Infinity Revisited

Infinity Revisited


As a child I seldom talked to adults in a conversational way but almost always in an information gathering mode. I had the naive belief that all the information I gathered from adults was accurate and the fact that it was not was often a reflection of the ignorance shared by the planet at large. So, when I learned about infinity it was much like learning about God in Sunday School. They both did things that defied my understanding but must be true because adults said they were.
I went through graduate school thinking that the infinite number of infinities were real in some way. I didn’t question their existence; I concentrated on how infinity behaved. Zorn’s Lemma was as real as a bacterium.
I will not dwell on the process that made me realize that the bacterium was Real and that infinity was not but I did make that realization. I am an infinity skeptic.

I suppose that the idea of infinity is an abstraction of “a whole lot".
I think of abstraction as a process that humans participate in and it is not necessarily harmful. I think that the abstraction from cow pie to manure to that which "happens" is useful.
That which is abstract to one person will be concrete to another. The concept associated with “one" is concrete to most animals if only because an individual animal forms a set exemplifying that concept. Most human animals find 1,2,3,4,5,6,7,8,9, and10 concrete. I suppose 1,2,3,4,5,6,7, and 8 are concrete to an octopus. I have read that crows count one, two, many. Homer says that the Greek troops at Troy were as many as the leaves on the trees or as many as the grains of sand on the beach; which are poetic ways of saying, “There were many Greeks.”
Natural numbers have varying degrees of abstraction and the taint of abstraction does not necessarily bar a number from the Real World. Four trillion is pretty abstract. I can’t picture 4 trillion bricks but I can conceive of 4 trillion bricks. My inability to count to 4 trillion lies in design flaws in the construction of my brain; the fault lies in me, dear Brutus, not in the 4 trillion.
By Arithmetic I mean classical arithmetic; I mean the arithmetic that uses the Natural Numbers of the Peano Axioms, the arithmetic that allows an infinite (as defined in the Ideal World) collection of Natural Numbers. If I say that the Natural Numbers form an ‘infinite set’ I mean that there is an unlimited supply of them; no matter how many I have, I can always find more.
Arithmetic doesn’t exist in the same world as stock markets, wars, starvation and Miami Beach but stock markets, wars, starvation and Miami Beach find that a finite part of it useful.



I think that there is no infinity so that the number of things in the universe can be counted and there is a largest counting number. Natural numbers larger than the largest counting number are in the Ideal World. The real numbers are all infinite decimals and hence in the Ideal World. “What about 2.0?” you say. 2.0 is really 2.000… . Every one of those zeros is necessary and should be included with the …. There are those who consider ½ and 1/3 as Real Numbers in which case they are not in the Real World. If I think of a half and a third as real numbers then I think of 0.500….and 0.333… Rational numbers are real numbers whose infinite decimals repeat. ½ and 1/3 are fractions, that is, ratios of Counting Numbers or proportions; they are in the Real World.
The Natural Numbers are used to count, the fractions give proportions and the Real Numbers measure. We picture the Real Numbers as the real line which has an uncountable infinity of points. None of the Real Numbers, however you define them, say Cauchy sequences, are in the Real World.
1.000… = 0.999… is a real number in the Ideal World and not in the Real World. 1 is a Counting Number and in the Real World.
These are two different 1’s, and the same symbol is often used for both; just as ½ is used for the proportion and the Real Number 0.50...
(Don’t blame me for the confusion; I didn’t design the number system.)
There is no board that is exactly 1 unit long. This ‘1’ is not counting anything. The standard meter in Paris is not exact. There is no concrete realization of any Real Number in the Real World and this is because of the stochastic nature of the Real World. No concrete realization, not in the Real World.









