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The Art of the Infinite: The Pleasures of Mathematics
Robert Kaplan's The Nothing That A Natural History of Zero was an international best-seller, translated into eight languages. The Times called it "elegant, discursive, and littered with quotes and allusions from Aquinas via Gershwin to Woolf" and The Philadelphia Inquirer praised it as
"absolutely scintillating."
In this delightful new book, Robert Kaplan, writing together with his wife Ellen Kaplan, once again takes us on a witty, literate, and accessible tour of the world of mathematics. Where The Nothing That Is looked at math through the lens of zero, The Art of the Infinite takes infinity, in its
countless guises, as a touchstone for understanding mathematical thinking. Tracing a path from Pythagoras, whose great Theorem led inexorably to a discovery that his followers tried in vain to keep secret (the existence of irrational numbers); through Descartes and Leibniz; to the brilliant, haunted
Georg Cantor, who proved that infinity can come in different sizes, the Kaplans show how the attempt to grasp the ungraspable embodies the essence of mathematics. The Kaplans guide us through the "Republic of Numbers," where we meet both its upstanding citizens and more shadowy dwellers; and we
travel across the plane of geometry into the unlikely realm where parallel lines meet. Along the way, deft character studies of great mathematicians (and equally colorful lesser ones) illustrate the opposed yet intertwined modes of mathematical the intutionist notion that we discover
mathematical truth as it exists, and the formalist belief that math is true because we invent consistent rules for it.
"Less than All," wrote William Blake, "cannot satisfy Man." The Art of the Infinite shows us some of the ways that Man has grappled with All, and reveals mathematics as one of the most exhilarating expressions of the human imagination.
"absolutely scintillating."
In this delightful new book, Robert Kaplan, writing together with his wife Ellen Kaplan, once again takes us on a witty, literate, and accessible tour of the world of mathematics. Where The Nothing That Is looked at math through the lens of zero, The Art of the Infinite takes infinity, in its
countless guises, as a touchstone for understanding mathematical thinking. Tracing a path from Pythagoras, whose great Theorem led inexorably to a discovery that his followers tried in vain to keep secret (the existence of irrational numbers); through Descartes and Leibniz; to the brilliant, haunted
Georg Cantor, who proved that infinity can come in different sizes, the Kaplans show how the attempt to grasp the ungraspable embodies the essence of mathematics. The Kaplans guide us through the "Republic of Numbers," where we meet both its upstanding citizens and more shadowy dwellers; and we
travel across the plane of geometry into the unlikely realm where parallel lines meet. Along the way, deft character studies of great mathematicians (and equally colorful lesser ones) illustrate the opposed yet intertwined modes of mathematical the intutionist notion that we discover
mathematical truth as it exists, and the formalist belief that math is true because we invent consistent rules for it.
"Less than All," wrote William Blake, "cannot satisfy Man." The Art of the Infinite shows us some of the ways that Man has grappled with All, and reveals mathematics as one of the most exhilarating expressions of the human imagination.
336 pages, Paperback
First published January 1, 1980
45 people are currently reading
571 people want to read
About the author
Robert M. Kaplan
6 books18 followersRobert and Ellen Kaplan have taught mathematics to people from six to sixty, at leading independent schools and most recently at Harvard University. Robert Kaplan is the author of the best-selling The Nothing That Is: A Natural History of Zero, which has been translated into 10 languages, and together they wrote The Art of the Infinite. Ellen Kaplan is also co-author of Chances Are: Adventures in Probability and Bozo Sapiens: Why to Err is Human, co-written with her son Michael Kaplan.
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Displaying 1 - 30 of 40 reviews
November 20, 2015
Well, simply put, this is NOT a book about mathematics. Sure, it has numbers and math as its subject matter, but what authors really wanted you to get out of this book is how wonderfully elitist their English language skills are. Combined with a narrator choice straight out of Downton Abbey (the upstairs kind, of course), this book is impossible to either read or listen to. Even as someone who has a degree in math and loves science, I could not hold on to this book. Long-winded Shakespearian tirades and philosophical comparisons are sprinkled throughout the chapters making this book VERY hard to comprehend. If authors wanted to wake our natural curiosity when it comes to numbers and math, travel with us from ancient times of Euclid and ratios all the way to modern times to show us amazing process of discovery and beauty and logic of math, they failed utterly. All you are left with after reading (or listening) to this book is their weird words salad.
