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The Great Mathematical Problems
Celebrated mathematician Stewart explains why mathematical problems exist, what drives mathematicians to solve them, and why their efforts matter in the context of science as a whole.
Great problems --
Prime territory : Goldbach Conjecture --
The puzzle of pi : squaring the circle --
Mapmaking mysteries : Four Color theorem --
Sphereful symmetry : Kepler Conjecture --
New solutions for old : Mordell Conjecture --
Inadequate margins : Fermat's Last Theorem --
Orbital chaos : Three-body problem --
Patterns in prime : Riemann Hypothesis --
What shape is a sphere? : Poincaré Conjecture --
They can't all be easy : P/NP problem --
Fluid thinking : Navier-Stokes Equation --
Quantum conundrum : Mass Gap Hypothesis --
Diophantine dreams : Birch-Swinnerton-Dyer Conjecture --
Complex cycles : Hodge Conjecture --
Where next? --
Twelve for the future.
Great problems --
Prime territory : Goldbach Conjecture --
The puzzle of pi : squaring the circle --
Mapmaking mysteries : Four Color theorem --
Sphereful symmetry : Kepler Conjecture --
New solutions for old : Mordell Conjecture --
Inadequate margins : Fermat's Last Theorem --
Orbital chaos : Three-body problem --
Patterns in prime : Riemann Hypothesis --
What shape is a sphere? : Poincaré Conjecture --
They can't all be easy : P/NP problem --
Fluid thinking : Navier-Stokes Equation --
Quantum conundrum : Mass Gap Hypothesis --
Diophantine dreams : Birch-Swinnerton-Dyer Conjecture --
Complex cycles : Hodge Conjecture --
Where next? --
Twelve for the future.
340 pages, Paperback
First published January 1, 2013
76 people are currently reading
857 people want to read
About the author
Ian Stewart
263 books756 followersIan Nicholas Stewart is an Emeritus Professor and Digital Media Fellow in the Mathematics Department at Warwick University, with special responsibility for public awareness of mathematics and science. He is best known for his popular science writing on mathematical themes.
--from the author's website
Librarian Note: There is more than one author in the GoodReads database with this name. See other authors with similar names.
--from the author's website
Librarian Note: There is more than one author in the GoodReads database with this name. See other authors with similar names.
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Displaying 1 - 28 of 28 reviews
March 7, 2013
As a science writer, whose only foray into maths has been to cover infinity – by far the sexiest and most intriguing mathematical topic – I am in awe of those who successfully popularize maths.
By comparison, science is easy. We all know from school that science can be dull, but if you go about it the right way, it is naturally fascinating, because it’s about how the universe we live in works. Admittedly maths has plenty of applications, but an awful lot of mathematics is about a universe we don’t live in. It can seem that many mathematicians spend their time doing the equivalent of arguing about the dietary habits of unicorns. Not really a proper job for a grown human being.
Probably the best of the current crop of popular maths writers is Ian Stewart. Certainly the most prolific – I don’t know how he finds the time for his day job. Stewart is decidedly variable in his books. Some of them are pure unicorn territory. I find myself turning page after page thinking ‘So what? I don’t care!’ But every now and then he gets it just right – and this is such an example.
Okay, there are occasional unicorn moments, where I had to skip through a page or two to avoid dropping off (when, for example, he gets altogether too excited about the prospect of constructing a regular 17 sided polygon using only a ruler and a pair of compasses), but they are rare indeed. Stewart takes on some of the greatest problems to face mathematicians through history – even the names are evocative, like Goldbach’s Conjecture and, of course, the Riemann Hypothesis. They sound like a Sherlock Holmes story. And Stewart makes them interesting. Which is truly wonderful.
In part the readability is because of a good smattering of stuff about the people – historical context is never more important than in popular maths – but he also pitches the mathematics itself at just the right level to keep our interest without going into mind-numbing detail, or being too summary. I am very wary of describing any book as a tour-de-force, but this one certainly comes close.
Even though Stewart does not keep things enthralling throughout – the dullest chapter is the one on Fermat’s Last Theorem, which I suspect is because Stewart focuses more on the maths here and less on the people, so excellently covered by Simon Singh – there is plenty in this book to keep the imagination alive. If you hate maths this is not going to make you a convert. But if, like me, you have a grudging admiration for maths but find a lot of it impenetrable or pointless, you should have a great time in Ian Stewart’s capable hands.
