15^th June 2006

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Controversy over the Foundations of Mathematics

 

 

Siamak Zandpour

 

Below I give a brief overview on the Controversy over the Foundations of Mathematics which is today almost forgotten but is of utmost importance from the aspect of the Philosophy of Universal Parallelism. As pointed out in this philosophy, the word and meaning of "infinity" is the most meaningful word in our vocabulary because the more we think and meditate on it, the more we become enlightened. A deep understanding of this word is essential to a deeper understanding of this philosophy.

 

So also the foundation of mathematics revolves around the concept of infinity. In the same way the concept of "infinity"  takes a central part in the above philosophy of  U.P., in the same way it takes the central part in the foundation of mathematics.

 

Cantor's infinite sets theory is a suitable entry point to present this brief outline on the Foundations of Mathematics with  maximum information on  least typing- space. As background information it is sufficient to know that Cantor's work was very influential, leading to point-set topology and other abstract fields in the mathematics of the twentieth century. But it was also very controversial. Some people said, it's theology, it's not real, it's a fantasy world, it has nothing to do with serious math! And Cantor never got a good position and he spent his entire life at a second-rate institution. I refer here to Cantor's infinite sets theory in order to exploit the stark contrast it has to the main qualities of the concept of "infinity" namely:

 

"Infinity is all consuming"

 i.e. it requires everything in order to be infinity.

Nothing can be Added or Subtracted  to Infinity

(or work with several infinities)

There can be only One Infinity

 

A deeper understanding of infinity is required to follow the logic of the  philosophy of  U.P.   This understanding will result in concepts which are of utmost importance and consequences of which most people are not aware and which have a direct relationship to our live philosophy, ideology, religion and so on. All these are explained in an easy language in the above Philosophy of Universal Parallelism.

Before going to Cantor's infinite sets theory here some generalities about infinity is given below:

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Some argue that thinking too much about infinity may make you mad and suicidal. while others think that it is quite liberating.

Some mathematicians  are frightened too think about, and deal with  infinity

Infinity is the background of all mathematics.

Someone said very candidly:  "if you have an infinite amount of monkeys and an infinite amount of type- writers,  than,  with an infinite amount of time the monkeys will type eventually by chance the complete works of Shakespeare."  [ i.e. if there were a finite amount of things to write, then the monkeys, typing  (playing)  randomly must strike on them eventually. Because all words (including punctuation and spaces), are just strings of letters, and all books are just strings of words, then given an infinite time, every possible combination would be exhausted, including all different combinations that make sense to our brain (words).]

Note: In the above comparison we can cancel the  statement "infinite monkeys and  infinite type- writers" and substitute them with just "one monkey and one type- writer"  and we would obtain the same result  because the single statement "infinite amount of time" is in its singularity all-consuming and  its very meaning and quality does not allow to add or subtract something to/ from it.

 

Or we could as well cancel the statement "infinite time" and retain only infinite monkeys with type writers. Here again we obtain the same result with the difference that one of these infinite monkeys would produce by chance all the works of Shakespeare in the very first try. For example we assume that all the works of Shakespeare consist of 30 million characters and spaces. Then with a certainty of hundred percent one of these infinite monkeys would produce all the works of Shakespeare in the very first try of typing 30 million characters.

 

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The below given excerpt was Originally published in C. S. Calude and G. Paun, Finite versus Infinite, Springer-Verlag, 2000, pp. 75-100

Lecture —
A Century of Controversy over the Foundations of Mathematics

…

 

Cantor's  infinite sets theory

 

 

The Crisis in Set Theory

So let me start roughly a hundred years ago, with Cantor...

Georg Cantor

    The point is this. Normally you think that pure mathematics is static, unchanging, perfect, absolutely correct, absolute truth... Right? Physics may be tentative, but math, things are certain there! Well, it turns out that's not exactly the case.

In this century, in this past century there was a lot of controversy over the foundations of mathematics, and how you should do math, and what's right and what isn't right, and what's a valid proof. Blood was almost shed over this... People had terrible fights and ended up in insane asylums over this. It was a fairly serious controversy. This isn't well known, but I think it's an interesting piece of intellectual history.

    More people are aware of the controversy over relativity theory. Einstein was very controversial at first. And then of the controversy over quantum mechanics... These were the two revolutions in the physics of this century. But what's less well known is that there were tremendous revolutions and controversies in pure mathematics too. I'd like to tell you about this. It really all starts in a way from Cantor.

What Cantor did was to invent a theory of infinite sets.

Infinite Sets

    He did it about a hundred years ago; it's really a little more than a hundred years ago. And it was a tremendously revolutionary theory, it was extremely adventurous. Let me tell you why.

Cantor said, let's take 1, 2, 3, ...

1, 2, 3, ...

We've all seen these numbers, right?! And he said, well, let's add an infinite number after this.

1, 2, 3, ... ω

He called it ω, lowercase Greek omega. And then he said, well, why stop here? Let's go on and keep extending the number series.

1, 2, 3, ... ω, ω+1, ω+2, ...

ω plus one, ω plus two, then you go on for an infinite amount of time. And what do you put afterwards? Well, two ω? (Actually, it's ω times two for technical reasons.)

1, 2, 3, ... ω ... 2ω

Then two ω plus one, two ω plus two, two ω plus three, two ω plus four...

1, 2, 3, ... 2ω, 2ω+1, 2ω+2, 2ω+3, 2ω+4, ...

Then you have what? Three ω, four ω, five ω, six ω, ...

1, 2, 3, ... 3ω ... 4ω ... 5ω ... 6ω ...

Well, what will come after all of these? ω squared! Then you keep going, ω squared plus one, ω squared plus six ω plus eight... Okay, you keep going for a long time, and the next interesting thing after ω squared will be? ω cubed! And then you have ω to the fourth, ω to the fifth, and much later?

1, 2, 3, ... ω ... ω² ... ω³ ... ω⁴ ... ω⁵

ω to the ω!

1, 2, 3, ... ω ... ω² ... ω^ω

And then much later it's ω to the ω to the ω an infinite number of times!

1, 2, 3, ... ω ... ω² ... ω^ω ... ω^ωωω...

I think this is usually called epsilon nought.

ε₀ = ω^ωωω...

It's a pretty mind-boggling number! After this point things get a little complicated...

And this was just one little thing that Cantor did as a warm-up exercise for his main stuff, which was measuring the size of infinite sets! It was spectacularly imaginative, and the reactions were extreme. Some people loved what Cantor was doing, and some people thought that he should be put in an insane asylum! In fact he had a nervous breakdown as a result of those criticisms. Cantor's work was very influential, leading to point-set topology and other abstract fields in the mathematics of the twentieth century. But it was also very controversial. Some people said, it's theology, it's not real, it's a fantasy world, it has nothing to do with serious math! And Cantor never got a good position and he spent his entire life at a second-rate institution ...

 

 

 

 

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