What? No Pizza???

Well sure, click.

A letter filched from the website of Manfred Börgens, a mathematician who collects postal souvenirs about mathematics. The letter, mailed in 1977, proudly boasts the meter stamp

FOUR COLORS SUFFICE

from the Mathematics Department of the University of Illinois at Champaign/Urbana, which used a supercomputer to prove the Four-Color Conjecture.

Below the map of the nations of the continent of Euforia on Planet Yobbo are two maps of the departments of Germany. Can you color them with just 3 colors? Can you color them with the proven minimum of 4 colors so no departments sharing a border are the same color?

Maybe you need 5 colors. Go ahead, use all the colors you want.

No. No Pizza for this one.

This is one of the most famous of what I call Mathematical Fairy Tales. Another, which Vleeptron has talked about before, is the 7 Bridges of Konigsberg.

(I'm not using Funny Euro Alphabet Letters anymore because they always show up on the blog and in my e-mail as Wrong Things. I search my Character Map for the correct, literate European rose, and it shows up on the other end looking like my toilet bowl plunger. I'm only using letters that I trust to look like they're supposed to look on the sending and receiving end. If this makes me look ignorant to European eyes -- So Be It. Vleeptron banishes your agues and your graves and your Umlauts, and they will stay banished until somebody fixes them so they work! But anyway, here's two dots .. and they go above the o in Konigsberg, stick them there yourself if you want them. And don't blame me if you get K[toiletplunger]nigsberg on your screen.)

A Math Fairy Tale has two beautiful virtues:

1. The Question is so clear and simple that a bright 10 or 12-year-old can understand it perfectly.

2. All the Ph.D. Math Einsteins, Gausses, Eulers, Newtons, Fermats and Reimanns can't figure out the Answer, sometimes just for centuries, sometimes Never.

And so one of the greatest things about these Fairy Tales is that, for all the years that no one can solve them, they inspire thousands of little kiddies to grow up to become mathematicians -- sometimes great mathematicians.

Andrew Wiles forgot the title of the book he read when he was 14, but it was something like "Every Boy's Book of Mathematical Fun and Puzzles," and in it he first encountered Fermat's Last Theorem.

Everybody knows the Pythagorean Theorem about right triangles:

a² + b² = c²

The sum of the squares of the two short sides always equals the square of the long side.

And in the same lesson, everyone learns that there are gazillions of solutions to this equation where a, b and c are positive whole numbers -- the smallest and most famous one being {3, 4, 5}.

Pierre de Fermat (1601 - 1665) wondered about an ancient and obvious Question

a^n + b^n = c^n
n > 2

where {a, b, c} AND n are ALL positive whole numbers, but n is bigger than 2.

Are there any solutions? Maybe a lifetime of laborious searching might discover something that looks like

32119 ^ 73 + 38014 ^ 73 = 55226 ^ 73

and is True!!! (This one's Not True.)

But if there are no solutions, a lifetime of laborious searching would cripple your pencil hand and make you blind (and crazy), but get you nothing for your labor. No one had ever found even one solution. But that's not the same as saying no solution exists.

Wiles grew up to become a top-tier mathematics professor at Princeton University. But he had a Deep Secret. He never forgot his 14-year-old thrill of encountering Fermat's Last Theorem. And he decided he would find The Answer. He worked on the problem for years in a little study in his attic, without a telephone and without a computer -- just pencil, paper and his wet human brain. He told none of his colleagues what he was doing in his spare time, because after 300 years, it was Standard Received Mathematical Wisdom that anyone who tries to solve FLT is a certifiable nutcase.

Consider trying to solve FLT B.C. and A.C. (Before and After the invention of the high-speed digital Computer). Suppose computers don't exist yet, but you strongly suspect

32119 ^ 73 + 38014 ^ 73 = 55226 ^ 73

is true. (If it's True, then Fermat would be Wrong, and the whole question would be immediately and forever settled.)

Now, with just paper and pencil, compute and verify it. And you better check your work, because just one addition or multiplication mistake makes all your effort worthless -- 50 sheets of toilet paper with numbers scribbled all over it.

Then do it again with your next 200 hunches of whole-number {a, b, c, n} .

Wiles encountered FLT as a B.C. boy, but grew into a professional mathematician after the invention and proliferation of powerful digital computers, and as a Princeton math professor, he would have had easy access to the world's biggest supercomputers.

But Wiles had a strong hunch that Pierre de Fermat was right when he theorized that if n > 2 , THERE ARE NO WHOLE NUMBER SOLUTIONS to

a^n + b^n = c^n

and then Fermat scribbled, in the margin of his father's copy of Diophantus' "Arithmetica" (written around 250 A.D.)

Cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.

(I have a truly marvelous proof of this proposition which this margin is too narrow to contain.)

Scholars had ripped through Fermat's papers for centuries trying to find his Truly Marvelous Proof, but without success. And the greatest mathematicians who studied the problem strongly suspected that his Truly Marvelous Proof, if it existed at all, was flawed.

CLARIFICATION & ADDENDUM: After peer review found some flaws in Wiles' proof, a colleague was able to patch up the flaws, and within about a year, the amended proof was universally accepted as an authentic Answer (and the first in history) to FLT.

