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Visions of Infinity: The Great Mathematical Problems
It is one of the wonders of mathematics that, for every problem mathematicians solve, another awaits to perplex and galvanize them. Some of these problems are new, while others have puzzled and bewitched thinkers across the ages. Such challenges offer a tantalizing glimpse of the field's unlimited potential, and keep mathematicians looking toward the horizons of intellectual possibility.
In Visions of Infinity , celebrated mathematician Ian Stewart provides a fascinating overview of the most formidable problems mathematicians have vanquished, and those that vex them still. He explains why these problems exist, what drives mathematicians to solve them, and why their efforts matter in the context of science as a whole. The three-century effort to prove Fermat's last theorem—first posited in 1630, and finally solved by Andrew Wiles in 1995—led to the creation of algebraic number theory and complex analysis. The Poincaré conjecture, which was cracked in 2002 by the eccentric genius Grigori Perelman, has become fundamental to mathematicians' understanding of three-dimensional shapes. But while mathematicians have made enormous advances in recent years, some problems continue to baffle us. Indeed, the Riemann hypothesis, which Stewart refers to as the “Holy Grail of pure mathematics,” and the P/NP problem, which straddles mathematics and computer science, could easily remain unproved for another hundred years.
An approachable and illuminating history of mathematics as told through fourteen of its greatest problems, Visions of Infinity reveals how mathematicians the world over are rising to the challenges set by their predecessors—and how the enigmas of the past inevitably surrender to the powerful techniques of the present.
In Visions of Infinity , celebrated mathematician Ian Stewart provides a fascinating overview of the most formidable problems mathematicians have vanquished, and those that vex them still. He explains why these problems exist, what drives mathematicians to solve them, and why their efforts matter in the context of science as a whole. The three-century effort to prove Fermat's last theorem—first posited in 1630, and finally solved by Andrew Wiles in 1995—led to the creation of algebraic number theory and complex analysis. The Poincaré conjecture, which was cracked in 2002 by the eccentric genius Grigori Perelman, has become fundamental to mathematicians' understanding of three-dimensional shapes. But while mathematicians have made enormous advances in recent years, some problems continue to baffle us. Indeed, the Riemann hypothesis, which Stewart refers to as the “Holy Grail of pure mathematics,” and the P/NP problem, which straddles mathematics and computer science, could easily remain unproved for another hundred years.
An approachable and illuminating history of mathematics as told through fourteen of its greatest problems, Visions of Infinity reveals how mathematicians the world over are rising to the challenges set by their predecessors—and how the enigmas of the past inevitably surrender to the powerful techniques of the present.
352 pages, Hardcover
First published January 1, 2013
About the author
Ian Stewart
240 books724 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 - 21 of 21 reviews
July 12, 2015
This book is trying to address a very difficult task: to present in a meaningful way, to the general public, complex open problems in modern mathematics. A huge challenge.
The author is trying too hard to indulge the mathematically challenged by refusing to use proper mathematical notation (and adopting natural language instead) even where proper mathematical notation would have made the treatment so much clearer. I found myself converting, in my mind, some long, cumbersome statements by the author into the corresponding maths.
And many statements by the author, even if devoid of any mathematical formula, do actually necessarily imply some good background mathematical knowledge anyway, so the lack of mathematical notation does not really make the treatment more accessible to the general user, only more confusing.
This is compounded by lack of mathematical derivations when actual formulas are presented.
Some parts are done nicely, such as the explanation of Fermat's last theorem, but other parts touch mathematics only very succinctly (example: P versus NP, Mass Gap Hypothesis, Navier-Stokes equation) and leave the reader wanting for more.
Overall, a mixed bag. Not too bad but I have seen better.
To really understand maths, you need maths - trying to explain, to the mathematically naive, the peculiarities of the Navier-Stocks equation, without using any maths whatsoever, is so ambitious as to be almost delusional. I understood exactly what the author meant in this particular chapter, but I had previously studied ordinary and partial differential equations of many types in the past, so I wonder how much a reader with no prior knowledge would be able to actually understand this subject.
I do not want to sound elitist, but there are limits to how widely intrinsically complex mathematical problems can be popularized - and this book achieves its objectives only to a limited extent, I am afraid. Pity, as the author is clearly competent and enthusiastic.
June 24, 2013
I'm a fan of Ian Stewart. I own many of his books, and he has a gift for explaining mathematical concepts in understandable ways.
In this book, he faced a huge challenge: To explain mathematics' most formidable problems. He has done an admirable job. Explaining the meaning of each problem, let alone why it might be important to us, and how one might approach solving the problem, is a major challenge.
