Visions of Infinity: The Great Mathematical Problems

Ian Stewart is a prolific writer of both books for math students as well for those with a casual interest. Visions of infinity falls into the latter category but has much content that can be much better followed by the former. Math is usually a difficult subject to translate for the non-expert as its problems are often very abstract (compared to say physics in which questions posed are on the fabric of reality). Visions of Infinity attempts to familiarize the non-expert with what working mathematicians today do and which problems grab their attention. The book is split into 17 chapters with the substance of the book being about 14 major problems in math (some solved, some remain open). The book is approachable for the non-expert and does give some very good insight into real math but given the complexity of every one of the problems the focus is on intuition from 30k feet with a bit of the detail which can quickly become incomprehensible.The book slowly takes the reader from problems which can be stated simply those which are pretty incomprehensible unless the reader has advanced mathematical knowledge. The author starts out with mathematical proof, Euclid and prime numbers. The first problem considered is the Goldbach conjecture (that every prime number greater than 2 is the sum of 2 primes) and starts getting into the reasons for its complexity. The author also discusses the classical Greek problem of constructing a circle with equal volume to a given square. Through math invented 2k years later this is shown to be impossible using straight edge and compass. The four colour theorem is discussed (any map can be filled in using only 4 colors such that no two colors are adjacent) and how computation has been introduced into what is now considered legitimate mathematical proof. Very intuitive yet complicated problems like Kepler's conjecture are given (the most space efficient method for packing regularly shaped spheres) and discussed in good detail. The author is good to use counterexamples that run against our intuition to show why the problems arent trivial. The author does not only tackle the problems that translate well to the non mathematician as i mentioned before, he includes the Mordell conjecture which is about rational solutions to elliptic curves. But goes into what these really are and where they come from so that the reader can appreciate the broad strokes. Given the origin of things like this come from diophantine equations of several thousand years ago, the big picture can be appreciated by the casual reader. Classical mechanics is considered and the fact that there arent explicit solutions to the 3-body problem in general is discussed. The qualitative theory of differential equations is introduced in this section and the visual aspects of what happens is really helpful for the reader base. The author discusses the Riemann Hypotheses and discusses aspects of complex function theory. These ideas are though and most of the math is hard to follow. But the nature of the distribution of primes is an idea that the non-mathematician can appreciate and as a result the reason for the problems importance. The author then discusses the Poincare conjecture (which was solved in 2002 though confirmed only many years after) and the ideas in topology and Ricci flow that were needed. The political controversy of the proving of the conjecture by Perelman is excluded. Computability is discussed with the P/NP problem. This is a little bit more technical then it needed to be and im surprised the author didnt focus more on the travelling salesman problem. The Navier-Stokes equation is discussed and the reader comes back to a place of stronger natural intuition, turbulence and fluid flow are all familiar and the authors discussion is interesting and relatively easy to follow. The author goes back to diophantine equations and considers the Birch-Swinnerton-Dyer Conjecture. This is related to the Mordell Conjecture and is pretty tough to follow. The author ends with the Hodge Conjecture, which is impossible to understand as a conjecture without prior knowledge. The author i think here shines in slowly translating the "On a projective non-singular algebraic variety over C, anyHodge class is a rational linear combination of classes cl(Z) of algebraic cycles" into something that can be at least put into some kind of context for the unfamiliar. It is a good end to the book as it ties various ideas introduced through the book together.Visions of Infinity is at times really engaging and readable and at times a bit overwhelming. It is inescapable when trying to introduce incredibly difficult (potentially unsolvable) abstract problems. Given the subject matter though on balance the author does a good job and I got a lot out of the book. The contents arent that unique and for many of the problems there are great books for the non-expert which give much more intuition and are much more complete ( The Poincare Conjecture: In Search of the Shape of the Universe , Four Colors Suffice: How the Map Problem Was Solved , In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation , and there are a number for the Riemann Hypothesis). Keith Devlin, another author of books for the mathematically interested wrote about the Millenium Problems for the non-expert and there is enourmous overlap and I prefer some of the writing there ( The Millennium Prize Problems ). But all in all there is good material here and most people will be able to take something out of the book, at the very least that mathematicians work on some pretty crazy problems, all in all the book is enjoyable though very complicated at times.

