Registered: Feb 2004
A comment about Page 204: multiway systems
Multiway systems - organic number generators?
Dear Mr Wolfram,
As far as I can see from searching the index of this well-organised electronic edition of your book, your interest in multiway systems seems to focus on the possibility of generating space-time systems or their analogues. However, as you note, there is an intuitive discrepancy between the features of a multiway-generated number space and what is observed in the real world: something that could be summed up as the absence of boundaries and starting points in real-world 'systems', if indeed it is appropriate to think of space-time as a system at all.
It is difficult to conceive of a cellular automaton with no starting point, even if one cheated and listed all its states in reverse so that it converged on a starting point from a boundary. However, your discussion of multiway systems suggests something else to me, which does not seem to be covered in the items listed under the related index entries.
You may be aware of research under way at the Centre for the Mind in Australia using data from autism, which is interpreted as supporting the hypothesis that the brain has precise computational capacities which are not explicable by standard (ie primitive) models of neurophysiology.
The basic idea is that the autistic children who can for instance multiply two seven-digit number instantaneously (but can't wash their hands) have, through developmental imperfections, gained access to mechanisms which exist in all individuals, but which are normally unconscious. So when we catch a thrown object, for instance, we are in fact making complex calculations of its trajectory by means of a number-crunching ability we have no awareness of in conscious life. This explanation seems to cover some very striking cases where autistic children out-perform normal human beings, and might well be of much more general application.
What I wonder is whether something like the multiway systems you describe might not be at the heart of such an organic number-crunching ability. More specifically, might it not be more fruitful to consider how these systems generate the intuitive number spaces considered by mathematicians, rather than looking for examples of physical systems with which they can be matched?
(This is an old hobby horse of mine: the idea that mathematicians actually investigate an internal, intuitive world, then hand their results over to physicists. Parenthetically, it seems to me that Mandelbrot and now you have reversed this arrangement, by using calculation systems to produce physical objects and then hunting for intuitions to explain them afterwards; but that is by the by.)
For instance, you note that multiway systems easily generate Euclidean spaces, and give examples of integers, positive and negative number lines, and so on. The feature I would focus on as evidence is the generation of a limited area of repetition, if I can put it like that.
Taking positive and negative integers as an example, we easily extrapolate from our intuitive grasp of counting to a number line extending to plus and minus infinity, and this does illustrate the quality of our intuition. However, in reality our intuitive grasp of number does not extend very far along the integer line. We immediately know the difference between six and seven, but even between 16 and 17 we have to stop and count (though there are autistic children who can look at 1356 and 1357 objects and see the difference).
Formally, there is no very important difference between this kind of multiway-generated number line and the other results available; but there might be constraints related to the physical nature of the brain structures involved, or the general logic of neurophysiological systems, and as far as I know one can only speculate so far about such matters.
Anyway, just one of many possible lines of thought to develop from this stimulating material.
With best wishes,
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