December 3, 2008
Wow - I loved this book. It really opened my mind around how numbers are represented and constructed.
This is one of the best books I have read that explores the foundations of numbers in a very understandable format. The approach/development of math knowledge in the book was entertaining, with a mix of history and inspiring proofs and examples. I wished I read this book when I was in high school.
Topics I really appreciated include the explanation of number fields. In particular, I really enjoyed the explanation of why a heptagon can't be constructed with compass and ruler through the previously developed understanding of fields. (Additional tools would be needed.)
Of course I've known a lot about numbers for years - and I use Real and Complex numbers every day as an engineer. But I never had such an appreciation for how to construct/represent numbers exactly.
As a side note: The book didn't talk about this, but floating point numbers are popular for approximating real numbers. Floating point numbers can be problematic because of truncation and rounding errors that occur in calculations. This book has inspired me to learn more about 'Exact Reals' and has given me the ability to follow/understand papers on various strategies for 'exact reals' as an alternative to floating point numbers.
This is one of the best books I have read that explores the foundations of numbers in a very understandable format. The approach/development of math knowledge in the book was entertaining, with a mix of history and inspiring proofs and examples. I wished I read this book when I was in high school.
Topics I really appreciated include the explanation of number fields. In particular, I really enjoyed the explanation of why a heptagon can't be constructed with compass and ruler through the previously developed understanding of fields. (Additional tools would be needed.)
Of course I've known a lot about numbers for years - and I use Real and Complex numbers every day as an engineer. But I never had such an appreciation for how to construct/represent numbers exactly.
As a side note: The book didn't talk about this, but floating point numbers are popular for approximating real numbers. Floating point numbers can be problematic because of truncation and rounding errors that occur in calculations. This book has inspired me to learn more about 'Exact Reals' and has given me the ability to follow/understand papers on various strategies for 'exact reals' as an alternative to floating point numbers.
June 10, 2025
prose so purple shit's asking me where's ronald
January 13, 2013
Somewhere between Flatland: A Romance of Many Dimensions and Cosmos, Art of the Infinite is a popularization of mathematics by way of extensive metaphor and cultured references to literature and philosophy. I thought this was just a bit cheesy at first, but after a few times around, I got why they were doing this. The chief metaphor is spatial: math is an exploration into these unimaginably vast vistas in which humans are comfortable with a negligible patch and can imagine only a fraction. The Kaplans assert that these vistas are full of "palaces and treasures" and that the pursuit is of some value to humanity.
The value they present is this heady experience in which the simple propagates along some logical line into infinity, like a door opening to reveal the set of all possible doors. These experiences are certainly delightful and make the book enjoyable to read, and I'd imagine this would be much more the case had I followed along with each chapter (which more or less corresponded to a continuous line of inquiry) trying my hand at the math and making sure I understand what the symbols in each section referred to. I'm sure I'm capable of this, especially since the Kaplans go out of their way to put things in comprehensible terms (though it is never oversimplified), but I was merely too lazy.
While the Kaplans present this as a sort of valuable service, math as a heady wonder-inducer, they don't skirt around the Lovecraftian implications of this. The last chapter, in which Cantor plumbs the nature of compound infinities (mind-blowing revelation in which he proves that there MUST be infinities larger than other infinities, as paradoxical as this is - and of course this is only the premise of the chapter!), ends up with his gradual descent to disgrace and insanity. The brilliant thing about this all is that whereas Cantor, it is implied, went insane because of his frustration with the unprovability of the Continuum theorem, i.e., because he hit some limit in the comprehensibility of this shit, the authors end by pointing out that Kurt Godel resolved Cantor's final issue rather handily (he proved that it was neither provable or disprovable).
