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Infinity: The Quest to Think the Unthinkable
It amazes children, as they try to count themselves out of numbers, only to discover one day that the hundreds, thousands, and zillions go on forever—to something like infinity. And anyone who has advanced beyond the bounds of basic mathematics has soon marveled at that drunken number eight lying on its side in the pages of their work. Infinity fascinates; it takes the mind beyond its everyday concerns—indeed, beyond everything—to something always more. Infinity makes even the infinite universe seem small; yet it can also be infinitesimal. Infinity thrives on paradox, and it turns the simplest arithmetic on its head, with 1 seeming feasibly to equal 0, after all. Infinity defies common sense. The contemplation of it has relieved at least two great mathematicians of their sanity. Thoroughly readable and entirely accessible, science writer Brian Clegg's lively history explores infinity in its many intriguing facets, from its ancient origins to its place today at the heart of mathematics and science. He examines infinity's paradoxes and profiles the people who first grappled with and then defined and refined them, offering information, mystery, and poetry to conceive the inconceivable and define the indefinable.
272 pages, Paperback
First published September 12, 2003
About the author
Brian Clegg
173 books3,302 followersBrian'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.
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.
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January 9, 2011
My two teachables, the subjects which I will be qualified to teach when I graduate from my education program in May, are mathematics and English. When I tell people this, they usually express surprise, saying something like, “Well, aren’t those very different subjects!”
And it irks me so.
They’re not, not really. Firstly, mathematics and English are both forms of communication. Both rely on the manipulation of symbols to tell a tale. As with writers of English, writers of mathematics have styles: some are elegant yet terse, seemingly expending little effort while getting their point across with an admirable economy of symbols; others are expansive and eloquent, elaborating at some length in order to furnish the reader with an adequate explanation. Secondly, as with English, mathematics is very much grounded in philosophy and history, and it is a subject that is open to deep, almost spiritual interpretation.
If you balk at that last idea, don’t worry. You’ve probably had it drilled into your head since elementary school that in mathematics there is only one correct answer! How could such a reassuringly logical subject be open to interpretation? Despite its apparent objectivity, mathematics is just another human endeavour, and like all our mortal works, it is vulnerable to our flaws, foibles, and fits of passion. Mathematicians can be just as stubborn and argumentative, if not more, than other people. There are many famous follies and feuds in the history of mathematics, and that is one of the reasons I enjoy learning about it so much.
Infinity is one of the mathematical concepts most central to those feuds. It’s one of the areas where math rubs up against the spiritual realm—for, as some mathematicians and philosophers have wondered, what is infinity if not God or some kind of greater being? So it seems natural to look at our shifting views on the infinite along the continuum of the history of maths. In A Brief History of Infinity, Brian Clegg does just that, following the classical, somewhat Eurocentric development of math from Greece to Rome, then zig-zagging down to the Middle East and India before flying back to Britain, France, and Germany.
As with most tricky math concepts, the trouble with infinity begins with its definition. One must be very careful with definitions in math—for example, it is not enough merely to say that infinity means “goes on without end”. After all, the surface of the Earth has no “end”, but that does not mean the Earth has infinite surface area! Rather, the surface of the Earth is unbounded. Grasping the idea of infinity as “not finite” is easy enough, though: there is no “last” counting number, because you can always add one to the largest number you can conceive, and suddenly you have a new largest number. So infinity is a quicksilver of a concept: intuitive and easy to grasp, yet also elusive and far too fluid for some mathematicians to handle. The Greeks, with their mathematics strictly confined to the geometric figure, would have no dealings with the infinite. Infinity confused Galileo, who nevertheless bravely meditated upon it in his final days. And the shadow of infinity hangs over the controversy of the calculus that caused the divide between Newton and Leibniz, and correspondingly, between Britain and the Continent.
The story of infinity gets even more interesting after that. In general, I love the history of mathematics during the 1700s and 1800s. So many brilliant minds pop up during that time: as Newton and Leibniz exit, Euler and Gauss enter. Later, Cauchy and Weierstrass formalize the concept of the limit, which does away with any need for infinity in calculus at all! There are plenty of names and plenty of stories—and this is where A Brief History of Infinity starts to lose its edge.
The first few chapters of this book are fascinating. Clegg devotes a lot more space to the Greek philosophers than others might, going so far as to mention some of the more obscure ones, like Anaxagoras. He provides a considerably detailed development of Zeno’s paradox (well, paradoxes) and a nice, if basic, grounding in the idea of an infinite series. Clegg lays the ground well for what will come in later chapters, all the while emphasizing the reluctance of the Greek philosophers to abandon the solidity of numbers found in the real world.
But as we get closer to those magical two centuries following the great Newton–Leibniz schism, the story of infinity gets more complicated as more people get involved. This book is very similar to Zero: The Biography of a Dangerous Idea . In my review of Zero, I praised the author’s ability to stay focused:
To be fair to Clegg, this book is almost as slim as Zero. And although he happens to go off on many a tangent, he at least has the ability to find his way back on track quickly enough—that is, his tangents are interesting and informative. He sometimes seems to go into more detail than is strictly necessary to get the point across, and once in a while he waxes melodramatic—as is the case when he links Cantor’s madness to his study of infinity. Overall, however, Clegg’s writing is crisp and clear.
I’m also impressed by the detail and depth of Clegg’s explanation of the math. He goes so far as to list and briefly elaborate upon each of the axioms of Zermelo-Fraenkel set theory! I was half expecting him to mention the Banach–Tarski paradox after that—he doesn’t quite get there, but he does explain the difference between ordinals and cardinals, develop the continuum hypothesis, and even mention Gödel’s Incompleteness Theorem. He tackles whether imaginary numbers are truly all they’re cracked up to be. And he even discusses nonstandard analysis—we didn’t even learn about that in university.
Don’t let my awe scare you away, though. Rather, think of it like this: if you are not particularly mathematical and read this book, you will gain a wealth of knowledge. You will be fun at parties! If you are particularly mathematical, then depending on how much you like the history of math, you might already be familiar with most of these anecdotes. But the book will still be fun to read, and chances are you will learn at least one or two new things.
So I would recommend A Brief History of Infinity to most people—perhaps not with the same zeal that I do Charles Seife’s Zero, but with a similar hope in mind. I hope this book, or at least my review of this book, demonstrates why I find math, as well as the history of math, so fascinating. It’s not just all about numbers, solving for x, and finding the One True Solution. Mathematics is a subject with a long and storied past, one that is fun to explore by looking at the humans who progressed—or regressed—throughout the centuries. A Brief History of Infinity is a book in this mould. While its organization and its focus leaves something to be desired, its scope and ambition do not.
And it irks me so.
They’re not, not really. Firstly, mathematics and English are both forms of communication. Both rely on the manipulation of symbols to tell a tale. As with writers of English, writers of mathematics have styles: some are elegant yet terse, seemingly expending little effort while getting their point across with an admirable economy of symbols; others are expansive and eloquent, elaborating at some length in order to furnish the reader with an adequate explanation. Secondly, as with English, mathematics is very much grounded in philosophy and history, and it is a subject that is open to deep, almost spiritual interpretation.
If you balk at that last idea, don’t worry. You’ve probably had it drilled into your head since elementary school that in mathematics there is only one correct answer! How could such a reassuringly logical subject be open to interpretation? Despite its apparent objectivity, mathematics is just another human endeavour, and like all our mortal works, it is vulnerable to our flaws, foibles, and fits of passion. Mathematicians can be just as stubborn and argumentative, if not more, than other people. There are many famous follies and feuds in the history of mathematics, and that is one of the reasons I enjoy learning about it so much.
Infinity is one of the mathematical concepts most central to those feuds. It’s one of the areas where math rubs up against the spiritual realm—for, as some mathematicians and philosophers have wondered, what is infinity if not God or some kind of greater being? So it seems natural to look at our shifting views on the infinite along the continuum of the history of maths. In A Brief History of Infinity, Brian Clegg does just that, following the classical, somewhat Eurocentric development of math from Greece to Rome, then zig-zagging down to the Middle East and India before flying back to Britain, France, and Germany.