Review first published on www.popularscience.co.uk and reproduced with permission
By comparison, science is easy. We all know from school that science can be dull, but if you go about it the right way, it is naturally fascinating, because it’s about how the universe we live in works. Admittedly maths has plenty of applications, but an awful lot of mathematics is about a universe we don’t live in. It can seem that many mathematicians spend their time doing the equivalent of arguing about the dietary habits of unicorns. Not really a proper job for a grown human being.
Probably the best of the current crop of popular maths writers is Ian Stewart. Certainly the most prolific – I don’t know how he finds the time for his day job. Stewart is decidedly variable in his books. Some of them are pure unicorn territory. I find myself turning page after page thinking ‘So what? I don’t care!’ But every now and then he gets it just right – and this is such an example.
Okay, there are occasional unicorn moments, where I had to skip through a page or two to avoid dropping off (when, for example, he gets altogether too excited about the prospect of constructing a regular 17 sided polygon using only a ruler and a pair of compasses), but they are rare indeed. Stewart takes on some of the greatest problems to face mathematicians through history – even the names are evocative, like Goldbach’s Conjecture and, of course, the Riemann Hypothesis. They sound like a Sherlock Holmes story. And Stewart makes them interesting. Which is truly wonderful.
In part the readability is because of a good smattering of stuff about the people – historical context is never more important than in popular maths – but he also pitches the mathematics itself at just the right level to keep our interest without going into mind-numbing detail, or being too summary. I am very wary of describing any book as a tour-de-force, but this one certainly comes close.
Even though Stewart does not keep things enthralling throughout – the dullest chapter is the one on Fermat’s Last Theorem, which I suspect is because Stewart focuses more on the maths here and less on the people, so excellently covered by Simon Singh – there is plenty in this book to keep the imagination alive. If you hate maths this is not going to make you a convert. But if, like me, you have a grudging admiration for maths but find a lot of it impenetrable or pointless, you should have a great time in Ian Stewart’s capable hands.
Review first published on www.popularscience.co.uk and reproduced with permission
October 29, 2017
Lamentablemente, es el libro más flojo de Stewart que he leído hasta el momento. Hace algunos años seguía su columna en una revista científica (Si no me equivoco era Scientific American) y el primer libro que compré a los 20 años con mi primera tarjeta de crédito fue Does God Play Dice?: The New Mathematics of Chaos.
Pero este libro no tiene el estilo de aquellos. Me pareció de un nivel excesivo para tratarse de un libro de divulgación, y requiere más que un poco de matemática universitaria: habiendo cursado hasta análisis con variable compleja y conociendo lo más básico de la mecánica cuántica y relatividad, lo que requiere una matemática algo pesada, sin embargo me encontré abrumada más de una vez por los innumerables principios y teoremas que nombra y que aparecen explicados muy brevemente. Confieso mi ignorancia total sobre formas modulares, análisis real funcional y otras cosas que aparecen en el libro, y debería dedicarle demasiado tiempo a ponerme al día para comprenderlo.
Otras partes me parecieron más llevaderas, generalmente las que hablaban de problemas de física (OBVIAMENTE, ya que son ramas de la matemática con las que estoy familiarizada) pero a la vez sentí que me contaba cosas que ya sabía, con lo cual el libro perdía la gracia (y además con un par de errores irritantes)
Punto extra por mencionar favorablemente a mi amor imposible de mis tiempos de estudiante, e incluir un gráfico hecho por este caballero y su equipo, me refiero al Dr. P. M. quien era ayudante de cátedra cuando cursé Física II y hoy es un físico importante. *insertar corazoncitos aquí*
Pero este libro no tiene el estilo de aquellos. Me pareció de un nivel excesivo para tratarse de un libro de divulgación, y requiere más que un poco de matemática universitaria: habiendo cursado hasta análisis con variable compleja y conociendo lo más básico de la mecánica cuántica y relatividad, lo que requiere una matemática algo pesada, sin embargo me encontré abrumada más de una vez por los innumerables principios y teoremas que nombra y que aparecen explicados muy brevemente. Confieso mi ignorancia total sobre formas modulares, análisis real funcional y otras cosas que aparecen en el libro, y debería dedicarle demasiado tiempo a ponerme al día para comprenderlo.
