Visions Of Infinity The Great Mathematical Problems Pdf

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· 198 ratings · 20 reviews

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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 aut

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.

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.

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Apr 09, 2013 Bryan Higgs rated it really liked it

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. H

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.

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.

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Jun 27, 2013 Jenae rated it really liked it

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 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. ...more

Apr 17, 2013 Jafar rated it really liked it

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.

Dec 30, 2013 Ben rated it it was ok

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 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. ...more

Feb 17, 2019 Beau rated it it was ok

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 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. ...more

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 t

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.

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Apr 17, 2018 Daniel rated it really liked it

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 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. ...more

Jul 17, 2019 Matt rated it really liked it

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 dif 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. ...more

Aug 23, 2020 albin james rated it it was amazing

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.

This review has been hidden because it contains spoilers. To view it, click here. 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.

Jun 06, 2019 Ken rated it it was ok

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. 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. ...more

May 05, 2015 Brett T rated it really liked it

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 use

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.

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Feb 12, 2017 Matt rated it it was ok

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.

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 au

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.

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.

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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' 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. ...more

Oct 06, 2013 Tart rated it it was ok

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.

Ian 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 au

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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Mary Roach is a science author who specializes in the bizarre and offbeat. With a body of work ranging from deep-dives on the history of...

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