The historicity of the discipline is heartening: it suggests that there is no aspect of this boggling system that is inherently beyond human grasp, that it isn't really the hostile and unmasterable force Lovecraft's nightmares conjure it as. Which doesn't help Cantor or any of us who exist at a fixed point on the timeline - we are stuck with the paradoxes of our day, not privy to the brilliant answers around the corner. I did think it was a bit unfortunate (if ultimately satisfying) that the Kaplans decided to end the book on a resolution, not its consequent question. It'd be interesting and tantalizing to know what problems are presently driving mathematicians insane.
And this is what I took away from the book: a greater sense of math as a discipline parallel to science. In the first chapter, the Kaplans lay out the plane of imaginary numbers intersecting with the plane of real numbers. They make the story of math equal in the same way to the history of science. Scientists have been doing things in the world to make things about it more clear, always clearing away more cobwebs and opening new doors, etc. Mathematicians have been doing the same thing, exploring the (seemingly real) properties of this other world, not precisely intersecting with ours, or exactly including it. It is just a different thing that they are investigating, with different tools, to be sure, but with the same process of discovery, despite the abstraction and lack of the empiricism we often define as integral to science.
The value they present is this heady experience in which the simple propagates along some logical line into infinity, like a door opening to reveal the set of all possible doors. These experiences are certainly delightful and make the book enjoyable to read, and I'd imagine this would be much more the case had I followed along with each chapter (which more or less corresponded to a continuous line of inquiry) trying my hand at the math and making sure I understand what the symbols in each section referred to. I'm sure I'm capable of this, especially since the Kaplans go out of their way to put things in comprehensible terms (though it is never oversimplified), but I was merely too lazy.
While the Kaplans present this as a sort of valuable service, math as a heady wonder-inducer, they don't skirt around the Lovecraftian implications of this. The last chapter, in which Cantor plumbs the nature of compound infinities (mind-blowing revelation in which he proves that there MUST be infinities larger than other infinities, as paradoxical as this is - and of course this is only the premise of the chapter!), ends up with his gradual descent to disgrace and insanity. The brilliant thing about this all is that whereas Cantor, it is implied, went insane because of his frustration with the unprovability of the Continuum theorem, i.e., because he hit some limit in the comprehensibility of this shit, the authors end by pointing out that Kurt Godel resolved Cantor's final issue rather handily (he proved that it was neither provable or disprovable).
The historicity of the discipline is heartening: it suggests that there is no aspect of this boggling system that is inherently beyond human grasp, that it isn't really the hostile and unmasterable force Lovecraft's nightmares conjure it as. Which doesn't help Cantor or any of us who exist at a fixed point on the timeline - we are stuck with the paradoxes of our day, not privy to the brilliant answers around the corner. I did think it was a bit unfortunate (if ultimately satisfying) that the Kaplans decided to end the book on a resolution, not its consequent question. It'd be interesting and tantalizing to know what problems are presently driving mathematicians insane.
And this is what I took away from the book: a greater sense of math as a discipline parallel to science. In the first chapter, the Kaplans lay out the plane of imaginary numbers intersecting with the plane of real numbers. They make the story of math equal in the same way to the history of science. Scientists have been doing things in the world to make things about it more clear, always clearing away more cobwebs and opening new doors, etc. Mathematicians have been doing the same thing, exploring the (seemingly real) properties of this other world, not precisely intersecting with ours, or exactly including it. It is just a different thing that they are investigating, with different tools, to be sure, but with the same process of discovery, despite the abstraction and lack of the empiricism we often define as integral to science.
April 13, 2012
Prose so purple I claim it was abused. This book needed an editor to cut out the blathering that the authors thought clever. The references to Rimbaud and Proust, to cite just a couple, were completely unnecessary and distracting.
I read the first 3 chapters and then skipped to the last, the chapter on Georg Cantor and aleph-null, aleph-one, and transfinite numbers. (Fun fact: Cantor was a conspiracy theorist!)
