Henri Poincaré called transfinite theory a disease—a disease he hoped would someday be cured. I must confess to having a certain amount of sympathy for his point of view, leaving the question of exactly how one would go about curing such an ailment of the mind.
Classical logicians of Poincaré's ilk would think that simply highlighting the paradoxes inherent in transfinite theory would be sufficient. However, classical logicians don't quite understand the sentiments of the modern man. From what I have seen, it is the very presence of such entrenched paradoxes that make the theory so attractive to many thinkers. Quite frankly, when you have a widely accepted mathematical theory that boldly proclaims "n^2 = 2^n" at infinity, then, you can basically do anything. The ability to justify any argument is the paradise that Georg Cantor created.
I admit. It is fun to write proofs that 1=2. It is fun to say: "this sentence is false." It is fun, but counterproductive. As I am writing this article, I am listening to news reports on the convolutions used by the accountants at Enron to mask debt and hide risks in a complex web of 5000 partnerships and subtle redefinitions of terms. When the game unraveled, the investors and employees lost everything. I have seen this same gamesmanship go on in many different companies. I've watched in horror as companies shifted assets and liabilities to pull strategic bankruptcies on their creditors. I have lost money on stocks in companies that cooked the books.
There are people in this world who love to mesh logic around and redefine terms. Such tactics can bring power in politics and money in business. Basing mathematics on the paradox ridden world of transfinite theory feeds this gamesmanship. Unfortunately, such dialectical gamesmanship is at best a zero sum game. It simply shifts wealth from the people who thought they understood the game to those with the power to redefine terms. The risk shifting games played by Enron simply removed the ability of the investor to understand the actual risks of their investment. Judging from the dramatic crash of Enron. All this redefining of liabilities and risk shifting wasn't just a zero sum game. It was a negative sum game.
A second negative effect of transfinite theory is that by positioning transfinite theory as the basis arithmetic and logic, transfinite theorist have accomplished the unintended goal of basing mathematics on mystical concepts: namely, the completed infinity. Such a result is highly attractive to the mystical mind. Again, classical logicians would abhor the idea of basing reasoning on mystical concepts, but the mystical mind adores it...gaining further acceptance for transfinite theory.
This leads to one of the weirdest twists in modern mathematics: The intuitionalist school, led by LEJ Brouwer actually demanded a higher degree of logical rigor than the Logicists school that was happy with a mathematics founded on the slippery slope of transfinite theory, completed infinities and paradoxes. Of course, we have seen the same twists in politics. The "people's" party is generally a pseudonym for a dictatorship. Tax reform almost always leads to higher taxes. Intuitionism in mathematics does not mean that people rely on their "intuition." Intuitionist school demands that all axioms be based on comprehendible ideas. The logistical schools holds that the axioms don't really matter, so long as symbolic logic used is consistent. The ideas expressed by the symbols can be insane.
A third negative result of transfinite theory is that, by replacing the syllogistic reasoning of Aristotle with set theory, transfinite theorists have created a culture where people are less skilled at analyzing the arguments of politicians, and are more willing to accept political views that separate people into classes and sets. It is not a coincidence that the generations in Germany and Russia that adopted transfinite theory also produced the most vile dictatorships in history.
In my opinion, the fact that schools no longer teach logic is the worst effect of transfinite theory. The democracy of the United States was based upon the idea that voters1 would analyze arguments, and not simply divide the country into classes and constituencies.
Once again the people want to build their power base by dividing the population into competing classes throw further acceptance to the transfinite theory over classical mathematics.
Classicists in the ilk of Poincaré, Kronecker and Brouwer misunderstood the sentiments of the "modern man" Pointing out the paradoxes and mysticism inherent in transfinite theory means nothing because it is the paradoxes and mysticisms that attract the audience. Transfinite theory gained it main adherents after Russell had shown Frege the problems with the reflexive paradox.
Curing the disease will take a different tack. It involves creating other ideas which can encapsulate transfinite theory while putting traditional logic back on track. Unfortunately, such a feat is beyond my meager means, but I might suggest the following direction. I would create a new theory based on large sets. I have penned the beginning of such an approach in the article titled rich theory. Rich theory is based upon the rich patterns found in arbitrarily large sets. Large sets can be immense, but they are not infinite.
By studying these sets, I believe that we can create a theory which can account for the symbolic manipulations of the Logicists, while creating a stronger intuitional understanding of the real numbers.
Of course, the real cure for transfinite theory has shaped up in a different discipline altogether. The true cure for transfinite theory is this magnificent little sitting before me today. In computer programming, we are restricted to the use of finite logic. As more and more people learn that he art and logic of computer programming, they will be less inclined to accept the mysticism of transfinite theory. Database programmers who are accustomed to defining one to many relations between sets have a harder time accepting that one to many relations turn into one to one correspondences.
Leopold Kronecker is often criticized for his constructivist stands, however, computing devices are limited to the mechanical logics championed by Kronecker.
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