, Mr. Conway.

1. Gardner, M. (1970). Mathematical games: The fantastic combinations of John Conway’s new solitaire game “life”. Scientific American, 223(4), 120-123. (pdf|html)

2. Conway, J., & Kochen, S. (2006). The free will theorem. Foundations of Physics, 36(10), 1441-1473. doi:10.1007/s10701-006-9068-6 (arXiv)

3. Dierk Schleicher (2013). Interview with John Horton Conway. Notices of American Mathematica Society, vol. 60, n. 5, pp.567-575 (pdf)