Otras partes me parecieron más llevaderas, generalmente las que hablaban de problemas de física (OBVIAMENTE, ya que son ramas de la matemática con las que estoy familiarizada) pero a la vez sentí que me contaba cosas que ya sabía, con lo cual el libro perdía la gracia (y además con un par de errores irritantes)
Punto extra por mencionar favorablemente a mi amor imposible de mis tiempos de estudiante, e incluir un gráfico hecho por este caballero y su equipo, me refiero al Dr. P. M. quien era ayudante de cátedra cuando cursé Física II y hoy es un físico importante. *insertar corazoncitos aquí*
January 27, 2018
Normally I'm quite choosy about the books I read. However, I made a fatal error when it came to this one. I was on a mission to buy my 2018 reading list at Waterstones on Piccadilly last December when I came upon this book sitting invitingly on a table, just begging to be bought. The topic, maths, is one of interest to me. All good so far. Thumbing through the book, I noticed the author (Ian Stewart) was unafraid to include formulas and diagrams - as someone who studied physics at university, this made the book even more appealing, despite the fact that IT WAS NOT ON MY READING LIST, WHICH WAS THE RESULT OF RESEARCH INTO GOOD BOOKS. But maybe I had just overlooked this one?
Then I made the fatal error - I read the reviews on the back cover. "Stewart's imaginative, often-witty anecdotes, analogies and diagrams succeed in illuminating" (they don't). And on that basis I bought it. Oops.
This book is just muddled. It throws layer upon layer of fog onto an already difficult subject. Aside from a few chapters, the explanations don't explain. Stewart could have written a book that talks about the historical context of some mathematical problems. That would have worked. The problem is he frequently tries to 'explain' with non-explanations - enticing you with arguments with missing parts or leaps of logic wider than the Grand Canyon, or throwing a multitude of briefly stated theorems at you that you are then assumed to 'know'. More than once I had to resort to the Web for an explanation, and more than once this turned out to be far simpler than reading the book!
Unless you already know the topic well, I suspect this book will only frustrate you.
Then I made the fatal error - I read the reviews on the back cover. "Stewart's imaginative, often-witty anecdotes, analogies and diagrams succeed in illuminating" (they don't). And on that basis I bought it. Oops.
This book is just muddled. It throws layer upon layer of fog onto an already difficult subject. Aside from a few chapters, the explanations don't explain. Stewart could have written a book that talks about the historical context of some mathematical problems. That would have worked. The problem is he frequently tries to 'explain' with non-explanations - enticing you with arguments with missing parts or leaps of logic wider than the Grand Canyon, or throwing a multitude of briefly stated theorems at you that you are then assumed to 'know'. More than once I had to resort to the Web for an explanation, and more than once this turned out to be far simpler than reading the book!
Unless you already know the topic well, I suspect this book will only frustrate you.
February 22, 2021
This book provides an illuminating tour of the breadth and complexity of mathematics, but each chapter consisted of my losing the thread of the mathematical problem by the second page. Interesting, but a frustrating read.
August 25, 2014
Yet another popular mathematics book about big solved and unsolved mathematical questions, and one in which Stewart combines his two worst traits (as a maths writer; he might beat his wife for all I know): the lies-to-children approach to instruction with which he fell in love during the writing of one of the Science of Discworld books, and his more recent tendency to write about subjects he doesn't quite understand himself in order to learn more about them.
The result is a book that's very nearly useless if you want to form a coherent picture of what any of the problems discussed are really about, and one that's very frustrating if you're already familiar with them, because you'll constantly be second-guessing his metaphors. Not his best.
The result is a book that's very nearly useless if you want to form a coherent picture of what any of the problems discussed are really about, and one that's very frustrating if you're already familiar with them, because you'll constantly be second-guessing his metaphors. Not his best.
December 16, 2014
Yet another very good book from Professor Stewart. Well written, engaging, and with loads of stuff in there that I had not read before. The biographical stuff is interesting, the explanations of the areas of mathematics in which the problems (or their solutions) reside are good, and the narrative histories of the problems and the efforts to solve them are good.
The only things I can say against it are that I did find the text veered from explaining really elementary stuff on the one hand to some very challenging explanations on the other. I have a mathematics degree (albeit a long, long time ago...) but I found a few of his demonstration steps quite opaque until I had worked out what they must have meant and then convinced myself they were valid. I also found the statements of some of the problems a bit hard to identify in the text, which seems an odd thing to get not quite right.
However, this is nit-picking. I thoroughly recommend the book although I think someone with a decent mathematical background will get more out of it. A strong 3 stars.
