On the intellectual front, I've started having the very strange feeling that it might actually be possible to refute the diagonal method.
From a political perspective, I know that such a thing is impossible. The fundamental dichotomy of transfinite theory exists because the political forces that dominates academic mathematics want such fundamental dichotomies to exist. Transfinite theory is essentially a political statement. It is impossible to refute a political statement.
Perhaps it would be better to refer to it as a religious statement.
Mathematicians love the paradoxes of transfinite theory because it gives them the ability to take a position on the foundations of mathematics. It is a theory that gives an immediate path into discussions of the philosophic.
The whole point of posting a Critique of the Diagonal Method was simply a quick jab at this boil on the face of mathematics, while I worked on the real contents of the site.
My primary hope in making a parody of the diagonal method was to give heart to the millions of students subjected to the method each year. The diagonal method has probably done more to drive people away from mathematics than any other single cause in the history of mathematics.
Transfinite theory is the beating heart of new math. Transfinite theory led directly to the current epidemic in mathematics illiteracy.
The theory is best described as a disease. The power of the dialectical methods of paradox is that, when you build paradoxes into the foundations of reason, anything you want can be true.
Transfinite theory is from the same generation of thinkers and has the same basic form of both communism and Freudian Psychology. You start by proclaiming the existence of a fundamental dichotomy. You stir in a number of profound insights and with dialectics of paradox, you can create an irrefutable intellectual paradox.
From that point forward, there is simply the matter of defending your interpretation of the paradox. Such activities are considered a plus by many thinkers. The process of defending interpretations of the paradox allows you to filter people by their politics. Such dialectical systems seem to be more about rewarding friends and punishing enemies than they are about the development of positive, productive discourse.
You can not refute theories such as Freudian Psychology or Transfinite theory. You simply have to sit and wait for the world to weary of the method.
The problem, of course, is that there is this massive body of literature on the fundamental dichotomy in transfinite theory. Although the very notion of trying to refute a belief system based on the dialectics of paradox is absurd. I wonder if it might be possible to write something that future historians to use as a basis to explain the disease that overtook mathematics in the modern era.
It really appears that people have tired of Freudian Psychology. There also seems to very small number of academics who question the logical foundations of communism.
Anyway, my mind has been burning with the idea of seriously trying to tackle the theory. Absurd, I know.
I recently read The Meaning of it All by Feyman. The classical analytic approaches used in science are so much more pleasing than the dialectical methods used in math and politics. I've read book after book after book on modern theories. It is gasp of fresh air to read to read about the scientific method...which is the absolute height of classical thinking.
The key to quality analytics is uncertainty. You create logical models that can explain phenomena, but must avoid the temptation of certainty...thinking that you have uncovered the invisible forms behind nature.
Set theory is this game where you wrap the whole intellectual universe in a cloud of ink and paradox so that you can feel smuggly secure in your intellectual invicibility.
I would rather live among the scientists who live with the specter that the next experiment just might prove them wrong, than with the mathematicians who have gained irrefutability by building paradox into the foundations of reason.