What We Cannot Know

Explorations at the Edge of Knowledge

Marcus du Sautoy

• Publisher: Fourth Estate
• Year: 2016
• eBook ISBN: 9780007576579
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

SEVENTH EDGE: THE CHRISTMAS CRACKER

13

Number is the ruler of forms and ideas and the cause of gods and demons.

Pythagoras

The statement on the other side
of this card is false

The statement on the other side
of this card is true

Bored of the uninspiring range of Christmas crackers available in the shops, I decided this year to treat my family to my very own home-made mathematical crackers. Each cracker included a mathematical joke and a paradox. My family were of the opinion that the jokes were stronger on the mathematics than the humour. I’ll let you decide … How many mathematicians does it take to change a light bulb? 0.99999 recurring. And if you’re not laughing, don’t worry. My family weren’t either. If you don’t get it, then – although you should never have to explain a joke – the point is that you can prove that 0.9999 recurring is actually the same number as 1.

The paradoxes were a little less lame. One contained a Möbius strip, a seemingly paradoxical geometric object which has only one side. If you take a long strip of paper and twist it before gluing the ends together, the resulting shape has only one side. You can check this by trying to colour the sides: start colouring one side and you soon find you’ve coloured the whole thing. The strip has the surprising property that if you cut it down the middle, it doesn’t come apart into two loops, as you might expect, but remains intact. It’s still a single loop but with two twists in it.

The cracker I ended up with wasn’t too bad, even if I say so myself. Even the joke was quite funny. What does the B in Benoit B. Mandelbrot stand for? Benoit B. Mandelbrot. (If you still aren’t laughing, the thing you’re missing is that Mandelbrot discovered the fractals that featured in my First Edge, those geometric shapes that never get simpler, however much you zoom in on them.) The paradox was one of my all-time favourites. It consisted of the two statements that opened this chapter, one on either side of a card. I’ve always enjoyed and in equal measure been disturbed by word games of this sort. One of my favourite books as a kid was called What Is the Name of This Book? It was stuffed full of crazy word games that often exploited, like the title, the implications of self-reference.

I have learnt not to be surprised by sentences formed in natural language that give rise to paradoxes like the one captured by the circular logic of the two sentences on my cracker card. Just because you can form meaningful sentences doesn’t mean there is a way to assign truth values to the sentences that makes sense.

I think the slippery nature of language is one of the reasons that I was drawn towards the certainties of mathematics, where this sort of ambiguity was not tolerated. But as I shall explain in this Edge, my cracker paradox was used by one of the greatest mathematical logicians of all time, Kurt Gödel, to prove that even my own subject contains true statements about numbers that we will never be able to prove are true.

SCIENCE V. MATHS

This desire for certainty, to know – to really know – was one of the principal reasons that I chose mathematics over the other sciences. In the sciences the things we think we know about the universe are models that match the data. To qualify as a scientific theory they must be models that can be falsified, proved wrong. The reason they survive – if they survive – is that all the evidence supports the model. If we discover new evidence that contradicts the model, we must change the model. By its very nature, a scientific theory is one that can potentially be thrown out. In which case, can we ever truly know we’ve got it right?

We thought the universe was static, but then new discoveries revealed that galaxies are racing away from us. We thought the universe was expanding at a rate that was slowing down, given the drag of gravity. Then we discovered that the expansion was accelerating. We modelled this with the idea of dark energy pushing the universe apart. That model waits to be falsified, even if it gains in credibility as more evidence is collected. Eventually, we may well hit on the right model of the universe, which won’t be rocked by further revelations. But we’ll never know for sure that we have got the right model.

This is one of the exciting things about science: it is constantly evolving – there are always new stories. We can feel rather sorry for the old stories that fade into irrelevance. Of course, the new stories grow out of the old. As a scientist you live with the fear that your theory may be flavour of the moment, winner of prizes, and then suddenly it is superseded. Plum pudding models of the atom, the idea of absolute time ticking away, particles having identifiable positions and momentum: these are no longer top of the science bestseller list. They have been replaced by new stories.

The model of the universe I read about as a schoolkid has been completely rewritten. The same cannot be said of the mathematical theorems I learned. They are as true now as the day I first read them and as true as the day they were first discovered. Sometimes that day was as long as 2000 years ago. As an insecure spotty teenager, I found the certainty it promised particularly attractive. That’s not to say that mathematics is static. It is constantly growing as unknowns become known, but those knowns remain known and robust, and become the first pages in the next great story. Why is the process of attaining mathematical truth so different from that faced by the scientist who can never really know?