It would be difficult for me to imagine how anyone else could do a better job than Stewart does. However, as I read deeper into this book, I found the material more challenging. I was trained as a physicist, and so do not have a fear of mathematics (although that is a long way from actually being a Mathematician -- they definitely think differently from most of us!). I might have had a harder time without that strong mathematical background.
Recommended for those with sufficient background and perseverance.
In this book, he faced a huge challenge: To explain mathematics' most formidable problems. He has done an admirable job. Explaining the meaning of each problem, let alone why it might be important to us, and how one might approach solving the problem, is a major challenge.
It would be difficult for me to imagine how anyone else could do a better job than Stewart does. However, as I read deeper into this book, I found the material more challenging. I was trained as a physicist, and so do not have a fear of mathematics (although that is a long way from actually being a Mathematician -- they definitely think differently from most of us!). I might have had a harder time without that strong mathematical background.
Recommended for those with sufficient background and perseverance.
June 27, 2013
This book is a beautiful glimpse into the wonders of mathematical discovery, from Euclid to the present. Initially, the author's familiar tone and witty asides irritated me, but somewhere along the way I grew to like his style. Stewart has a gift for taking seemingly simple problems and unfolding the surprising complexity within them in a very accessible way. I started to glaze over on some of the more esoteric problems (and that is with the advantage of an undergrad math minor), so I'm not sure how accessible this book would actually be for those who aren't at least a bit fond of math.
May 12, 2013
This book can be a bit too technical for some people, but it’s a great survey of the famous solved and unsolved problems in the history of mathematics. It was a good reminder to me of how truly beautiful mathematics is. I feel sad when I read a book like this and realize how much math I’ve forgotten.
December 30, 2013
I had hoped this book would be "pop math" - that is, would describe the great mathematical problems throughout history with the level of detail and accessibility of a Malcolm Gladwell book. But it got way too deep into the weeds for me to attempt to follow with any degree of interest. Maybe it's just too hard to write about great mathematical problems like Fermat's last theorem and the Higgs boson in an accessible way - as the author acknowledges, it's exceedingly difficult even to describe the PROBLEM to be solved, let alone the solution.
February 17, 2019
Ian Stewart is clearly very knowledgeable in many different branches of mathematics and it was inspirational to read about the history of some of the greatest and most difficult problems in mathematics, but this book (along with other books of Stewart's I have read) suffers from two main problems. First, his writing is too dry; I don't really feel any excitement and enthusiasm for the math coming through his writing. Second, he has an audience problem. It is difficult to tell whom he is writing for. On the one hand, he seems to be writing for those with no mathematical background beyond high school because he avoids almost all mathematical detail and notation, and he often says things like: "Here comes a little bit of algebra. But don't worry, you don't have to understand it, just enjoy the story around it." I mean, come on, don't apologize for mathematics in a book that is supposed to be popularizing it! And don't insult your audience. On the other hand, he seems to be writing for people with advanced degrees in mathematics! The amount of technical detail and specialized vocabulary is sometimes quite overwhelming. I mean, why is he telling the reader the meaning of simple high school topics on the one hand, and not explaining huge swaths of advanced ideas and terminology on the other? On top of all of this, I think the book might paint the immensely exciting subject of mathematics as a bit dull. He talks a lot about the long, grueling toil the mathematical community undergoes across decades and centuries in coming up with partial results on these great problems--refining bounds from 6.0183 to 6.0182, dispensing with one case at a time in a list of infinitely many cases, using computers to check trillions of cases. Maybe I'm wrong (not being a mathematician), but I think mathematics is more beautiful than that.
June 6, 2017
A grade above most popularization books on technical subjects in that there is no hope of understanding the issues involved in 90% of the problems Stewart addresses. At least with string theory, genetic engineering, chaos theory etc. the layman has an intuitive first step on the ladder to understanding. Some of these problems are unimaginable to the non-mathematician and cannot be explained to any real depth.
But what Stewart is good at is explaining the steps that mathematicians take to tackle these problems and the intuition and drudgery involved. Sometimes inhuman genius is needed, sometimes years of calculation and minute bookkeeping. Stewart knows the life of the mathematician and can explain that world very well - not a gift that a high functioning academic always has!
Stewart is also very good at what makes a "great" problem. It has to have a lot of substance in solving it and the solution opens up vistas that weren't even known to exist before. Sometimes a trivial problem (like Fermat's "last" theorem - trivial because nothing really "hung" on its solution, other than a lot of historical momentum) becomes great through all the diverse areas of mathematics it took to solve it.
But what Stewart is good at is explaining the steps that mathematicians take to tackle these problems and the intuition and drudgery involved. Sometimes inhuman genius is needed, sometimes years of calculation and minute bookkeeping. Stewart knows the life of the mathematician and can explain that world very well - not a gift that a high functioning academic always has!
