Introducing Infinity: A Graphic Guide

Introducing Graphic Guides

by Brian Clegg, Oliver Pugh (Illustrator)

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.

Genres: Science, Nonfiction, Mathematics, Philosophy, Comics, Stem, Graphic Non Fiction

179 pages, Kindle Edition

First published December 11, 2012

About the author

Brian's latest books, Ten Billion Tomorrows and How Many Moons does the Earth Have are now available to pre-order. He has written a range of other science titles, including the bestselling Inflight Science, The God Effect, Before the Big Bang, A Brief History of Infinity, Build Your Own Time Machine and Dice World.

Along with appearances at the Royal Institution in London he has spoken at venues from Oxford and Cambridge Universities to Cheltenham Festival of Science, has contributed to radio and TV programmes, and is a popular speaker at schools. Brian is also editor of the successful www.popularscience.co.uk book review site and is a Fellow of the Royal Society of Arts.

Brian has Masters degrees from Cambridge University in Natural Sciences and from Lancaster University in Operational Research, a discipline originally developed during the Second World War to apply the power of mathematics to warfare. It has since been widely applied to problem solving and decision making in business.

Brian has also written regular columns, features and reviews for numerous publications, including Nature, The Guardian, PC Week, Computer Weekly, Personal Computer World, The Observer, Innovative Leader, Professional Manager, BBC History, Good Housekeeping and House Beautiful. His books have been translated into many languages, including German, Spanish, Portuguese, Chinese, Japanese, Polish, Turkish, Norwegian, Thai and even Indonesian.

Reviews

[Sam · Rating 4 out of 5]

Easily readable. The last 40 odd pages were too clever for me!

[Samane · Rating 2 out of 5]

اگر بی نهایت یک لامپ را روشن و خاموش کنیم، نهایتاً لامپ روشن می ماند یا خاموش؟ هر کدامش ممکن است پیش بیاید. البته این پاسخ ریاضی دان هاست. فیزیک دان به شما می گوید دست آخر لامپ خاموش می شود چون می سوزد.

[Sabrina]

Very interesting, but ngl, i started reading this because I was panicking about my calculus class

[Lloyd Downey · Rating 5 out of 5]

I’ve recently read a few books on the concept of infinity. But still find I’m somewhat “boggled” by the implications of the infinite (and the infinitesimally small). But I think this book does a really good job of explaining the concepts and some of the reasoning and math behind it. Especially Hilbert’s hotel with the infinite rooms and infinite guests.
Did I understand it all? Well kind-of.....just don’t ask me to teach it to you. I read recently in New Scientist that some mathematicians were having a second look at concepts involving infinity and maybe replacing it with just very large numbers. That seems to be eminently sensible to me. Here’s a few extracts from the book...unfortunately, without the wonderful graphics.
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:
1, 2, 4, 8, 16, 32, 64...
Or you can have sequences with a back-and-forth alternation of steps, for example:1, 3, 2, 4, 3, 5, 4, 6, 5, 7...
There's the Fibonacci sequence, and others relying on adding previous numbers:
1,2, 3, 5, 8, 13, 21...
Or sequences where multiplication is involved:
1, 2, 2, 4, 8, 52, 256, 8192...
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:
1, 1/2, 1/3, ¼, 1/5, 1/6,........

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:
1-1 + 1-1+1-1...
It's hardly rocket science. Each 1 is cancelled out by a -1, so the total of the series is 0:
(1-1) + (2-1) + (1 - 1)...=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:
1(-1+1)(-1+1)(-1+1)...=1

Infinity had to exist, Aristotle decided, because time did not have a beginning or an end. Nor did the counting numbers stop..... Yet a body is defined by its bounds - that is how you distinguish it from everything else. So an infinite body can't exist.
Aristotle decided that infinity was a potential state. This might be difficult to grasp, but Aristotle gave us a beautiful picture to understand it,
Think, he said, of the Olympic Games. They exist... Where is it?". Well, it's not there. I can't show you it. The Olympic Games exist, but they aren't something we can point to (except for two weeks every four years). They are potential, just as infinity is potential.

