Welcome God!

'Common Notions'

Euclid developed his theory of plane geometry from 5 axioms, of which the Parallel Postulate is merely the most famous, and five ‘common notions.’

The five ‘common notions’ are:

   “1:Things that are equal to the same thing are also equal to one another.

     2:If equals are added to equals, then the wholes are equal.

     3:If equals are subtracted from equals, then the remainders are equal.

     4:Things that coincide with one another equal one another.

     5:The whole is greater than the part.”

http://en.wikipedia.org/wiki/Euclidean_geometry

Four of the common notions are statements about equality.   The fifth is not.

We do not care to take issue with the first four of Euclid’s common notions, here, but we do wish to discuss the fifth common notion:  “The whole is greater than the part.”  This seems self-evident, certainly in what we think of as the physical world.  But just because it seems self-evident is no reason to suppose it is the only possibility.  After all, Euclidean geometry seems to be the ‘only’ description of reality, until one leaves the comfort zone of the near and the slow, near being less than a few miles, slow being much slower than the speed of light. 

So the world we can measure and call physical is not the whole world.  Consider instead the mathematical world of the infinitely divisible.  There any whole is infinitely divisible, but so is any part of that whole.  As a particular example, in mathematical analysis, any line segment is identical in every way to any smaller line segment that is a part of it.  This suggests that the fifth common notion may, in the description of the world, be not the only true one.

Consider then instead the notion 5’: The whole is equal to the part.  The whole is equal to the part.   Supposing this common notion is true implies that all parts of the whole are equal to each other. This idea seems degenerate, but let’s pause a moment.  From this common notion, we can deduce the idea of identical particles. We cannot deduce identical particles from notion 5. In fact from notion 5, by itself, we have an inchoate whole with parts in no particular relation to each other, whose only common feature is they are all interior to the whole. Indeed, from 5’ we have a sort of explanation for the existence of identical particles, where otherwise we have none.   Of course, in physics, we have different kinds of identical particles.  But: All electrons are alike.  All protons are alike.  All photons are alike. Etc. There are only a relatively few kinds of particles, certainly in the world of our mundane experience, and it is easy to view them as different aspects of one unified whole, the universe.

But it seems to also imply that each particle is, in some sense, as big as the universe. And indeed, as we explore the parts of these particles, or get closer to their core, it seems that ever larger amounts of energy are required to discern the increasingly fine details of their structure. Which brings us to:

5”:  The whole is less than the part.  Every day, there are physicists blasting particles apart into pieces, many pieces of which are greater in mass than the particles from which they are originally blasted. And in these pieces, these particles, this mass is ever more tightly concentrated.   That is, we may derive the idea that it is the minimization of energy that holds particles together, but that, ‘inside’ of each particle is an infinite amount of energy, or at least perhaps as much as there is in the entire universe. Or even more.  And thus we come to the idea that the structure of the universe in a sense is an inversion about the ‘boundaries’ of particles, that the very small, as much smaller than these particles as the whole of the universe seems to be larger, is somehow identified with the very largest. 

And from 5 the we have the idea that things are made of different collections of (identical, from 5’) particles, and larger things, larger collections of particles, from smaller things, smaller collections of particles.  Chemistry, and the world of our casual observation.

So we have the idea that the different common notions themselves cover different aspects of the physical world. 

Mathematical aside:  Compare how we treat these common notions to the parallel postulate: Euclid’s parallel postulate is just one of three, the one which describes a flat plane. It is equivalent to the statement:  Through any point on the plane, not on a line, exactly one line may be drawn, parallel to the first one, (parallel in the Euclidean plane meaning the distance between them does not change.)

The parallel postulate for Spherical (or Elliptical) geometry is that through a point, on the sphere, a line, (which in Spherical geometry is a great circle, like the equator of the earth) no other line may be drawn, parallel to the first one. That is each line, through a point not on the first line, intersects the first line exactly two times.  (Or just one time, in the case of Elliptical geometry) 

And the parallel postulate for Hyperbolic geometry is that through a point, on the Hyperbolic ‘surface,’ (a small area of which looks like a saddle) not on a line, infinitely many parallel lines exist, (of which two are asymptotic, (That is they grow ever closer to the first line as infinity is approached, one in each direction.) and infinitely many are ultraparallel. ‘Parallel’ here refers to both.) 