Stewart is also very good at what makes a "great" problem. It has to have a lot of substance in solving it and the solution opens up vistas that weren't even known to exist before. Sometimes a trivial problem (like Fermat's "last" theorem - trivial because nothing really "hung" on its solution, other than a lot of historical momentum) becomes great through all the diverse areas of mathematics it took to solve it.
June 20, 2023
I'm always interested in reading books that attempt to push the edges of the audience for mathematics a little further. I wanted desperately to love this one, as I have many others. And I did!
Unfortunately, I think that the author may have fallen victim to the dilution that can come about by attempting to join methods and concepts that are ill-suited for each other. It is, perhaps, an essentially impossible task to explain the Hodge conjecture to someone with no math background, since that succinct statement would likely be at best poorly understood by a bright student just graduated from their undergrad in mathematics. What is certain is that gigantic, page-filling blocks of text with little to no delineation between concepts and a great number of digressions is not a valuable method for elucidating these ideas.
The author shines when explaining the idea-genealogy of these problems, which makes for some of the most interesting reading of the book.
Unfortunately, I think that the author may have fallen victim to the dilution that can come about by attempting to join methods and concepts that are ill-suited for each other. It is, perhaps, an essentially impossible task to explain the Hodge conjecture to someone with no math background, since that succinct statement would likely be at best poorly understood by a bright student just graduated from their undergrad in mathematics. What is certain is that gigantic, page-filling blocks of text with little to no delineation between concepts and a great number of digressions is not a valuable method for elucidating these ideas.
The author shines when explaining the idea-genealogy of these problems, which makes for some of the most interesting reading of the book.
April 18, 2018
I enjoyed reading about all the math problems in this book. I wish I understood the problems better. Math is sure exciting. What else can I write. I sure have a long way to go to be good at math. What can I say I enjoyed the book. I am not much of a book reviewer. It difficult for me to understand math but, I think I leaned something by reading this book. I never practiced math or spent much time working on math when I was growing up. This book makes me wish I had so I could attempt to sole any of the unsolved math problems in the book. I still have time. Maybe I can get better at math.
July 17, 2019
This book is not for the math layperson, but it was an intriguing read. Stewart discusses some of the great problems in mathematics, both solved and unsolved, while also painting a broad picture regarding the beauty, elegance, and interconnectivity of mathematical problems. He does a good job of explaining many of the problems, but I found his explanations of the more challenging problems harder to follow. However, this is not a fault of the book; the chapters I am referring to describe very difficult nuances than only a mathematician would understand. Overall - I really enjoyed the book.
August 23, 2020
It covers the important aspects of 14 “Great problems” with great clarity, and provides extra details in the Notes section. The writing is as direct as possible with a pragmatic and a slightly chill disposition. The personal stories of the contributing mathematicians are covered with sensitivity and just the right amount of reverence. A few sections could be shortened but that is a subjective opinion and may reflect this reviewer’s lack of patience/focus.
September 3, 2018
between poorly written and hard subject matter. Authors tries but cannot disconnect himself from years of his education. His simple explanations go over my head and I would consider that I have an above average understanding of math.
This entire review has been hidden because of spoilers.
June 7, 2019
Did not finish. Either my brain couldn't comprehend a lot of the chosen problems or the writing wasn't simplified enough for me and I consider myself to be a math person. I largely enjoyed what I did understand though.
January 1, 2016
An hoary old philosophical conundrum asks if God could create a rock so heavy even he couldn't lift it. Mathematicians never have to ask if they could come up with problems so tough to solve that they could never figure them out, because they work with and around such problems all the time. Ian Stewart lays out several of them in Visions of Infinity; some were solved after much work and some remained unsolved when he wrote the book in 2013. Some might never be solved.
The way "solve" is being used here is slightly different than the way it might be used in other circumstances. Solving a math problem means finding what happens to the numbers after they've been processed according to the rules represented by the symbols accompanying them: 2+2 "solved" is 4. Solving an equation with a variable in it means processing those numbers in a way that will transform an unknown variable into a known number: Solving for x in "x+2=4" means finding out that x=2.
But solving the problems that Stewart talks about means demonstrating that certain math statements with nothing but unknown variables will always be true, no matter what numbers are plugged into them. Or that they won't be, by finding a set of numbers for which the equation will be false.
Visions is a brief look at several such problems and the story either of how they came to be and why they are still mysterious or how they were solved. It is not a quick read; some of the problems are interconnected and Stewart has a habit of dragging concepts from earlier chapters up without much of a signal or refresher of what they might entail. Some of his explanations of where the equations came from are as head-scratching as the equations themselves and furnish a reader with some seriously dense slogging.