As with most tricky math concepts, the trouble with infinity begins with its definition. One must be very careful with definitions in math—for example, it is not enough merely to say that infinity means “goes on without end”. After all, the surface of the Earth has no “end”, but that does not mean the Earth has infinite surface area! Rather, the surface of the Earth is unbounded. Grasping the idea of infinity as “not finite” is easy enough, though: there is no “last” counting number, because you can always add one to the largest number you can conceive, and suddenly you have a new largest number. So infinity is a quicksilver of a concept: intuitive and easy to grasp, yet also elusive and far too fluid for some mathematicians to handle. The Greeks, with their mathematics strictly confined to the geometric figure, would have no dealings with the infinite. Infinity confused Galileo, who nevertheless bravely meditated upon it in his final days. And the shadow of infinity hangs over the controversy of the calculus that caused the divide between Newton and Leibniz, and correspondingly, between Britain and the Continent.
The story of infinity gets even more interesting after that. In general, I love the history of mathematics during the 1700s and 1800s. So many brilliant minds pop up during that time: as Newton and Leibniz exit, Euler and Gauss enter. Later, Cauchy and Weierstrass formalize the concept of the limit, which does away with any need for infinity in calculus at all! There are plenty of names and plenty of stories—and this is where A Brief History of Infinity starts to lose its edge.
The first few chapters of this book are fascinating. Clegg devotes a lot more space to the Greek philosophers than others might, going so far as to mention some of the more obscure ones, like Anaxagoras. He provides a considerably detailed development of Zeno’s paradox (well, paradoxes) and a nice, if basic, grounding in the idea of an infinite series. Clegg lays the ground well for what will come in later chapters, all the while emphasizing the reluctance of the Greek philosophers to abandon the solidity of numbers found in the real world.
But as we get closer to those magical two centuries following the great Newton–Leibniz schism, the story of infinity gets more complicated as more people get involved. This book is very similar to Zero: The Biography of a Dangerous Idea . In my review of Zero, I praised the author’s ability to stay focused:
The story intersects with the lives of many famous mathematicians, but the obvious slimness of this book testifies that Seife managed to distill only what was necessary about their lives in his quest to explain the mystery of zero.
To be fair to Clegg, this book is almost as slim as Zero. And although he happens to go off on many a tangent, he at least has the ability to find his way back on track quickly enough—that is, his tangents are interesting and informative. He sometimes seems to go into more detail than is strictly necessary to get the point across, and once in a while he waxes melodramatic—as is the case when he links Cantor’s madness to his study of infinity. Overall, however, Clegg’s writing is crisp and clear.
I’m also impressed by the detail and depth of Clegg’s explanation of the math. He goes so far as to list and briefly elaborate upon each of the axioms of Zermelo-Fraenkel set theory! I was half expecting him to mention the Banach–Tarski paradox after that—he doesn’t quite get there, but he does explain the difference between ordinals and cardinals, develop the continuum hypothesis, and even mention Gödel’s Incompleteness Theorem. He tackles whether imaginary numbers are truly all they’re cracked up to be. And he even discusses nonstandard analysis—we didn’t even learn about that in university.
Don’t let my awe scare you away, though. Rather, think of it like this: if you are not particularly mathematical and read this book, you will gain a wealth of knowledge. You will be fun at parties! If you are particularly mathematical, then depending on how much you like the history of math, you might already be familiar with most of these anecdotes. But the book will still be fun to read, and chances are you will learn at least one or two new things.
So I would recommend A Brief History of Infinity to most people—perhaps not with the same zeal that I do Charles Seife’s Zero, but with a similar hope in mind. I hope this book, or at least my review of this book, demonstrates why I find math, as well as the history of math, so fascinating. It’s not just all about numbers, solving for x, and finding the One True Solution. Mathematics is a subject with a long and storied past, one that is fun to explore by looking at the humans who progressed—or regressed—throughout the centuries. A Brief History of Infinity is a book in this mould. While its organization and its focus leaves something to be desired, its scope and ambition do not.
July 7, 2019
Be sure you know what you're buying!
Perhaps I should state what I think should have been made a little more obvious. Clegg's A BRIEF HISTORY OF INFINITY is not a mathematics book. It is definitely a history book. In fact, it outlines the history of man's struggle to come to grips with the exceedingly complex and devilishly bewildering concept of infinity. Of necessity, of course, it touches on matters mathematical but the meat of this book is the history.
A BRIEF HISTORY OF INFINITY delves into man's contemplation of matters infinite from the earliest days of its discussion by Greek philosophers, to St Augustine's theological musings of creation, to Leibniz and Newton battling over bragging rights for the creation of calculus, to Cantor's transfinite numbers and even to the implications of infinity in quantum physics.
Having noted that the book is more focused on history than mathematics, it's definitely worth pointing out that the mathematics would still be daunting for a complete neophyte. That said, my hope was for somewhat more mathematics and a little less of the historical background. For example, I found the section on Leibniz and Newton's battles with Bishop Berkley over infinitesimals quite dreary and plodding.
But, the misunderstanding as to the exact nature of the book can probably be laid more at my doorstep. A more careful examination of previous reviews and the marketing info on the book cover would have better informed me as to what I was stepping into.
Recommended.
Paul Weiss
Perhaps I should state what I think should have been made a little more obvious. Clegg's A BRIEF HISTORY OF INFINITY is not a mathematics book. It is definitely a history book. In fact, it outlines the history of man's struggle to come to grips with the exceedingly complex and devilishly bewildering concept of infinity. Of necessity, of course, it touches on matters mathematical but the meat of this book is the history.
A BRIEF HISTORY OF INFINITY delves into man's contemplation of matters infinite from the earliest days of its discussion by Greek philosophers, to St Augustine's theological musings of creation, to Leibniz and Newton battling over bragging rights for the creation of calculus, to Cantor's transfinite numbers and even to the implications of infinity in quantum physics.
Having noted that the book is more focused on history than mathematics, it's definitely worth pointing out that the mathematics would still be daunting for a complete neophyte. That said, my hope was for somewhat more mathematics and a little less of the historical background. For example, I found the section on Leibniz and Newton's battles with Bishop Berkley over infinitesimals quite dreary and plodding.
But, the misunderstanding as to the exact nature of the book can probably be laid more at my doorstep. A more careful examination of previous reviews and the marketing info on the book cover would have better informed me as to what I was stepping into.
Recommended.
Paul Weiss
October 17, 2021
Brilliant, but I could never quite get to the end...
April 20, 2020
This book was my break from the seemingly endless amount of novels waiting for me. I can't believe i let it gather dust in my bookcase for a year. It was a graphic guide on infinity for dummies. See, i am a whore for the logically absurd, you know... the whole shebang. So you can only imagine how i devoured this book. While reading some parts, my heart literally throbbed as if im reading the best romantic novel in the world (aka PnP). It was soooo interesting. It made me wanna delve deeper into the subject. It dabbed ideas of infinity from mathematics, science, philosophy, theology, etc- the ultimate crash course. I was introduced to some of the most important pioneers of, basically, all human knowledge present. From St. Augustine to Newton to Leibneiz to Hume to Turing to so on and so forth. To be honest though, i was hoping i'd see a female pioneer, but the author wasnt able to do so (even though there are lots of em). So.... I dont know... Anyways, I read it slowly on purpose because i wanted to suck it all in. I constantly stopped to write interesting topics on my commonplace book. I didnt want it to end immediately. I wanted more words. I wanted more explanations. Its just that good. Brian Clegg's writing wasnt too technical but it also wasnt unscholarly. It was, at times, humorous even. It wasnt one of those incomprehensible book for the brainwacks so extra points for that lol. Id love to add more but im currently dumbfounded. Maybe tomorrow : )
May 11, 2016
I can find my way around a darts board rather well and have never had a problem with watching the runs tick over while watching the cricket. Other than that maths just is not my strong point. But when a complete maths fool such as myself enjoys a book like this then there has to be something going for it. Infinity? Of course, how could there not be. Read and enjoy!
November 30, 2022
Who would have thought that infinity, which is synonymous with mathematics, has a close relationship with philosophy and theology.
