(E. T. Bell, in Men of Mathematics, tells us that, during the 1920s, Albert Einstein awakened from meditations on unified field theory and cosmology to ask, anent a current controversy within mathematical logic, "What is this frog and mouse battle between the mathematicians?")

The mouse was a German mathematics professor, David Hilbert, who had accomplished many feats of creativity and order along the Stream of Mathematics. On sunny mornings, Hilbert the Mouse would sit amidst patterns of twigs and gleaming pebbles and propound, to the gathered assembly of admiring creatures, "We must use logic to prove the consistency of mathematics, since -- squeek!-I've already shown" (here, he adjoined a squared stone, to norm the sym- metry of a miniature Hilbert space) "that we can map all of mathematics onto arithmetic."

And his hearers, bobbing enthusiastically, would scamper off, crying, "Yes! To work! To work!"

The frog was a Dutch mathematician, Luitzen Egbertus Jan Brouwer. One blue-strewn day, discarding his tadpole tail with a grunt, Brouwer the Frog swam to a lily pad, assumed the lotus position, and delivered, to the shocked passersby on the bank, his doctoral thesis: "The Lord helps those who help themselves! Brek-kek-kek-kek! Hilbert the Mouse and Russell the Rabbit and other logicians and mathematicians think they can cop out in their proofs. Brek! When it's too hard to prove a statement directly, they assume the contradictory of this statement, then seek a contradictory of the contradiction. Broax-broax! They suppose that the Good Lord, like a naive, orderly German professor, or an eccentric English aristocrat, has providently bundled all possible statements into neat pairs of contradictories. And, when any sagacious thinker finds an earthly copy of one of a given pair of statements to be defective, he can forward it to heaven in sacrificial smoke, and the Lord sends back its negation by return zephyr. Brek-kek-kek! Such nonsense! As every sensible frog knows, the Lord is an industrious ruler, who loves a constructive worker with crafty hands. If you wish a proof of a statement, you must construct it, directly, of your own initiative, and then the Lord will approve. Henceforth, I shall accept nothing as rigorously established in mathematics in absence of its construction. We may assume nothing more, except, as Kronecker the Shrew noted, the intuition of the natural numbers, passing endlessly by, like the ripples of this cool, sweet stream."

With that, croaking a loud "Broax!", Briuwer the Frog hopped into the Stream of Mathematics with a splash and swam to the bank. Assuming the lecture position, Brouwer directed the student tadpoles to determine the fix point in the widening circles on the Stream.

When the creatures round about had recovered their wits, they scampered off to tell Hilbert the Mouse of this new heresy. A grasshopper vaulted after them, spitting tobacco juice and declaring: "There's gonna be a fight! Sputt! Betcha the Mouse and the Frog will get into a scrap as soon as they meet! Sputti"

And fight they did! -- verbally, in loud angry tones that reverberated through the Academic Grove, beyond the Stream of Mathematics.

On one occasion, Hilbert squeaked, to the assembly: "There have been two great crises in the Foundations Of Mathematics. First, when the Pythagoreans discovered the irrationality of the square root of two -- or, equivalently, the incommensurability of the diagonal of the square with its side. Second, when the philosophical Bishop, Berkeley the Bear, pointed out the contradiction of dividing by zero, and the associated infinitesimals on which Newton the Lion based his calculus. But Cantor the Goat, by his General Set Theory, has shown us how to resolve these problems, and to dispel crisis from the Foundations of Mathematics! And now we shall go forward to--."

"Don't listen to that quietist propaganda!" boomed a voice from the rushes. "Cantor's set theory attacked those problems only to create a third crisis in the Foundations of Mathematics! Hilbert is quite familiar with the paradoxes of set theory!"

"But they've been taken care of!" squeaked Hilbert, in shrill protest. "And mathematics is the richer for Cantor's General Set Theory, and for the chain of infinite aleph numbers he derived therein. No one," he snapped, whiskers twiddling, eyes blazing, "no one shall drive us from the Paradise Cantor created for us!"