To find the diagonal, the Pythagoreans were looking for a ratio of
two numbers a and b where a/b = V2. With basic logic, depending only on a knowledge of odd and even numbers, it's possible to show that the square root of 2 is not any ratio of two whole numbers.........The diagonal of a square is a relatively tame irrational, We can easily write down a formula that describes it. But the Greeks were also aware of less tractable irrationals. The obvious example is the ratio of the circumference of a circle to its diameter......... We now know what the problem with squaring the circle is. The irrational number at the heart of the circle is pi (m) - 3.14159...
Unlike the square root of 2, this is a number that doesn't have a simple relationship to whole numbers........ Pi embodies a kind of infinity. You would have to write out an infinitely long decimal to capture its value exactly. It has now been calculated to many millions of places....... This doesn't mean that it's impossible to calculate pi using a formula - such methods have been available since the 16th cen-tury. But unlike v2, the formula for pi (and other transcendental numbers) depends on the sum of an infinite series, rather than a finite equation that can be fully written down.

Hume's argument was flawed, equating the capabilities of the senses with reality. German mathematician David Hilbert (1862-1943) would suggest that the thought process could not be separated from reality. This being the case, he suggested, when we think we're dealing with infinity, in fact we're just thinking of something very, very big........
Not everyone agrees. Shaughan Lavine, Associate Professor of Philosophy at the University of Arizona, points out a very simple way that anyone can envisage infinity. As long as you can grasp the meaning of "finite" and the meaning of "not", he says, you should have a simplistic picture of the infinite.
Even so, some serious mathematicians never accepted the reality of infinity, even as a mathematical concept. The great German mathematician Johann Carl Friedrich Gauss (1777-1855) was convinced that infinity was an illusion, like the end of the rainbow, that could never be reached
Around this time, the idea of indivisibles became popular. This was a similar approach to the ancient Greek idea of atoms. Ancient Greek atoms were the result of cutting something up so small that it was no longer possible to cut any further (a-tomos means "uncuttable). The use of indivisibles involved dividing an object into smaller and smaller pieces, but not necessarily in all three dimensions. Take the example of the area of a circle.
Antiphon was a contemporary of Socrates, born in the 5th century BC. He suggested that you could work out the area of a circle by drawing a regular polygon inside it and gradually increasing the number of sides until it was closer and closer to the circle itself. But 15th-century philosopher Nicholas of Cusa went further. He imagined stacking segments of a circle on top of each other, alternating direction. This made something very close to a rectangle that would be pi x r in height and r in width, making its area pi x r2.
One reason why Leibniz's calculus has proved more popular than Newton's fluxions is that the notation Leibniz invented was so much more practical than Newton's.

Once you divide zero by zero, all bets are off.Anything with zero on top should be zero. Anything divided by zero should be infinite. you only have to look at the attempts of two early Indian mathematicians to explain this ratio:
“ZERO DIVIDED BY ZERO IS ZERO!”
“NO! ANYTHING DIVIDED BY ZERO, INCLUDING O/O, IS INFINITE.”
In practice, 0 over 0 is indeterminate - it doesn't have a result.

Cantor spent all his working life at the university in Halle. This is a German town famous for music, but not for maths. Cantor thought he would soon move on - and he probably would have done, had he not come up with some conclusions that were so mind-bending that at least one mathematician would set out systematically to ruin Cantor's career.
Cantor's first great contribution was to formalize the mathematics of sets*. Sets had been around really as long as people conceptualized - but Cantor embedded them firmly into mathematics.......... For our purposes we need to pick out one aspect of set theory, called cardinality*. Let's think of two very simple sets. The first is the set of legs on my dog. The second is the four horsemen of the apocalypse. These two sets have the same cardinality if I can pair off the members of the
sets on a one-to-one basis......... Before Cantor got involved, Italian mathematician Giuseppe Peano (1858-1932) had already used the cardinality of a set to define the cardinal numbers* - the counting numbers.

Russell then looked at the set "Sets that aren't members of themselves". This set would include, for instance, the set "All pieces of music".
IS THE SET "SETS THAT AREN'T MEMBERS OF THEMSELVES" A MEMBER OF ITSELF?........ If it is a member, then it isn't a member. If it isn't a member, then it's not a set that isn't a member of itself - so it should be a member. It's a bit like trying to work out if the statement "This is a lie" is true. Russell showed that this paradox was fundamental to set theory.