I was excited when I read this in the introduction:
"Many small things estrange math from its proper audience. One is the remoteness of its machine-made diagrams. These reinforce the mistaken belief that it is all very far away, on a planet visited only by graduates of the School for Space Cadets. Diagrams printed out from computers communicate a second and subtler falsehood: they lead the reader to think he is seeing the things themselves rather than pixellated approximations to them."
An insightful remark, if somewhat overwrought. Hand-drawn diagrams will make the text feel less imposing and let the reader know that, however true, they are humble human calculations. Brilliant, right?
Unfortunately, the text is ridiculously remote. A sample:
"Have negative numbers definitively moved mathematical thought into abstraction, where the dance of symbols becomes vivid instead of figures? Or do you find the visual proof in the appendix to this chapter not only convincing but illuminating? Notice that in our dances the same steps--axioms of additive and multiplicative inverses, and distributivity--occur again and again. This is because, like squaredances in the confines of a barn, little room to maneuver leads to intricate patterns. The more elaborate these become, each linking onto the last, the more such patterns will all seem to lodge in a sense at once more ancestral and more abstract than sight. It is as if the predominance in our brains of the visual cortex masked a different, deeper apprehension--of time, then, or something akin to music: structure itself."
There's a lot to make fun of in that paragraph, but for me, I have to say: Squaredances? Really? This was published in 2003!
Note: I am apparently the 3rd person to have checked out this copy from the library. The person before me checked it out on 16 August 2004 and it was marked returned on 30 March 2007. I'm going to assume that was a faculty member.
I read the first 3 chapters and then skipped to the last, the chapter on Georg Cantor and aleph-null, aleph-one, and transfinite numbers. (Fun fact: Cantor was a conspiracy theorist!)
I was excited when I read this in the introduction:
"Many small things estrange math from its proper audience. One is the remoteness of its machine-made diagrams. These reinforce the mistaken belief that it is all very far away, on a planet visited only by graduates of the School for Space Cadets. Diagrams printed out from computers communicate a second and subtler falsehood: they lead the reader to think he is seeing the things themselves rather than pixellated approximations to them."
An insightful remark, if somewhat overwrought. Hand-drawn diagrams will make the text feel less imposing and let the reader know that, however true, they are humble human calculations. Brilliant, right?
Unfortunately, the text is ridiculously remote. A sample:
"Have negative numbers definitively moved mathematical thought into abstraction, where the dance of symbols becomes vivid instead of figures? Or do you find the visual proof in the appendix to this chapter not only convincing but illuminating? Notice that in our dances the same steps--axioms of additive and multiplicative inverses, and distributivity--occur again and again. This is because, like squaredances in the confines of a barn, little room to maneuver leads to intricate patterns. The more elaborate these become, each linking onto the last, the more such patterns will all seem to lodge in a sense at once more ancestral and more abstract than sight. It is as if the predominance in our brains of the visual cortex masked a different, deeper apprehension--of time, then, or something akin to music: structure itself."
There's a lot to make fun of in that paragraph, but for me, I have to say: Squaredances? Really? This was published in 2003!
Note: I am apparently the 3rd person to have checked out this copy from the library. The person before me checked it out on 16 August 2004 and it was marked returned on 30 March 2007. I'm going to assume that was a faculty member.
December 21, 2010
Though I was only able to follow about two thirds of the math in The Art of the Infinite, it was extremely informative. I found most interesting the principles of shifting from perspective to perspective, using techniques and processes of one branch of mathematics to interpret techniques and processes of another, the use of mathematical substitutions from seemingly unrelated contexts to sidestep mathematical deadends and the varying styles of thought used to approach math. I especially enjoyed the revelations of deep pattern throughout mathematical concepts very satisfying. Finally, a great moment for me: many years ago I came across the term "Cantorian transfinites" in a science fiction short story.
