Fermat's Enigma

by Simon Singh

He realized that numbers exist independently of the tangible world and therefore their study was untainted by the inaccuracies of perception. This meant he could discover truths that were independent of opinion or prejudice and that were more absolute than any previous knowledge.

He hoped to find a plentiful supply of freethinking students who could help him develop radical new philosophies, but during his absence the tyrant Polycrates had turned the once liberal Samos into an intolerant and conservative society.

Pythagoras was also intrigued by the link between numbers and nature. He realized that natural phenomena are governed by laws, and that these laws could be described by mathematical equations. One of the first links he discovered was the fundamental relationship between the harmony of music and the harmony of numbers.

Pythagoras had uncovered for the first time the mathematical rule that governs a physical phenomenon and demonstrated that there was a fundamental relationship between mathematics and science. Ever since this discovery scientists have searched for the mathematical rules that appear to govern every single physical process and have found that numbers crop up in all manner of natural phenomena.

Pythagoras realized that numbers were hidden in everything, from the harmonies of music to the orbits of planets, and this led him to proclaim that “Everything Is Number.”

By exploring the meaning of mathematics, Pythagoras was developing the language that would enable him and others to describe the nature of the universe. Henceforth each breakthrough in mathematics would give scientists the vocabulary they needed to better explain the phenomena around them. In fact developments in mathematics would inspire revolutions in science.

“Although this may seem a paradox, all exact science is dominated by the idea of approximation.” Even the most widely accepted scientific “proofs” always have a small element of doubt in them. Sometimes this doubt diminishes, although it never disappears completely, while on other occasions the proof is ultimately shown to be wrong. This weakness in scientific proof leads to scientific revolutions in which one theory that was assumed to be correct is replaced with another theory, which may be merely a refinement of the original theory, or which may be a complete contradiction.

Science is operated according to the judicial system. A theory is assumed to be true if there is enough evidence to prove it “beyond all reasonable doubt.” On the other hand, mathematics does not rely on evidence from fallible experimentation, but it is built on infallible logic.

“Do you know,” the Devil confided, “not even the best mathematicians on other planets—all far ahead of yours—have solved it? Why, there’s a chap on Saturn—he looks something like a mushroom on stilts—who solves partial differential equations mentally; and even he’s given up.” Arthur Porges, “The Devil and Simon Flagg”

Three centuries later Bertrand Russell would comment on this apparent oxymoron: “How dare we speak of the laws of chance? Is not chance the antithesis of all law?”

Probability problems are sometimes controversial because the mathematical answer, the true answer, is often contrary to what intuition might suggest

A talent for analyzing probability should be part of our genetic make-up, and yet often our intuition misleads us.

He then argued that the possible prize of eternal happiness has an infinite value and that the probability of entering heaven by leading a virtuous life, no matter how small, is certainly finite. Therefore, according to Pascal’s definition, religion was a game of infinite excitement and one worth playing, because multiplying an infinite prize by a finite probability results in infinity.

In particular Euclid exploited a logical weapon known as reductio ad absurdum, or proof by contradiction. The approach revolves around the perverse idea of trying to prove that a theorem is true by first assuming that the theorem is false. The mathematician then explores the logical consequences of the theorem being false. At some point along the chain of logic there is a contradiction (e.g., 2 + 2 = 5). Mathematics abhors a contradiction, and therefore the original theorem cannot be false, i.e., it must be true.

“Reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.”

The riddle’s status has gone beyond the closed world of mathematics. In 1958 it even made its way into a Faustian tale. An anthology entitled Deals with the Devil contains a short story written by Arthur Porges. In “The Devil and Simon Flagg” the Devil asks Simon Flagg to set him a question. If the Devil succeeds in answering it within twenty-four hours then he takes Simon’s soul, but if he fails then he must give Simon $100,000. Simon poses the question: “Is Fermat’s Last Theorem correct?”

One of Euler’s greatest achievements was the development of the algorithmic method. The point of Euler’s algorithms was to tackle apparently impossible problems. One such problem was predicting the phases of the moon far into the future with high accuracy—information that could be used to draw up vital navigation tables.

But how can something that is undeniably smaller than an infinite quantity also be infinite? At the beginning of the twentieth century the German mathematician David Hilbert said: “The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite.”

Hilbert’s Hotel seems to suggest that all infinities are as large as each other, because various infinities seem to be able to squeeze into the same infinite hotel—the infinity of even numbers can be matched up and compared with the infinity of all counting numbers. However, some infinities are indeed bigger than others. For example, any attempt to pair every rational number with every irrational number ends in failure, and in fact it can be proved that the infinite set of irrational numbers is larger than the infinite set of rational numbers. Mathematicians have had to develop a whole system of nomenclature to deal with the varying scales of infinity, and conjuring with these concepts is one of today’s hottest topics.

“Logic is the hygiene the mathematician practices to keep his ideas healthy and strong.”

Hilbert believed that everything in mathematics could and should be proved from the basic axioms. The result of this would be to demonstrate conclusively the two most important elements of the mathematical system. First, mathematics should, at least in theory, be able to answer every single question—this is the same ethos of completeness that had in the past demanded the invention of new numbers like the negatives and the imaginaries. Second, mathematics should be free of inconsistencies—that is to say, having shown that a statement is true by one method, it should not be possible to show that the same statement is false via another method. Hilbert was convinced that, by assuming just a few axioms, it would be possible to answer any imaginable mathematical question without fear of contradiction.

Then in 1931 an unknown twenty-five-year-old mathematician published a paper that would forever destroy Hilbert’s hopes. Kurt Gödel would force mathematicians to accept that mathematics could never be logically perfect, and implicit in his works was the idea that problems like Fermat’s Last Theorem might even be impossible to solve.

Gödel had proved that trying to create a complete and consistent mathematical system was an impossible task.

Just four years before Gödel published his work on un decidability, the German physicist Werner Heisenberg uncovered the uncertainty principle. Just as there was a fundamental limit to what theorems mathematicians could prove, Heisenberg showed that there was a fundamental limit to what properties physicists could measure.

Pierre de Fermat’s casual jotting in the margin of Diophantus’s Arithmetica had led to the most infuriating riddle in history. Despite three centuries of glorious failure and Gödel’s suggestion that they might be hunting for a nonexistent proof, some mathematicians continued to be attracted to the problem.

A problem worthy of attack Proves its worth by fighting back. Piet Hein

The proof was a gigantic argument, intricately constructed from hundreds of mathematical calculations glued together by thousands of logical links. If just one of the calculations was flawed or if one of the links became unstuck then the entire proof was potentially worthless.