"Broax! and nonsense! You prate constantly about building a neat little pen of axioms to protect the innocent sheep of your doctrinal theses from the wolves of contradiction. But you build by General Set Theory. And, as Poincaré the Otter so aptly asked: What good is it to build a pen for your sheep if you corral the wolves of contradiction inside with them?"

At another time, Brouwer the Frog sat on a lily pad and dispensed the new reformist gospel to a fascinated but uneasy crowd of creatures on the banks of the Stream of Mathematics.

"Hilbert defends the use-in order to prove any mathematical theorem -- of the so-called rule of logic, tertium non datur: Either a statement or its negation is true, with no third possibility. But logic is not a set of commandments handed down from heaven -- or a rock you may skip from one side to another of the Stream of Mathematics. No! Logic is a set of (possibly) useful rules of thought or argument, abstracted from our experience with mathematics -- principally from number theory and arithmetic. As I pointed out in my doctoral thesis, we areallowed to abstract a limited form of tertium non datur from our experience with finite collections, since we know that, should we doubt the conclusion derived by means of this, the conclusion can always be checked out by a finite sequence of trials! But we have no experienced assurance of the safety of this procedure with infinite collections! So we cannot, from tertium non datur, devise an existence (or reductio ad absurdum) proof to posit properties of infinite collections--of which we know so little!"

But Hilbert the Mouse was now jumping up and down on the bank in anguished distress. "To deny tertium non datur, or the law of the excluded middle, to the mathematician is like taking the telescope from the astronomer, or the microscope from the biologist! Don't listen to Brouwer! You've seen what shambles there would remain of mathematics if you try to follow constructivist ?nethods!"

Croaked the Frog: "It wouldn't be shambles! But it also wouldn't be the Cloud-Cuckoo-Land you create with your Axiomn of Choice -- which is the same as the Multiplicative Axiom of Russell the Rabbit or the Lemma of Zorn the Muskrat. You creatures! Do know what delusions you can arrive at by means of the Axiomof Choice?"

"No," they cried. "What?"

"Behold! By appealing to the Axiom of Choice, I can claim that I can dissect the very moon into but five parts, then put it back together and pop it into my mouth -- like this fly! Gulp!"

"Shazammm!" they gasped, "and what is this Axiom, of Choice which counsels us thusly?"

"What is the Axiom of Choice?" boomed the Frog. "Nothing less than a blank check on the Bank of Cardinality. This Axiom assures us that every collection -- be it finite or infinite -- even the set of points in the incomensurable diagonal that deflated the Pythagoreans! -- is as countable as the ripples of this cool, sweet stream!"

"Shazammm!" gasped the creatures. "And how do we do that?"

"Hump! Don't ask Hilbert the Mouse, or Russell the Rabbit, or Zorn the Muskrat, for they don't know how. They claim assurance only by appeal to the kind of reasoning that produced tertium non datur! And -- to be fair -- a little more. They claim that the Lord is their Thesis Adviser and has neatly filed away all finite and infinite collections -- into file drawers. Whenever an example, or counterexample, is required in a proof, they've only to appeal to the Lord, their Thesis Adviser, and He will beneficently retrieve it from a neat file of ordered manila folders and dispatch it to earth -- in the latest mathematical journal."

"Shazamm!" gasped the creatures.

"Shazamm, nothing! Hoakum!"

At this moment, Brouwer filled a great bubble with air in his throat, and boomed it forth in angry explosion. "If a criminal, in committing destroy the only evidence that could be used to convict him -- must I, therefore, accept the verdict that he's innocent? No! Either he can be proven guilty as charged, or innocent of charge, or we rest with the verdict of unproven by existing evidence. Similarly, in mathematics, a thesis can be proven or disproven constructively, or we must declare it unproven. Brek-kek-kek!"

Other proponents of differing schools of thought would quarrel for hours on the banks of the Stream of Mathematics. Thus, argument would break out about Cantor's continuum hypothesis: There is no (infinite) aleph number (cardinal) between the one representing the set of nutural numbers ("stream numbers") and the higher infinite cardinal representing the set of real numbers ("diagonal numbers"), providing for the continuum as foundation for analysis, which includes the calculus. Some believed that this hypothesis must follow from the axioms of logic or of set theory; still others, that the matter could be definitely settled one way or another.