In intervening years other reading confirmed the existence of a mathematician named Cantor and my suspicion that the transfinites were real, and, as they hinted at numbers beyond infinity, I was fascinated. Math teachers I questioned were unfamiliar with the idea through fourteen years of teaching. But, there it is in the last chapters of The Art of the Infinite.
The text is challenging at best and can be overwhelming to the layman, but the patterns, principles and often bizarre coincidence of numbers makes it quite worthwhile.
In intervening years other reading confirmed the existence of a mathematician named Cantor and my suspicion that the transfinites were real, and, as they hinted at numbers beyond infinity, I was fascinated. Math teachers I questioned were unfamiliar with the idea through fourteen years of teaching. But, there it is in the last chapters of The Art of the Infinite.
The text is challenging at best and can be overwhelming to the layman, but the patterns, principles and often bizarre coincidence of numbers makes it quite worthwhile.
June 27, 2014
Mathematics is something that I find interesting, but definitely wish I knew more about. So, I went to my local library looking for a good book on math to give me an introduction to the subject. When I found this book, I thought I'd found what I was looking for. Boy, was I wrong.
The book is written in this weird, florid prose, and just the way it was written made it impossible to read. I tried really hard, but I couldn't get past the first few pages. Finally, I took it back to the library and looked for a better alternative.
I gave it two stars because it probably would have been a decent book had it not been for the weird writing style. I just needed something less flowery and more technical. If you can handle the flowery prose, then you might enjoy this book, but I just couldn't handle it.
The book is written in this weird, florid prose, and just the way it was written made it impossible to read. I tried really hard, but I couldn't get past the first few pages. Finally, I took it back to the library and looked for a better alternative.
I gave it two stars because it probably would have been a decent book had it not been for the weird writing style. I just needed something less flowery and more technical. If you can handle the flowery prose, then you might enjoy this book, but I just couldn't handle it.
May 22, 2021
A great read, and having finished the library copy, I'll probably go buy a copy of my own. Certainly I'll want to re-read it, and there are any number of passages that deserve prolonged contemplation, if not downright study.
One minor tech note: this book did expose a shortcoming of the Kindle experience, at least as far as using the Kindle app on my iPad Mini goes. The typographical characters appearing throughout the book that are not part of the standard set of letters, particularly those that had sub- or superscripts, were pretty much illegible. I boosted the font size, which helped a bit, but in order to get them to where I could comfortably discern them, the plain text had to be ludicrously huge. I don't think this is a failing of what's stored in the electronic file itself, because if I look at this Kindle book in a web browser, the problem is not there. It seems more like it's a limitation of the app itself -- something do with not enough care having been put into the implementation of extended character sets, is my guess.
Granted, there are not so many books where there is a need to display, say, aleph-null or e to the pi*i, but I thought I'd put this report out there anyway. Suggestions about where it might be better directed are most welcome.
[ETA] I should add that this problem is not unique to the Kindle app. For example, if I put in an aleph right here: ℵ, it displays as way too small. To my mind, at least, it should be the same size as a capital letter from the standard character set. (I couldn't add the zero subscript, because that, sadly, is not part of the HTML that Goodreads supports, in comments.)
One minor tech note: this book did expose a shortcoming of the Kindle experience, at least as far as using the Kindle app on my iPad Mini goes. The typographical characters appearing throughout the book that are not part of the standard set of letters, particularly those that had sub- or superscripts, were pretty much illegible. I boosted the font size, which helped a bit, but in order to get them to where I could comfortably discern them, the plain text had to be ludicrously huge. I don't think this is a failing of what's stored in the electronic file itself, because if I look at this Kindle book in a web browser, the problem is not there. It seems more like it's a limitation of the app itself -- something do with not enough care having been put into the implementation of extended character sets, is my guess.
Granted, there are not so many books where there is a need to display, say, aleph-null or e to the pi*i, but I thought I'd put this report out there anyway. Suggestions about where it might be better directed are most welcome.
[ETA] I should add that this problem is not unique to the Kindle app. For example, if I put in an aleph right here: ℵ, it displays as way too small. To my mind, at least, it should be the same size as a capital letter from the standard character set. (I couldn't add the zero subscript, because that, sadly, is not part of the HTML that Goodreads supports, in comments.)