June 29, 2017
A great read detailing the history of some of the great problems that were solved, and with some still to be solved! AS always Ian Stewart captures the reader and keeps them interested by what some might say is a boring subject. As far as popular mathematics books go, this is one of the more human-side-of-the-story ones.
May 25, 2025
There's definitely something to be said for having on your bookshelves a guide to all of the great buzz phrases in mathematics ('three body problem', 'P vs NP', 'Riemann Hypothesis', and so on). On the other hand, trying to do justice to the range of problems the author tackles in less than three hundred pages was always going to be a bit ambitious.
Although there are one or two passages comprising what looks suspiciously like extraneous detail, in general the treatment of each topic felt a bit too brief to me, in the sense that key steps in the explanation were effectively hand-waved away. Maybe it's just me. I'm no mathematician, but having been weaned on Martin Gardner books back in the seventies, I'm accustomed to coming away from a popular mathematical account with at least a sense of coherent insight, and much of the time it didn't happen for me here.
Some chapters were better than others on this score (the earlier ones, mainly). Maybe it's just too difficult.
On the other hand, the author's easy writing style and weaving in of historical narrative make for an agreeable reading experience. I enjoyed the book, and it does succeed in at least providing a summary of what the key problems in mathematics are about.
Although there are one or two passages comprising what looks suspiciously like extraneous detail, in general the treatment of each topic felt a bit too brief to me, in the sense that key steps in the explanation were effectively hand-waved away. Maybe it's just me. I'm no mathematician, but having been weaned on Martin Gardner books back in the seventies, I'm accustomed to coming away from a popular mathematical account with at least a sense of coherent insight, and much of the time it didn't happen for me here.
Some chapters were better than others on this score (the earlier ones, mainly). Maybe it's just too difficult.
On the other hand, the author's easy writing style and weaving in of historical narrative make for an agreeable reading experience. I enjoyed the book, and it does succeed in at least providing a summary of what the key problems in mathematics are about.
March 19, 2024
L'argomento è affascinante e la trattazione efficace. Non sono sempre riuscito a seguire tutti i ragionamenti, ma l'autore ha talento nel dare un'idea generale di concetti molto complicati per renderli almeno parzialmente comprensibili a chi non è del mestiere.
Rende il fascino della matematica come continua scoperta piena di risultati sorprendenti, senza passare da metafore vaghe o storielle, ma entrando nella materia senza paura di spiegare cose difficili in modo semplice.
Fa venire voglia di continuare ad imparare, che penso sia il miglior complimento possibile per un libro del genere. Eccellente lavoro, adatto però a chi è abituato a ragionare matematicamente (direi almeno a livello del liceo scientifico).
Rende il fascino della matematica come continua scoperta piena di risultati sorprendenti, senza passare da metafore vaghe o storielle, ma entrando nella materia senza paura di spiegare cose difficili in modo semplice.
Fa venire voglia di continuare ad imparare, che penso sia il miglior complimento possibile per un libro del genere. Eccellente lavoro, adatto però a chi è abituato a ragionare matematicamente (direi almeno a livello del liceo scientifico).
August 30, 2019
This book is good to get an overview of each problem. However, as expected, you will by no means be an expert on the issue at the end of the chapters. The author is successful though to present the progression (attempts) in an academic way (through giving references).
Some problems have connections between as the author suggests. However, a single fast read was not enough for me to appreciate the noted interconnections between the problems. Perhaps, more theoretical background is needed for such appreciation.
Overall, it is a nice feeling to have a reference that contains some notable (and all of the millennium problems) in a single place.
Some problems have connections between as the author suggests. However, a single fast read was not enough for me to appreciate the noted interconnections between the problems. Perhaps, more theoretical background is needed for such appreciation.
Overall, it is a nice feeling to have a reference that contains some notable (and all of the millennium problems) in a single place.
March 24, 2022
Антология решенных и решаемых математических задач от древности до наших дней и от квадратуры круга и Великой теоремы Ферма до гипотезы Ходжа. Саму проблематику автор попытался изложить доступным языком по крайней мере для того, кому хорошо давалась математика в школьной программе, и кто не растерял этих знаний с возрастом, и, как правило, это лица инженерных профессий. Попытка вульгаризации строгих истин автору удалась на мой взгляд на четыре с плюсом по старой системе оценок (пятибальной). Потому что на пять баллов это вряд ли кому-то под силу сделать.
