Introducing Infinity E-Book

Brian Clegg

• Herausgeber: Icon Books
• Kategorie: Wissenschaft und neue Technologien
• Serie: Graphic Guides
• Sprache: Englisch

Seitenzahl: 108

Veröffentlichungsjahr: 2014

Beschreibung

Infinity is a profoundly counter-intuitive and brain-twisting subject that has inspired some great thinkers – and provoked and shocked others. The ancient Greeks were so horrified by the implications of an endless number that they drowned the man who gave away the secret. And a German mathematician was driven mad by the repercussions of his discovery of transfinite numbers. Brian Clegg and Oliver Pugh's brilliant graphic tour of infinity features a cast of characters ranging from Archimedes and Pythagoras to al-Khwarizmi, Fibonacci, Galileo, Newton, Leibniz, Cantor, Venn, Gödel and Mandelbrot, and shows how infinity has challenged the finest minds of science and mathematics. Prepare to enter a world of paradox.

Leseprobe

Published by Icon Books Ltd, Omnibus Business Centre, 39–41 North Road, London N7 9DP Email: [email protected]

ISBN: 978-184831-406-1

Text copyright © 2014 Icon Books Ltd

Illustrations copyright © 2014 Icon Books Ltd

The author and illustrator has asserted their moral rights

Originating editor: Richard Appignanesi

No part of this book may be reproduced in any form, or by any means, without prior permission in writing from the publisher.

Contents

Cover

Title Page

Copyright

Big numbers

Googoled

Symbols from India

The Book of Calculation

0, a powerful tool

Archimedes: The Sand Reckoner

The poetry of infinity

Number sequences

Strange sequences

The infinity machine

Zeno’s paradoxes

Achilles and the tortoise

Apeiron

Aristotle

Potential infinity

Left brain/right brain

The power of algebra

Visual thinking

Pythagorean perfection

Diagonals of a square

Drowning by numbers

Squaring the circle

Transcendental pi

Infinity of pi

Omega

Not really numbers at all?

God and the infinite

The human perspective

“Only a manner of speaking”

Galileo

Infinity on wheels

Back to geometry

The normal rules don’t apply

The infinity of 1?

A common error

The indivisibles

Newton and potential infinity

Fluxions

From o to 0

Leibniz’s calculus

Newton vs. Leibniz

Notation

Differential and integral calculus

Battling Bishop Berkeley

The infidel mathematician

Dividing zero by zero

Flow and change

Tending towards zero

Finding a symbol

The Möbius strip and Klein bottle

Bolzano and real infinity

Cantor: mind-bending infinity

The joy of sets

Venn diagrams

Boolean algebra

Making sets of the world

Peano and the cardinals

Russell’s paradox

Cantor and subsets

Imaginary numbers

Aleph null

Cardinals and ordinals

Ordinal infinity

Countably infinite

Cantor’s elegant proof

Covering the number line

Another Cantor proof

Points in space

The shock of the infinite

Power sets

Cantor under attack

Cantor succumbs

Gödel’s shocking proof

Back to the continuum hypothesis

Does infinity exist?

Fractal infinity

Recursion

Fractals in nature

Measuring the coastline

The Cantor set

An infinite universe?

Edge of the universe?

Quantum infinity

The slit experiment

Spin

The infinitesimal

Non-standard analysis

Infinitesimals and Brownian motion

Hilbert’s Hotel

Gabriel’s Horn

The jungle of infinity

Glossary

Further Reading

Author’s acknowledgements

Artist’s acknowledgements

About the Authors

Index

Big numbers

Infinity, as no end of people will tell you, is a big subject. It will take you into history, philosophy and the physical world, but is best first approached through mathematics. It makes sense to ease into it via big numbers.

By giving a lengthy number a name you seem to demonstrate your power over it – and the bigger the number is, the more impressive your ability. This is reflected in the reported early life of Gautama Buddha. As part of his testing as a young man in an attempt to win the hand of Gopa, Gautama was required to name numbers up to a huge, totally worthless value. Not only did he succeed, but he carried on to bigger numbers still.

100,000,000,000,000,000? EASY, THAT’S ACHOBYA.

Googoled

It’s fine giving names to numbers we encounter every day, but how many of us will ever use this number?

As it happens, it does have a name, one that proved a problem for the unfortunate Major Charles Ingram when it was his million-pound question on TV show Who Wants to be a Millionaire? He was asked if the number – 1 with 100 noughts after it – was a “googol”, a “megatron”, a “gigabit” or a “nanomol”. Major Ingram favoured the last of these, until a cough from the audience prompted him towards googol. To be honest, who can blame him? “Googol” sounds childish.

