Monday, May 03, 2010

Bits and pieces

I have been quiet for a while as we have just had a month long visit from one of my sons and a granddaughter. Very enjoyable, interesting and distracting from normal activities. I want to make some brief comments which maybe I will enlarge upon later.

1. Why is probability so unintuitive? Peter Cameron has an interesting post about probability. I am particularly amused by his admission that after five years of teaching probability (something I have never done) he had a recurring nightmare that a student might ask him "What is probability?" and that he would be unable to answer.
He implicitly raises the question I began with, by describing a very simple problem the usual solution to which he finds unconvincing. Recently a whole book has been written about a famous simple question, the Monty Hall Problem.
I have always had similar difficulties about the meaning of probability theory, but recently in our work on algebras of processes I have come to the conclusion that one of the main reasons for the difficulty is that probability theory, and in particular the set theoretical formulation of Kolmogorov, is too abstract. Probabilities should not be associated with events, but instead to transitions or processes. Conditional probability then arises from communication between processes. (Peter starts to describe protocols. In fact, he shows that different protocols for his problem yield different results.) I think the mathematical model needs to be richer. The lack of intuition is related to the fact that problems are presented in too abstract a context: ask an Italian about Monty Hall and he will imagine any kind of imbroglio. We wrote a little paper in this direction, and are currently writing another.

2. How can people write programs without having a precise semantics for the language? I have been working for some years on the theory of programming languages. I have the view that the design of a programming language should involve first having an idea of an algebra of systems, and that the language should arise from the notion of free algebra. This doesn't seem to be an idea shared by many. (As a crude example of the idea, the integers form a ring, and elements of a free ring Z[x,y,..] are programs for making calculations with numbers.)
It seems however that programmers don't need to know mathematics and can write programs that are more or less correct without knowing really what the language means. (Also I write programs like that.) How is it possible?
To make a comparison, students have the greatest difficulty making proofs. Very few succeed.

3. Absurdities. I think there are some mad ideas being presented in science these days. Just two examples: multiverses, and Max Tegmark's classification of these; the aggresive insistence of David Deutsch on the Everett interpretation of quantum theory. More worrying is that both of these are being supported at the highest level of the scientific establishment. The Edge also seems to me to be full of dubious ideas.
I read in an article by Deutsch that people like me who express doubts about many worlds are like those who doubted the motion of the earth, on common sense grounds, at the time of Galileo. Looks like I am in bad company.

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