Local machines

Suppose there is a deep conjecture which haunts the imagination of a part of the mathematical community. By the common work of many, maybe even spread over several centuries and continents, slowly a solution emerges and the conjecture becomes a theorem. Beautiful, or at least horrendously complex theoretical machinery is invented and put to the task. Populations of family members experienced extreme boredom when faced to the answers of the question “what are you thinking about?”. Many others expressed a moderate curiosity in the weird preoccupations of those mathematicians, some, say, obsessed with knots or zippers or other childish activities. Finally, a constructive solution is found. This is very very rare and much sought for, mind you, because once we have a constructive solution then we may run it on a computer. So we do it, perhaps for the immense benefit of the finance industry.

Now here is the weird part. No matter what programming discipline is used, no matter which are programmers preferences and beliefs, the computer which runs the program is a local machine, which functions without any appeal to meaning.

I stop a bit to explain what is a local machine. Things are well known, but maybe is better to have them clear in front of the eyes. Whatever happens in a computer, it is only physically local modifications of it’s state. If we look at the Turing machine (I’ll not argue about the fact that computers are not exactly TMs, let’s take this as a simplification which does not affect the main point), then we can describe it as well as a stateless Turing machine, simply by putting the states of the machine on the tape, and reformulating the behaviour of the machine as a family of rewrite rules on local portions of the tape. It is fully possible, well known, and it has the advantage to work even if we don’t add one or many moving heads into the story, or indirection, or other ingredient than the one that these rewrites are done randomly. Believe it or not (if not then read

Turing machines, chemlambda style
http://chorasimilarity.github.io/chemlambda-gui/dynamic/turingchem.html

for an example) but that is a computer, indifferently of what technological complexities are involved into really making one.

times_only_bb_short_opt

(this is an animation showing a harmonious interaction between a chemical molecule derived from a lambda term, in the upper side of the image, and a Turing machine whose tape is visible in the lower side of the image)

Let’s get back to the algorithmic form of the solution of the mathematical problem. On the theoretical side there are lots of high meanings and they were discovered by a vast social collaboration.

But the algorithm run by the computer, in the concrete form it is run, edits out any such meaning. It is a well prepared initial tape (say “intelligently designed”, hope you have taken your daily dose of humour), which is then stupidly, randomly, locally rewritten until there’s no more reaction possible. Gives the answer.

If it is possible to advance a bit, even with this severe constraint to ignore global semantics, then maybe we find really new stuff, which is not visible under all these decorations called “intelligent”, or high level.

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