Remark
There are some interesting attempts at conceptualizing very large integers. In India there is the length of time it takes to make a cubic mile of wool if one fiber is added every century.
The one I like best goes as follows:
I define an ‘a’ inside a triangle to equal ‘a’ raised to the power ‘a’. So a 2 inside a triangle equals 2*2 = 4.
I define an ‘a’ inside of a square to equal an ‘a’ inside of ‘a’ triangles. So a 2 inside of a square would equal a 2 inside of 2 triangles. The 2 inside the inner triangle equals 4 and we are left with a 4 inside of a triangle = 4*4 = 256.
I define an ‘a’ inside of a pentagon to be an ‘a’ inside of ‘a’ squares. So, a 2 inside of a pentagon equals a two inside of 2 squares. The ‘a’ inside the inner square equals 256 so we have 256 inside a square which equals 256 inside of 256 triangles. At this point I stop. This number is so large that I can’t comprehend it. The only way I can write it is 256 inside a square.
When I first read this scheme (The Mathematical Experience by Phillip Davis and Reuben Hersh) I took out a piece of paper, drew a pentagon and wrote a 2 inside of it. I stared at the marks I had put on the paper, my mind a blank. I drew a hexagon and wrote a two inside of it. It was like looking at an unexploded bomb. I turned the paper over and left the room.
I think that a 2 inside a hexagon does not exist in the Real World because it is too big. It exists in the made up world of mathematics. Putting a 2 inside of a hexagon doesn’t mean that it represents a number in the Real World any more than a picture of a unicorn represents a Real World animal.

Remark
Why can’t I add one to the largest Natural Number in the Real World and get a larger Natural Number in the Real World? The old “you can always add 1” ploy. There may be no largest Natural Number in the Ideal World but there is in the world I live in, the Real World. I’m going to shorten the symbol, 2 inside a hexagon, to 2-H. I think that 2-H is an upper bound of the counting numbers considered as a subset of the Natural Numbers.
All Natural Numbers (which are in the Ideal World) that are bigger that 2-H are not in the Real World. I hypothesize that the largest Counting Number is less than 2-H. For good measure make it 2-M, a 2 inside a megagon.
The denominator of a proportion can’t be larger than the Counting Number associated with the set of all subsets of the elementary particles. The denominator of a proportion is a Counting Number that counts the sample space which must be less than the Counting Number that counts EVERYTHING.
This idea of being able to add 1 and get a bigger Natural Number comes from the standard ways of defining the Natural Numbers, say the Peano axioms, which implicitly assume that the Natural Numbers are unbounded. I can always add ‘1’ in the Ideal World but not in the Real World. If I add one to the largest Natural Number in the Real World, I get a Natural Number that doesn’t count anything and is therefore not in the Real World.
I have been considering large Natural Numbers that have left their good homes in the Real World. What about numbers, in particular Fractions, that are very small? How small can they get and still be in the Real World? I haven’t used “zero” because I don’t know there is a “zero”.



I always cringed when I let Δ x get arbitrarily small in a calculus class. It was especially inappropriate if x represented a number of light bulbs. In the Ideal World I can let Δ x get arbitrarily small without a qualm. But in the Real World there is a smallest number. The reciprocal of 2-M is smaller than anything in the Real World (This needs a proof.).

If Achilles is trying to catch a hare in the Real World by cutting the distance between them into approximately two equal pieces, there comes a point where the distance can’t be cut in half. There is no ‘in between’ the hare and Achilles.

Zero is a little tricky. Being broke is a Real World concept and if I say I have 0 dollars, I’m saying that I have nothing to count. I see 0 as short hand for “I got plenty of nothin’”.
But zero as a distance between two objects doesn’t exist in the Real World. There are no instants of time whose duration is zero. It may seem that the distance between the sole of your shoe and the floor is 0.00..., but it’s not. The distance is small but greater than zero. 0.00... considered as a Real Number isn’t in the Real World because in the Real World there is a smallest positive rational number. In the Real World there is no ‘nothing’. There is no concrete realization of the Real Number 0.00... in the Real World. There is a Rational Number, considered as a Real Number, that is smaller than the mass of an elementary particle, that is smaller than a quantum of distance, that is smaller than a quantum of time and so there is nothing small enough for this or any smaller Rational Number (considered as a Real Number) to measure, nothing small enough to be a concrete realization of it in the Real World.
The symbol 0 in the Mathematical World has the same relation to small Rational Numbers in the Ideal World as the symbol ∞has to large Natural Numbers.