Googol is childish – for a good reason. In 1938, according to legend, mathematician Ed Kasner was working on some numbers on his blackboard at home. His nephew, nine-year-old Milton Sirrota, was visiting. Young Milton spotted the biggest number and is supposed to have said: “That looks like a googol!”

This isn’t a very convincing story, though. There’s no reason why Kasner would bother to write such a number on a blackboard.

WHAT WOULD YOU CALL A REALLY, REALLY BIG NUMBER (SAY 1 WITH 100 NOUGHTS AFTER IT)? A GOOGOL!

Symbols from India

To deal with any number we need symbols that represent numerical values. The symbol equivalents of the words “one”, “two”, “three” and so on (1, 2, 3…) arrived in the West from India via the Arabic world. The oldest known ancestors of the modern system were found in caves and on coins around Bombay dating back to the 1st century AD.

The numbers 1 to 3 were based on a line, two lines and three lines, like horizontal Roman numerals, though they can still be seen with some imagination in the main strokes of our modern numbers. The markings for 4 to 9 are closer ancestors of the symbols we use today.

The Indian symbols were adopted in the Arabic world, coming to the West in the 13th century thanks to two books, written by a philosopher in Baghdad and a traveller from Pisa. The earlier book, lost in the Arabic original, was written by al–Khwarizmi (c. 780–850) in the 9th century. The Latin translation of this, Algo-ritmi de numero Indorum, was produced around 300 years later, and is thought to have been considerably modified in the process.

The version of al-Khwarizmi’s name in the title is usually given as the origin of the term “algorithm”, though it’s sometimes linked to the Greek word for number, arithmos.

The Book of Calculation

The traveller from Pisa was Leonardo Fibonacci (c. 1170–1250). (His father, a Pisan diplomat, was Guglielmo Bonacci, and “Fibonacci” is a contraction of filius Bonacci, son of Bonacci.) He travelled widely in North Africa and became the foremost mathematician of his time, his name inevitably linked to the Fibonacci numbers (see here), which he popularized but didn’t discover. Although Numero Indorum was translated into Latin a little before Fibonacci’s book Liber abaci came out in 1202, it seems that Liber abaci (“The Book of Calculation”) had the bigger influence in introducing the Indian system to the West.

ON MY TRAVELS I WAS INTRODUCED TO THE ART OF THE INDIAN’S NINE SYMBOLS.

0, a powerful tool

The symbols we use for numbers are arbitrary. ¶, β, √, π, ԓ would do as well as 1, 2, 3, 4, 5. However, the new Indian numerals brought with them a very powerful tool. Earlier systems from Babylonian through to Roman were tallies, sequential marks to count objects. We’re most familiar with Roman numerals – the tally sequence is obvious in I, II, III, IV, V – where V is effectively a crossed through set of IIII and IV is one less than V. But the trouble with such systems is that there’s no obvious mechanism to add, say, XIV to XXI.

THE NEW SYSTEM USED COLUMNS WITH A PLACE-HOLDER O FOR EMPTY SPACES, TRANSFORMING ARITHMETIC.

Archimedes: The Sand Reckoner

But whatever symbols are used, big numbers kept their appeal. In a book called The Sand Reckoner, ancient Greek philosopher Archimedes (c. 287-212 BC) demonstrated to King Gelon of Syracuse that he could estimate the number of grains of sand it would take to fill the universe.

We don’t know a lot about Archimedes, but we do have a number of his books, which show him to be a superb mathematician and a practical engineer. He is said to have devised defence weapons for Syracuse ranging from ship-grabbing cranes to vast metal mirrors to focus sunlight and set ships on fire.

Unlike many of Archimedes’ other works, The Sand Reckoner wasn’t exactly practical. But there was a serious point behind this entertaining exercise. What Archimedes set out to do was to show how the Greek number system, which ran out at a myriad myriads (100 million), could be extended without limit. He first estimated the size of the universe at around 1,800 million kilometres (just outside the orbit of Saturn).

“UNIVERSE” IS THE NAME GIVEN BY MOST ASTRONOMERS TO THE SPHERE WHOSE CENTRE IS THE CENTRE OF THE EARTH.

He then decided how many sand grains are needed to be the size of a poppy seed, how many of these fit in a sphere of finger’s breadth, and so on up to fill the universe, using his newly designed system. His final count suggested that the universe should hold around 1051 sand grains (1 with 51 zeros after it).

