Fermat’s Last Theorem: The compelling biography and history of mathematical intellectual endeavour

by Simon Singh

focus and determination that is hard to imagine.

intention of founding a school devoted to the study of philosophy and in particular concerned with research into his newly acquired mathematical rules.

It is the same with life. Some are influenced by the love of wealth while others are blindly led on by the mad fever for power and domination, but the finest type of man gives himself up to discovering the meaning and purpose of life itself. He seeks to uncover the secrets of nature. This is the man I call a philosopher for although no man is completely wise in all respects, he can love wisdom as the key to nature’s secrets.

The very latest quantum theories of gravity are tied to the development of mathematical strings, a theory in which the geometrical and topological properties of tubes seem to best explain the forces of nature.

The idea of a classic mathematical proof is to begin with a series of axioms, statements which can be assumed to be true or which are self-evidently true. Then by arguing logically, step by step, it is possible to arrive at a conclusion. If the axioms are correct and the logic is flawless, then the conclusion will be undeniable. This conclusion is the theorem.

Cylon.

there are an infinite number of Pythagorean triples.

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

Calculus is the ability to calculate the rate of change, known as the derivative, of one quantity with respect to another.

Economics is a subject heavily influenced by calculus. Inflation is the rate of change of price, known as the derivative of price, and furthermore economists are often interested in the rate of change of inflation, known as the second derivative of price.

While calculus has since been used to send rockets to the moon, and while probability theory has been used for risk assessment by insurance companies, Fermat’s greatest love was for a subject which is largely useless – the theory of numbers.

Ptolemy

Ptolemy’s dream of building a treasure house of knowledge lived on after his death, and by the time a few more Ptolemys had ascended the throne the Library contained over 600,000 books. Mathematicians could learn everything in the known world by studying at Alexandria, and there to teach them were the most famous academics. The first head of the mathematics department was none other than Euclid.

Euclid devoted much of his life to writing the Elements, the most successful textbook in history. Until this century it was also the second best-selling book in the world after the Bible.

approximation.

reductio ad absurdum

Mark Antony realised that the way to an intellectual’s heart is via her library,

The Hindus recognised that zero had an independent existence beyond the mere spacing role among the other numbers – zero was a number in its own right. It represented a quantity of nothing. For the first time the abstract concept of nothingness had been given a tangible symbolic representation.

Once we have defined i as being the square root of –1, then 2i must exist, because this would be the sum of i plus i (as well as being the square root of –4). Similarly i⁄2 must exist because this is the result of dividing i by 2. By performing simple operations it is possible to achieve an imaginary equivalent of every so-called real number. There are imaginary counting numbers, imaginary negative numbers, imaginary fractions and imaginary irrationals.

Figure 10. The introduction of an axis for imaginary numbers turns the number line into a number plane. Any combination of real and imaginary numbers has a position on the number plane.

The first woman known to have made an impact on the subject was Theano in the sixth century BC, who began as one of Pythagoras’ students before becoming one of his foremost disciples and eventually marrying him. Pythagoras is known as the ‘feminist philosopher’ because he actively encouraged women scholars, Theano being just one of the twenty-eight sisters in the Pythagorean Brotherhood.

the enchanting charms of this sublime science reveal themselves only to those who have the courage to go deeply into it.

In mathematics a property which always holds true no matter what is done to the object is called an invariant.

The challenge was to match the creativity with which Fermat used his techniques.

These fundamental assumptions are the axioms of mathematics. One example of the axioms is the commutative law of addition, which simply states that, for any numbers m and n, m + n = n + m

First, mathematics should, at least in theory, be able to answer every single question – this is the same ethos of completeness

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.

Principia Mathematica as a guide for establishing a flawless mathematical edifice, and by the time Hilbert retired in 1930 he felt confident that mathematics was well on the road to recovery.

Then in 1931 an unknown twenty-five-year-old mathematician published a paper which would forever destroy Hilbert’s hopes. Kurt Gödel would force mathematicians to accept that mathematics could never be logically perfect,

First theorem of undecidability If axiomatic set theory is consistent, there exist theorems which can neither be proved or disproved. Second theorem of undecidability There is no constructive procedure which will prove axiomatic theory to be consistent.

Decades later, in Portraits from Memory, Bertrand Russell reflected on his reaction to Gödel’s discovery: I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure.

Fortunately, as with Russell’s paradox and the tale of the librarian, Gödel’s first theorem can be illustrated with another logical analogy due to Epimenides and known as the Cretan paradox, or liar’s paradox. Epimenides was a Cretan who exclaimed: ‘I am a liar!’

This statement does not have any proof. If the statement were false then the statement would be provable, but this would contradict the statement. Therefore the statement must be true in order to avoid the contradiction. However, although the statement is true it cannot be proven, because this statement (which we now know to be true) says so. Because Gödel could translate the above statement into mathematical notation, he was able to demonstrate that there existed statements in mathematics which are true but which could never be proven to be true, so-called undecidable statements.

Heisenberg showed that there was a fundamental limit to what properties physicists could measure. For example, if they wanted to measure the exact position of an object, then they could measure the object’s velocity with only relatively poor accuracy. This is because in order to measure the position of the object it would be necessary to illuminate it with photons of light, but to pinpoint its exact locality the photons of light would have to have enormous energy. However, if the object is being bombarded by high-energy photons its own velocity will be affected and becomes inherently uncertain.

In 1944 John von Neumann co-wrote the book The Theory of Games and Economic Behavior, in which he coined the term game theory. Game theory was von Neumann’s attempt to use mathematics to describe the structure of games and how humans play them. He began by studying chess and poker, and then went on to try and model more sophisticated games such as economics.

It used to be said that the First World War was the chemists’ war and that the Second World War was the physicists’ war. In fact, from the information revealed in recent decades, it is probably true to say that the Second World War was also the mathematicians’ war – and in the case of a third world war their contribution would be even more critical.

You have to really think about nothing but that problem – just concentrate on it. Then you stop. Afterwards there seems to be a kind of period of relaxation during which the subconscious appears to take over and it’s during that time that some new insight comes.’

instead I would go for a walk down by the lake. When I’m walking I find I can concentrate my mind on one very particular aspect of a problem, focusing on it completely. I’d always have a pencil and paper ready, so if I had an idea I could sit down at a bench and start scribbling away.’

Wiles’s lectures at the Isaac Newton Institute

‘The first seven years that I worked on this problem I enjoyed the private combat. No matter how hard it had been, no matter how insurmountable things seemed, I was engaged in my favourite problem. It was my childhood passion, I just couldn’t put it down, I didn’t want to leave it for a moment. Then I’d spoken about it publicly, and in speaking about it there was actually a certain sense of loss. It was a very mixed emotion.

‘It was so indescribably beautiful; it was so simple and so elegant.

‘I had this very rare privilege of being able to pursue in my adult life what had been my childhood dream. I know it’s a rare privilege, but if you can tackle something in adult life that means that much to you, then it’s more rewarding than anything imaginable.