April 24, 2020
Beautiful Mathematic Book for college students. THE GREAT CONTROVERSE is more important to know than all other stuff !!! The question: Where do we get our knowledge from and how do we know, that that is it ?????
John von Neuman said: "In Mathematics we never understand things but we just get used to them."
(That can’t be quite right — yet our understanding must be stretched to the breaking point before it becomes flexible enough to adjust to the unthinkable.) (page 024)
Paul Erdös said about The Book: "“You don’t have to believe in God, but you do have to believe in The Book.” I myself understand these proofs !
The New Scientist meant: "This is mathematics for the Soul - just the way it should be." I can only agree.
John von Neuman said: "In Mathematics we never understand things but we just get used to them."
(That can’t be quite right — yet our understanding must be stretched to the breaking point before it becomes flexible enough to adjust to the unthinkable.) (page 024)
Paul Erdös said about The Book: "“You don’t have to believe in God, but you do have to believe in The Book.” I myself understand these proofs !
The New Scientist meant: "This is mathematics for the Soul - just the way it should be." I can only agree.
March 15, 2025
I love language and I'm fascinated by mathematics and how it objectively explains the world around us. At first I thought I was missing the point of this book, so went looking to see what others thought of it. I'm glad I'm not alone! Reading it feels like being at dinner with a bunch of people and not being in on the joke. It's like the secret to decoding the prose is arrogantly hidden while the author/s prattle on in a cutesy tone as if to say 'how clever are we'. Time is precious and there's so much to read, so I decided to ditch this book because nothing should be so difficult to chew and digest.
If you want a great book on mathematics and the art of it, May I suggest Max Tegmark's 'Our Mathematical Universe'. That one is superb!
If you want a great book on mathematics and the art of it, May I suggest Max Tegmark's 'Our Mathematical Universe'. That one is superb!
March 4, 2019
"The essence of mathematics lies entirely on its freedom", said Georg Cantor. THE man who figured out that there are countable and uncountable infinities. Wow. Infinite freedom? Well, at least freedom within a playground nested with infinities (Hilbert’s paradise), as you are about to learn in this book. Hey, Robert and Ellen managed to make number theory palatable for the layperson with an entire chapter on Cantor’s work. An infinite gold mine as a chapter. And there is much more. The whole thing is a vivid narrative about mathematical beauty everywhere; playful, with subtle humor and deep erudition. That is, history + geometry + philosophy + poetry + … = infinite entertainment.
July 29, 2020
This book was in my math teacher father’s collection when he died. I always thought I was good at math, but much of this text exceeded my capacity. I loved the parts about prime numbers (and the unexpected lack of pattern for finding the next one) and sums of series (and the unexpected discovery that some such series of ever declining fractions converge (e.g., 1 + 1/2 + 1/4 + 1/8 + 1/16 +.... = 2) and some diverge (e.g., 1 + 1/2 + 1/3 + 1/4 + 1/5 +...)). I was reminded how much I enjoyed doing proofs—especially geometric proofs. The latter part of the book was over my head, but I didn’t mind reading it.
April 18, 2023
This could have been a very good book if not for the authors trying to be to cute times a million. Too much flowery language and oh-so-clever phrases. I don’t like being patronized when I read a book.
May 14, 2023
Probably the best book about mathematics I have read. Beautifully written about fascinating fundamental concepts. Luckily the authors are not afraid to use formulas and proofs, not always easy, but understandable with some basic math background.
February 7, 2024
Pure Mathematics Art
It's not just about set theory or infinity but it deals with the whole evolution of maths in it's most beautiful art form. A must read for all maths students and enthusiasts.
It's not just about set theory or infinity but it deals with the whole evolution of maths in it's most beautiful art form. A must read for all maths students and enthusiasts.
May 26, 2020
One of my favorite books on math.
January 5, 2022
I prefered his book on 0 (zero). This one I found too mathematical. Infinity proves to be much more boring then zero.
