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A Brief History of Infinity: The Quest to Think the Unthinkable

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'Space is big. Really big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the street to the chemist, but that's just peanuts to space.' Douglas Adams, Hitch-hiker's Guide to the Galaxy We human beings have trouble with infinity - yet infinity is a surprisingly human subject. Philosophers and mathematicians ha 'Space is big. Really big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the street to the chemist, but that's just peanuts to space.' Douglas Adams, Hitch-hiker's Guide to the Galaxy We human beings have trouble with infinity - yet infinity is a surprisingly human subject. Philosophers and mathematicians have gone mad contemplating its nature and complexity - yet it is a concept routinely used by schoolchildren. Exploring the infinite is a journey into paradox. Here is a quantity that turns arithmetic on its head, making it feasible that 1 = 0. Here is a concept that enables us to cram as many extra guests as we like into an already full hotel. Most bizarrely of all, it is quite easy to show that there must be something bigger than infinity - when it surely should be the biggest thing that could possibly be. Brian Clegg takes us on a fascinating tour of that borderland between the extremely large and the ultimate that takes us from Archimedes, counting the grains of sand that would fill the universe, to the latest theories on the physical reality of the infinite. Full of unexpected delights, whether St Augustine contemplating the nature of creation, Newton and Leibniz battling over ownership of calculus, or Cantor struggling to publicise his vision of the transfinite, infinity's fascination is in the way it brings together the everyday and the extraordinary, prosaic daily life and the esoteric. Whether your interest in infinity is mathematical, philosophical, spiritual or just plain curious, this accessible book offers a stimulating and entertaining read.


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'Space is big. Really big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the street to the chemist, but that's just peanuts to space.' Douglas Adams, Hitch-hiker's Guide to the Galaxy We human beings have trouble with infinity - yet infinity is a surprisingly human subject. Philosophers and mathematicians ha 'Space is big. Really big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the street to the chemist, but that's just peanuts to space.' Douglas Adams, Hitch-hiker's Guide to the Galaxy We human beings have trouble with infinity - yet infinity is a surprisingly human subject. Philosophers and mathematicians have gone mad contemplating its nature and complexity - yet it is a concept routinely used by schoolchildren. Exploring the infinite is a journey into paradox. Here is a quantity that turns arithmetic on its head, making it feasible that 1 = 0. Here is a concept that enables us to cram as many extra guests as we like into an already full hotel. Most bizarrely of all, it is quite easy to show that there must be something bigger than infinity - when it surely should be the biggest thing that could possibly be. Brian Clegg takes us on a fascinating tour of that borderland between the extremely large and the ultimate that takes us from Archimedes, counting the grains of sand that would fill the universe, to the latest theories on the physical reality of the infinite. Full of unexpected delights, whether St Augustine contemplating the nature of creation, Newton and Leibniz battling over ownership of calculus, or Cantor struggling to publicise his vision of the transfinite, infinity's fascination is in the way it brings together the everyday and the extraordinary, prosaic daily life and the esoteric. Whether your interest in infinity is mathematical, philosophical, spiritual or just plain curious, this accessible book offers a stimulating and entertaining read.

