Remarks on The Philosophy of Mathematics

Remarks on The Philosophy of Mathematics

I have been reading “What Is Mathematics, Really?” by Reuben Hersh. This book is about The Philosophy of Mathematics”, I guess. I’m not really sure what The Philosophy of Mathematics is; but I know it when I see it.
My answer to his question is that Real World Mathematics is a tool; Ideal World Mathematics is a game.

After reading Prof. Hersh’s book and thinking about it for a while, I realized that The Philosophy of Mathematics wasn’t much different than Theology, The Philosophy of Religion, and they both study Ideal World Games.
I liken The Philosophy of Mathematics to The Philosophy of Religion and mathematics to religion. There are religionists who work in the trenches trying to make the lives of their flocks make sense on a day to day basis. There are mathematicians who work in the trenches of applied mathematics and trying to understand the Real World and teachers of mathematics trying to pass that knowledge along to the next generation.
And then there are the elites who count angels on the heads of pins and talk about The Trinity; who untie knots in six dimensions and try to well-order the real numbers.

Talking about a god for whom everything is possible is like talking about the set of all sets.
They both claim to seek truth without knowing what truth really means in the Real World and then defining truth in the Ideal World by fiat. They both ask questions that can’t be answered in the Ideal World and have no relevance in the Real World. Paradoxes are in the Ideal Worlds of both religion and mathematics.
Both disciplines build very complicated edifices out of Ideal World bricks, so complicated, indeed, that only the very few, the very elite, can understand the marvelous constructions. Both disciplines develop a shaman class to tell the ignorant masses what they need to know about janitorial jobs.
As a child I took what I heard in Sunday School as truth and was amazed. As a beginning graduate student I took the theorems I studied as truth and was amazed. Disillusion has followed in both cases.

Real World Mathematics is a tool that humanity has developed to help solve the problems of survival, like a scraper or a club. Societies developed the tools that were useful for their particular society and didn’t make tools that were not utile. Tools that are not useful are called objects of art. The wheel was not found useful by the Native American so they didn’t invent it; although I have seen circles in Pre-Columbian art.
Some societies were interested in counting things like money, wives and sheep and numbers were developed to make this sublime goal a reality. They needed to make buildings that didn’t fall down so geometry and a way to measure was developed.
Some societies felt that trees had to be chopped down so they sharpened stones and made axes. In order to chop the trees down faster they developed bronze, then iron, then steel, then chainsaws.
But as swords were developed to be more deadly, they were also developed as an art to be worn in dress parade and not on the battle field.
As axes were developed for a variety of uses, the single bitted axe, the double bitted axe, the hatchet, the machete, the meat cleaver, so was Real World Mathematics.
A machinist works within an error tolerance and the dimensions on blueprints are given ‘plus or minus’. The blueprint is a Real World document that tells how make a Real World object on a Real World lathe. If error tolerances are not given in the blueprint, then it is a Real World document that is a picture of an Ideal World object, like the ceiling of The Sistine Chapel.
A differential equation is a Real World picture of an Ideal World object. A harmonic oscillator is in the Ideal World. x``+x = 0 is a Real World picture that tries to express an Ideal World rock bouncing on an Ideal World spring in an Ideal World vacuum. (Like the ceiling tries to express an Ideal World God giving Ideal World life to an Ideal World Adam.) Does the picture of the Ideal rock and spring give any insight into a Real rock and spring? Experimentation has shown that for some Real rocks and Real springs, it does. Experimentation is still being done to see if the ceiling gives any insight into the origin of life.

Real World Mathematics is a tool. Ideal World Mathematics is a game. It turns out that parts of the game of mathematics model parts of the Real World; not model it exactly but close enough to be useful. How well the model fits the Real World is determined through experimentation. End of story; what’s to get philosophical about?