But what a proof DOESN'T contain is its own proof that it's the simplest, shortest, or only possible proof of the original question.

There are about ten mathematically distinct proofs of the Pythagorean Theorem (including a unique 1876 proof by US President James Garfield). They all "work," but some are simpler or more elegant than others, and different proofs may prove the original question using entirely different logical and mathematical arguments.

Until someone discovers a different and simpler proof to FLT, scholars wondering about Fermat's Truly Wonderful Proof have to assume that ANY proof would have to involve Wiles' groundbreaking argument centering around elliptic curves. Elliptic curves were not analyzed or mathematically developed until well after Fermat's death. So Fermat, using the math of his day (much of it of his own invention), may have sincerely believed he had a proof -- but he probably didn't, because he didn't have a sufficiently modern grasp of elliptic curves.

But if Fermat's unproven hunch was right, then supercomputers would be useless in trying to prove it. So Wiles never used one. His 1993 proof of Fermat's Last Theorem is long, difficult, and can only be understood by other mathematicians who specialize in some very arcane mathematical topics -- elliptic curves, mainly -- but Wiles' Proof of FLT doesn't use or depend on computers!

~ ~ ~

Surprisingly, these amazing computer thingies which reliably annihilate monstrous volumes of computations without ever stopping to sleep or pee or drink coffee or celebrate religious holidays or rush to the hospital because their kid broke her arm haven't made very many substantial contributions to solving The Big, Tough Mysteries of Mathematics.

It's as if The Big Tough Mysteries always knew computers were coming, and made their Mysteries so profoundly hard that they were computer-proof or highly resistant to assaults by computers. No dumb fucking $3,000,000 box of transistors was going to slay Fermat's Last Theorem or the other great Mathematical Fairy Tales.

These are sucky times for decent human behavior on Planet Earth -- wars, ethnic and religious hatred and violence, human-caused Global Climate Change, bigger and more deadly weapons ...

But the reason you catch me escaping from Planet Earth to my vacation condo near the Shoe Mirrors Underway station in Ciudad Vleeptron, or, as now, fleeing into the Mysteries and Beauties of Mathematics, is that these are very exciting and superPositive times for Human Beings on Planet Earth. In Mathematics, the sentients of Earth can be quite proud.

In Iraq, in "The Holy Land," in Myanmar, in Zimbabwe, in Washington DC, in the Arctic and Antarctic, in Greenland and Brazil and Indonesia, all of us should be deeply ashamed of the way we've pissed on the Earth and harmed and murdered and insulted our sisters and brothers and neighbors.

In 1993, Wiles -- who was afraid to tell anyone he was trying because they'd think he was quite loony -- PROVED Fermat's Last Theorem after about 320 years.

And just a few years before that, in 1976, two math professors and a grad student PROVED this Beautiful Mathematical Fairy Tale about the Colors of All Bordered Entities/Regions/Countries on All Possible geopolitical Maps.

This one didn't resist being solved as long as FLT. In 1852, a math student, Francis Guthrie, was doodling with the colors of a map of the counties of England, and wondered how few colors would do the job. He told his brother, who told a brilliant math professor, Augustus de Morgan.

So it only took 124 years for Earth's best mathematicians to slay this sucker. One day the brilliant mathematician Hermann Minkowski (1864 – 1909) told his Gotingen University grad class that the reason nobody had solved the 4-Color Conjecture is because "only second-rate minds" had worked on the problem. "I believe I can prove it."

A few months later, Minkowski sheepishly told the class: "Heaven is angered by my arrogance; my proof is also defective."

Anyway, the reason I'm not dangling any Pizza about the Map Color Problem is that it's a hugely famous Fairy Tale, and every truly civilized woman, girl, boy and man on Earth SHOULD know all about it -- including its famous and Highly Controversial Proof!

Even Klaas in Rotterdam knows about the 4-Color Conjecture (high-class guess), now called the 4-Color Theorem.

Why controversial?

Because it was the first proof in pure mathematics that relied on the exhaustive computations of a high-speed digital computer. Only a computer could do the enormous volume of computations.

And only ANOTHER computer can verify the first computer's huge proof!

Many mathematicians reject the 1976 proof, and refuse to acknowledge it as a valid mathematical proof.

A mathematical proof may be long and hard and complicated, but these skeptics believe that One Human Being should be able, step by step, to read a valid proof and eventually to understand every step of it. Until this one, all mathematical proofs could be comprehended and contained in one single human being's brain.

As one critic said: A proof should be like a poem. This proof is like a telephone book.

Using the discoveries of a century of earlier investigators, Professors Kenneth Appel and Wolfgang Haken realized that the 4CC could be boiled down to 1,936 special cases, each of which would require complicated analysis and verification. They begged, borrowed and stole a huge amount of CPU time from the supercomputer at the University of Illinois at Champaign/Urbana. Supercomputer time is worth a lot of money, and the supercomputer's owners are very cheap about giving CPU time away. Solving a Map Color Mystery was not high on their priorities for the best way to use the university's precious supercomputer. Like all good scientists and mathematicians, Appel and Haken didn't care and stole the time any way they could get away with it.

For years their math department's outgoing mail boasted

FOUR COLORS SUFFICE