Gull, and Gill, each claim to have proved theorems concerning distributed computing. Their assumptions are slightly different, and their conclusions are slightly different. But they have the same "executive summary": it is not possible to write computer programs which run on two completely separated computers, and which do the following:
- each computer repeatedly asks a user to submit an angle between 0 and 360 degrees
- it then outputs a +/-1
- when the input angles are alpha and beta the average over many trials of the product of the outcomes is - cos(alpha - beta)
Gill's proof is built around the case when the users use fair coin tosses to each repeatedly select from just one pair of angles. Alice uses only the angles 0 and 90 degrees, Bob uses only the angles 45 and 135 degrees.
Gull's proof is built around the case that Alice uses angles chosen uniformly at random between 0 and 360 degrees, and Bob's angles are the same angles, shifted by an amount delta. Gull moreover imagines the whole experiment repeated many times with different values of delta.
I'm really grateful to Fred whose insistent criticism helped me and my student Dilara figure out what Gull was probably actually talking about. I'm really grateful to Joy for challenging us to come up with a proof of Bell's theorem which did not use probability or statistics (I think that was his challenge). Since all mathematics is intimately connected I guess that is an impossible challenge: he wants a proof without mathematics. Imagine that - physics without mathematics.
There is a connection with Bell's theorem. Suppose, in contradiction to what Bell claimed, that functions A(alpha, lambda) and B(beta, lambda) existed, taking the values +/-1, and a probability distribution rho over values of lambda in some set Lambda, such that
,
and suppose one could program those functions A and B, and simulate a long sequence of independent drawings of lambda from the probability distribution rho, then one could easily write those two computer programs to do that distributed Monte Carlo simulation task. The many independent drawings lambda_1, lambda_2, lambda_3, ... would simply be performed in advance and saved on hard disks of both computers.
I suggest Fred stops saying "blah blah blah" and then writing some nonsense which shows he hasn't read the posts of others. Let him get down to work. If Joy Christian's mathematical claims are correct and if Fred is such a great computer programmer, he can easily go ahead and win the 64 thousand Euro challenge by writing those two computer programs.