Remark
Since I have cast ∞and 0 out of the real world I have effectively tossed out continuity as well.
Since the eye was the premier measuring device for much of the development of the basic physical ideas and motion ‘looks’ continuous it is not surprising that continuity was assumed. But movies ‘look’ continuous even though movie time passes in 1/32 second quanta. I can be on one side of a fence in one frame, on the other side in the next frame which violates continuity.

Remark
Suppose I drop a rock to the floor and want to describe, to model, the motion mathematically.
The picture I see is that of two strings of beads. One string has the time quanta beads and the other string has the path beads, i.e. the distance quanta beads on the path that leads from my hand to the floor. As the rock falls, the path beads light up indicating where the rock is and time beads light up indicating when it got there. Note that the path beads are distance quanta beads and are dependent on how the time beads are lighting up. The falling rock connects path beads to time beads to make velocity beads. The relation between time and distance is what I want to use in the description of the rock’s motion. I pick a time bead and when it lights up I’ll note the path bead that lights up and connect the string between them.
Could more than one path bead light up when the time bead lights up, leaving me in a quandary as to where to tie the string? But if more than one path bead lit up, the rock would be at more than one place at the same time. So I suppose that only one path bead lights up and in my model I can run a string from each time bead to exactly one path bead.
Could strings run from several time beads to one distance bead? The problem with this possibility is that if, say, four consecutive time beads lit and only one distance bead it would mean that the rock had stopped for those four time beads and I don’t want a model where the rock stops for four time beads on the way down to the floor..
So when I make my model, I’m going to tie a string from exactly one time bead to exactly one path bead. As the rock falls to the floor, all the time beads and all the path beads light up. There is a 1 to 1 relation between all time beads and all path beads. Velocity is the ratio of the length of a path bead to the length of a time bead.

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My point is that motion, like time and distance, comes in quanta.
This discrete model can be made to work, more or less. It has to be made to work because we live in the Real World which is discrete.
I read where an applied mathematician commented that in his universe the only numbers that existed were finite decimals that fit in a computer and I took his comment to heart.
When the number π is used, the author is leaving it for someone else to drag it kicking and screaming into the Real World. Nobody in the Real World can use π in their Real World computations. You don’t build an altar bowl using π, you use 3.
In a way, my Real World is the right one by default. The only numbers ever used in a computation are my Real World numbers. We all live in an everyday world that is finite.

Back to the Model of the Falling Rock

How can I keep my strongly felt intuition that the rock doesn’t stop and can’t be in two places at once? How can I make a model of the falling rock that keeps the 1 to 1 relation between time beads and path beads? .



I can bop over to the Ideal World and use infinite sets. I model a path where the path beads have reduced to points and the time beads have reduced to instants. I model in a world where paths have an infinite number of points; intervals of time have an infinite number of instants.
But I have lost velocity. I can't take the ratio of the length of a point to the length of an instant. I will leave this for the derivative to fix.

While this model may not be in my Real World, it comes close enough for government work and to my way of thinking it’s a whole lot easier than the discrete model. I can visually see the Ideal World model by drawing “unicorn” pictures often called graphs.
I don’t say that modeling in the Ideal World isn’t elegant or doesn’t give results that are useful in my Real World. Basic physics is modeled in the Ideal World and has given Real World numbers that have worked well enough to get to my Real World moon. But no matter how elegant the formulae, at the end of it all somebody has to set a dial to a number that is a finite decimal small enough to fit on the display.

Remark
I am forced to consider what difference it makes whether infinity is in the Real World or not. Infinity is used to solve Real World problems and the answers so obtained seem to give reasonable answers, so who cares if Infinity is Real or not?
I’m not sure what does make a difference. Does it make a difference to a farmer whether the earth turns or the sun rises?
Does it make a difference that √2 is not in the Real World? I guess it makes a difference in the sense that it would be a very different Real Universe if √2 were in it.