These geometries may also be characterized by the angles of a triangle on their respective surfaces: On the flat plane, the sum of the three angles of a triangleis always exactly equal to180⁰; on the Spherical and Elliptical, the sum of the angles is always greater than 180⁰; on the Hyperbolic, the sum of the angles is always less than 180⁰. 

Now each of these geometries describes, (or more precisely models,) an aspect of the physical universe.  Spherical geometry (in two dimensions) models the geometry of the surface of planets and stars.  (Elliptical geometry electrons?) Hyperbolic geometry models the shape of relativistic space-time.  And Euclidean plane geometry models the behavior, in the limit, of the very small scale, on the sphere, and in hyperbolic geometry the behavior of the nearly massless and very slow. (But not light itself, per se, which is massless, but travels at- light speed. 

What do we call Euclid’s geometry with instead common notion 5’.  Or Hyperbolic geometry with notion 5”.  Are they nonsensical? Or are these new fields?  Or are they already understood, but the underlying ‘common notion’ not recognized as such? 

In any case, the only reasons we might have for accepting just common notion 5, and rejecting 5’ and 5”, is our limited powers of observation, and our limited habits of conception.

So what does this have to do with God?  Well, God is everything, and therefore as all three of the parallel postulates apply to some aspect of the universe, and to God, all three of the fifth common notions, 5, 5’, and 5”, also apply. 

Let us consider 5”. God, the whole, is less than the part.  What we have is an explanation for the creation.  God wanted to be more than He was. Indeed, if the Heisenberg Uncertainty Principle applies, then a being infinite and changeless in time is infinitesimal in space.  Similarly, as that being becomes more defined/restricted in time, as the instant of time itself becomes more defined in terms of itself, its past,  (The instant of time becomes shorter with respect to the whole of time, that is the past, as 1/n,  as n goes to  infinity. However, the instant of creation becomes, in observation, n intervals long.)  we would expect it to become ever larger in space.

5’ then is: God the whole is equal to each particle, and each particle is equal to God. But the only thing equal to God is God, therefore each particle is God.  So, depending on however many particles make up the whole, we have a very large number of Gods, all God.  But since we have different kinds of particles, we have facing us different aspects, faces, of God, depending on the kind of particle. But each particle is God, in His entirety, if you dig deep enough. And of course, every one of all the particles of the same type is also each a different aspect of God, depending on its location, even though they are the same. Each particle has a unique experience, depending on its location in the universe.   

For 5 we have the various Theisms, where God is the greatest being, being greater than all parts, thus necessarily equal to the whole.  Of course, in any theology with an all- powerful creator God, the world is just an idea in His mind.  With pantheism, though, the world takes on more substance, having the nature of His body, though not separate from His mind.  Indeed, being God’s mind, the universe acquires a certain intransigence, a rigidity, which it would not have were it separate from God, and merely God’s whimsical creation.

 Why doesn’t He seem to interfere?  Well, we have it that His mind is the laws of nature.

Now we have glossed over applying 5’ to combinations of particles.  Consider that each particle is God.  Now consider two particles. Clearly, this combination is Greater than God, by 5. But it is also less than God, by 5. And by 5”, the same conclusion applies, though in opposite order: The combination of two particles is both less than God, and greater than God. 5’ apparently also applies to parts of particles, all parts of particles equal to God, since the other two, 5 and 5”, also seem to apply, assuming their validity.  Each part is less than God, by 5, but greater than God by 5”.  However, assuming their validity, we see both of these common notions apply to all cases.  Therefore there is no reason to suppose that 5’ does not apply to all cases, as well. So we assume it does apply to all cases. So by 5’, any combination of particles, (which is still a part of the whole,) is equal to the whole. 

You are a combination of particles.  You are equal to the whole, which is God.  You are also greater than God, and less than God. 

You are God.  But don’t let your head get over-inflated with the idea.  After all, You are also surrounded by God.