I studied in mathematics and all I know about infinity is about a number that goes on and on forever. When you divide any number with 0, then you will get infinity. In my mind, what does that infinity means? How can a number become infinity when it was divided by a zero? How powerful is zero when it can turn any number to infinity? Then, another question arises. How long it takes to reach 1 from 0? Can you imagine that you can put infinity amount of numbers between 0 and 1. You can even put a huge amount of numbers as wide as the universe between 0 and 1. Then, you will have a universe between only 2 numbers, 0 and 1. That's how beautiful infinity, and mathematics, itself.
When talking about infinity, we will not be able to escape paradoxes. The most famous paradoxes including Zeno's paradox and Cantor's paradox. Of course, when we think about infinity, that even our mind cannot properly comprehend, paradoxes and fallacies will occur. That's why, when you learn and think about infinity, it is a quest to think the unthinkable.
Infinity is explored under theology. Is infinity the absolute thing in the universe? Is infinity is part of the god or the infinity is the god? I do believe that infinity is not the god. God (الله) is the Almighty and infinity is one of the knowledge for mankind to be explored. I don't want to discuss about infinity in theology context further.
Overall, this book give some useful insight on what is infinity and how people discovered the thing that unthinkable. Brian Clegg use many parable and mathematical explanation to give better exposure about infinity. Throughout the book, my head goes 🗿🤯 and I went 🥴 real quick.
It's a good book.
♾️
I studied in mathematics and all I know about infinity is about a number that goes on and on forever. When you divide any number with 0, then you will get infinity. In my mind, what does that infinity means? How can a number become infinity when it was divided by a zero? How powerful is zero when it can turn any number to infinity? Then, another question arises. How long it takes to reach 1 from 0? Can you imagine that you can put infinity amount of numbers between 0 and 1. You can even put a huge amount of numbers as wide as the universe between 0 and 1. Then, you will have a universe between only 2 numbers, 0 and 1. That's how beautiful infinity, and mathematics, itself.
When talking about infinity, we will not be able to escape paradoxes. The most famous paradoxes including Zeno's paradox and Cantor's paradox. Of course, when we think about infinity, that even our mind cannot properly comprehend, paradoxes and fallacies will occur. That's why, when you learn and think about infinity, it is a quest to think the unthinkable.
Infinity is explored under theology. Is infinity the absolute thing in the universe? Is infinity is part of the god or the infinity is the god? I do believe that infinity is not the god. God (الله) is the Almighty and infinity is one of the knowledge for mankind to be explored. I don't want to discuss about infinity in theology context further.
Overall, this book give some useful insight on what is infinity and how people discovered the thing that unthinkable. Brian Clegg use many parable and mathematical explanation to give better exposure about infinity. Throughout the book, my head goes 🗿🤯 and I went 🥴 real quick.
It's a good book.
♾️
April 30, 2024
Math, philosophy and history mixed together make for a terrific read!
I mean, who knew!!
Embark in a journey through the thought processes of the greatest men and polymath and their struggles, and suddenly, a lot start to makes sense.
The solutions are a journey and mind bending!!
To infinity ♾️ and beyond ✨️ !!!!!
Buy it on Audible.com, the narrator is remarkable 👌 and makes this book so enjoyable!!!!
I mean, who knew!!
Embark in a journey through the thought processes of the greatest men and polymath and their struggles, and suddenly, a lot start to makes sense.
The solutions are a journey and mind bending!!
To infinity ♾️ and beyond ✨️ !!!!!
Buy it on Audible.com, the narrator is remarkable 👌 and makes this book so enjoyable!!!!
June 4, 2016
Good book overall, but oftentimes the author manages to contradict himself in the same phrase or paragraph, and uses hopelessly confusing and sometimes inappropriate illustrations to get the point across. I wouldn't have understood him had I not encountered the concepts before, and he really lost me on the one thing I hadn't.
And most annoyingly, mathematical concepts are so loosely used that would make serious mathematicians cringe - among many other things, calling irrational numbers "irrational fractions" was maddening. While toning it down so it would be readable and accessible to any interested person, it shouldn't have been so hard to keep it rigorous and mathematically correct.
Besides, introducing new people and relating the story of their lives was done in such a way that you were wondering what could possibly be their connection to the subject, which makes the reading anything but smooth.
And most annoyingly, mathematical concepts are so loosely used that would make serious mathematicians cringe - among many other things, calling irrational numbers "irrational fractions" was maddening. While toning it down so it would be readable and accessible to any interested person, it shouldn't have been so hard to keep it rigorous and mathematically correct.
Besides, introducing new people and relating the story of their lives was done in such a way that you were wondering what could possibly be their connection to the subject, which makes the reading anything but smooth.
February 17, 2013
Poorly written, both as regards actual language use and overall structure. Clegg seems to be confused about what he's actually trying to communicate, and doesn't appear to be able to distinguish actual mathematicians (or proto-mathematicians) exploring the concept of infinity from crackpots merely abusing language.
The result is something that lacks both a historical narrative and enough rigour to pass even as popular mathematics. And there's no shortage of better books on the same subject; just read Taming the Infinite or The Infinite Book or something instead.
The result is something that lacks both a historical narrative and enough rigour to pass even as popular mathematics. And there's no shortage of better books on the same subject; just read Taming the Infinite or The Infinite Book or something instead.
January 23, 2016
Not a bad overview of the history of infinity, though it sits on the fence a lot, and is a little prone to classic pop science micro biographies to limp along (hey here is Godel, he was brilliant but he was nuts...) Infinity is a subject I used to know a lot about, and since I have been out of the incomprehensible big stuff game I wondered if much new had come up. One bit of quantum computing aside, not really, and the book isn't strong on some of the philosophical implications, but it is a brief history.
May 29, 2021
This work felt a little anticlimactic; a lot of build up and history for...nothing? No real pay off, certainly no solutions. So why did we spend all this time looking back at the history of numbers, mathematics, theology, and philosophy? Perhaps there is a lesson about infinity even in this experience. No matter how far you move into the infinite, infinity still lurks beyond your next step. Clegg did do a good job of keeping this exploration simple. While I did feel that he spent a bit too much time on the biographies of some of the historical figures whose views he discussed, but all in all this is an approachable introduction to a strange, mind-bending topic.
September 4, 2022
There are so many solvable problems here but the main issue is a thought experiment listed in Robert Anton Wilson's "Quantum Psychology." The illustration is that of a monk contemplating in a cell that he is a cow and after a some time he is unable to leave his cell because he his horns won't fit through the door. The problems in this book are similar a many mathematicians are have their mental hands tied by others mental ropes and thus claim something unthinkable by not allowing the ropes to be untied or just admitting they don't exist. So here are a couple of issues that can be fixed by removing some mental shackles.
Cantor's Insanity for a Decimal Based Infinity with Alpha Null - One of the most heart breaking stories is Cantor's suicide due bulling and the right idea but with a corrupted math language... he was on the right track with his geometric factorial sets (line ratios) but the decimals threw him. So with a little logic of primes, the base ten system is good for numbers composed of factors of 2 and 5. 1/2 is .5 and 1/5 is .2... second powers are the same with 1/4 being .25 and 1/25 being .04... the problem comes with a prime factor that is not a 2 or a 5 like 3. 1/3 is .33333333333333... and on but a pie can be cut in three pieces with no reminder (as long as one isn't to the molecular anal level) but this can be proven with a base 30 system. The Hebrews use there 22 letter alphabet as a number system as well so lets make a base 30 system with the roman alphabet plus 4 other characters. ABCDEFGHIJKLMNOPQRSTUVWXYZ$#@0 would be the unique numerals and would repeat in trecmals instead of decimals so "A0"would be 30 and "B0" would be 60... "CE" would be 95. So in a lower trecmal place A (1) divided by C (3) or A/C would be ".J" and end there being the 10/30th of the way through the system with no remainder. All that is to say that fractions as nesting functions are accurate and true to measure the space from 0/0 to 1/1 rather than the corruption of decimal strings which are handy for less or greater than comparisons (anyone who's uses metric and imperial standard fractions wrenches knows the difference as the denominators is change so using 16 or 32 transforms makes it a little easier). Whats more one can see a cool stacking phenomena in pie if you count the overlap frequency as 1/1, 1/2 - 2/2, 1/3 - 2/3 - 3/3, then 1/4 -2/4 -3/4 - 4/4 etc as 0/0, 0/1, 0/1 are later. But the inverse of N/1 will be it's frequency in above and below 1/1... every other number has at least on 2 so half of infinity is "even" and the 4s nest on ever other 2 like wave intersecting... the same for 8s as every third 2 or it's exponent counting within frequencies... but it's the same between 0/0 and 1/1. So 1/2 of the fractions will have a redundancy on the 1/2 line in terms of a bar graph. The same with exponents of 3 in 1/3 or exponents of synthetic of primes like 1/6 landing on both their primes and unique marks. Yet the main reason for misunderstanding infinite sequences is due to the base ten decimal strings that hide the fractional frequency overlaps above and below the gauge of one over one and there's plenty to explore in this space but Cantor couldn't get it with those horn on his head and stubborn peers fencing him in.
