The Mathematical Century, by Piergiorgio Odifreddi, subtitled The 30 Greatest Problems of the Last 100 Years, is what I would call the best book of its kind that I have ever read. The meaning of the phrase "of its kind" comes from one simple fact, quite obvious if you would see a page or two, that this is a very technical, mathematics-oriented book. It is not like the average storybook of fictional characters going through an archetypal journey, but it does tell the story of how these 30 problems have been analyzed by real mathematicians and solved over the years. Considering the sheer size and original complexity of some of these ideas and problems, Odifreddi does an excellent job of explaining, although you may need to stop and really think abstract for a few minutes. For those of you who enjoy abstract thinking, interesting ideas, or just mathematical concepts, this book is a rare treat. Since this quality of a book is often a self-judging trait, a sample of the book is necessary. In this passage, Odifreddi's way of presenting and narrating the history of the problems and their very concept is captivating:
"One of the great achievements of nineteenth-century mathematics was the classification of the 2-dimensional surfaces from the topological point of view, that is, by regarding them as rubber sheets that can be deformed at will provided they are not torn apart. From this abstract point of view, an inflated ball and a deflated one are the same surface, even if ..." (Odifreddi, 78)
For each problem presented, there are about six pages about its origin, nature, history, attempts at solving, and, for all but the open problems (ones yet to be answered), the solution. The problems include many old questions of geometry, algebra, calculus, and number theory while several more modern ones as well. An interesting factor of many of these problems is how much computers show up in this storyline, as tools that themselves prove too incapable to display or solve some problems in full completeness. To me, this was a little surprising, considering I usually see computers able to do just about anything that programmers tell them to do. Often, these problems are solved with what seems at first the most irrelevant equation or concept that happens to tie in exactly to complete the puzzle. All of this Odifreddi packs efficiently into short sections whose length compares to pamphlets, keeping this brief but informative rather than something of a textbook. With the style of light reading, even the depth of these questions keeps reading this from being overly involved.
In addition, the capacity this book has for stimulating abstract thinking far outweighs anything else. To begin with, the very nature of many of the ideas presented begs for abstraction, like presenting a geometric surface that intersects itself and thus cannot have constructed model to represent it. Take note that in this 180 page book there are thirty problems and even more concepts, the greater half of those could use some abstraction. A critical backbone in this abstraction capacity comes from the fact that this is not science fiction, this deals with real mathematical and physical concepts that exist, whether or not we care to notice, with proof and some given trends. The inability to dismiss these abstract concepts, in my opinion, makes them all the better and more interesting.
Overall, this is not a book that I would recommend to buy without flipping through it and reading a page or two. After doing that, if The Mathematical Century suits your liking, it should be a great book to read. One nice property for people like me who sometimes wish to read in a random order is the effective ease of doing that in this book. Few, if any, sections of this book require any reading of the pages before it, which will be relieving to readers who might dislike or completely not understand one part but love the rest. Personally, I would rate this book as a five out of five for those of you who like this type of material, although those who lack an interest in such a subject as this would probably give this book a low rating.