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April 21st, 2012

I have for some while been working on computer code that, incidentally, would be relevant to this, which is just a modern way of talking about Minkowski world: http://en.wikipedia.org/wiki/Spacetime_algebra

I’m more interested in typeface design, however. I basically want a more abstract way to talk about (mostly plane) geometry than is the common practice. I have shown a lot of interest in this: http://en.wikipedia.org/wiki/Geometric_algebra#Conformal_geometric_algebra_.28CGA.29 However it is likely I will end up using a purely Euclidean metric, leaving the expansion to non-euclidean metric a mere framework for possible expansion.

Bell's handwaving

This page from John Bell’s book (the earlier edition -- a posthumous second edition recently came out) illustrates his handwaving argument and confusion of correlation and causation:

The talk about ‘residual fluctuations’ being assumed independent is pure handwaving; there is no justification for it. We aren’t even justified in calling lambda ‘fluctuations’; they are simply facts we happen not to know. Calling them ‘fluctuations’ shows you how physicists tend to think of probability expressions as ‘nature making choices’ (fluctuating) when, in fact, a probability expression can arise in perfectly deterministic, utterly predictable situations, where you simply happen not to know the details (Ideal Gas, random number generators, etc.).

Then there is this sentence, oddly written in the form of bluster: ‘‘Note well that we already incorporate in (10) a hypothesis of ‘local causality’ or ‘no action at a distance’.’’ But if you look at (10) there is nothing in there about causality at all. It is simply an expression of conditional probabilities, agnostic as to how the correlations come about. Indeed, the subject of the expression is (by my impression) epidemiological statistics, which by design have nothing directly to do with what causes people to have heart attacks. They are, at most, suggestive of where to look. To know what causes people to have heart attacks you do research into mechanisms of causation -- exactly what Bell is trying to convince you is impossible, hopeless, and not-to-be-undertaken in the field of physics.

Since Bell, the handwaving arguments have gotten more sophisticated and full of terms derived from Greek, but that is probably all that has happened.

(Postscript: Indeed, if the temperatures in Lille and Lyons are not correlated, due to shared causal factors, then I want to know the color of the sky is on your planet. Because the temperatures are correlated, it doesn’t matter whether the temperature in Lille affects the heart attack rates in Lyons; if the temperature in Lyon affects heart attack rates in Lyon, and temperatures in Lille are correlated with temperatures in Lyon, then temperatures in Lille may be correlated with heart attack rates in Lyons. This somehow escaped Bell’s notice.)

Why my revival of interest in QM folly

I had my interest in QM folly revived due to barking_iguana’s recent lament about the terminology of statisticians being an impediment to the analysis of baseball. I saw a similarity in the refusal by professionals dealing with probability to be ‘abstract’ about what a probability expression represents, the way one is ‘abstract’ in the use of symbolic logic or integral calculus.

When I was in graduate school, in my stochastic signals course we were assigned homework to give an example of a ‘random variable’. I went by the book, and gave some kind of mapping employing the ‘modern definitions’ described at http://en.wikipedia.org/wiki/Probability_theory and that build up probability theory as a kind of integral calculus. It turned out neither the TA nor the professor could understand what I wrote, and had expected an answer like ‘rolled dice’. I got credit, eventually, mainly because I cried for them and wouldn’t go away, and because they liked me. I was very proud to have figured out what the book was talking about.

Since then I have become attached to another abstract view of probabilities, which is that, if you have certain excellent requirements of what a generalization of Aristotelian logic to numbers between 0 and 1 would look like, then the only possible solution turns out to be probability theory. Aristotelian logic is completely agnostic about its subject matter, so this form of probability theory is exactly the same. It has the advantage over the ‘integral calculus’ approach of being more obviously applicable to logic-like problems, such as pattern-detection; however, anything calculus-like tends to be easier to visualize, if you ask me, and so perhaps may be more productive for theoreticians.
When I learned a little probability theory in school, we arrived at bell-shaped ‘gaussian’ curves as the result of mixing a bunch of ‘random variables’ together. Theorems showed they tended to spread out into that well known bell shape. At least, that’s how I remember the theory going.

But probability theory as logic has no ‘random variables’, so how come it ends up with the same ‘gaussian’ functions? It turns up as a way of saying that some facts are unknown. If you think about it, this is exactly what the idea of a ‘random variable’ is trying to capture -- facts that are there but which for some reason we don’t know exactly, but can only make an educated guess. it is not the rolling of the dice that makes them a ‘random variable’, but the fact that we don’t know what sides will end up on top until after we have rolled the dice. This is what makes an abstract view necessary. Suppose, for instance, that we know there are 5d6 lying somewhere, and someone has arranged them purposely rather than rolled them, but hasn’t told us the arrangement. Then the same exact ‘random variable’ applies here as in the case of rolled dice; it is the not-knowing that makes a probability problem what it is, not the fact that rolling was involved. (Rolling of dice is probably about as deterministic a process as one can imagine, anyway, akin to shooting pool. Few would doubt that it is simply an intractable problem in Newtonian dynamics.)

Why not PRNG?

All of the preceding shows why I prefer ‘random number generator’ to ‘pseudo-random number generator’. The only essential difference between ‘pseudo-random numbers’ and ‘true random numbers’ is that we have a good idea of the process by which the first kind come about; we simply choose to ignore that process; whereas for the second kind it is impossible or intractable to know how the numbers came about. They may still be fully determined in ways we do not know and/or cannot reproduce. This is what makes them essential to good cryptography, where the goal is explicitly nothing more than to make decryption intractable.

Better distinguishing terms might be something like ‘algorithmically generated random numbers’ versus ‘random numbers from real-world sampling’, etc.

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