Dividing by Zero and the Infinitesimals or Fluctuation - The Author mentions Bonaventura Cavalieri's work on infinitesimals for one dimensional lines or two dimensional planes one dimension thick. The same thing appears with Newton's fluctuations or Leibniz's infinitesimals and the symbols to describe them in the joint discovery of Calculus as well as Bishop Berkeley's commentaries. So once again there is a clear phenomena but because it doesn't have a mental reference, it is said not to exist with a name alone as math requires numbers. So let's give it a number but seeing how far away an infinitesimal is from itself: 1/1 - 1/1 = 0/1. So it is a zero point but if you keep the denominator it is differentiated from plain zero... and has a point place on a line or plane or volume for any number referenced from the origin. Thus a line is 0/1 thick and a 2D area is 0/1 deep... they can be summed but that requires more out of the box thinking with the denominator and frequency spacing as described with the fractions above. The main point is that you can divide by zero if you think about the problem and add more detail so information is not lost. Say you have an area that is 1*1 which is 1^2 in the X and Y dimensions so that 1^2 = 1 (X) * 1(Y). Now if we count the zero point lines in each direction there should be 1/0 lines of 1/1 with respect to X's vertical lines in Y and Y's horizontal lines in X each a 0/1 thick (1/0*0/1 = 1/1 *0/0 for no information loss with 0). So if we break it down, the zero point lines can be stack on top of each other to make a line (1/0)*(1/1) tall (in Y) or wide (in X) and (0/1) thick in (in X or Y respectively) and still have the same amount (0/0) of area in terms of the (1/1)*(1/1) by keeping track of 1s and 0s and the amount of dimensions as well as the total stuff that is zeroed in. You could do the same for a different area or by adding another dimension for a volume. Granted, new notation is needed but that's not hard so long as it always describe the phenomena accurately instead of the vain task of making the phenomena fit the arbitrary notation. This can also be applied to the example of Gabriel's Horn in the books as scaling there infinities and zero point is the basis of the fundamental theorem of calculus with differentiation and integration. The slope (dualistic) at a point (non dualistic zero point) would be similar to a 1/2 offset with a grid and it's delta or measure from the origin (0/0) zero point in left,right,up, down,back, front in terms of delta zero points so they are offset by 1/2 a zero point or it polarities... thus a point overlapping its delta on a line slope would have certain amount of zero points to one side and so many to the other for a differentiated slope at a point (dual and non dual).
Heterogeneous Fractions Vs Homogeneous Fractions - This is more of an addendum to make the new numbers add up but it requires unique rules for unique numbers. As a visual, think of a box 1/1 by 1/1 and 1/2 of it is blue and 1/2 is red. So with a homogeneous fraction there would be 2 rectangles 1/1x1/2 one red and one blue. In an as above so below, lets imaging all even fractions denominator points (N/2, N/4, N/6... etc) on the one side of 1/2 as blue and all the odd fraction denominator points (N/1, N/3, N/5... etc) so while the points can be counted the can be order below the 1/1 gauge from 0/0 with all the points. Now imagine the square is purple and thus heterogeneous fractions so that ever other point is red (0/1), blue (0/2), red (0/3), blue (0/4)... etc. So these are the points Zeno would count by an at the lowest level then can be tangent and the full standard infinity will end at 1/1 as a 1/0 of 0/1s. So a heterogeneous 0/3 in 3 would be three distinct 0/1s (yellow, black, white) and a homogeneous 0/4 would be a solid set of 0/1s. So they can be added but in the denominators and additional language is required... like with waves instead of marks on a ruler... so a 0/5 is five time the wavelength of a 0/1 (with the duality of units). Additional 0/2*2/0 = 2/2 * 0/0 reduces to 1/1 but that hides information... the units are the same because 0/2 is twice the length of 0/1 but 2/0 is twice the length of 1/0... so it is 2/1 as long compared to 1/1 but 1/2 as dense since it's zero points are larger... so the math can work and expand capabilities reliant to light waves and infinities in black holes for physics and other features.
Bertrand Russel's Logical Book Shelf - So this one morose has to to with an "All" set and a logical syntax that can handle various classes. Symbolic logic is fun but it often runs into issues with defining necessary and sufficient conditions due to complexities that would take to long to define down to the atomic level. So in abstract a classic is "all trees are plants" so her being a plant is a necessary condition to be a tree but not a sufficient one because a thing could be a plant and not a tree by being a flower or grass. Now if a thing is a tree, it is sufficient to conclude that it a plant... however, the math of primes is much easier to use and would allow for complex categorical groupings. Instead of a plant/tree example, lets look at the number 30... logically we can say that 30 is sufficient in saying that it's primes are 2, 3, 5 (1,6,10,15, and 30 as more ambiguous degrees). 2, 3, and 5 are necessary for 30 but not sufficient as 2 could be present but so could 7 for 14 and all three of them need to be present without additional prime factors. So it wouldn't be hard to assign values and orders to categories based on primes but not allowing them to collapse so that 4*8 would not be the same as 2 * 16 even though both would make 32... the same would be true of an area but a 4x8 rectangle is not the same as a 2x16 rectangle even if the area is equal... information is lost in the system. So for his lack of an "all quality" for a category we could use the power zero, a power of 1 for an infinitive category (a something verse the something), and an ordered list of from powers 2 and beyond. So for a library of all music we could use 2 as a prime category and power zero for the full set where as 4 could be a specific song, 8 another, 16 another, and so on. Synthetics of categories could also be used as he mentions a bookshelf not belonging to the category of books. So the category of book could be 3 and the category shelf could be 5 so an infinitive book shelf would be 3*5... It could be shelf 125 of a list of all shelves and contain books 9, 81, etc.. on the list of books. It's a clunky system but more of an abstract way to group and nest categories withing the All of All class of zero and carve out ordered space in infinity in a logical manner.
This is a wonderful book filled with the drama, sorrows, and egos of mathematicians grappling with the infinite nature of numbers as they underpin science and thus human knowledge and progress. The mathematics education is out done by the vivid narratives that went into the discoveries.
Cantor's Insanity for a Decimal Based Infinity with Alpha Null - One of the most heart breaking stories is Cantor's suicide due bulling and the right idea but with a corrupted math language... he was on the right track with his geometric factorial sets (line ratios) but the decimals threw him. So with a little logic of primes, the base ten system is good for numbers composed of factors of 2 and 5. 1/2 is .5 and 1/5 is .2... second powers are the same with 1/4 being .25 and 1/25 being .04... the problem comes with a prime factor that is not a 2 or a 5 like 3. 1/3 is .33333333333333... and on but a pie can be cut in three pieces with no reminder (as long as one isn't to the molecular anal level) but this can be proven with a base 30 system. The Hebrews use there 22 letter alphabet as a number system as well so lets make a base 30 system with the roman alphabet plus 4 other characters. ABCDEFGHIJKLMNOPQRSTUVWXYZ$#@0 would be the unique numerals and would repeat in trecmals instead of decimals so "A0"would be 30 and "B0" would be 60... "CE" would be 95. So in a lower trecmal place A (1) divided by C (3) or A/C would be ".J" and end there being the 10/30th of the way through the system with no remainder. All that is to say that fractions as nesting functions are accurate and true to measure the space from 0/0 to 1/1 rather than the corruption of decimal strings which are handy for less or greater than comparisons (anyone who's uses metric and imperial standard fractions wrenches knows the difference as the denominators is change so using 16 or 32 transforms makes it a little easier). Whats more one can see a cool stacking phenomena in pie if you count the overlap frequency as 1/1, 1/2 - 2/2, 1/3 - 2/3 - 3/3, then 1/4 -2/4 -3/4 - 4/4 etc as 0/0, 0/1, 0/1 are later. But the inverse of N/1 will be it's frequency in above and below 1/1... every other number has at least on 2 so half of infinity is "even" and the 4s nest on ever other 2 like wave intersecting... the same for 8s as every third 2 or it's exponent counting within frequencies... but it's the same between 0/0 and 1/1. So 1/2 of the fractions will have a redundancy on the 1/2 line in terms of a bar graph. The same with exponents of 3 in 1/3 or exponents of synthetic of primes like 1/6 landing on both their primes and unique marks. Yet the main reason for misunderstanding infinite sequences is due to the base ten decimal strings that hide the fractional frequency overlaps above and below the gauge of one over one and there's plenty to explore in this space but Cantor couldn't get it with those horn on his head and stubborn peers fencing him in.
