first draft...i need to add references.
Paradoxes of the infinite have plagued thinkers from ancient times to the present. One of the most notable encounters with the infinite came with the realization by members of the Pythagorean Society that the square root of two could not be expressed as a ratio of two numbers. Pythagoras lived from aproximately 540 BC to 522 BC.
Other philosophers tried their brains against the infinite, and tried to construct large logical edifaces to contain the infinite, but usually failed. Zenos of Elea made great fun of the atomists by pointing out paradoxes that took several millennia to resolve.
After the debacle of Pythagorean numerology and with the mounting number of known paradoxes, Greek mathematicians chose to pursue the study of geometry.
Aristotle, Eudoxes and others put the issue to rest largely by making the infinite taboo. Concentrated on the more fruitful field of finite logic. When they employed arguments about infinity, they were extremely careful to separate the concept of a potential and an actual (or completed) infinity. They automatically rejected any argument that employed the latter.
Contrary to popular beliefs, Aristotle did not stop the world. A large number of great minds contemplated the infinite. In his Treatise on Two World Systems, Galileo put forward the supposition that, since you won't run out of integers before you run out of squares the two sets must be the same size.
The most important contribution to the study came in the 1600s. The Calculus developed by Isaac Newton and Leibniz made implied use of the infinite, although neither approaches succeeded in creating a strong logic foundation for their work. So here was a problem: The calculus worked astoundingly well, but it lacked the rigorous logical foundation of traditional geometry.
Mathematicians, one after an other, threw their minds against this problem. For the most part they simply got caught in the same paradoxes and absurdities discovered by the ancient Greeks.
<< Need to add Michael Rolles bio here. >>
The controversies on the subject finally began to abate when Augustin Louis Cauchy developed rigorous definitions for limits, the derivative and continuity. Quite frankly, I think mathematicians were tired of the endless debates on infinity. Much of the resistence Cantor felt toward his work was the negative reactions mathematicians had to the inane arguments made on the subject in the past.
Transfinite theory is not only tied up in the history of the infinite. It plays an important role in many other branches of mathematics, logic and philosophy.
Cantor was a disciple of Immanual Kant, and like Schopenhaur, Hegel et al, he was eager to topple the ediface of Aristotlean logic and replace it with the new German Kantian style dialectic.
Transfinite theory is intimately tied to set theory. Basically, Cantor refined the principles of modern set theory in his attempts to develop a logical foundation for his definition of the real numbers and transfinite theory.
Gotlobb Frege borrowed Cantor's work on set theory and transfinite theory, and used it as a foundation for a grandeous treatise on the foundation of mathematics. A treatise that he had hoped would serve as both the complete and final foundations of both mathematics and logic.
Historians like to portray Cantor and Frege as the hapless victims of evil the evil designs of Henry Poincare and Leopold Kronecker. In reality Cantor and company were performing a blitzreig on the sciences. They were actively trying to topple the last two thousand years of evolved thinking, and replace it with a new language and dialectic based upon the transfinite.
Cantor was trying to flip the entire world of math and logic on its head. Before his work, logicians were content with discrete mathematics, and relegated the paradoxes of the infinite to the fringes of the subject.
The followers of Cantor bulldozed two thousand years of evolved thought and replace it with a new dialectic that placed incomensurables and the infinite at the foundations of all mathematics and logic.
Notice how the Principea Mathematica addresses the development of the Cardinal Numbers before the Ordinals. Transfinite theory has flip flopped our thinking. We are studying the infinite before the finite.
Notice also that introduction of new math into schools ended the study of logical discourse.
Hilbert may have called Cantor's work a new paradise. But you have to understand the paradise that was thrown away in its making. Yes, the strip mall is a shopping paradise, but the farm it replaced had its merits as well.
The ultimate irony is that, despite the fact that Cantor's followers ultimately toppled the old guard, they failed to rid transfinite theory of its paradoxes. They simply created a new crisis for the world of mathematics. The paradoxes that were once on the fringe of mathematics now sit squarely the middle.
Transfinite theory, set theory and new math all have an extremely interesting history. In studying the history you can see clearly the conflicts between ideas that occurred throughout the centuries.
Richard Dawkins would speak about the different theories as mimes with different survival strategies. If you stand on the sideline, you will see a wonderful show of gladiators battling in our minds for control of our thoughts and sentiments.
You will find references in hundreds of books about the persecution Cantor by Kronecker. However, the proponents of transfinite theory have been as, if not more vicious in the defense of their territory than Kronecker. The life of LEJ Brouer reads like the life of Georg Cantor, but in reverse.
Today the meme of transfinite theory is actually pitted against a foe far more dangerous than the occasional dissenter. It is faced against the ubiquitous forces of the personal computer.
Computers are more in line with the world view of Kronecker. Computers with finite memory cannot hold infinite decimals. As powerful as our machines might be, they are confined to this restrictive world of discrete mathematics. Computer can handle arbitrarily large numbers, and their operators are more than content to stay with in those bounds.
Computers are bringing back discrete mathematics and classical logic. Why do we need to go through all the hoops needed to discuss the transfinite and the analog world when we have an even greater power at our finger tips.
All that is left for transfinitist to make some desparate and pathetic plea that their theory is the true foundation of logic.
But, no, people know that thoughts evolve. Ideas will continue to battle for a place in our minds. What will decide the battle is not some pathetic plea that transfinite theory is a high truth than classical mathematics. What will win out is the theory that is of the most benefit to its host.
The convolutions of transfinite theory make it a poor parasite, and if evolution is kind, it will be cast aside as Poincare as a simple disease.
Of course, I will make a further prediction. Thoughts will continue to evolve. but maybe I can give them a push.
Index - - Cure? - - Rich