September 7, 2023
Fully agree with other reviewers, if you are looking for a comprehensible overview of great math problems, this book is not for you. The best pages are the outline. If you really want to understand anything, pick the problems that interest you based on the outline and choose other books and online sources to grasp them.
June 30, 2022
Es un poco más pesado en teoría, personalmente lo que busco en un libro de matematicas, pero no recomendable para quienes quieren interesarse en matematicas (como el libro intenta venderse), porque probablemente será abrumante y/o aburrido.
June 5, 2023
Appears to be friendly to non-university level maths students, but sometimes does not explain details well enough or just omitts ideas. Oftentimes provides unnecessary details and can get confusing, but the ideas were useful anyway and some value was still gained from each chapter
February 29, 2024
Not what I expected and quite frustrating. With every chapter, it looks like you will be given a history of a math issue and how humanity dealt with it, but then you don't get it, actually. I felt like there's no cause and effect sequence. Disappointing.
September 2, 2017
Цікаво. Хоча деякі терміни/ ідеї доведень мені було важко зрозуміти
April 29, 2018
Close to 4 stars anyway. Didn't understand much of this but I enjoyed picking up bits here and there.
October 13, 2021
Решил "теорему" Римана, подробности чуть позже
August 13, 2025
Podobała mi się 😍, ale średnio co rozdział (jeden problem) robiłam sobie przerwę, bo wymagała skupienia 😅
May 3, 2020
I've enjoyed the chapter on the poincare conjecture, because of Donald O'Shea's book and i expected those book to be similar. But this book was such a chore to read. I love pure mathematics and applied math and expected this book to be a sort of combined history lesson and tour on the evolution of mathematical techniques and how they were used to solve these problems and more interestingly the people who solved them and the era in which these solutions were developed. After all, he[Ian Sterwart] begins with the story of Andrew Wiles. I loved the documentary on fermats last theorem, i expected this book to be a paperback version of that documentary, it wasn't. Instead It was more of a series of (i felt) vaguely connected essays on what the problem is and its solution story. Not the story of the people who solved it.
September 8, 2023
El libro recorre los problemas matemáticos más importantes de la actualidad, los que están resueltos y los que aún están pendientes. Los analiza de forma histórica, viendo cómo aparecieron y cómo se han atacado por los matemáticos. Además del interés de los problemas, de sobra conocidos, está el de seguir el proceso que ha llevado primero a plantearlos y después a intentar, y en algunos casos conseguir, dejarlos resueltos. A cualquier aficionado a las matemáticas le gustará este libro, pero también a todo el que tenga interés en entender cómo funciona el proceso del descubrimiento científico.
January 15, 2015
Mimo, że autor i tłumacz pewne matematyczne zagadnienia chcą przedstawić w prosty sposób, niestety do przeczytania książki jest potrzebna trochę większa wiedza matematyczna, niż ta którą wynieśliśmy z podstawówki. Przy paru zagadnieniach rozbolała mnie głowa ;) Ale doczytałem do końca i jestem zadowolony. Ponieważ czytałem ebooka to dopiero na końcu zobaczyłem słownik pojęć!! Niby ebook nie powinien mieć nic z tym wspólnego, ale ma. Mając papierową wersję przewertowałbym i zobaczyłbym, a tak jakoś słownik pojęć mi się nie pokazał mimo że widziałem spis treści. Trochę to bez sensu, ale tak mam ;)
July 25, 2017
Unclear exposition of almost everything except Kepler conjecture and Three-body problem. Even in the end when the author "introduces" 12 unsolved problems of his own selection, he seems to be confused about what the actual problems are and what the actual conjectures say. No matter how many times you read the poorly-written chapters and section, you won't gain clear insight into the problems. 2 stars only because the chapters on Kepler conjecture and Three-body problem are fairly clear, should actually deserve 1.
October 4, 2015
This book manages to be both too easy and too hard at the same time.
The book has mathematical concepts worded in a simple way, which only makes it frustrating when you fail to understand it.
The book has mathematical concepts worded in a simple way, which only makes it frustrating when you fail to understand it.
October 17, 2019
Everything about math is pure beauty but lacks enthuasiasm to distant looks, this book is successful to some extend to show it.
January 6, 2021
Terrible. Unless you are working your way to a maths A-Level or degree, give this a miss. Very inaccessible.
April 14, 2023
«Доказано, что log log n стремится к бесконечности, но никто никогда не видел, как он это делает».
December 16, 2017
struggled through this book.
Displaying 1 - 28 of 28 reviews