Dividing by Zero and the Infinitesimals or Fluctuation - The Author mentions Bonaventura Cavalieri's work on infinitesimals for one dimensional lines or two dimensional planes one dimension thick. The same thing appears with Newton's fluctuations or Leibniz's infinitesimals and the symbols to describe them in the joint discovery of Calculus as well as Bishop Berkeley's commentaries. So once again there is a clear phenomena but because it doesn't have a mental reference, it is said not to exist with a name alone as math requires numbers. So let's give it a number but seeing how far away an infinitesimal is from itself: 1/1 - 1/1 = 0/1. So it is a zero point but if you keep the denominator it is differentiated from plain zero... and has a point place on a line or plane or volume for any number referenced from the origin. Thus a line is 0/1 thick and a 2D area is 0/1 deep... they can be summed but that requires more out of the box thinking with the denominator and frequency spacing as described with the fractions above. The main point is that you can divide by zero if you think about the problem and add more detail so information is not lost. Say you have an area that is 1*1 which is 1^2 in the X and Y dimensions so that 1^2 = 1 (X) * 1(Y). Now if we count the zero point lines in each direction there should be 1/0 lines of 1/1 with respect to X's vertical lines in Y and Y's horizontal lines in X each a 0/1 thick (1/0*0/1 = 1/1 *0/0 for no information loss with 0). So if we break it down, the zero point lines can be stack on top of each other to make a line (1/0)*(1/1) tall (in Y) or wide (in X) and (0/1) thick in (in X or Y respectively) and still have the same amount (0/0) of area in terms of the (1/1)*(1/1) by keeping track of 1s and 0s and the amount of dimensions as well as the total stuff that is zeroed in. You could do the same for a different area or by adding another dimension for a volume. Granted, new notation is needed but that's not hard so long as it always describe the phenomena accurately instead of the vain task of making the phenomena fit the arbitrary notation. This can also be applied to the example of Gabriel's Horn in the books as scaling there infinities and zero point is the basis of the fundamental theorem of calculus with differentiation and integration. The slope (dualistic) at a point (non dualistic zero point) would be similar to a 1/2 offset with a grid and it's delta or measure from the origin (0/0) zero point in left,right,up, down,back, front in terms of delta zero points so they are offset by 1/2 a zero point or it polarities... thus a point overlapping its delta on a line slope would have certain amount of zero points to one side and so many to the other for a differentiated slope at a point (dual and non dual).
Heterogeneous Fractions Vs Homogeneous Fractions - This is more of an addendum to make the new numbers add up but it requires unique rules for unique numbers. As a visual, think of a box 1/1 by 1/1 and 1/2 of it is blue and 1/2 is red. So with a homogeneous fraction there would be 2 rectangles 1/1x1/2 one red and one blue. In an as above so below, lets imaging all even fractions denominator points (N/2, N/4, N/6... etc) on the one side of 1/2 as blue and all the odd fraction denominator points (N/1, N/3, N/5... etc) so while the points can be counted the can be order below the 1/1 gauge from 0/0 with all the points. Now imagine the square is purple and thus heterogeneous fractions so that ever other point is red (0/1), blue (0/2), red (0/3), blue (0/4)... etc. So these are the points Zeno would count by an at the lowest level then can be tangent and the full standard infinity will end at 1/1 as a 1/0 of 0/1s. So a heterogeneous 0/3 in 3 would be three distinct 0/1s (yellow, black, white) and a homogeneous 0/4 would be a solid set of 0/1s. So they can be added but in the denominators and additional language is required... like with waves instead of marks on a ruler... so a 0/5 is five time the wavelength of a 0/1 (with the duality of units). Additional 0/2*2/0 = 2/2 * 0/0 reduces to 1/1 but that hides information... the units are the same because 0/2 is twice the length of 0/1 but 2/0 is twice the length of 1/0... so it is 2/1 as long compared to 1/1 but 1/2 as dense since it's zero points are larger... so the math can work and expand capabilities reliant to light waves and infinities in black holes for physics and other features.
Bertrand Russel's Logical Book Shelf - So this one morose has to to with an "All" set and a logical syntax that can handle various classes. Symbolic logic is fun but it often runs into issues with defining necessary and sufficient conditions due to complexities that would take to long to define down to the atomic level. So in abstract a classic is "all trees are plants" so her being a plant is a necessary condition to be a tree but not a sufficient one because a thing could be a plant and not a tree by being a flower or grass. Now if a thing is a tree, it is sufficient to conclude that it a plant... however, the math of primes is much easier to use and would allow for complex categorical groupings. Instead of a plant/tree example, lets look at the number 30... logically we can say that 30 is sufficient in saying that it's primes are 2, 3, 5 (1,6,10,15, and 30 as more ambiguous degrees). 2, 3, and 5 are necessary for 30 but not sufficient as 2 could be present but so could 7 for 14 and all three of them need to be present without additional prime factors. So it wouldn't be hard to assign values and orders to categories based on primes but not allowing them to collapse so that 4*8 would not be the same as 2 * 16 even though both would make 32... the same would be true of an area but a 4x8 rectangle is not the same as a 2x16 rectangle even if the area is equal... information is lost in the system. So for his lack of an "all quality" for a category we could use the power zero, a power of 1 for an infinitive category (a something verse the something), and an ordered list of from powers 2 and beyond. So for a library of all music we could use 2 as a prime category and power zero for the full set where as 4 could be a specific song, 8 another, 16 another, and so on. Synthetics of categories could also be used as he mentions a bookshelf not belonging to the category of books. So the category of book could be 3 and the category shelf could be 5 so an infinitive book shelf would be 3*5... It could be shelf 125 of a list of all shelves and contain books 9, 81, etc.. on the list of books. It's a clunky system but more of an abstract way to group and nest categories withing the All of All class of zero and carve out ordered space in infinity in a logical manner.
This is a wonderful book filled with the drama, sorrows, and egos of mathematicians grappling with the infinite nature of numbers as they underpin science and thus human knowledge and progress. The mathematics education is out done by the vivid narratives that went into the discoveries.
January 20, 2024
I picked up A Brief History of Infinity out of pure curiosity (and maybe a little motivation to understand Gojo’s Infinity better—Jujutsu Kaisen fans, you get it). While the book does a decent job of explaining the historical and philosophical development of the concept of infinity, it didn’t completely blow me away.
Brian Clegg takes readers through an intellectual journey, from ancient Greek philosophers grappling with paradoxes to modern-day mathematics and physics. The narrative is accessible and entertaining at times, particularly when delving into figures like Zeno or Cantor, whose ideas laid the foundation for how we think about infinity today.
However, as much as I appreciated the history and anecdotes, the explanations often felt surface-level. For a concept as complex and mind-bending as infinity, I wanted more depth and less repetition. At times, it felt like the book skimmed over the truly fascinating bits, favoring simplicity over tackling harder questions.
That said, it was an informative and enjoyable read for what it is—a primer on the history and ideas surrounding infinity. While it didn’t fully unlock Gojo’s powers for me (still working on that), it gave me a better appreciation of how infinity has puzzled and inspired humanity throughout history. If you’re casually interested in math or philosophy, this book is worth a look.