Weyl the Badger, long considered a disciple of Hilbert the Mouse, was so persuaded by the arguments of Brouwer the Frog that he ventured the following pessimistic opinion: "We must learn a new modesty. We have stormed the heavens but have succeeded only in building fog upon a fog, a rnist which will not sup- port anybody who earnestly desires to stand upon it."

Listening to arguments about the issue of constructive proof versus existence proof in geometry, a toad interrupted: "As for me, in my school, we are persuaded by the wisdom of Whitehead the Woodchuck. As you may recall, he said that the negative judgment is the height of perception -- we often know what is not, but rarely what is. Hilbert the Mouse sometimes makes truth searching seem as easy as borrowing a cup of sugar from a neighbor. I doubt if truth is just next door -- or just a few doors down the street!"

EventuaUy, the different facets of the controversy factioned the creatures of the region into many little bands of quarreling doctrinaires: Formalists, Logicists, Intuitionists, Neo-Intuitionists, Finitists, etc. It just wasn't possible to keep up with the issues and actions percolating out of so many storm centers. So, the Grasshopper decided it was time to bring the controversy to focus by staging a great debate between the two principal (and original) adversaries, Hilbert the Mouse and Brouwer the Frog.

On the morning of the great event, all the creatures who lived along the Stream of Mathematics, and many from the neighboring Forest of Logic, were crowded under the great oak tree, in the Academic Grove, to hear the opening arguments.

But the discussion was chaired by the Ant, who announced that his neighbor, the Grasshopper, unexpectedly called away, had asked him to substitute. So they began. Within an hour, the fervor of the two speakers, and the intellectual passion surging through the crowd, set off an angry antiphonal exchange between tne two proponents of totally different philosophies of mathematics.

Suddenly, into the midst of the shouting, there sprang the giddy Grasshopper, ablaze with excitement: "Wait! Hold it! This is even better than a debate! Gödel the Fox has achieved a completely unexpected result: He has proven that you cannot prove the consistency of arithmetic!"

"What?" exclaimed a stunned Hilbert the Mouse.

"Oh, did he really?" boomed a doubtful Brouwer the Frog.

"Yes, he did, Brouwer. Indeed, indeed. Gödel the Fox used methods you cannot quarrel with. You see, the tools of his proof are constructed according to a model he abstracted from the structure of the natural numbers, your paradigm for all of mathematics. And -- Hilbert, my friend -- Gödel proved that any axiomatic system, such as you favor (and use), must either fail to capture all essentials of arithmetic, or else it must already have penned into it at least one of the wolves of contradiction!"

The pervasive silence that smothered the assembly under the oak tree was broken by the voice of the practical Ant: "Then -- the debate is over. So, let us get back to our labors."

One by one, the creatures straggled off, leaving a sad Grasshopper to muse over a fading summer, with so many passionate days.

A voice nearby awoke him to the sunlight and the cheerful chatter of the Stream of Mathematics: "Hi, there, Neighbor. What was the big meeting about? An election?" It was a strange young beaver, come downstream to found a new colony.

"Oh, no, no. Even better than that. Come up here, young fellow, while I show you along our Stream of Mathematics. And, as we go, I'll tell you the amazing story of the battle between the Mouse and the Frog. You see, one day...."

And, out in the Meadow, led by a (classicist mathematical) shepherd, the sheep blea@ted as they ran:

The Lord is my Thesis Adviser; I shall not err.
He arranges for me to be published
  in the respectable journals; he teaches me how
  to use the reductio argument.
He enshores my validity; he leads me by
  the classical logic, for the truth's sake.
Yea, though I walk through the valley of
the existence proofs, I will fear
  no contradiction;
  for he edits my work.
The Axiom of Choice and Zorn's Lemma, they comfort me.

He invites a couoquium on classical analysis,
  for my participation, in the absence
  of the constructivists;
He frequently and approvingly abstracts me
  in Mathematical Reviews;
Surely, honors and grants shall follow me
  all the days of my career,
And I shall rise in the ranks of the Department,