I've pre-read the soon to be released "golden ticket" book --ala Willy Wonka's candy bars--  The Golden Ticket: P, NP, and the Search for the Impossible -- which explores P-NP, and along with Ian's Infinity book here, these are really the only two UP TO DATE books on the millenium and other "tough" problems in math and computing, both solved and awaiting new approaches. Although P-NP is technically a computational complexity problem, knowing which problems can and can't be solved in polynomial vs. exponential time is widely believed to hold the key to "all" the other problems in computing AND math! An astonishing prospect.Infinity is a GREAT, fun read, with just enough math to please, but not intimidate an advanced high school level reader. It is not just another of the numerous "here are the heroes --heroines-- of math" books that dwell on older, solved problems. The history and problems given lead us into both unsolved CURRENT problems, and the birth of many of the newer --over 450 at present writing-- subspecialties of analysis, math, tricky current theorems, etc. As usual, Stewart doesn't "talk down" to his readers, and you get a real "brain workout" feeling after reading this wonderful survey of the "top 14" toughest solved and unsolved, and their real-life ramifications as solved and when solved.One of the really amazing side shows in this exciting field is the nature of humans vs. computers that is quietly raging in math. P-NP --although mostly thought to be unequal--, for example, even peeks into quantum observer and quantum computing issues about how "great" even average human brains might potentially be. It pulls back a curtain in which a variety of "modern" brute force, algorithmic, sieve and spectral methods and other algorithmic approaches to problems are finding unexpected barriers.This is non trivial, because a modern operating system like Windows with hundreds of thousands of lines of code is "controlled" by tiny algorithms of two or three lines, as are million line fly by wire jets. Our planet is rapidly becoming algorithm driven, to the extent that most new math methods and problems are described in algorithmic terms. But P-NP is only a little slice of this wonderful book, which covers a vast number of newer and less known stumbling blocks within the top 14, including the better known millenial problems. In Systems Engineering, we know that infinite and tiny problems are much easier than the "middle" complexity problems.Just as dynamic systems can "settle down" into chaos/fractals, strange attractors or an oscillation, the book, after taking us on a fascinating journey through the known and unknown, gives us a great, up to date feel for which problems are in which category of difficulty and likely vs. unlikely to be solved in our times. The "toughest" problems are the stuff of cryptanalysis and are "good" from the standpoint of providing security, but Ian also shows the many possible openings at the back of the tent in addition to the door, by suggesting possible "close enough" solutions and directions that are worth pursuing.Great survey, with just enough depth in each area to tease the reader into looking further. Because of Ian's usual "analogy" explanations and diagrams, very tough problems are explained intuitively, but the actual math is sneakily grad level. You WILL hunger for more if any of these tantalizing puzzles hit home. If you're a P-NP enthusiast, you should know that Ian doesn't think it is solvable in anyone reading this books lifetimes! But don't get discouraged, he could be wrong, but does make a strong and thoughtful case.In the golden ticket book above, Fortnow also says P/NP won't be solved in --possibly-- thousands of years, due to the kind of "proving a negative" of "showing" that EVERY potential algorithm HAS to fail. The cool thing mentioned by both Ian and Fortnow is that these REALLY tough problems tend to bring vast areas of science together that were silos in the past. If you look up Systems Science on Wiki, you'll get an idea of the missing link in a lot of these discussions. The unstated speculation is whether the human brain "perceives" higher dimensions than algorithms can reach in some fashion, and P/NP has areas as strange as the quantum observer for the possibly unexpectedly great potential of humans, as yet unknown.Library Picks always buys the books we review, and has nothing to do with Amazon, the authors or publishers, and our comments are solely for the benefit of Amazon shoppers.

This book shows that mathematics is a fascinating human adventure, most of the time a collective one, where great mathematicians, standing in the shoulders of other giants, finally solve a famous problem. Sometimes there are heroes that work for years in isolation (Wiles and Perelman, for instance), although they still base their work on the results of others. Some of the famous problems are still unresolved and Ian Stewart even suggests that some will be proved undecidable if we use the standard ZFC axioms(such as the Continum Hypothesis was proven to be by Gödel and Cohen). It really moved my soul reading how some of the hard problems had been solved through decades or centuries one o various exponents at a time (Fermat's Last Theorem) or one or various dimensions at a time (Poincare's conjecture)Generally speaking Stewart is a very good communicator although the difficult chapters 15 and 16 could perhaps be improved. Chapter 15, for instance, on the Birch-Swinnerton-Dyer conjecture is probablty more condensed than the equivalent chapter in Keith Devlin's "The Millenium Problems". If you didn't major in math in college you will probably have some hard time in reading these two chapters. So, why am I rating it 5 stars? Because in some chapters I was strongly moved by the story in the same way as when I read an excellent novel or watch a very good movie.

A good read but intensive study required.