Brian Clegg takes readers through an intellectual journey, from ancient Greek philosophers grappling with paradoxes to modern-day mathematics and physics. The narrative is accessible and entertaining at times, particularly when delving into figures like Zeno or Cantor, whose ideas laid the foundation for how we think about infinity today.
However, as much as I appreciated the history and anecdotes, the explanations often felt surface-level. For a concept as complex and mind-bending as infinity, I wanted more depth and less repetition. At times, it felt like the book skimmed over the truly fascinating bits, favoring simplicity over tackling harder questions.
That said, it was an informative and enjoyable read for what it is—a primer on the history and ideas surrounding infinity. While it didn’t fully unlock Gojo’s powers for me (still working on that), it gave me a better appreciation of how infinity has puzzled and inspired humanity throughout history. If you’re casually interested in math or philosophy, this book is worth a look.
June 15, 2014
This was a very interesting exploration of a concept we bandy around every day but have no real understanding of. By first exploring the philosophical and religious approaches through the millennia toward infinity, Mr. Clegg provided a surprisingly well-rounded view of humanity's quest to grasp the ungraspable. The mathematical and scientific approach to infinity eventually enters the story, and Mr. Clegg reveals that many today accept infinity as a useful tool in their calculations without really bothering to understand what it means. Infinity in modern physics is often treated like the "weirdness" of quantum physics: it makes no sense, but it works. This book doesn't offer (nor does it promise) a true understanding of infinity, but it is a fascinating look at how great minds try to tackle one of the many concepts that may be forever beyond human understanding.
April 6, 2010
This book is more about the people who thought about infinity than it is about infinity itself. As such, it's sort of an orthogonal projection of the history of science onto one particular subject, so it includes Newton vs. Leibniz and the infinitesimals and limits of calculus, as well as the feud between Cantor and Kronecker over Cantor's infinite sets. For someone who doesn't know the theory (I didn't run across anything I wasn't already familiar with--I can prove that the integers, whole numbers, and rational numbers are smaller sets than the real numbers without working too hard), it should be fairly approachable.
June 5, 2013
Dammit. I wanted to like this book; I really did. I'm fascinated by the topic of infinity and enjoy a well-written non-textbook discussion of math, science, and/or history. Sadly, this isn't one of those. The author uses far too many swathes of other texts (and then summarizes them) and has an odd writing style that simply puts you to sleep. I gave it five chapters before I had to put it down. Life's too short to read boring books.
April 17, 2012
This book is a very boring read. The main problem is that some verses from other books are simply copy-pasted at various junctures. This makes the book very hard to read. One can read one, two, three ... infinity by George Gamow or some of the Simon Singh's or Carl Sagan's books instead of wasting time on this book.
September 14, 2010
A largely tepid affair. Another history book masquerading as a math book, with all the usual fear of actually attempting to explain some harder concepts. That's a little harsh, I think; there were some decent bits, but nothing that isn't covered better elsewhere.
October 23, 2013
Badly written and without a clear idea or point.
December 11, 2020
I got interested in the problem of infinity when I first learned about renormalization in quantum mechanics. See: The Infinity Puzzle: Quantum Field Theory and the Hunt for an Orderly Universe. The idea that physicists "divided out" infinity and got reasonable answers to actual questions never bothered me since I was a chemistry major who just wanted to be able to calculate the binding energy of our latest drug candidate with the protein target. All was good. The fact that mathematicians thought that this was ridiculous and beyond contempt didn't enter my mind until I stopped caring about drug discovery and started caring about what is real and what is not. Spoiler: Most drug discovery is about as credible as a Dan Brown novel. I spent 15 years doing this, so...flame away.
I started this book because I came across the book Number: The Language of Science, The Masterpiece Science Edition. Which, by the way, is the only book I own that has an endorsement from Einstein on the jacket, which is amazing in its own right. "Number" describes how the numbers depend on infinite processes and made the point that without infinity, math mostly can't work. Spoiler: If you limit the number line then multiplication breaks at some point...so it can't be limited, and you have to accept infinity.
I also decided on listening to this book because the author did such a good job with Professor Maxwell's Duplicitous Demon: How James Clerk Maxwell unravelled the mysteries of electromagnetism and matter. Clegg is a good science writer - not great, but good. The Douglas Adams quotes in this book could have pushed him into the "very good" category if he could have avoided the Pixar quote...alas
Here's something to think about: If the square root of two is really infinity long and comes from no algebraic equation, then that means every song you have ever heard, or will ever be written is encoded in every digital format ever invented or will ever be invented somewhere in the digits of that real number. So will every DNA sequence of every living thing on earth now and during the entire existence of the earth. In fact, all the RNA sequences (all the rage these days) are included too! All in the square root of two. Oh! and also somewhere in Pi and also in e... This is the conclusion of Tegmark's book Our Mathematical Universe: My Quest for the Ultimate Nature of Reality. In a multiverse our current universe is just an address. Maybe in Pi, I am Alexander the Great, in e I have more Goodread votes than Manny and Not combined.
Unless you are currently very high, none of this is very interesting. But, if you are not, then the fact that all of the calculus depends on the infinite is seriously interesting. Further, only recently has the infinitesimal been put on a solid mathematical footing.
This brings me to the point of this review. Books like Clegg's make me start reading real textbooks on the subject. I hope that pop-science does this for you too. It is amazing that a little Dover book like Infinitesimal Calculus, which is mentioned in concept by Clegg, can explain calculus without needing a rigorous definition of infinity (which apparently drove Cantor mad). The whole thing is about 100 pages long!
I really recommend this book, but only if you are willing to read more difficult texts. Otherwise, when you talk about it in Zoom happy hours, you are likely to get a "What? Meh."
I started this book because I came across the book Number: The Language of Science, The Masterpiece Science Edition. Which, by the way, is the only book I own that has an endorsement from Einstein on the jacket, which is amazing in its own right. "Number" describes how the numbers depend on infinite processes and made the point that without infinity, math mostly can't work. Spoiler: If you limit the number line then multiplication breaks at some point...so it can't be limited, and you have to accept infinity.
I also decided on listening to this book because the author did such a good job with Professor Maxwell's Duplicitous Demon: How James Clerk Maxwell unravelled the mysteries of electromagnetism and matter. Clegg is a good science writer - not great, but good. The Douglas Adams quotes in this book could have pushed him into the "very good" category if he could have avoided the Pixar quote...alas
Here's something to think about: If the square root of two is really infinity long and comes from no algebraic equation, then that means every song you have ever heard, or will ever be written is encoded in every digital format ever invented or will ever be invented somewhere in the digits of that real number. So will every DNA sequence of every living thing on earth now and during the entire existence of the earth. In fact, all the RNA sequences (all the rage these days) are included too! All in the square root of two. Oh! and also somewhere in Pi and also in e... This is the conclusion of Tegmark's book Our Mathematical Universe: My Quest for the Ultimate Nature of Reality. In a multiverse our current universe is just an address. Maybe in Pi, I am Alexander the Great, in e I have more Goodread votes than Manny and Not combined.
Unless you are currently very high, none of this is very interesting. But, if you are not, then the fact that all of the calculus depends on the infinite is seriously interesting. Further, only recently has the infinitesimal been put on a solid mathematical footing.
This brings me to the point of this review. Books like Clegg's make me start reading real textbooks on the subject. I hope that pop-science does this for you too. It is amazing that a little Dover book like Infinitesimal Calculus, which is mentioned in concept by Clegg, can explain calculus without needing a rigorous definition of infinity (which apparently drove Cantor mad). The whole thing is about 100 pages long!
I really recommend this book, but only if you are willing to read more difficult texts. Otherwise, when you talk about it in Zoom happy hours, you are likely to get a "What? Meh."
