Registered: Mar 2005
Jason Cawley, in the post before last, objected to the use of the word, 'religious', to describe physicists' belief in continuous mathematics. I would assume those posting on this site are not religious, and the word was intended and understood as being used in the perjorative sense. I also get the impression that jason is playing devil's advocate with his objections, or just trying to calm the tone of the debate. However, I think this is actually a good analogy on many levels, and is of some use in illustrating why the majority of scientists and mathematicians do maintain a strong belief in the real number line as being very much a real object, and as natural as the natural numbers, and why a minority of enlightened NKSers do not share this belief. It is a similar situation to the position faced by atheists, and for similar reasons: If one approaches with an open mind just that fraction of human scientific knowledge that one would consider 'commonly known', the existence of a god seems highly unlikely, and the specific details added by the most popular religions, even less likely. But most people do have some sort of religious beliefs, and in fact we can identify particular features of human brain functioning (such as the feeling of sometimes being watched and judged, that even atheists get), as well as details of human culture and history, that make people likely to invent god and religion. Similarly, if you write down a set of axioms defining the continuous manifolds used in the current most popular 'Theories Of Everything' in physics (or just axiomatise the real number line, with the standard topology induced by the euclidean metric), they would appear rather contrived and unwieldly, and one would suspect that if one were to systematically probe different mathematical structures in the hope of finding the simplest one that could describe our universe, they would not be amongst the first to be tested. Nevertheless, even a child can have a strong sense of what properties a continuous line should have, so that when one finally learns a system of axioms that yields these properties, one has the feeling of atruth being revealed. However I would consider that these 'defining properties' of manifolds are seen as having a special status among mathematical objects because they coincide with rules applied by the human visual system to break up objects into regions and boundaries, and with other systems involved generating a sense of spacial awareness. For instance, the sharp and fuzzy boudaries one envisages when thinking about (topologically) open and closed sets suggest the origin of these concepts is in objects being in the foreground or background of our field of vision.
My own views on the matter are that the real number line is that they are very much a product of our imaginations, invented by man rather than by nature. A reference I would recommend for the sceptical is the reprint volume of Gregory Chaitin's papers, 'Information, Randomness and Incompleteness' from World Scientific. All the papers in it seem rather like variations on a single theme, and in some senses its content could be seen as a special case of the mathematical universality described in ANKOS. However, I believe it is a very important special case: He focuses a large amount of intellectual power on unravelling the real numer line, and it left me with the feeling that after 'computing' the natural numbers (a task many insomniacs have undertaken, with varying results), and then computing that fraction of the real numbers one could possibly compute, even using any number of different rulaes and languages with which to perform the computations, the remainder, the uncountable continuum of reals, would not really add anything new: They can in fact all be described by a single operation: Toss a coin: If heads, write a 1; if tails, a zero. Carry on forever. Any way in which such a number could affect a physical system in which it was used as a parameter would be indistinguishable from just making a random choice 'within the program' (or making both choices, if you believe in many worlds theories, which i do, for pretty much this reason: that fundamental mathematical systems are deterministic but quantum mechanics isn't). To say that there was actually a number all along that determined this outcome, but one couldn't possibly have known it, is very much a matter of faith, and in no way comparable to the physically real 'unpredictability' displayed by computationally irreducible systems which, after all, could in fact be predicted if sufficient computational muscle were applied (Even with such irreducibility in fundamental physics, one could use a large part of the universe to model a smaller one).
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