30 review for A Brief History of Infinity: The Quest to Think the Unthinkable

  1. 5 out of 5

    B Schrodinger

    When I saw that Brian Clegg had written a book about infinity and the audiobook was narrated by Gordon Griffin I lined this up as a monthly download from Audible. It was a wonderful listen. Lots of history, lots of philosophy, lots of maths and a sprinkling of science all narrated by one of the most pleasurable narrators I have listened to. There were parts that amazed me, parts that made me laugh, parts that made me sad and parts where I could not quite follow audibly and probably needed a diag When I saw that Brian Clegg had written a book about infinity and the audiobook was narrated by Gordon Griffin I lined this up as a monthly download from Audible. It was a wonderful listen. Lots of history, lots of philosophy, lots of maths and a sprinkling of science all narrated by one of the most pleasurable narrators I have listened to. There were parts that amazed me, parts that made me laugh, parts that made me sad and parts where I could not quite follow audibly and probably needed a diagram or the written book. While Brian Clegg is not the most awe inspiring and beautiful science writer out there, he is an under-appreciated work-horse kind of writer that consistently produces books that are fun, fascinating and accessible. Part of his style that I like is that he is so sensible. There are no large and outrageous claims in his books, they are based on well-researched and well-established material. Probably the best way I can put it is that he isn't out to write sexy books. To some this may seem boring, but to geeks like me it is refreshing and like a comfortable companion that is not out to prove anything, just to tell you a great yarn. Also his books are not aimed to be definitive volumes on the subject, he always refers the reader on to where they may find out more. These are more like a well-produced documentary, with interesting stories and overviews of deeper concepts. So once again Brian has produced an entertaining little volume. I'd recommend his works to both the layman and the professional. The layman gets an interesting introduction to the material and the professional find out many little interesting historical facts and stories along the way.

  2. 5 out of 5

    Kara Babcock

    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: 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: 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.

  3. 5 out of 5

    fourtriplezed

    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!

  4. 5 out of 5

    Peter Baran

    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 brie 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.

  5. 4 out of 5

    Bettie

    FMI - Human beings have trouble with infinity - yet infinity is a surprisingly human subject. Philosophers and mathematicians have gone insane contemplating its nature and complexity - yet it is a concept routinely used by schoolchildren. Presented by the renowned astronomer Heather Couper, these programmes take the listener on a journey with an endless audio horizon and feature contributions from musicians who write endless music; science fiction authors, who create infinite worlds and timeless FMI - Human beings have trouble with infinity - yet infinity is a surprisingly human subject. Philosophers and mathematicians have gone insane contemplating its nature and complexity - yet it is a concept routinely used by schoolchildren. Presented by the renowned astronomer Heather Couper, these programmes take the listener on a journey with an endless audio horizon and feature contributions from musicians who write endless music; science fiction authors, who create infinite worlds and timeless beings; theologians; Buddhist lamas; astro-physicists and mathematicians. It is infinity... in a nutshell. Part One: Space and the Universe -------------------------------- This first programme takes us outside the known universe, and tries to measure whether it has a beginning and end. If a star is a fixed number of light years away, how do we know if there is anything lurking beyond the stars which will not show itself to a human being on earth for several more light years? Are we more comfortable with the finite? Is this why we developed theories like Big Bang to explain things we don't know in sizes which we do know? A group of experts show how the human mind prefers to stop short of a total acceptance of the truly "without end". Supersize, it seems, has to stop somewhere! Part Two: Mathematics --------------------- Imagine a hotel with an infinite number of rooms. Even if the hotel is completely full and there are no vacancies, another guest can be easily accommodated... This is just one of the fascinating insights that Heather Couper will be exploring in the second part of her quest to think the unthinkable. She is joined by mathematicians, psychologists, Sci Fi writers, theologians and a room full of monkeys to try and grasp a handle on how infinity can be seen in our world. The programme looks at how infinity has troubled some of the greatest minds and thinkers, with sometimes deadly consequences. We also learn how Infinity drove one such mathematician to the brink of sanity. Scary stuff indeed. But the entire concept of infinity in mathematics is maddening, and the deeper we go, the stranger the results become. Infinity has also played an important part in the arts. Heather explores the music of Steve Reich, Michael Nyman and Philip Glass to see how endless repetition has inspired these great composers. Join Heather on this tour of an endless audio horizon, where 1=0 and 2 is just a number bigger than 1.

  6. 5 out of 5

    Mary

    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 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 : )

  7. 4 out of 5

    Simona

    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 "irration 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.

  8. 4 out of 5

    Koen Crolla

    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 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.

  9. 5 out of 5

    Bennett Coles

    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 real 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.

  10. 4 out of 5

    Aaron Humphrey

    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 numb 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.

  11. 5 out of 5

    Ellie Julio

    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. 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.

  12. 4 out of 5

    Shanmuganathan

    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. 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.

  13. 5 out of 5

    Mark Schnitzius

    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. 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.

  14. 4 out of 5

    Marco Hokke

    Badly written and without a clear idea or point.