January 9, 2019
"Do numbers go on forever?" my 4-year-old daughter asked me the other day. Infinity, or at least the idea of very big numbers, grips the mind at a tender age. But little did I know, until I read this book, that the numbers that go on forever are just one form of infinity, and that there is a 'bigger' infinity, one that is paradoxically compressed into the tiny gap between 0 and 1. This was the brilliant insight of Georg Cantor, whose pioneering work on infinity in the late nineteenth and early twentieth century helped drive the poor fellow mad. The proof is an elegant one: think of a list of every number between 0 and 1, such as 0.493683037468, 0.876806059576 and so on. Now construct a new number, taking the first digit of the first number in the table, the second digit of the second number and so on. Then change every digit in this number (for example by adding 1, shifting 9 up to 0). The number produced logically can't be the same as any number in the list - even though that (infinitely long) list included every number between 0 and 1. It was as if Cantor had discovered a new 'dimension' of infinity. No wonder he lost his grip on sanity - although opposition from a vindictive rival, Leopold Kronecker, was perhaps what drove him over the edge.
Cantor's story is just one of the fascinating episodes in this book, which tells of the mathematicians and philosophers who have pondered infinity and how perceptions of this impossible number have evolved over the centuries. On the way, there are several interesting diversions, including one on how the Ancient Greeks managed without having a concept of fractions. Rather than thinking of a "half", they would visualise a shape that was smaller than another by a factor of 2, thinking of the relationship purely in terms of whole numbers.
In many ways, this book reminded me of Robert Kaplan's excellent The Nothing That Is: A Natural History of Zero, except that Clegg tilts more towards maths and Kaplan towards history. And though the concept of zero can be problematic, that of infinity is more mind-bending by far. There's something about infinity we can't get our head around, almost by definition, but in Brian Clegg's genial company, it is fun to try. My only slight reservation was that while most of the mathematical concepts were excellently explained, other parts of the book made me painfully aware of the lacunae in my knowledge, especially the chapters which touched on calculus. That said, the non-mathematical reader shouldn't be scared of attempting this book, which is a fascinating journey for the mind.
"You will catch a glimpse of beauty that stops you in your tracks, but moments later you are not sure if you saw anything at all. Then, quite unexpectedly, the magnificent animal stalks out into full view for a few, fleeting seconds..."
Cantor's story is just one of the fascinating episodes in this book, which tells of the mathematicians and philosophers who have pondered infinity and how perceptions of this impossible number have evolved over the centuries. On the way, there are several interesting diversions, including one on how the Ancient Greeks managed without having a concept of fractions. Rather than thinking of a "half", they would visualise a shape that was smaller than another by a factor of 2, thinking of the relationship purely in terms of whole numbers.
In many ways, this book reminded me of Robert Kaplan's excellent The Nothing That Is: A Natural History of Zero, except that Clegg tilts more towards maths and Kaplan towards history. And though the concept of zero can be problematic, that of infinity is more mind-bending by far. There's something about infinity we can't get our head around, almost by definition, but in Brian Clegg's genial company, it is fun to try. My only slight reservation was that while most of the mathematical concepts were excellently explained, other parts of the book made me painfully aware of the lacunae in my knowledge, especially the chapters which touched on calculus. That said, the non-mathematical reader shouldn't be scared of attempting this book, which is a fascinating journey for the mind.
"You will catch a glimpse of beauty that stops you in your tracks, but moments later you are not sure if you saw anything at all. Then, quite unexpectedly, the magnificent animal stalks out into full view for a few, fleeting seconds..."
September 25, 2025
I loved many of Brian Clegg's other books, and as a Mathematician, I wanted to love this one - which is about mathematics. But I had a difficult time. This book is about infinity - and how our understanding of it changed through the ages, culminating in mathematical theories such as calculus (and its "infinitesimals"), set theory with its infinite sets, and Cantor's transfinite numbers.
However, it turns out that this is less a book about mathematics than a book of history of everything that has to do with this topic, and even things that don't have anything to do with this topic. On the quest to explain our current understanding of infinity, Clegg mentions everyone that said anything remotely related (or not even remotely related) to this topic. Those "mentions" are actually more-like mini-biographies of dozens of individuals. Many times, I found this writing style tedious - for example, although it makes sense to mention calculus - and the difference between the original infinitesimals and the later "limit" approach, there was no need to write several pages on the minute (and irrelvant) details between the slightly different (to modern ears) approaches of Newton, Leibniz, and other contemporaries (each, again, receives a mini-biography). And later, when Clegg's "stream of consciousness" brings him to discuss complex numbers, do we really need the biographies of the different people that invented them? Complex numbers are an interesting topic by itself, but has nothing to do with infinity. Later in the book still, when Clegg nicely explains why the two-dimensional plane has the same cardinality as one-dimensional line, he refers to Cartesian coordinates, and instead of just assuming that any reader with high-school math knowledge (or just, has ever seen a map...) would find it obvious - takes it as an opportunity to also give a biography of Decartes.
I found some of these mini-biographies to be interesting, and learned interesting things I didn't know about the theologist Augustine of Hippo, the mathematicians Weierstrass, Bolzano, and others. But I deducted stars in my review because too many times I found myself bored when reading this book, reading minute details of some random historic figure's personal life or details of theories that have long ago been disproven or fallen out of fashion. I would have personally preferred a book that skips most of the mis-steps on the way to the current accepted theory, and and don't spend pages on pages on the personal life of dozens of different people.
However, it turns out that this is less a book about mathematics than a book of history of everything that has to do with this topic, and even things that don't have anything to do with this topic. On the quest to explain our current understanding of infinity, Clegg mentions everyone that said anything remotely related (or not even remotely related) to this topic. Those "mentions" are actually more-like mini-biographies of dozens of individuals. Many times, I found this writing style tedious - for example, although it makes sense to mention calculus - and the difference between the original infinitesimals and the later "limit" approach, there was no need to write several pages on the minute (and irrelvant) details between the slightly different (to modern ears) approaches of Newton, Leibniz, and other contemporaries (each, again, receives a mini-biography). And later, when Clegg's "stream of consciousness" brings him to discuss complex numbers, do we really need the biographies of the different people that invented them? Complex numbers are an interesting topic by itself, but has nothing to do with infinity. Later in the book still, when Clegg nicely explains why the two-dimensional plane has the same cardinality as one-dimensional line, he refers to Cartesian coordinates, and instead of just assuming that any reader with high-school math knowledge (or just, has ever seen a map...) would find it obvious - takes it as an opportunity to also give a biography of Decartes.
I found some of these mini-biographies to be interesting, and learned interesting things I didn't know about the theologist Augustine of Hippo, the mathematicians Weierstrass, Bolzano, and others. But I deducted stars in my review because too many times I found myself bored when reading this book, reading minute details of some random historic figure's personal life or details of theories that have long ago been disproven or fallen out of fashion. I would have personally preferred a book that skips most of the mis-steps on the way to the current accepted theory, and and don't spend pages on pages on the personal life of dozens of different people.
February 9, 2023
Not an easy book to review. Clegg ranges over so many historical developments as he explains how concepts of infinity were developed and as techniques emerged to deal with the innately large and the infinitely small. One of the things that I found most fascinating was his short biographies of some of the characters along the way. Cantor, for instance, being opposed and persecuted by his former mentor, Kronecker, and, eventually, Cantor suffering from mental breakdowns. Were they directly caused by Kronecker? Maybe. And his equally delightful pen-picture of Kurt Godel...the party boy...who later turned into a very strange old man suffering from paranoia. And, of course, Galileo...suffering at the hands of the "defenders of the faith". Faith has a lot to answer for! And having faith never seems to have turned out correct when it didn't support objective facts.
I've been struggling with writing reviews of several books recently and this is one of them. There is a heck of a lot of solid information there and Clegg ranges over so many subjects from Indian concepts of zero divided by zero, to Newton and Leibnitz's competition over the invention of calculus; Cantor's techniques for dealing with multiple infinities; Bertrand Russell and set theory (and Venn diagrams .....did Euler actually invent them?). Yes, there is certainly a lot to digest there though Clegg writes well and the historical material is quite fascinating. But if you want to get a better grasp on the nature of infinity I'd recommend "Beyond Infinity" by Eugenia Cheng. She goes into a lot more detail about how one might go about filling "Hilbert's Hotel".....with infinite rooms when every room is full and a coach load with infinite guests arrives. I enjoyed Clegg's book but don't pretend to understand all the concepts. I give it 4 stars.