The all-important ingredient in the mathematician’s cupboard is proof.

PROOF: THE PATH TO TRUTH

There is evidence of people doing maths as far back as the second millennium BC. Clay tablets in Babylon and papyri in Egypt show sophisticated calculations and puzzles being solved: estimates for π; the formula for a volume of a pyramid; algorithms being applied to solve quadratic equations. But in general these documents tell of procedures that can be applied to particular problems to derive solutions. We don’t find justification for why these procedures should always work, beyond the convincing evidence that it’s worked in the thousands of examples that have been documented in the clay tablets to date. Mathematical knowledge was based on experience and had a more scientific flavour to it. Procedures were adapted if a problem cropped up that wasn’t amenable to the current algorithm.

Then around the fifth century BC things began to change as the ancient Greeks got their teeth into the subject. The algorithms come with arguments to justify why they will always do what it says on the tin or tablet. It isn’t simply that it’s worked the last thousand times, so it will probably work the next time: the argument explains why the proposal will always work. The idea of proof was born.

Thales of Miletus is credited with being the first known author of a mathematical proof. He proved that if you take any point on the circumference of a circle and join that point to the two ends of a diagonal across the circle, then the angle you’ve created is an exact right angle. It doesn’t matter which circle you choose or what point on the circle you take, the angle is always a right angle. Not approximately, and not because it seems to work for all the examples you’ve drawn. But because it is a consequence of the properties of circles and lines.

Thales’ proof takes a reader from things they are confident are true and by a clever series of logical moves arrives at this new point of knowledge, which is not one that you would necessarily think obvious just from looking at a circle. The trick is to draw a third line going from the initial point B on the circle to the centre of the circle at O.

Why does this help? Now you have two triangles with two sides of equal length. This means that in both triangles the two angles opposite the centre of the circle are equal. This is something that has already been proved about such triangles. Take the larger triangle you originally drew. Its angles add up to 2α + 2β. Combine this with knowledge that a triangle has angles adding up to 180 degrees, and we know that α + β must be 90 degrees, just as Thales asserted.

When I first saw this proof as a kid it gave me a real thrill. I could see from the pictures that the angle on the circle’s edge looked like a right angle. But how could I know for sure? My mind searched for some reason why it must be true. And then as I turned the page and saw the third line Thales drew to the centre of the circle, and took in the logical implications that flowed from that, I suddenly understood with thunderous clarity why that angle must indeed be 90 degrees.

Note that already in this proof you see how the mathematical edifice is raised on top of things that have already been proved, things like the angles in a triangle adding up to 180 degrees. Thales’ discovery in turn becomes a new block on which to build the next layer of the mathematical edifice.

Thales’ proof is one of the many to appear in Euclid’s Elements, the book which many regard as the template for what mathematics and proof are all about. The book begins with basic building blocks, axioms, statements of geometry that seem so blindingly obvious that you are prepared to accept them as secure foundations on which to start building logical arguments.

The idea of proof wasn’t created in isolation, out of nowhere. Rather, it emerged alongside a new style of writing that developed in ancient Greece. The art of rhetoric, as formulated by the likes of Aristotle, provided a new form of discourse aimed at persuading an audience. Whether in a legal context or political setting, or simply the narrative of a story, audiences were taken on a logical journey as the speaker attempted to convince listeners of his position. The mathematics of Egypt and Babylon grew out of the building and measuring of the new cities growing up round the Euphrates and Nile. This new need for logic and rhetorical argument emerged from the political institutions of the flourishing city-states at the heart of the Greek empire.

Rhetoric for Aristotle was a combination of pure logic and methods designed to work on the emotions of the audience. Mathematical proof taps into the first of these. But proof is also about storytelling. And this is why the development of proof at this time and place was probably as much to do with the sophisticated narratives constructed by the likes of the dramatists Sophocles and Euripides, as with the philosophical dialogues of Aristotle and Plato.

In turn, the mathematical explorations of the Greeks moved beyond functional algorithms for building and surveying to surprising discoveries that were more like mathematical tales to excite a reader.

