ART MATRIX PO 880 Ithaca, NY 14851-0880 USA (607) 277-0959, Fax (607) 277-8913 'The Paths of Lovers Cross in the Line of Duty.' THE WHY IS GOD Copyright (C) 1990 Homer Wilson Smith All Rights Reserved One of the great discoveries of the 19th century is that everything in the physical universe (except maybe consciousness) follows mathematical laws. All of the relationships between objects in space and time can be measured in terms of numbers, and all of the causal influences between them can also be described in terms of equations and mathematical relationships. Apparently God was a mathematician first, and a purveyor of damnation second. Thus the study of math becomes of paramount importance in the process of understanding and controlling the universe around us. It goes without saying that if the universe is controlled by mathematical laws then the qualities and conditions found in those mathematical laws will be manifest in the universe that is controlled by them. Thus the study of pure mathematics is beneficial to us all, for as we gain deeper insight into the nature of the math we use, we are more able to find these same things manifested in the exterior physical universe. If it were not for the mathematician making the discovery on the purely theoretical level, these manifestations would go forever unnoticed or unexplained in the physical universe. It is quite apparent that the mathematics that governs the macro scale world is based mostly on the simple force laws of Newton, Gauss, Maxwell and Co. namely the 1/R*R laws that relate to the propagation of force lines through out the surface of a surrounding sphere. This one idea alone brought science into the 20th century and with good reason. It is basically correct enough. We find this simple mathematical equation expressed not only in the laws of gravity, but also in electromagnetism and electricity without which our world would still be cold and dark. Thus it would seem that if the mathematicians are to stay ahead of the game they should spend some time studying the ins and outs of the 1/R*R laws and their related equations, such as derivatives and integrals. For the longest time, the missing part to this whole story was the concept of iteration. For example in the force law F = GMM/R*R we see Force expressed as a function of mass and position, but we do not see POSITION expressed as a function of mass and POSITION. Thus the use of the F=GMM/R*R law is strictly on an evaluation basis. You find the inputs and you get one output. There is no place to stick the output back into the input because the Force term is not in the input, and the input terms are not in the output. Most of physics went this way for eons although it need not have. People have long known of differential equations wherein the present CHANGE in state is a function of the present state. But really it took some very bright people looking into living biological systems to start coming up with equations that expressed the PRESENT STATE directly in terms of the prior PRESENT STATE (just a moment before!) This of course gives the needed conditions for ITERATION, or repeated evaluation of the equations involved. It is quite possible to then go back to all the old familiar stuff, like planets moving around the sun, and REWRITE the equations in terms of iteration, wherein the present position is a function of past position. In fact the rocket guys do it all the time, as simple algebraic solutions to multi bodied planetary systems are either impossible or difficult. Actually the use of iterated differential systems has been with us for a long time. So if we go back and look at the world in this new light, all of a sudden we find we have a plethora of iterated expressions with little to no understanding of what their MATHEMATICAL nature is. So once you start looking into the matter you find a wonderful world of structure and 'nature' in the subject. You find there are fixed points, and period cycles, and chaotic cycles, there are repulsive and attractive points with basins of attraction, and there are open and closed Julia Sets which give rise to all sorts of fantastic Mandelbrot Sets. One finds that all these things depend on the values of the parameters involved in the equations, and that all non linear equations manifest these things in one form or another. One also finds that for many of them the various boundaries surrounding basins of attraction or what have you, are quite FRACTAL, an incidental but very important new thing to know about the matter. By fractal we mean infinitely convoluted, not a straight line in sight, no matter how small. IT COULD NOT BE SO PROMINENT AND NOT BE IMPORTANT, DON'T YOU SEE? Not as long as the equations which manifest such things are key causal descriptors in the physical universe in question. First you find out what is true MATHEMATICALLY, and THEN you find out where those things are applied in the real universe. I assure you these things do not exist in a vacuum of vanity. If they exist mathematically, they ARE applied. Now the important equations of the physical universe have not changed, Y = X*X + C is still Y = X*X + C, but the CONCEPT of what you are going to DO with it has changed, from simple evaluation which produces parabolas, to multiple iteration which produces Mandelbrot Sets and Julia Sets. And all of a sudden this entire new landscape of structure and nature appears to unfold as if out of nowhere. It may seem like it is new, but you and I know it has been there all along. AND YOU CAN BET THE PHYSICAL UNIVERSE KNOWS IT TOO. The physical universe has been using these same equations to march its wheels of time forever, and you can be sure that if these equations show some inner structure or 'nature' then that same nature will be manifested outwardly in the physical universe expression of that equation. The physical universe is basically an iterated system, so actually it is surprising we have made the progress we have, using only simple evaluation. The equations have been around forever. The physical universe has been USING them almost forever. The equations have as part of their nature things like fixed points, period cycles, chaotic cycles, basins of attraction, etc., so you can be sure all these things are manifested in the physical universe INCLUDING FRACTALNESS. To say therefore that fractals have nothing to with anything and have not explained or proven useful in our understanding of the universe is more a statement about the people who are working with fractals rather than a statement about the pertinence of fractals to the world at large. Fractals are so pertinent to the universe no one can see it yet. Long time ago, they thought math did not pertain either. The 'why' was God. The 'why' may still be God, but if it is, then clearly God is a mathematician of significant merit, and no doubt a fractal enthusiast. Homer