Tantalizingly, Archimedes also mentions the work of the philosopher Aristarchus, who had written a book (now lost) that put the Sun at the centre of the universe rather than the Earth. Archimedes calculated the size of this universe too, which he made significantly bigger than the traditional model. Aristarchus’ fanciful Sun-centred universe would hold around 1063 grains.

THERE ARE SOME, KING GELON, WHO THINK THAT THE NUMBER OF THE SAND IS INFINITE IN MULTITUDE…

The poetry of infinity

Archimedes’ feat was later celebrated by the poet John Donne (1572–1631), who commented: “Men have calculated how many particular graines of sand, would fill up all the vast space between the Earth and the Firmament.” Donne used this huge number to emphasize that it was negligible in contrast with the limitless nature of infinity and eternity.

BUT IF EVERY GRAIN OF SAND WERE THAT NUMBER, AND MULTIPLIED AGAIN BY THAT NUMBER, YET ALL THAT MADE UP NOT ONE MINUTE OF THIS ETERNITY …

Number sequences

In practice, Archimedes had used only a tiny fraction of his system, but the Greeks were also aware of sequences of numbers that went on for ever. Number sequences are part of every culture. Most children recite counting rhymes (“One, two, buckle my shoe …”) to help recall the sequence of the counting numbers.

Once children learn the basic structure of the numbers and the way the sequence of integers* works, they often count up and up interminably. But where do the numbers stop? Children often seem to be trying to find the biggest number. But they’ll never get there. They could count for the rest of their life, and there would still be as many numbers to go as there were to start with. Imagine there were a biggest number, let’s call it max. What’s to stop us adding max+1, max+2 and so on? The dance never ends.

Of course the counting numbers aren’t the only simple number sequence that most of us would recognize. You can make a sequence by doubling the previous number:

Or you can have sequences with a back-and-forth alternation of steps, for example:

There’s the Fibonacci sequence, and others relying on adding previous numbers:

Or sequences where multiplication is involved:

And there’s no need to stick to whole numbers. As far back as the ancient Greeks there has been an awareness of sequences of fractions, such as:

Strange sequences

At first sight, chains that go on for ever seem harmless, but it doesn’t take long to find some that are strange. In a series* we add the numbers up as we go along to produce a sum. Take a very simple series, alternating 1 and –1:

It’s hardly rocket science. Each 1 is cancelled out by a –1, so the total of the series is 0:

Or is it? Just shift the brackets and we still have a series that cancels out, but now we’ve got a 1 left over:

So the same series adds up to both 0 and 1. This has been rephrased as: “If you turn a light bulb on and off an infinite set of times, does it end up on or off?” It could be either. This is a mathematician’s answer – a physicist will tell you that it’s off, because the bulb has blown.

Or take another simple series where each item is half the last:

It seems, as we add in element after element …

… that it’s going to eventually reach 2:

… though in practice with any particular number there’s always a little gap left:

You could say that the series adds up to 2 if you have an infinite set of components – but what does that mean? And how can an infinite number of things add up to a finite quantity?

The infinity machine

In 1949, the German mathematician and physicist Hermann Weyl (1885–1955), a contemporary of Einstein, devised an imaginary “infinity machine”, inspired by this sequence. Such a machine would carry out a sequence of steps, taking (say) 1 second for the first step, ½ second for the second step, ¼ second for the third step and so on. In principle it could undertake an infinite sequence of steps in a finite time.

There seem to be two difficulties in practice, though.

ONE IS MAKING ANYTHING PHYSICAL HAPPEN IN AN INCREASINGLY SHORT TIME … … AND THE OTHER IS WHETHER TIME CAN TRULY BE SPLIT INTO INFINITELY SMALL SEGMENTS.

Zeno’s paradoxes

This series 1 + ½ + ¼ + 1/8 + 1/16 … was the basis of one of Zeno’s famous paradoxes. Greek philosopher Zeno of Elea (c. 490–430 BC) belonged to the school of Parmenides, which considered reality to be unchanging and movement to be an illusion. Zeno knocked up a number of entertaining examples to demonstrate the faulty nature of our attitude to change and motion. Probably the best known is the arrow that encourages us to imagine two arrows. One floats stationary in space. The other is flying at full speed. Now catch them at a snapshot in time.

HOW DO WE TELL THE DIFFERENCE? HOW DOES ONE ARROW KNOW TO MOVE IN THE NEXT FRACTION OF TIME WHILE THE OTHER DOESN’T?

Achilles and the tortoise