  15. 4 out of 5

    Randy

    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 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."

  16. 4 out of 5

    Nick

    "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 t "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..."

  17. 5 out of 5

    Paul Weiss

    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 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

  18. 5 out of 5

    Usman Baig

    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 concep 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.

  19. 4 out of 5

    Stephanie

    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 fasci 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.

  20. 4 out of 5

    Alex Chan

    Good mixture of the mathematical bits and history of the relevant people. Covers a lot of ideas in a fairly compact way. I don’t know how accessible the mathematical explanations are (I skipped them; I’ve already come across most of it), but the historical details all worked. And the cast in each chapter is usefully small – there’s not much skipped around “oh, this name was last mentioned three chapters ago, who was that again”? So that was nice. Glad to have read, now going on the charity shop pi Good mixture of the mathematical bits and history of the relevant people. Covers a lot of ideas in a fairly compact way. I don’t know how accessible the mathematical explanations are (I skipped them; I’ve already come across most of it), but the historical details all worked. And the cast in each chapter is usefully small – there’s not much skipped around “oh, this name was last mentioned three chapters ago, who was that again”? So that was nice. Glad to have read, now going on the charity shop pile for the next person.

  21. 5 out of 5

    The Tick

    There were a lot of things I really liked about this book, so I was waffling about whether to give it a 3 or 4. In the end, though, there were too many quotes (which didn't add as much as I guess the author thought it did), and there was also almost no mention of anything beyond Europe and how they thought of infinity, but I have a hard time believing nobody outside of Europe thought about it at all. I guess they messed with the narrative he was going for. There were a lot of things I really liked about this book, so I was waffling about whether to give it a 3 or 4. In the end, though, there were too many quotes (which didn't add as much as I guess the author thought it did), and there was also almost no mention of anything beyond Europe and how they thought of infinity, but I have a hard time believing nobody outside of Europe thought about it at all. I guess they messed with the narrative he was going for.

  22. 4 out of 5

    Hannele Kormano

    Three stars for a couple of reasons: very eurocentric (I am so tired of every science history book starting with the ancient Greeks), and also I found it weirdly difficult to focus on the book if the concept wasn't tied to a mathematician's personal story. On that note though, it was fun to read more details about Leibniz vs Newton, and other similar personal struggles that scientists and mathematicians had in introducing ideas that challenged the status quo. The sections on Cantor and sets were Three stars for a couple of reasons: very eurocentric (I am so tired of every science history book starting with the ancient Greeks), and also I found it weirdly difficult to focus on the book if the concept wasn't tied to a mathematician's personal story. On that note though, it was fun to read more details about Leibniz vs Newton, and other similar personal struggles that scientists and mathematicians had in introducing ideas that challenged the status quo. The sections on Cantor and sets were also very interesting. I might try to find more pop science books in that vein.

  23. 5 out of 5

    Ashar Malik

    This book is a nice read. The author lists chronologically the developments from ancient times to modern day in different areas where infinity has popped up and how understanding has evolved on this subject as humans have struggled with it. Will definitely recommend it to everyone. Infinity is a fascinating subject.

  24. 4 out of 5

    Anas

    At first, I was not sure about this book but turns out it is really that good. More than expected. Brian Clegg summarizes up how we as human beings, a finite creature try to grasp the concept of infinity through philosophy, theology, and (of course) mathematics.

  25. 4 out of 5

    Ryan

    I just can't get interested. And this is the second time I've tried to read this book I just can't get interested. And this is the second time I've tried to read this book

  26. 5 out of 5

    James Wheat

    I was hoping for a socio-cultural history of the concept of infinity but this was very heavily mathematical.

  27. 4 out of 5

    Johnny T

    A very entertaining blend of math and history.

  28. 4 out of 5

    Marcus

    Interesting read. Brought back lots of discussions from university days.

  29. 4 out of 5

    Auceanne

    I wasn’t smart enough for this book.

  30. 5 out of 5

    Arun Babji

    It presents an entertaining history along with the background motivation and life story of great people that contributed to our ideas of Infinity.

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