With set theory in place, Cantor was ready to build on Galileo's observations on the integers and the squares......The infinite set of counting numbers has the same cardinality as the set of squares, (every positive integer squared) because we can pair them off like dog legs and horsemen. And the squares are a subset of these integers.
"Subset" is a spot of set theory that has become common® usage. Here, it means that all the squares are members of the set "positive integers", but they aren't the full set. Cantor used this behaviour to define an infinite set........... An infinite set has a one-to-one correspondence with a subset. It has the same cardinality as its subset.

We are looking for the number that, multiplied by itself, gives -1. But we know that both 1 x 1 and -1 x -1 are 1. Neither is the square root of -1. So mathematicians assign a value of i to the square root of -1. A whole structure of mathematics has been built on these imaginary numbers, and on complex numbers, combining real and imaginary, such as 3 + 2i.

Once we're taking a set theory approach to infinity - true infinity, rather than Aristotle's potential infinity - it needs a different symbol. Cantor chose aleph, the first letter of the Hebrew alphabet, and specifically he called the infinity of the counting numbers aleph zero or aleph null א0. Aleph null is the basic infinity, the infinity of the positive integers.

Aleph null has some strange qualities. We can add 1 to it and still end up with the same value. You can see why this happens if you imagine putting the series 1, 2, 3 in one-to-one correspondence with x, 1, 2, 3... So 1 in the first series corresponds with x in the second, 2 in the first series with 1 in the second, and so on. You can go through the whole lot matching them off, so they have the same cardinality. What's more, add aleph null to itself and you get aleph null. (Because 1, 2, 3... and 1a, 1b, 2a, 2b... have the same cardinality.) For that matter, you can multiply aleph null by itself - and still get aleph null. Yet really it hardly seems surprising that this happens, because infinity is infinity.

Cantor wanted to check how flexible aleph null really was. 0א, is the cardinality of the counting numbers and the squares, or for that matter the odd or even positive integers. These sets are called countably infinite* or denumerable. At first sight this term "countably infinite" is an oxymoron. By definition, something infinite can't be counted. Apart from anything else, as we saw with the integers and the squares, such a set can be put in a one-to-one correspondence with a subset. How can you count anything like that? But countable just means having the same cardinality as the counting numbers.

In a square that, like our original number line, runs from 0 to 1 on both X and Y sides, we can refer to any point using two numbers between 0 and 1, the X coordinate and the Y coordinate. What Cantor spotted was that we don't need two numbers to refer to that point. We can create a new number by alternating the digits in the two values.
So, for instance, a point identified as 0.5921 on the X axis and 0.2843 on the Y axis is uniquely identified by 0.52982413, where the odd decimal places identify the X axis location and the even the Y.
When they told you at school that you needed two numbers to identify a point on a two-dimensional plane, they were wrong.
Every point is identified by a single value. Of course there's more information in the new number - it has twice as many decimal places - but it's still a single number. And so the cardinality of points in a square is the same as that of the continuum between 0,1, cא (the infinity of the continuum....all the fractions between 0 and 1). The same argument applies to the points in a cube, or an n-dimensional hypercube.
Under constant, renewed attack from Kronecker, trapped in the mathematical backwater of the university of Halle, unable to prove the so-called continuum hypothesis, Cantor's mind was shattered.
Kurt Gödel (1906-78) devised the most shocking proof in mathematics. His masterpiece, the incompleteness theorem, states that in any system of mathematics there will be some problems that it's impossible to solve........ A crude approximation to Gödel's theorem is to imagine dealing with the statement: "This system of mathematics can't prove that this statement is true." Is this statement true?

The French-born mathematician Andre Weil (1906-98), summed up the situation: with after Gödel and Cohen's work on the continuum hypothesis. It is not inconsistent with set theory, and vet it can never be linked with set theory. It can neither be proved nor disproved. As long as we stick with the same axioms used as the foundation for set theory, it will never be possible to make any further progress.