A proof is a logical story that takes a reader from a place they know to a new, unvisited destination. Like Frodo’s adventures in Tolkien’s Lord of the Rings, a proof is a description of the journey from the Shire to Mordor. Within the boundaries of the familiar land of the Shire are the axioms of mathematics, the self-evident truths about numbers, together with those propositions that have already been proved. This is the setting for the beginning of the quest. The journey from this home territory is bound by the rules of mathematical deduction, like the legitimate moves of a chess piece, prescribing the steps you are permitted to take through this world. At times you arrive at what looks like an impasse and need to take a lateral step, moving sideways or even backwards to find a way around. Sometimes you need to wait for new mathematical characters like imaginary numbers or the calculus to be created so you can continue your journey. The proof is the story of the journey and the map charting the coordinates of that journey. The mathematician’s log.

To earn its place in the mathematical canon it isn’t enough that the journey produce a true statement about numbers or geometry. It needs to surprise, delight, move the reader. It should contain drama and jeopardy. Mathematics is distinct from the collection of true statements about numbers, just as literature is not the set of all possible combinations of words or music all possible combinations of notes. Mathematics involves aesthetic judgement and choice. And this is probably why the art of the mathematical proof grew out of a period when the act of storytelling was flourishing. Proof probably owes as much to the pathos of Aristotle’s rhetoric as to its logos.

NUMBERS AT THE EDGE

While many of the first geometric proofs are constructive, the ancient Greeks also used their new mathematical tools to prove that certain things are impossible, beyond knowledge. The discovery, as we saw, that the square root of 2 cannot be written as a fraction is a striking example.

The proof has a very narrative quality to it, taking the reader on a journey under the assumption that the length can be written as a fraction. As the story innocently unfolds, one gets sucked further and further down the rabbit hole until finally a completely absurd conclusion is reached: odd numbers are even and vice versa. The moral of the tale is that the imaginary fraction representing this length must be an illusion. (If you would like to take a trip down the rabbit hole, the story is reproduced in the box on page 370.)

For those who encountered a number like the square root of 2 for the first time, it must have seemed like something that by its nature was beyond full knowledge. To know a number was to write it down, to express it in terms of numbers you knew. But here was a number that seemed to defy any attempt to record its value.

Proof of the irrationality of the square root of 2

Let L be the length of the hypotenuse of a right-angled triangle whose short sides are both of length 1 unit. Pythagoras’ theorem implies that a square placed on the hypotenuse will have the same area as the sum of the area of squares placed on the two smaller sides. But these smaller squares each have area 1 while the larger square has area L². Therefore L is a number which when you square it gives you 2.

Suppose L is actually equal to a fraction L = p/q.

We can assume that one of p or q is odd. If they are both even we can keep dividing both top and bottom by 2 until one of the numbers becomes odd.

Since L² = 2, it follows that p²/q² = 2.

Multiply both sides by q²: p² = 2 × q².

So is p odd or even? Well p² is even so p must be even because odd times odd is still odd. So p = 2 × n for some number n. Since p is even that means that q must be the odd number. But hold on …

2 × q² = p² = (2 × n)² = 2 × 2 × n²,

so we can divide both sides of the equation by 2 to get

q² = 2 × n².

Remember that we’d worked out that q was the odd number.

So q² is also odd. But the right-hand side of this equation is an even number! So if L is expressible as a fraction it would imply that odd = even. That’s clearly absurd, so our original assumption that L can be written as a fraction must have been false.

I think this is one of the most stunning proofs in mathematics. With just a finite piece of logical argument we have shown there is a length that needs infinity to express it.

It was an extraordinary moment in the history of mathematics: the creation of a genuinely new sort of number. You could have taken the position that the equation x² = 2 doesn’t have any solutions. At the time, the numbers that were known could not solve this equation precisely. Indeed, it was really only in the nineteenth century that sufficiently sophisticated mathematics was developed to make sense of such a number. And yet you felt it did exist. You could see it – there it was, the length of the side of a ...

Table of contents

• Title Page
• Copyright
• Dedication
• Contents
• Edge Zero: The Known Unknowns
• First Edge: The Casino Dice
• Second Edge: The Cello
• Third Edge: The Pot of Uranium
• Fourth Edge: The Cut-Out Universe
• Fifth Edge: The Wristwatch
• Sixth Edge: The Chatbot App
• Seventh Edge: The Christmas Cracker
• Further Reading
• Index
• Acknowledgements
• Illustration Credits
• Also by Marcus du Sautoy
• About the Publisher