Re: [math-fun] Mathematical cosmos article by Max Tegmark
A logician's view of a "theory" is that it is simply the shortest (in bits) description that fits (more or less) the data. We can use any computational engine we want to simulate the universe from the description, but since they are all equivalent to Turing Machines (modulo the number of bits required to describe the interpreter), it doesn't much matter which one we use. Fredkin is fond of "Life"-like cellular automata, but those models may require substantial modifications depending upon the "geometry" that fits the universe the best. The job of physics/chemistry/biology/etc. is the "compress" the experimental data into as few bits as possible. We are allowed an arbitrarily high degree of cleverness in coming up with such descriptions, but clearly new data may require arbitrarily large adjustments to the existing descriptions. It is interesting that physics loves "continuity" so much that they embed this concept into their view of physical models. Thus, when Einstein showed that Newton was wrong, physics explained this by saying that the previous models & data were correct, but just not for the precision required at large distances, high velocities, large masses, etc. I am particularly interested in why the simplistic models of the ancients _ever_ worked. There seems to be no a priori reason why _circular_ orbits should work as well as they do; if most of the planetary orbits had been highly elliptical, it might have required another several millenia for the mathematicians to come up with a proper description. Furthermore, if the planetary orbits had been closer together, the resulting chaos would have made even elliptical orbits an impossible model. (Using some of the web sites that allow experimentation with Newtonian gravity, it appears to me that _most_ gravitational systems with a relatively small number of objects "blow up" (i.e., kick one or more of their objects off to infinity) after not very much time. So perhaps a planetary system with widely spaced (minimally interacting) planets in nearly circular orbits is characteristic of long-lived configurations.) So we fall back on the fact that the fact that we are here, may imply that highly chaotic planetary orbits are inimical to human existence. We now have the circular argument that the universe is understandable (partially decomposable/factorable) simply because that is the only universe that supports human life. Alternatively, of all of the chaos in our lives, only the pitifully small percentage that is amenable to simple patterns is actually visible/"understandable" to us. So, yes, Max is correct, but only vacuously so.
On 10/1/07, Henry Baker <hbaker1@pipeline.com> wrote:
So we fall back on the fact that the fact that we are here, may imply that highly chaotic planetary orbits are inimical to human existence. We now have the circular argument that the universe is understandable (partially decomposable/factorable) simply because that is the only universe that supports human life. Alternatively, of all of the chaos in our lives, only the pitifully small percentage that is amenable to simple patterns is actually visible/"understandable" to us.
I agree - in some cases, we live in the "regular" places because that's what makes those places hospitable to complex life forms evolving, and in other cases we theorize about the regular places because those are the places where it's easiest to develop theories. But in that latter case, our theories do keep extending in places (e.g. climate and weather) that were once thought to be entirely beyond the grasp of our theories: while we know there's chaos there, there's still a lot of predicting that can be done. I think neuroscience is another such field: we are learning about regularities in brain function in all kinds of new ways, where once the brain was thought to be just too complicated for that type of low-level interpretation. So I don't think it's vacuous entirely - the scope of our theory does keep increasing, uncovering regularities in what previously seemed to be chaos (in either or both of the colloquial or mathematical senses of that word). --Joshua Zucker
More thoughts on whether the cosmos is mathematics ____________________________________________ It's been my conviction for as long as I can remember ["How long's that?" Uh... how long's what?] that a firm distinction must be made between pure mathematical modelling --- a cosy activity within a comforting, disciplined world whose confines I greatly appreciate --- and applied engineering --- a nasty, unpredictable business demanding unquantifiable intuition based on considerable experience. I see no convincing reason why cosmology should not also be expected to observe this distinction. But the second stage of the process may involve a considerable psychological hurdle: for the whole point of designing a model is for its user to be able to identify model with application; yet this confusion is precisely what its designer must resist! I can't make up my mind just now whether or not the following anecdote really does support rejection of Tegmark's thesis. But the point it makes seems worthwhile --- and anyway it gives me a pretext to go boring on again about Geometric Algebra. Clifford algebras are a more general concept embracing complex numbers, quaternions, and some other more exotic species of "number": the basic idea is to start with the real numbers, then attach a fixed set of "imaginary" generators. Some of these square to 0, some to -1, some to +1; any two distinct generators anti-commute; otherwise they observe the usual laws of algebra. Several theoretical physicists have utilised CA's for building cosmological models. My own aspirations are more humble: I just want to do ordinary geometry in Euclidean 3-space, hopefully in a more systematic fashion than it is usually hacked about in present-day computer graphics etc. For this I employ an algebra I call DCQ (dual complex quaternion), with 4 generators o,x,y,z such that o^2 = 0, x^2 = y^2 = z^2 = 1, and o x = - x o, x y = - y x, etc. [Several alternatives have been proposed by other authors, notably David Hestenes.] Having chosen a particular "pure model" --- the algebra, a matter of mathematics --- I am faced with an entirely separate decision regarding its application to the "real world" --- projective geometry. In this case the decision is so apparently straightforward that it seems usually to have been made unconsciously: obviously, the point with real Cartesian coordinates (a,b,c) will be represented (homogeneously) by a x + b y + c z + 1 o --- yes? The coordinate system which results from this naive identification is quite useful for some things --- for instance, subspaces and their meet and join operations are representable --- but frankly it's barely worth the effort. So, actually --- no! In fact, an immensely superior application identifies instead the "number" a x + b y + c z + d o with the _plane_ a x + b y + c z + d = 0 (in the usual notation). The unexpected bonuses of this alternative include metrical quantities and proper and improper isometries, the combination of which permit the design of algorithms in robotics which do not at present appear to be possible to motivate or formulate by conventional means, however determined. Finally, the point of this little saga is to illustrate that --- even within this very restricted universe --- it is not possible to ditch the "baggage": the pure model may be the same in each case; but lacking any identification, the real worlds may nonetheless be distinct. OK, I'll shut up and go away now. Fred Lunnon
participants (3)
-
Fred lunnon -
Henry Baker -
Joshua Zucker