A contemporary once remarked that Cantor's ideas "appear repugnant to the common sense". In the end we're always plagued with the uncertainty: does infinity truly exist, or is it merely a convenient - as Aristotle would have it, a potential - concept?....... It's hard to say if there's a true infinity in the real world...... Like Aristotle, we have to ask: does time stop or end? Is there a point where we can no longer divide time or space?

It looks like everything's fine, but then a special coach turns up. It's a coach with an infinite set of passengers on board. Again, the desk clerk has to apologize. The hotel is entirely full. Luckily, you're still at reception and able to take charge. "No problem",
, you say. Move the person in room 1 to room 2......Move the person in room 2 to room 4. .....Move room 3 to room 6....... And so on, doubling up room numbers throughout the hotel.......Now all the odd-numbered rooms - an infinite set of them – are free for the new guests.

What’s my overall take on the book? I really liked it, I think they have done a first rate job of explaining some notoriously difficult concepts. Five stars from me.

[Caitlin · Rating 4 out of 5]

A fun overview

This was an enjoyable, quick read with a lot of fun and interesting discussions. Good even if you're not a math person, I think, and an enjoyable review if you are.

[Trever · Rating 2 out of 5]

This book was nothing new, I have learned all the information before. This book would be good for my students who don't know anything about the history of mathematics and how what they are learning came to be. It feels like they don't appreciate math for math, they hate math because of the math.

[Monark Trivedi · Rating 5 out of 5]

Nullfinity Infinull

It covers infinite content in finite words. If this sounds paradoxical, you have to read it to create more paradoxes out of it. Fascinating read!

[Roger · Rating 4 out of 5]

This short book, written by Brian Clegg and illustrated by Oliver Pugh, provides a thought-provoking introduction to the concept of infinity, including how the idea has developed since philosophers first mulled over the problem in the time of Ancient Greece. As short books go, this is extremely short and the Kindle edition I read had large print yet few of its pages were completely full of text whereas the illustrations often filled an entire page. As a consequence, it didn’t take long to read.
Yet, despite its small size, it covers a very wide field and seems to touch on all of the key areas related to infinity. The obvious downside is that nothing is written about in great detail and it leaves a lot of loose ends to be followed up by those whose interest has been piqued. With that in mind, it usefully contains a list of recommendations for further reading, one of these being an earlier work by Clegg entitled “A Brief History of Infinity” – while that may also be “brief” I assume it is longer than this book.
I enjoyed Clegg’s style, which used subtle humour in places, and I like how he often threw in fascinating historical titbits. Some snippets that I enjoyed:
“Numbers have no physical reality”.
“… we have to remember that mathematics is not about the real world”.
Quoting Gauss, who was convinced that infinity was an illusion: “The infinite is only a manner of speaking, in which one properly speaks of limits to which certain ratios can come as near as desired, while others are permitted to increase without bound”.

[Bill · Rating 3 out of 5]

A rather good introduction to the concept of infinity, starting with just enough of the history of mathematics to set the foundation. Gradually moves forward through the growth of mathematics and the place of infinity within it, spending a good deal of time on its pivotal role in calculus, and the work of Cantor to comprehend and define this challenging concept. Goes off on an occasional tangent and doesn't always clearly explain the mental gymnastics required to grasp the modern concepts of infinity; that said, some of those tangents (especially those focusing on the politics, grudges, and rivalries in the history of mathematics) are quite entertaining. Where else are you going to see a drawing of Newton giving side-eye and angrily growling at Leibniz to illustrate their dispute over the invention of calculus?

[Stefan · Rating 3 out of 5]

A neat visual introduction

It's a good overview of the relevant subtopics. Unfortunately, some of the explanations were oversimplified to the point that a reader with no other experience would be left with an inaccurate understanding of the issue discussed. For example, describing classical Greek mathematics as "right-brained" perpetuates multiple popular misconceptions at once.

This is the second graphic guide I have read. I will probably continue with more books in the series. They are entertaining and include content that I haven't seen in other introductory materials. Following up with the books in the further reading lists would be a good next step for readers of this series.

[A. · Rating 4 out of 5]

Infinity is like a wild animal, spotted in the depths of a forest. Well-written book, clear overview of infinity through the ages, helpful illustrations. The high point of the book is the build up to and focus on Cantor, on cardinals and ordinals, which is also the part where I was hanging to comprehension by the skin of my teeth. I’m now better in tune with the infinite, the transfinite, the infinitesimal. Who knows? Maybe the Wallace book will make sense too.

