Infinity Part I

Does anybody know what reality is? Does anybody really care?

Even though I speak as though my words represent truth, I mean them to only represent my opinions. For almost everything I assert, I can think of counter-arguments, and counter-counter-arguments.
I am rather in the position of Aquinas trying to prove the existence of God. I don’t know that the totality of natural numbers exists in the universe...or not, any more than I know that God exists...or not.
I do believe, which is a stronger verb than think, that there are only finite sets in the universe and that there is both a largest counting number and a smallest fraction number.
I’m an advocate of Lord Occam and infinity is too complicated and too far out for me to take seriously.
Evidently the latest concept of the universe is that it is a finite volume that has no outside. That’s pretty weird when you stop and think about it, but I don’t find it as weird as infinity.
I think of the Ideal World as a fantasy world like Alice’s Wonderland. Games are invented in Wonderland where the game pieces are Mad Hatters and disappearing cats. In the Ideal World games are played with derivatives and infinite series. In the Ideal World there are the ideal geometric figures of Plato. But these game pieces are not in the Real World.
Some the games played in the Ideal World seem to model the Real World well enough to use in the description of the Real World. The Pythagorean Theorem is an example. Even a fairly crude 3-4-5 triangle gives a right angle close enough to build a square based burial tomb.
I think the world is finite and stochastic. There is no such thing as an exact measurement beyond the simple counting of small sets. Any number given for the number of baseballs in a 100 yard cube is part of a probability distribution. We can conceive of an exact number of baseballs, so it’s a Real World concept, but it can’t be known for sure.
I think that because of the nature of space and matter at atomic sizes there is no exact distance in the Real World, only a statistical distance. In the mythical, Ideal Mathematical World boards are exactly two feet long.
There is no number in either world, except a small counting number, that is realized exactly in the Real World.
In the argument that follows my universe is finite because I make assumptions that imply that it is finite. I say what my assumptions are and try to explain why I make them. I’m not trying to convert anyone but I do think that spreading the Mathematical Ideal World into the Real World is mysticism.

A “concept” is a mental construct. This definition ties the idea of concept to humans; well, to intelligent life. “Unicorn” is a mental construct that is a concept of something in Wonderland, not the Real World. If there was no brain to think of a unicorn then a unicorn wouldn’t exist in any world.
I say that a concept “exists” in the Real World if I can conceive of it being realized by a concrete example.
I think that courage is a Real World concept. I can think of occasions when I have observed courage so the concept of courage is realized in my Real World. Courage may not be fully understood but I claim that the concept exists in my Real World.
Concepts like ‘Courage’ and ‘Mother Love’ are tricky and I will leave this kind of stuff for another day.
I say that a counting number “exists” if there is something in the Real World for it to count, that is, if I can conceive of it being realized by a concrete example. The counting number 5 exists in my Real World as a counting number because it is realized by the fingers on my right hand. I can conceive of the counting number four trillion. I can’t picture our national debt but I can conceive of it.
I will say that a set of objects is finite if it is the realization of a counting number.
When is a natural number a counting number? A natural number is a counting number when the natural number can be realized by a finite set of objects in the Real World. That is, when it purports to count something in the physical world.
Are there arbitrarily large natural numbers that are also counting numbers? The answer is, “No”.

My definition of Real World implies that every living organism has its own personal Real World so there is no general consensus on what the Real World includes. This is what happens when you start talking about mysticism. I thank God that I’m not deciding which world She lives in.

Assumption 1. The universe began with a set of identical elementary particles each of which has the same finite, non-zero mass. The universe has to make do with a fixed set of elementary particles; after the beginning it doesn’t get any more elementary particles and it doesn’t lose any.
A particle has to have something for it to exist. I take it that mass and energy are the same thing and if the particle has no mass, in what sense does it exist? I can’t think of any so an elementary particle has mass. I can’t conceive of an elementary particle having ‘infinite mass’; I don’t have a glimmer of an idea of what ‘infinite mass’ could even mean.