Even though some of the math Stewart talks about may have even less of a "real world" application than algebra does in the eyes of a middle-schooler, he argues that it's still very important. Attempts to solve several of the problems in Visions led to many other mathematical breakthroughs and even failures often helped bring about a clearer understanding of the way the world works. And even if they did not, exploring math's outer reaches is no less a voyage of discovery than those taken the ancients who first ventured out of sight of land. Should human beings who seek to satisfy their curiosity about the physical world stop just because the frontiers are in the minds of the explorers? Stewart's answer in the different chapters of Visions may be complicated and take a long time to understand, but it boils down to, "No, they shouldn't," which sounded right to me before I read his book and still does afterwards.
Original available here.
The way "solve" is being used here is slightly different than the way it might be used in other circumstances. Solving a math problem means finding what happens to the numbers after they've been processed according to the rules represented by the symbols accompanying them: 2+2 "solved" is 4. Solving an equation with a variable in it means processing those numbers in a way that will transform an unknown variable into a known number: Solving for x in "x+2=4" means finding out that x=2.
But solving the problems that Stewart talks about means demonstrating that certain math statements with nothing but unknown variables will always be true, no matter what numbers are plugged into them. Or that they won't be, by finding a set of numbers for which the equation will be false.
Visions is a brief look at several such problems and the story either of how they came to be and why they are still mysterious or how they were solved. It is not a quick read; some of the problems are interconnected and Stewart has a habit of dragging concepts from earlier chapters up without much of a signal or refresher of what they might entail. Some of his explanations of where the equations came from are as head-scratching as the equations themselves and furnish a reader with some seriously dense slogging.
Even though some of the math Stewart talks about may have even less of a "real world" application than algebra does in the eyes of a middle-schooler, he argues that it's still very important. Attempts to solve several of the problems in Visions led to many other mathematical breakthroughs and even failures often helped bring about a clearer understanding of the way the world works. And even if they did not, exploring math's outer reaches is no less a voyage of discovery than those taken the ancients who first ventured out of sight of land. Should human beings who seek to satisfy their curiosity about the physical world stop just because the frontiers are in the minds of the explorers? Stewart's answer in the different chapters of Visions may be complicated and take a long time to understand, but it boils down to, "No, they shouldn't," which sounded right to me before I read his book and still does afterwards.
Original available here.
February 12, 2017
Though specifically stated to be accessible to the layman, I found it inaccessible. Unless you are in the field or a true mathematics buff, choose a different book.
June 21, 2020
I really wanted to like this book. I read Ian Stewart's column in Scientific American as a kid, and it helped me fall in love with math.
But I don't know whom this book is for, or rather if many such people exist. It is way too technical for the lay audience and way too slow for mathematicians, who are already convinced in the book's main thesis -- that math is interesting. I guess there may be some sufficiently technical folks out there who missed out on the beauty of math, who are the target audience and who would appreciate this book more, but it's hard to imagine there are many of them.
I am in the category of people who do math for a living, so I can speak to my experience of reading this book. I kept waiting for Stewart to get to the main mathematical points and to provide rigorous definitions, which often never came. The book did have some interesting explanations and historical gems that I enjoyed, but it was too much tedium to get to them.
But I don't know whom this book is for, or rather if many such people exist. It is way too technical for the lay audience and way too slow for mathematicians, who are already convinced in the book's main thesis -- that math is interesting. I guess there may be some sufficiently technical folks out there who missed out on the beauty of math, who are the target audience and who would appreciate this book more, but it's hard to imagine there are many of them.
I am in the category of people who do math for a living, so I can speak to my experience of reading this book. I kept waiting for Stewart to get to the main mathematical points and to provide rigorous definitions, which often never came. The book did have some interesting explanations and historical gems that I enjoyed, but it was too much tedium to get to them.
August 25, 2014
I haven't actually finished this yet, and I'm not sure I'm going to. Although the various problems are interesting to read about, the level of knowledge required to understand this book is steep. This books is supposed to be for laymen, but even with a minor in mathematics I'm having a devil of a time keeping up with all the ins and outs of the problems. I felt the author could have gone into less detail on the nuts and bolts of the problems and touched more on the significance of them. Maybe I'll come back to it later, but I doubt it.
Want to read
April 13, 2013Recommended by Catherine Weller
October 13, 2014
I thought this one was interesting and I love reading about unsolved or long unsolved math problems, but the technical knowledge required for this one was just too far above my skill level.
Read
March 7, 2014510 S8497v 2013
Displaying 1 - 21 of 21 reviews