I've been struggling with writing reviews of several books recently and this is one of them. There is a heck of a lot of solid information there and Clegg ranges over so many subjects from Indian concepts of zero divided by zero, to Newton and Leibnitz's competition over the invention of calculus; Cantor's techniques for dealing with multiple infinities; Bertrand Russell and set theory (and Venn diagrams .....did Euler actually invent them?). Yes, there is certainly a lot to digest there though Clegg writes well and the historical material is quite fascinating. But if you want to get a better grasp on the nature of infinity I'd recommend "Beyond Infinity" by Eugenia Cheng. She goes into a lot more detail about how one might go about filling "Hilbert's Hotel".....with infinite rooms when every room is full and a coach load with infinite guests arrives. I enjoyed Clegg's book but don't pretend to understand all the concepts. I give it 4 stars.
July 22, 2018
The concept of infinity invariably comes up in countless places when one contemplates life, be it mathematical or otherwise. What is the biggest number? What is the length of the longest line? These and other such questions have fascinated mathematicians for centuries but as Brian Clegg shows, infinity as a mathematical concept was not properly dealt with until very recently. Even though the infinitely small is essential to the working of calculus, there was a certain vagueness around the concept due to mathematicians resorting to ‘potential infinities and infinitesimals’ in order to avoid the paradoxes that this subject throws up. This book with its slow pace and engrossing style, treats this concept and its history in a way that will allow all readers , even the ones unfamiliar of basic calculus to enjoy the read. That is a tall task for a book that aims to explore the understanding of the infinite through the centuries. On the other hand, the reader more adept at Maths may feel a little unsatisfied by the depth provided in the discussion of Set Theory and George Cantor’s contributions. A stand out feature of book is the author’s brief introduction to each new mathematician that played a role in the history of the subject. These pieces not only put those specific individuals into the academic context of their time but also provides us with tidbits of their personal lives so that we may appreciate their contributions on a higher level. All in all, this is a highly readable book and a suitable primer to more advanced studies of the infinite in the mathematical world.
January 5, 2024
Book #2 of 2023. "Infinity" by Brian Clegg. 2/5 rating. This book was an absolute slog to get through. Brian took an incredibly complex topic and did not make it any easier to understand.
I literally teach math and theoretically would be in the audience of people most likely to understand what was being discussed in this book. Instead, Brian used a whole bunch of hard-to-grasp examples from mathematical history and tried to explain them in words without many visuals. He did offer some history on the views of infinity over time, but even here, I felt like it was a whole bunch of random facts about people in history. There was next to no throughline, story-arc or anything else.
Apart from this, he uses a lot of jargon for seemingly no reason, which just makes the topic seem more impossible to grasp. What made me laugh is I wrote a note that he was doing this, then he ironically wrote in the next chapter that some people use jargon for no reason and it "can be just as confusing as if it were intended to conceal". Hmmmm, sounds somewhat familiar...
The book just seemed like a winding, somewhat nonsensical meandering through religion and vague history on infinity.
The only really interesting thing I got from this book is that Cartesian coordinates were named after Rene Descartes. Now that I actually thought was a fun fact.
Don't read this book, it's a waste of your time.
I literally teach math and theoretically would be in the audience of people most likely to understand what was being discussed in this book. Instead, Brian used a whole bunch of hard-to-grasp examples from mathematical history and tried to explain them in words without many visuals. He did offer some history on the views of infinity over time, but even here, I felt like it was a whole bunch of random facts about people in history. There was next to no throughline, story-arc or anything else.
Apart from this, he uses a lot of jargon for seemingly no reason, which just makes the topic seem more impossible to grasp. What made me laugh is I wrote a note that he was doing this, then he ironically wrote in the next chapter that some people use jargon for no reason and it "can be just as confusing as if it were intended to conceal". Hmmmm, sounds somewhat familiar...
The book just seemed like a winding, somewhat nonsensical meandering through religion and vague history on infinity.
The only really interesting thing I got from this book is that Cartesian coordinates were named after Rene Descartes. Now that I actually thought was a fun fact.
Don't read this book, it's a waste of your time.
October 22, 2025
Infinite jest
I bought this for my mathematician son-in-law a couple of years ago and pulled it off his shelf to read last week. It's an entertaining, discursive account of the concept of infinity: something we all think we know a bit about (it's the biggest number you can imagine, it's one divided by zero, it's a measure of the powers of the deity, etc), but have trouble pinning down.
As the title suggests, the approach here is historical: starting with Zeno and his paradoxes, successive ideas about infinity are presented (and you find yourself nodding in agreement) before being cast aside in favour of more enlightened and subtle notions. By the end, I wasn't sure whether I knew more or less than I started, but there are some good stories along the way: Russell's set paradox, Gabriel's horn, Hilbert's hotel, to name but a few.
Because (I think) infinity also encompasses the infinitely small, there's also an account of the development of the calculus (including the Newton-Leibniz dispute), which was helpful to have in the context of the wider theme. I thought the digression on the dispute between Kronecker and Cantor (whilst of human interest), was more of a distraction, however.
Originally reviewed 3 September 2019
I bought this for my mathematician son-in-law a couple of years ago and pulled it off his shelf to read last week. It's an entertaining, discursive account of the concept of infinity: something we all think we know a bit about (it's the biggest number you can imagine, it's one divided by zero, it's a measure of the powers of the deity, etc), but have trouble pinning down.
As the title suggests, the approach here is historical: starting with Zeno and his paradoxes, successive ideas about infinity are presented (and you find yourself nodding in agreement) before being cast aside in favour of more enlightened and subtle notions. By the end, I wasn't sure whether I knew more or less than I started, but there are some good stories along the way: Russell's set paradox, Gabriel's horn, Hilbert's hotel, to name but a few.
Because (I think) infinity also encompasses the infinitely small, there's also an account of the development of the calculus (including the Newton-Leibniz dispute), which was helpful to have in the context of the wider theme. I thought the digression on the dispute between Kronecker and Cantor (whilst of human interest), was more of a distraction, however.
Originally reviewed 3 September 2019
July 12, 2018
I was going to give this book 4 stars only because I thought there could've been more content, but when I thought about it, there really is so much to say about infinity. But in order to maintain the audience's attention, and to attract a wider audience, I would have to say that this book did an excellent job. Virtually anyone can read it. It does help to have even a minimal understanding of calculus, but Clegg does a good job of explaining the concepts that involve infinity.
It's truly so fascinating how complex this topic is. I really enjoyed the different accounts of men who tried to tackle this concept, some had nervous breakdowns, some got into pretty intense encounters with other men doing the same thing. It was really something to read about.
Would recommend this book to anyone who wants to whet their appetite regarding infinity.
It's truly so fascinating how complex this topic is. I really enjoyed the different accounts of men who tried to tackle this concept, some had nervous breakdowns, some got into pretty intense encounters with other men doing the same thing. It was really something to read about.
Would recommend this book to anyone who wants to whet their appetite regarding infinity.
January 11, 2025
I first read A Brief History of Infinity while completing my undergraduate teaching degree, and it left a lasting impression. Brian Clegg’s exploration of infinity is both accessible and fascinating, offering a deep dive into one of the most mind-boggling concepts in mathematics and philosophy.
What I really appreciated was how the book managed to explain such complex ideas in a way that was both engaging and understandable. Clegg blends history, science, and thought-provoking concepts seamlessly, making this a perfect read for anyone curious about the limits of human understanding.
If you're interested in philosophy, mathematics, or the mysteries of the universe, this book is a must-read.
What I really appreciated was how the book managed to explain such complex ideas in a way that was both engaging and understandable. Clegg blends history, science, and thought-provoking concepts seamlessly, making this a perfect read for anyone curious about the limits of human understanding.
If you're interested in philosophy, mathematics, or the mysteries of the universe, this book is a must-read.
April 29, 2022
A good history of infinity and the mathematical concepts that are tied to it and/or arose from its conception as an idea. Some of the concepts are explained well and are understandable, and some are not (requiring further googling to comprehend better). Honestly, the best part of this book is how it worked as a companion piece for me as I studied calculus. It made calculus more interesting as I learned about the history of the people and ideas behind its development. My favourite parts were the story of the conception and proof of finding the volume of a circle via infinitesimal wedges and the chapters about Cantor and aleph-null!
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