Last week I laid out the foundations for one of the most complex, subtle and compelling arguments than any human being has ever made, based on the work of Irish physicist John Bell, who unfortunately died before he could be awarded a Nobel for his genius.
This week I have covid.
So I'm going to try to write the follow-up I promised, but it might not be up to my usual standard of semi-coherence.
Our public health authorities in here British Columbia have completely abandoned us, deeming a few thousand more dead to be acceptable losses as compared to the harm to their political fortunes that a more robust response might yield. As such, the personal behaviours and precautions that have kept us safe for the past two years when mask mandates and other protections were in place are no longer working. Because out "public health" authorities are also violating our human right to the data being used to govern us, it has been very difficult to estimate the prevalence of viral spread in the community. It turns out to be higher than I'd estimated. /rant
In any case, last week I laid out the foundation of Bell's argument. There's a lot of history that goes with it that I skipped, which may not have been the best rhetorical move ever. As may be, I'll do a quick recap here and then show what Bell demonstrated.
The argument concerns a pair of peculiar coins which are repeatedly flipped under any one of three sets of circumstances that can be thought of as being on a line with two end-points and one middle point: "same", where they are individually flipped on the same point on the line, "opposite" where they are individually flipped on different endpoints, and "mixed" where one is flipped on an endpoint and one is flipped on a middle point.
They show the following behaviours:
1) If flipped under "same" circumstances they always yield the same result. If one comes up heads, so does the other. If one comes up tails, so does the other.
2) If flipped under "opposite" circumstances they always yield different results. If one comes up heads the other is tails.
3) All subsets of flips are fair: it doesn't matter how you slice and dice the data, the individual coins always have a 50% chance of heads and a 50% chance of being tails, within statistical error.
Given these results, what can we say when the circumstances are "mixed" given that the coins are truly independent and have no way of communicating with each other?
This is the point of Bell's argument: in the early 20th century it became apparent to physicists that what we would like to think of as "different parts of reality" like two electrons that had scattered off of each other were not in fact independent if our descriptions of it were correct, but the universe was structured in such a way that we could never observe the fact of its connectedness directly.
We knew that once two particles had interacted the results of measurements on them would be correlated. This is just classical physics: if one billiard ball hits another and I know the initial conditions and the outgoing trajectory of one ball I can predict the outgoing trajectory of the other based on nothing but conservation of energy and momentum. So far, so good.
But quantum theory doesn't describe a single outgoing trajectory for two electrons that scatter off each other: it describes a set of pairs of correlated trajectories, without saying anything about which trajectory "actually exists".
There were several different ways of thinking about this. One was that all the trajectories were followed, but the system "collapsed" into one pair or another when a measurement was made on one of the electrons. Another--favoured by Einstein and Schrodinger, among others--was that there was one real trajectory, but we didn't know what it was. A third was the Copenhagen interpretation, and decades later we added the Many Worlds interpretation.
Each of these interpretations had problems:
The Collapse interpretations have difficulty specifying what it is that leads to the process of collapse, and collapse itself has to happen instantaneously, in violation of the notion that causality in our universe moves at the speed of light.
The "It's Real But We Don't Know" interpretations are at a loss to explain interference phenomena unless they make claims about the wave aspect of particles that are pretty ad hoc. David Bohm's "quantum potential" work tried to do this many years later, and while it isn't exactly ad hoc it is still pretty damned strange.
It has been said that the Cophenhagen interpretation makes the other interpretations look good because they at least have the virtue of not being completely incoherent word salad.
And Many Worlds says simply that everything happens, we just only see one branch for some reason, where the reason is as mysterious as whatever causes collapse in Collapse interpretations. Many Worlds people will claim this isn't so, but their argument is circular. No one looking at the solutions to Schrodinger's equation would ever guess that anything like the classical world even existed. It is only by knowing that it does--for some reason--that we can "explain" it, like financial pundits "explaining" the motion of the market after the fact. We don't actually know the reason classical reality exists (or that we exist as classically-aware conscious beings to experience it) and would never predict it from the underlying equations.
This was the situation for some decades after Einstein, Podolsky, and Rosen published their famous "EPR" paper that pointed out the paradoxical implication of the quantum description of reality: it implied there was necessarily some kind of "spooky action at a distance" that kept the books of momentum, energy, and angular momentum balanced, or that quantum mechanics was just an approximate description of reality.
John Bell was interested in showing we could know which was the case, and to that end he created a physically realistic version of the situation I'm describing here in schematic form.
Without going into the detailed physics of that, which I will in future, the argument he made was simple: if the coins behave as described above, then we can predict how they will behave in the "mixed" case, and if they deviate from that prediction there must be communication between them, or one of the assumptions the argument rests on must be wrong.
What I suggested last week was we could fully characterize the coins with a simple list of outcomes for each case along the line: HHH would mean no matter where a coin was flipped it will come up heads. HTT means the coin will come up heads at one end of the line, tails in the middle, and tails at the other. And so on.
Give the perfect correlation between outcomes in point 1 above, the two coins must always be programmed identically, and given the perfect anti-correlation between flips under opposite conditions the only allowed programs are:
Coin1/Coin2
HTT/HTT
HHT/HHT
THH/THH
TTH/TTH
Since the coins are fair under all conditions we have to take at least to of these programs in equal proportion.
If we take both HHT/HHT and TTH/TTH in equal measure, for example, we can get fair flips for any subset of the data, and opposite outcomes under opposite circumstances. Some other combinations won't work to reproduce the observed statistics.
Taking this into account, the allowed combinations are:
HHT/HHT and TTH/TTH
THH/THH and HTT/HTT
or all four together.
But notice: each of those cases implies that the the "mixed" condition, where one coin is flipped in the middle position and the other at one of the ends must have 50/50 outcomes: half the time they will be identical, half the time they will be different. Consider HTT/HTT and the various ways they could be flipped:
HTT/HTT
HTT/HTT
HTT/HTT
HTT/HTT
So we can say that if there is a "fact of the matter" about which outcome the coins will have in any given case, and they behave as described above, then the "mixed" outcome, where one coin is flipped at the end of a line and one is flipped in the middle, must be the same half the time and different half the time.
Bell demonstrated how this argument could be made to run for real experiments, and further refinements reduced these experiments to practice. There are probably undergraduate labs doing this now, which is about as dumbed-down as an experiment can possibly be.
In the real case the prediction is an inequality, not a simple value, due to the complexities of real systems that my magic coins don't face. My point here is to give the flavour of Bell's argument, to show how it is possible "know something about the world we cannot know".
What experimental realizations of this set-up have shown is that quantum mechanics is an accurate description of reality: outcomes of arbitrarily distant measurements on systems that have previously interacted are correlated in ways that are not allowed if those systems are in fact independent of each other.
We can't use these correlations for communication: the "fair coin" aspect of the statistics ensures this, to the extent that assuming no communication is possible. But using Bell’s indirect inference we can never-the-less know.
Having now done a mediocre job of establishing what the fuss is about, I'm next going to swing back and talk about the larger picture concerning metaphysics, epistemology, physics, and ontology... which may take some time. I'll try to keep it both entertaining and comprehensible, and with any luck end up back at the work I've introduced over the past two weeks with a view of the place that'll be more easily understood.
I have only read through this once, so far. It is very difficult to understand. It seems what you are trying to show is a situation in which all the 'laws' of randomness are maintained, but their is intuitively something causal going on. The bigger problem is what this means for laws that imply causal mechanisms. Is this correct? Actually, writing this out has helped me understand the problem. I still think I have to read it once more time.