May 5, 2022
the hand drawn graphs make a lot of the math easier to grasp, but then so little devoted to explanation of notation you need a strong background up to calculus to understand many of the concepts.
June 4, 2011
Husband and wife team Robert Kaplan and Ellen Kaplan have written a rich exploration of several aspects of the expansive field of mathematics. Through the book they cover the foundations of number and arithmetic, the rigors of mathematical proof, the nature of mathematical insight, the primes, infinite sequences, Euclidean geometry, building algebra from geometry, complex numbers, projective geometry (this was completely new for me), and finish with the nature of different infinities and the life of Georg Cantor.
In many ways, this book is a nice companion to Jerry P. King's books The Art of Mathematics and Mathematics in 10 Lessons. In The Art of Mathematics, King argues that mathematics is an art-form that can be appreciated through its own theory of aesthetics built on precision and concise elegance. Both [Mathematics in 10 Lessons] and this book attempt to give to the interested but lay reader a taste of the art-form.
I really enjoyed both of King's books, but found The Art of the Infinite a little less accessible. First, the Kaplan's write with a style rich in complex, ornamental language, a bit surprising for a math text, but apparently they both studied linguistics at one point. What this means is there are lots of literary references and clever metaphors. It took me a while to get used to this style of writing, and found it distracting in parts. Particularly, when it comes to my second criticism. That is I had a hard time understanding a few of the chapters because the math was new to me and more complex requiring me to reread sections to make sure I followed. Not fully following all of the intricacies of the mathematics is frustrating, but I generally feel like if I spend enough time with it I'll start to get it. With this work, it took longer because of the gloss of the language.
I'm happy to have read The Art of the Infinite, but I can't say that it will be a book I reference when I need an explanation of some bit of math. That said, they have a good bit on the nature of play and creative exploration in mathematics, and this insight alone is worth the reading of this book.
In many ways, this book is a nice companion to Jerry P. King's books The Art of Mathematics and Mathematics in 10 Lessons. In The Art of Mathematics, King argues that mathematics is an art-form that can be appreciated through its own theory of aesthetics built on precision and concise elegance. Both [Mathematics in 10 Lessons] and this book attempt to give to the interested but lay reader a taste of the art-form.
I really enjoyed both of King's books, but found The Art of the Infinite a little less accessible. First, the Kaplan's write with a style rich in complex, ornamental language, a bit surprising for a math text, but apparently they both studied linguistics at one point. What this means is there are lots of literary references and clever metaphors. It took me a while to get used to this style of writing, and found it distracting in parts. Particularly, when it comes to my second criticism. That is I had a hard time understanding a few of the chapters because the math was new to me and more complex requiring me to reread sections to make sure I followed. Not fully following all of the intricacies of the mathematics is frustrating, but I generally feel like if I spend enough time with it I'll start to get it. With this work, it took longer because of the gloss of the language.
I'm happy to have read The Art of the Infinite, but I can't say that it will be a book I reference when I need an explanation of some bit of math. That said, they have a good bit on the nature of play and creative exploration in mathematics, and this insight alone is worth the reading of this book.
June 20, 2015
Un excelente libro que une las “dos culturas”: exactas y humanidades. No es un libro de divulgación típico, por eso resalta. Los autores además de esposos son matemáticos, políglotas y como si no fuera suficiente ella historiadora y bióloga. ¿Qué puede salir de esa colaboración? Pues oro puro.
Entre pantallazos que muestran cómo funcionan las matemáticas, cómo se adentran en conceptos tan desentrañables como el infinito hay constantes referencias a la poesía y a la historia. Es en realidad un relato inusual por esa especie de autismo que provoca la sobre-especialización del mundo académico. Es fascinante que junto a citas de Julio César se entremezcle en el relato la distribución de los números primos y una analogía con la búsqueda del tiempo perdido de Proust. El capítulo de Cantor y los números transfinitos es brutal, de las mejores explicaciones del tema.