[Aaron B · Rating 5 out of 5]

With apologies to Borges, The Aleph explained

I enjoyed this book and found it to be an excellent companion to volumes in the same series on Fractals, Chaos, and Quantum Physics. It introduced different ways that infinity can be considered; for example, consider logical ways of finding accommodations at the fully-booked Hilbert's Grand Hotel or explaining the finite volume of Gabriel's Horn despite its infinite volume. Despite occasional brain cramping, I recommend this book.

[David Odum · Rating 4 out of 5]

Fascinating

A fascinating and mostly understandable introduction. I wish the author had "shown his work" a little more instead of sometimes skipping ahead in an algebraic problem. It was also uncomfortably irreverent at places, with the author loving to not only show classic nude male statues, but also to render his own drawings of them instead for the next illustration.

[Sayani · Rating 3 out of 5]

Another exciting volume in the Introducing Books series. One of the books in the mathematics theme this is a quick tour through the history of calculus, the advent of the lemniscate sign of infinity, Cantor's set theory, Russell's paradox and many other people who have contributed to the study like Gödel. I wish there was volume on geometry but there isn't one.

[Rob · Rating 2 out of 5]

Has a wide breadth of content on infinity throughout history, but it was a little too scattered for me. I know it is titled as introducing infinity, but it actually goes into some deep ideas which I think is tough for people that might not have been exposed to some of the maths behind it before.

[geoff lewis · Rating 4 out of 5]

good introductory text for the concepts of infinity.

Written in an informal style with little previous knowledge assumed.
Good for none maths people looking for something interesting to say at a maths party.

[Happy Reader · Rating 1 out of 5]

The math is simplified to the point of introducing errors or misleading claims.

The other thing I did not like is that it promotes the misconception that pondering infinity too much can lead to mental illness.

[Joan MS · Rating 2 out of 5]

Interessant però una mica desordenat i no gaire clar en alguns moments. Puntualment entretingut, això sí. No sé tampoc si s'aprofita prou la ilustració com a valor afegit. Conté algun error (transcripció?).

[Bob · Rating 3 out of 5]

This book does go on longer than you would think. Despite trying to make some of the theoretical math concepts easier to understand, I missed out on a lot of them. But not all of them. I have my limits.

[William Melcher · Rating 5 out of 5]

Short but thorough history

This is a short but packed history of infinity and mathematics in general, as well as touching on topic such as quantum physics and the universe.

[Andrew Noselli · Rating 3 out of 5]

Touched on Augustine, whose work dedicated to elucidating Jesus Christ's Sermon on the Mount I am reading next. A curious synchronicity.

[Brian · Rating 4 out of 5]

If this book has taught me anything, it’s that problems of infinity are not problems of mathematics, but problems of language.

[John Chambers · Rating 3 out of 5]

Infinite fun

First maths book i have enjoyed all the way through. I have read other books but they were hard work.

[Len · Rating 2 out of 5]

my dumb ass had to read every line twice before my brain can take it in LOL. the "graphic" part of 30% illustrating the theories and 70% caricature and jokes so it's not much helping.

[Sky · Rating 5 out of 5]

Ik snap er nu minder van dan ik dacht dat ik er over begreep. Maar ik weet wel meer nu. Zeker een keer herlezen als ik wat ouder ben😱

[Suzie Phillips · Rating 4 out of 5]

Another nerdy maths comic book, need I say more

[Arvind · Rating 4 out of 5]

Not the best written book. Lots of explanations were ambiguously brief (e.g. the section on infinite ordinals was rather confusing). But I still learned a lot.

[Devashish Sharma · Rating 4 out of 5]

It will give you many things that you need to think about.
Nice illustrations to support arguments.
A fun read.

[Steffen · Rating 4 out of 5]

Verrassend onderwerp en goed leesbaar voor breed publiek hoewel er passages in zitten waar net een stap teveel wordt overgeslagen bij de uitleg. Dat zorgt af en toe voor hoofdbrekens, maar dat is voor degenen die dit soort boeken leuk vinden ook geen probleem denk ik :-)