MATHEGENESIS

THE BOOK OF MATHEMATICAL FOUNDATIONS

In the beginning there was Cantor, and he gave us Ordinals and Cardinals and numerable and non denumerable sets. And the multitude looked on in wonder and marveled, for the vistas opened up were uncountable. But Kronecker cried out, "This is sin and wickedness, God has made the integers, all else is the work of the competition." And some heeded him, but the multitude was beguiled by the infinite pleasures of Cantor's new theory, and they mocked Kronecker, and drove him into the wilderness.

And it came to pass that in those days there was a man named Frege, a most wondrously subtle and wise man. And he said, behold, I have seen a vision. It has been revealed unto me that all mathematics and all reason are but a species of logic. And he built a mighty palace that reached up into the heavens without limit. And within that palace there were no end of wondrous things, including contradictions.

And the Lord said, I will cast down this palace, for it is not mete that a mortal should thus storm the heavens. And the Lord called unto him a man named Russell, and said unto him, "Behold, this man Frege hath displeased me, for he hath made much out of nothing. Thou art my instrument to cast him down. I shall arm you with a dox, nay with a pair of dox, and set you forth". And Russell set forth with his paradox and Frege was cast down.

The multitude saw this and lamented, for they loath to leave garden of paradise of absolute infinities, yet the garden was infested with paradox.

And Russell saw this and felt shamed for he had destroyed much and had built nothing. So he set about to restore the palace of Frege, free of paradox, wherein the Garden of Cantor might be placed. And he labored mightily, and brought forth a new palace. And he guarded against paradox by all manner of walls and barriers, types and orders.

Now there came unto Russell a Spanish Barber, saying, "Thy work is wondrous, though somewhat tedious, but where is this marvelous garden that Cantor gave us." And Russell replied, "Behold, I have added a path within the palace from which all can be reached, for it is of infinite length." And a great cry went up, for this path was a bit hard to swallow.

Whilst Russell was raising the palace that he had razed, a new prophet arose, saying, "Cantor gave unto a thing of mathematics, and Frege hath made of it a thing of logic. Let us hark unto the vision of Cantor." And he said, "The false prophet, Frege, gave us law. And he said it was good, for law is surely good. But the dox, both of them, hast shown that Frege's law was false law. Now I say that Cantor was a man of truth, and his vision was true. But truth needs law, as law needs truth. So let us lay down law for Cantor's truth." And he did. And his name was Zermelo.

Now Zermelo was a man of great order, yea beyond all measure. But some saw that he was arbitrary in the ways he chose to attain that order, and they rejected his order.

And it came to pass that the spirit of Kronecker whispered unto two men and prevailed on them to take on the quest of restoring the spirit of Mathematics. And each did so, each in his own way, taking separate paths. And their names were Hilbert and Brouwer, and the names of their paths were Formalism and Intuitionism. And they labored mightily, and they traveled far. But in the end, nothing much came of it.

The law of Zermelo took hold in the land, and the people praised it, Theologians formulated it and elaborated it and tested it, and created all manner of axiomatic set theories. And their names were Frankel, and Bernays, and Skolem.

But Skolem was a holy man, and there came unto him a vision. And he saw that the great garden of Cantor was an illusion, a great shadow cast by countable models. And he preached that it was so. And the multitude said, yes it is so, and we will make it so in our proof theory. But their agreement was with their lips and not in their hearts. And they said, "None-the-less the continuum is not countable", and they hearkened unto Platonism.

Now the Lord looked down, and he saw that many believed that Man is the measure of all things, and that all problems could be solved. And the Lord grew wroth, for this sort of thing was lese majeste, and He called unto Him a man named Goedel, and He armed him with theorems, and He set Goedel forth to preach.

And Goedel preached unto the multitude, saying "Whatever thou knowest, if it be truly worth knowing, thou canst never be sure of". And he said unto them, "For every truth that thou canst prove, there is another that thou cannot prove." And the multitude was abashed and sore perplexed.

In the wilderness of Harvard there came a man named Quine. And he surveyed the ruins of the palace that Russell had built on the ruins of the palace that Frege had built. And he said, "I can repair this mess." And he did. And he called his labors, New Foundations, and he cast them unto the multitude. And the multitude said, "Can we do Mathematics in the ways of our Fathers?" And Quine said, "Not exactly." And the multitude asked, "Can we follow your law and the law of Zermelo at the same time?". And Quine replied, "Not completely." And the multitude rejected Quine, and Quine returned to the wilderness of Harvard and wrote books.

The spirit of Kronecker was wroth, for the multitude followed the law of Zermelo and frolicked in the garden of Cantor. So he appeared unto a man named Turing, and a man named Church, saying unto them, "The multitude hath donned all manner of fantastic garments, but they can do no more than their forefathers did, for the integers are all there really is." And Turing and Church hearkened unto the spirit of Kronecker and they devised theories of computability. And Church laid down a great Thesis, saying that all theories of computability are equivalent, and that what ye may do is forever limited. And he laid down a universal law prescribing what Man may or may not do. And he shewed that the great law could never be proven.

And thus it came to pass that inconsistency was removed from set theory in many divers and inconsistent ways. And the Lord looked upon this and saw that it was good. And the multitude wrote papers and taught courses and garnered grants, and saw that that was good too.

interlude

I rather suppose that most of the humor of this little piece will be lost on my the majority of my readers -- if you aren't a mathematician, take my word for it that it is quite amusing to mathematicians with a sense of humor. That, by the way, is not an oxymoron. Mathematicians often have a lively sense of humor. Rather recondite, but lively.


This page was last updated July 3, 1996