I realize that my last post was very contemptuous. Temerity might be a better description of amateur daring to speak of what many consider to be the greatest triumph of modern thought...transfinite theory.
More than once, people have told me that I am a contemptuous little snit who does not recognize his betters. Most of the comments I have received about investigations into the subject point out that I am too small and petty of an individual to even utter the great words “diagonal method.”
The theory was not intended for the common man or for small minded rabble who drown their pettiness in pubs (people like me). It was meant for that small tiny transcendental elite who flutter in the upper heavens of thought and spirituality
My actions do wreak of contempt. When I speak of transfinite theory, I feel the part of an uncouth gentile soiling sacred temple grounds.
In truth, I do have a great deal of respect for the time and effort that scholars apply to their respective fields of study. I’ve always thought that it is more likely that I am wrong in my beliefs than it is likely that the great mathematicians of the modern era were misled. This assumption that I am wrong in my thoughts is a driving force in both my academic and professional life.
The idea that I am wrong drives my own personal investigations.
What I do in my research in start with the assumption that I am wrong. I assume that the great mathematicians who hold to the theory are right. Somehow, set theory really is the hidden forms that Plato saw when exited the cave of temporal illusion. I assume that I just somehow misunderstood the diagonal method.
The way I would approach the issue is to put the theory aside for several months or years. I would then pick up a few books on Set Theory. I would read through, doing exercises, etc.. I am usually impressed with the astute logic and structure of discrete set theory. Frege’s propositional and prepositional logics are a grand contribution in the history of thought.
Then I would get into the sections on transfinite theory. All of my old objections would flood back into mind...generally the objections would be added with new objections. I find myself wanting to take an even stronger stand against transfinite theory, transcendental logic and all of the silliness of the modern tradition.
Anyway, during the last iterations of this process. My mind was burning on the issue of how one could rid mathematics of this idea that mathematics is driven by a fundamental dichotomy.
A year ago, I lighted on the idea of creating something called “Rich Theory.”
Transfinite theory essentially says that there are only two layers of infinity. It starts with the claim that all denumerable sets are the same size. The diagonal method creates a set which is larger than this denumerable infinity. Transfinite theory holds the following. It holds that n = n+1 where n is infinity, 2*n = n, when n is infinity and n^2 = n when n is infinity. However, it states that 2^n is greater than n.
Rich Theory notes that n < n+1 < 2n < n^2 < 2^n when n is a large number. We can actually imaging each of these operations creating a different level of infinity. Essentially, infinity is a rich space.
For various reasons that will take a long time to explain, what we see in infinity is largely a reflection of what we see in the finite. When we look at complex issues in the finite world, we see that there’s a large number of different things going on (wheels within wheels). Transfinite Theory’s effort to sum up mathematics with a simple dichotomy between denumerable and nondenumerable was as off base as Marx’s attempt to sum up the economy in terms of a single conflict between the bourgeoisie and proletarian classes.
Transfinite theory does not open a paradise. It impoverishes a paradise. It wants to reduce the fantastically complex nature of infinity into two piles...denumerable and nondenumerable.
Rich Theory says that infinity is rich. We can find all sorts of wonderful things reflected in the study of infinite sets.
Calling this view of infinity “rich” is really a mean thing.
The theory was not satisfactory because it did not address the reason that attracted attention in the first place. Mathematicians want a way to address the important distinction between discrete and continuous mathematics.
So, my mind had been racing on how the issue of the distinction between discrete and continuous mathematics. The distinction of course, pretty much falls from definitions. Discrete mathematics is reduced to finite elements. Continuous mathematics allows of the use of infinite length entities.
My current thinking is that an approach that the best way to approach transfinite theory would be a discussion of binary strings. Such a course could present better and clearer models for discussing the infinite. Likewise it would fit in well with computer science.
Which, of course, brings me back to the problem that I am a contemptuous little snit.
There is a possibility that I have a good idea that could help students understand mathematics. The problem is that, even if I have a good idea, my being a contemptuous little snit immediately nullifies anything that I might say.
In classical mathematics, there would be a hope that serious mathematicians could look beyond my personality defects. If they found something worthwhile, they would run with it and discard the inane things I say.
Unfortunately, we live in the world of modern mathematics. Modern mathematics is driven by who a person is and not what they say. The Hegelian heart of modern mathematics has the world spirit realized in the writings of a few super intellectuals (philosopher kings).
The very act of speaking, researching or trying to understand mathematics is an act of contempt.
The modern world has created a culture of silence. In the days of classical mathematics, you saw people in all walks of life engaged in the discourse of mathematics.
The set theoretic model discards input from people judged as lesser men. They drove the classical mathematicians and scholastic scholars from the schools. Today all mathematics and logic is written in terse, arcane symbolic form. Commoners are simply to worship our transfinite betters and stay silent. To even speak or think is an act of contempt.
It is a trap that is extremely difficult to break.