El libro es ordenado, claro y hay que leerlo como cualquier libro de matemáticas: con lápiz y papel. Para los que no son de exactas a desempolvar los libros de la escuela, tendrán que trabajar y seguir las demostraciones que si bien son simples necesitan de algún nivel de esfuerzo. Al final no todos somos Gauss.
Entre pantallazos que muestran cómo funcionan las matemáticas, cómo se adentran en conceptos tan desentrañables como el infinito hay constantes referencias a la poesía y a la historia. Es en realidad un relato inusual por esa especie de autismo que provoca la sobre-especialización del mundo académico. Es fascinante que junto a citas de Julio César se entremezcle en el relato la distribución de los números primos y una analogía con la búsqueda del tiempo perdido de Proust. El capítulo de Cantor y los números transfinitos es brutal, de las mejores explicaciones del tema.
El libro es ordenado, claro y hay que leerlo como cualquier libro de matemáticas: con lápiz y papel. Para los que no son de exactas a desempolvar los libros de la escuela, tendrán que trabajar y seguir las demostraciones que si bien son simples necesitan de algún nivel de esfuerzo. Al final no todos somos Gauss.
July 3, 2007
Man, I used to consider myself decently intelligent when it came to math, but I'm telling you, some of these mathy books I've read this year and last year are making me feel REALLY stupid. I understood about 1/2 of this book, and really *got* it; the other 1/2 went over my head. :( (But I'm consoling myself by saying that the fact that I'm still willing to read them means that I really am still a math geek.) However, the writing is good, the diagrams are good, and I think the authors did a good job of explaining the concepts, even if *I* didn't 100% understand them.
June 26, 2008
Although some of the math stuff was over my head (or patience level), I felt like I understood a remarkable amount of what the authors had to say about the history and nature of infinity. This book blew my mind in several places and gives a nice and readable history of the people and ideas involved in studying the nature of the infinite, which is what you might expect from a guy who wrote a very readable book about the history and nature of zero.
January 18, 2010
The narrative got a bit flowery and hard to follow at times, and the mathematics likewise (toward the end): I'm not sure I particularly enjoyed the chapter on Projective Geometry, and the infinites material was probably more than I was ready for. Still, I learned a few things, and am probably a better person for having gone through this book.
August 16, 2011
Mathematics is such a beautiful art that is often not appreciated as it should. I never liked how it was taught in schools, but his book reveals the artistry of the world's mathematical geniuses and with child-like wonder, I devour their formulae as if deprived of much-needed nutrition. It does get heavy-going at some points but it is okay to skip these if a general idea is what you are after.
August 24, 2012
Enjoyable book with a good mix of math concepts and math history. For example, is was very interesting to follow the path over hundreds of years of discoveries that are covered in one semester of abstract algebra. I also enjoyed the creative and philosophical language the authors used to explore the topics with different perspectives.
August 1, 2015
Not as overwhelming as Journey through Genius, but this book is full of little gems of mathematics. Nearly all the "classic" proofs make an appearance.
The prose is flowery and full of allusions, which I like, but it could make some sections (already challenging due to the math in them) less accessible.
The prose is flowery and full of allusions, which I like, but it could make some sections (already challenging due to the math in them) less accessible.
April 7, 2008
Well, this is one of my math nerd books, so I obviously enjoy it. It explains a lot of history of Mathematics and gives some alternative ways of looking at proofs. The Calculus portion helped me a lot through my hellish nightmare!
July 11, 2008
Although the grandiose language in this book is a bit much at times, this is a really great book to appreciate the beauty of mathematics. The author does a great job of making the math accessible without dumbing it down.
February 6, 2015
Radiant, luminous, poetic, lovely book, written like an adventurous novel, taking the reader through the broad vistas and hidden valleys of mathematics, lifting ancient stones of thought, revealing the natural beauties and deep humanity of numbers.
March 6, 2017
The book is a bit technical and the language used by authors make it even more difficult to understand. I enjoyed only the last chapter (set theory by Cantor and infinities involved). Overall an avoidable book.
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