For example, consider the Banach-Tarski Paradox game. I, the gullible graduate student, was told that Banach and Tarski had proved that you can take a pea, cut it up into a finite number of pieces and reassemble the pieces into a pea the size of a basketball, or the sun for all that matter. Well, that seemed pretty cool.
Now, the way the Greeks found the area of a circle was to approximate that area by the areas of inscribed regular polygons and then approximate the circle’s area by the areas of circumscribed regular polygons. Since regular polygons were made up of triangles, any damn fool could find the area of a triangle, the areas of the triangles could be added together to find the areas of the polygons and get a lower and upper estimation of the circle’s area. If the outer approximations by circumscribed polygons get close to the inner approximations by inscribed polygons as the number of sides of the polygons increases, the Greeks figured that the common value of that the approximations were approaching ought to be the area of the circle. This is a little down and dirty but most high school sophomores can straighten out the demonstration.
The idea is that you approximate the area of a Region by the areas, which you know, of regions that contain the Region; call the number these approximations get close to the outer measure of the Region. Then you approximate the area of the Region by the areas, which you know, of regions contained by the Region; call the number these approximations get close to the inner measure of the Region.
If the inner measure of the Region equals the outer measure of the Region, then the Region has an area equal to the common value. This was the case with a circle.
Does there exist a region whose inner measure doesn’t equal its outer measure? Such a set, if it existed would be said to be unmeasurable. Well, you won’t find one at Wal-Mart but not to worry, mathematicians claim to be able to make them.
To get an unmeasurable region you need to use The Axiom of Choice. Being young and naive, I accepted The Axiom of Choice. What the hell, wasn’t it called an axiom?
As it turns out, The Axiom of Choice is obviously true in the Real World. When it was idealized and made a game piece it was like a football. (The genius of the ovoid shape of a football is that it bounces erratically and not true as a round ball would. This was pointed out to me by Sue Heim nee Smith. In support of my hypothesis that every number is really equal to 12, it was she who pointed out that spiders have 12 legs.)

My examples have been in two dimensions but roughly the same ideas work in three dimensions, where peas and basketballs live.
But if I break the pea up into unmeasurable sets, I have lost control over the size of the unmeasurable component parts so it hardly seems surprising to me, being no longer young nor naive, that I can put the parts together in such a way as to be any size I want. Duh.
I have never gone through a proof of the Banach-Tarski Paradox and my remarks don’t prove anything, but my point is that I am no longer in awe; it seems quite possible, even plausible.
It seems plausible in Wonderland, the world of unmeasurable sets and the Axiom of Choice, because it seems to fit within the rules of Wonderland; even Wonderland has laws which must be obeyed.

Because there are mushrooms and cats in my world, I can see where I might make up a game where the mushrooms were magic and the cats were Cheshires.
Because I can cut up peas and oranges in my world, I can see where I might make up a game where the parts of the peas and oranges were unmeasurable.
We invent a world and its natural laws, objects in the world and rules that govern how the objects interact with the world and each other and voila, we have a game.
Some games are refinements of other games. I think of the progression of games from Wolfenstein to Doom to Doom II to Quake; or the progression of games from the line to the plane to 3-space to finite vector spaces to Hilbert spaces.
Some games are made up and have little or no contact with the Real World, like Tetrus or Banach Algebras.

Another example of a game is Goedel’s Incompleteness Theorem. This theorem says that there are true statements that can’t be proved. On the face of it this seems like a surprising result until you realize that it is part of a game whose rules preclude planting the “provable flag” on some “mountains”. You can never plant the “Nineteen Point” flag on a cribbage hand.
The objects and rules in Goedel’s game are those of a carefully prescribed logic and the natural numbers. Some of these objects are called statements and some of the rules allow connecting statements to make chains. Any statement that is the tail of such a chain is said to be provable.
Some statements that are marked red (false) and some are marked blue (true). The object of the game is to start on a blue statement and build a chain of statements to any other blue statement.
Considering the huge number of blue statements in the version of the game Goedel was playing, it really doesn’t surprise me that there was a blue statement that wasn’t the end of any chain.
I readily admit that Goedel’s accomplishment was a tour de force of gamesmanship; certainly the equal of DiMaggio’s consecutive game hitting streak. I have gone through Goedel’s demonstration and I was impressed at his ingenuity.

I feel that The Philosophy of Mathematics is akin to studying the computer game Civilization and thinking that the game is reality. So why am I talking about it. Am I joining the ranks of those who count angels?
I have a real problem with the idea that Ideal World constructs are real and should be addressed as such. It is true that the Ideal World of Mathematics hasn’t been the justification something as high in body count as The Crusades, or the St. Bartholomew’s Day Massacre, it was used to justify “The New Math” in the 1960’s and early 1970’s.
There is nothing inherently wrong about constructing new games or refining old ones in the Ideal World. I do it often myself. I have several Ideal World theories; that we are devolving back to apes; that existence needs an observer and God was the primal observer; that the extinction of the dinosaurs was a mass suicide due to 100 million years of boredom; that God arose from a stable configuration of fields that occurred in The Big Bang. I have already referred to my theory that every number is really twelve. (See The Calculus: An Opinion; a link can be found at mathematicsteacher.org. Of course, if you are reading this, you know about this link, so I’m telling you something you already know. Is this Real World or what?)
But I don’t wish harm to come to anyone to thinks that the dinosaurs died because of the consequences of a celestial body crashing into the earth.