(I don’t know that cosmologists think there was just one elementary particle, they think there are quarks at the bottom, but it makes sense to me. If God made the primordial ball, why would She start with two elementary particles when one would do. Certainly we wouldn’t want one of the characteristics of God to be sloppiness. And if She didn’t make the primordial ball, if the ball was “just there”, why would the ball be made of a variety of different particles that just happened to be there?
The beginning of the universe is also the beginning of nature and Mother Nature is well known for her parsimony.)

Assumption 2. All of the objects in the universe are finite sets of elementary particles.
Definition 1. I will say that an object in the universe is finite if the set of particles it contains is the realization of a counting number.

There are several equivalent assumptions I could make at this point
Assumption 3. I assume that the universe began with a finite mass.

Since each particle has an identical, positive mass and the total mass is finite, I conclude that initially there was a counting number of elementary particles. I will denote this number by P which is both a natural number and a counting number.
The natural number associated with the set of all subsets of a finite set (of elementary particles in this case) can be computed. I’m going to denote this natural number by E. All of the objects in the universe are subsets of the elementary particles; the “totality of all the objects in the universe” has an associated counting number, which is also a natural number, less than or equal to E.
Every natural number greater than E is not a counting number; everything has already been counted. Since natural numbers greater than E don’t count anything, they have no concrete realization in the Real World and hence they are in the Mathematical Ideal World but don’t exist in the Real World. So the set of natural numbers that are also counting numbers is bounded. A bounded set of natural numbers has a maximum. This is the largest natural number in the Real World.
Note that since the building blocks of objects are finite in number there can’t be an unbounded set of physical objects.

(I am often asked if a thought is an object since it doesn’t appear to have mass. But a thought is a set of arrangements of particles in the brain so that a thought is a finite collection of finite sets of elementary particles.
Thoughts seem to flow continuously and continuity would seem to imply the existence of infinity. But movies seem to flow continuously and the images on the screen change every 1/32 second.
I think of time as coming in positive quanta so that in the time it takes for a thought, there only a finite number of sets of elementary particles. Over a fixed span of time, humanity has a finite number of thoughts and since everybody seems to agree that humans have only existed for a finite amount of time, humanity has had a finite number of thoughts, period. Eventually humans will all be dead or they will have changed to something else so the span of time that humanity is on the stage is finite. The totality of all human thoughts is finite. Thus the set of things humans can dream up is finite.)

When I first encountered the concept of the natural numbers and infinity, it seemed I was being asked to believe in God. How else could I lose every hand of poker but end up with all the money?
I finally decided that I was playing poker in Wonderland where poker games could go on forever. Unicorns and Medusa and Isis live in Wonderland, not in the Real World. If I can deal with talking caterpillars in Wonderland, I can deal with an unbounded set of natural numbers in the Mathematical World.
My Wonderland seems akin to Plato’s ideal world where the ideal geometric figures reside. The difference seems to be that Platonists believe their ideal squares and what not, actually exist; I don’t know what Plato’s definition of exist is so I’m not sure in what sense they are supposed to exist. Ideal squares are invisible to start with. Its sides are line segments that consist of an infinite number of points. I can’t conceive of a concrete realization of an ideal square. The square I draw is in the Real World, but I can’t conceive of a concrete realization of an ideal square. The jump between the Real World square and the ideal square is too great.
The square I draw is a picture of an ideal square and is like the picture I draw of a unicorn. A picture doesn’t imply the existence of either an ideal square or a unicorn.
Infinity is not in my Real World because I can’t conceive of a concrete realization for infinity.
I don’t allow arbitrarily small numbers in the Real World. For small numbers in the Ideal World I’ll use the reciprocals of natural numbers. I have absolutely no conception of what distance means at the size of Plank’s constant. But I do think that a journey from A to B requires at most a finite number of legs. I think time is like my old grammar school clock, it was 3:29 or 3:30, nothing in between, only the time quanta are much smaller than a minute.
I realize that relativity raises its head when considering distance and time and I have more or less ignored that aspect of space-time. But I don’t see how the properties of relativity would give birth to more elementary particles.