The Fundamental Nature of God
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Start with a physicistís koan:
1. God is the boundary condition of the Universe.
2. The Universe has no boundary.
Contrary to what you might think, that is not a statement of atheism. It is certainly an invitation to think further into the subject.
In the Paradiso of his Commedia (The Divine Comedy, 1320) Dante Alighieri wrote a remarkably modern description of God in Its relation to the physical Universe. He imagined seeing God as a single point surrounded by the angels and the cosmos and, at the same time, enclosing the angels and the cosmos. Itís strange to think that we can see an accurate description of God in a work of Fourteenth Century Italian science fiction, but we can prove and verify that proposition.
Begin with a boundary. By definition, a boundary separates two different entities. If one of those entities is nothingness, then the other must be somethingness. That boundary necessarily has zero thickness, because any non-zero thickness would have to be part of somethingness (with regard to nothingness and somethingness there is no in-between). The boundary thus has the same geometric properties on both sides. Extent and duration do not exist in nothingness; therefore, the boundary must exist as a single mathematical point on which time does not elapse.
Given that the boundary is manifested as a single point, we infer that the minimum requirement for something to exist is a set of additional points. There is no basis for the existence of some specific number of points, so the number of points must be indeterminate; that is, the points must comprise an infinite set. In order to be multiple points, the elements of the set must differ from one another in some way. That way must constitute an infinite set of mutually different possibilities, so we conceive it as analogous to the real numbers, another infinite set, which we use to represent those possibilities by assigning them to the points.
The boundary must be the absolute zero of that set and each and every point differs from it by some function of the pointís index, which function we call distance. To go from the boundary to any given point, we must cross all of the intervening points (points bearing index numbers less than that of the given point), thereby tracing a line. Now imagine going to the point farthest from the boundary.
When we come to a point on that traverse, we have just crossed a point with a slightly smaller index number and we are about to cross a point with a slightly larger index number. When we come to the farthest point, the point with the maximum index number, we have just crossed a point with a slightly smaller index number but there can be no point with a slightly greater index number. We seem to have come to an end, but the line canít simply end: that would make the farthest point a second boundary, which is impossible. To provide proper closure for our set of points, we must infer the existence of at least one second point with a slightly lower index number touching our farthest point.
Thus there must exist a subset of points going from the boundary to the farthest point and at least one second subset of points going from the farthest point back to the boundary. In order to distinguish the points of the two subsets, we must give them all a second index number. We thus infer that each and every one of the points in our infinite set has two index numbers. Instead of a one-dimensional figure (a line), our infinite set of points appears as a two-dimensional figure (a plane). From the farthest point, the boundary appears as a circle completely enclosing that plane.
An observer at the farthest point can identify what appear to be two separate points on that circle and, using surveying techniques, can measure the two distances between those points. Of course, itís an illusion: the boundary has zero actual extent, so the distances must both equal zero. The ratio of the distances, zero divided by zero, is undefined and makes the distances themselves indeterminate. Through the two illusory points, then, must pass an infinite set of circles whose points all appear to lie the same distance from the farthest point. (Imagine two points on a sphere with all possible circles drawn on the sphereís surface passing through them.) The boundary of space must thus appear to the observer as a spherical shell completely enclosing a space whose points all bear three index numbers (three dimensions).
Each circle on the boundary appears to have some finite, non-zero length, though its actual length must equal zero. The circle consists of an infinite set of illusory points, so its apparent length is indeterminate. In order to manifest the different values of length that the indeterminacy requires, we must give all of our real points a fourth index number, which number represents what we call time.
With space and time we have the possibility of motion. An object can go from one point to another as time elapses (that is, the temporal index changes its value, usually increasing). A point can be one such object. On that basis we assert that space consists of an infinite set of inertial frames of reference, each moving at a uniform velocity relative to any of the others and consisting of an infinite set of points that are all motionless relative to each other, extending to the boundary.
We can deduce two facts pertaining to motion.
First, the Universe can have no motion (thereís nowhither for it to go); therefore, all of the motions in the Universe must always add up to a net zero. If some object changes its motion by some amount, then some other object must necessarily change its motion by the same amount in the opposite direction. Thatís a statement of Newtonís third law of motion, otherwise known as conservation of linear momentum. We can deduce a similar rule for rotary motion, conservation of angular momentum.
Second, no motion can ever come to the boundary of space. The boundary is necessarily frozen at time zero. Any motion that comes to the boundary would constitute an event, which marks the elapse of time. There can be no events at the boundary, so nothing can have even the possibility of reaching the boundary. Thus, the boundary must occupy a state of motion that no object can ever achieve.
All inertial frames of reference must be indistinguishable from one another except by the velocity between them, so the unachievable state of motion must be definite and must be the same in all of them. We infer from that statement that the boundary appears to move away from any given point in any inertial frame at the same speed in all directions. We can subsequently deduce the fact that the speed of the boundary coincides with the speed of light. If an object goes from one inertial frame to another (i.e. changes its motion), the motion of the boundary away from that object will remain unchanged.
From that latter fact (essentially Einsteinís second postulate of Relativity) and the indistinguishability of inertial frames (the first postulate of Relativity) we can deduce the four equations of the Lorentz Transformation, the centerpiece of the theory of Special Relativity. We thus work out a complete four-dimensional geometry of space and time. Through that geometry we deduce the basic rules of relativistic dynamics.
If we combine relativistic dynamics and four-dimensional geometry, we can deduce the principle of least action. From that principle we deduce the other basic laws of physics, including the second law of thermodynamics and the basic rules of the quantum theory. So the most fundamental laws of Nature come from an application of axiomatic-deductive logic starting from a few simple axioms and the statement that a boundary exists between something and nothing. That fact necessarily means that, at the very least, the boundary condition of the Universe is rational.
Is that boundary condition sentient? Does It intervene in human affairs as an arbiter of morality? Those questions take us beyond the scope of this little essay, so we will put them aside... for now.
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