Popular history of various concepts of the infinite, from the ancient Greeks up through Cantor. This has some real math in it, but it's all explained very well, and I found it all accessible. Very full of Wallacisms, however, comprising abbreviations, asides, and footnotes galore. I actively enjoy that, however, so for me this is one of the best pop math books I've ever read. If you're inclined to pick up such a book in the first place, that will probably be true for you as well, provided his style doesn't get on your nerves. ( )

In his Foreword, the author calls this book “a piece of pop technical writing” for “readers who do not have pro-grade technical backgrounds and expertise.” Well, maybe. But some concepts do not lend themselves to easy explanations. This book is pretty tough, even for someone like me who has a decent background in college math. Nevertheless, if you can make it through the book (which I did over a period of months, in small snippets), you will indeed find, as he claims in the Foreword, that math is in fact very beautiful.

He begins with a discussion of the difficulties of abstraction. Children understand the idea of “five oranges” much better than the concept of “five” removed from “concrete particularity.” He compares the problem to philosophical theories: what do we mean by “existence” or “motion” or “knowledge”? Indeed, as the physicist genius Richard Feynman responded, when asked about the mechanism of magnets, “I can’t explain that attraction in terms of anything else that’s familiar to you.”

He also said, ““I think I can safely say that nobody really understands quantum mechanics,” and yet, scientists can use the theory successfully to account for a wide range of physical phenomena, from how transistors work to why stars shine.

Similarly, Wallace argues, you don’t need to understand how the car works to be able to drive, just as you don’t need to understand mathematical concepts intuitively to be able to use them and extrapolate from them to generate a variety of useful applications. Mathematics, Wallace avers, is “thoroughly private-sector and results-oriented.”

Yet, he allows,”there is nothing more abstract than infinity.” He then proceeds to go into the idea of infinity in depth, observing that “the important thing to keep straight is that the problems and controversies about infinity that are going to concern us here involve whether infinite quantities can actually exist as mathematical entities.”

Mankind has long been fascinated by the concept of infinity, both as it relates to the very large and to the very small. The Greeks struggled with the concept and produced some interesting ruminations, the most famous of which is Zeno’s paradox of Achilles and the tortoise. Zeno wrote that Achilles, who was reputed to be very fleet of foot, could never catch a slow moving tortoise because he would first have to cover the original distance between himself and the tortoise—but the tortoise would have moved ahead during the time it took Achilles to reach its original starting place. Then Achilles would traverse the distance to the tortoise’s last location only to discover it had moved ahead again, and this partial catch up would continue ad infinitum without ever overtaking the tortoise. This may seem like sublimely stupid physics, but it was something Greek mathematics, with its emphasis on rigor and proof, was unable to account for.

It was not until Newton and Leibniz invented the calculus that math had tools to deal effectively with Zeno’s paradox. And even then, their efforts, using “fluxions” and “infinitesimals,” lacked the degree of rigor demanded by today’s mathematicians. It was not until the late 19th century than mathematicians achieved what was to them a satisfactory grounding of the basis of analysis of the infinitely large and infinitely small.

Wallace identifies the chief character in the history of infinity as Georg Cantor, with some assistance from Richard Dedekind, R. Cauchy, and others.

Galileo had been puzzled by the fact that there seemed to be as many perfect squares (1, 2, 4, 9, 16, 25, 36, . . .) as there were counting numbers (1, 2, 3, 4, 5, 6. . . .). He demonstrated this curiosity by matching every counting number with its appropriate perfect square. The same technique was used to prove that the number of even numbers was equal to the number of all the digits, both odd and even. Then Cantor showed that all the rational numbers (proper fractions as well as digits) could be paired on a one-to-one basis with the counting numbers. Thus, however counterintuitive it may seem, the set of all rational numbers is no greater than the set of counting numbers! This particular infinity (there would be more), he labeled “Aleph 1.”

Cantor went further. He demonstrated that the number of points between any two points on Dedekind’s infinitely dense number line was greater (in fact, infinitely greater) than all the counting numbers! This higher infinity he labeled ‘c’. In addition, he showed that it was possible to generate an infinite number of infinities, each (infinitely) larger than the last one. These he called “Aleph 2,” “Aleph 3, ” . . . etc.

Therefore, the set of all real numbers is uncountable; that is: while both the set of all natural numbers and the set of all real numbers are infinite sets, there can be no one-to-one function from the real numbers to the natural numbers.

One problem that he was unable to solve, and which may have driven him mad, was the question of whether there were any, or even an infinite number of, infinities that lay between Aleph 1 and c. I’ll leave that one to the reader to ponder, but please, maintain your sanity.

Evalation: Everything and More, is a very abstruse and challenging book, but well worth the effort required to finish it. My summary may not do it justice in that many more problems are discussed and analyzed therein. I only hope I have not mischaracterized some of the concepts discussed. Note: This may be, as Wallace asserts, a "pop science" book, but it is not for the non-mathematically adept.

(JAB) ( )

High-wire all the way. More DFW or math in either direction and the thing falls to it's death. If you would like a glimpse of what Real math is like and don't hate Mr. Wallace, this is worth it. ( )

Partiamo dal principio. La nuova copertina della nuova edizione del libro è favolosa. La traduzione è quella corretta (ma lo era già nelle ristampe del 2011), salvo gli eventuali svarioni di DFW e due o tre tecnicalità di cui non vi accorgerete se non sapete già di che si sta parlando. Ma la cosa più bella del libro è la sua genesi. L'edizione originaria faceva infatti parte di una collana "Great Discoveries", pubblicata da Norton, nata come serie di biografie tecniche di scienziati. DFW non era un matematico, anche se aveva studiato abbastanza per poterne parlare con cognizione di causa, e questo lo si vede dal modo in cui approccia il tema, con una forte componente metafisica che in genere viene trascurata quando si arriva all'Ottocento. Ma era per l'appunto DFW, il che significa che il testo non è per niente lineare e parte per la tangente con le note NCVI ("Nel Caso Vi Interessi", in originale IYI, If You"re Interested) che naturalmente sono imprescindibili, e una serie di rimandi incrociati. Poi lo stile è al solito scanzonato, il che darà al lettore la falsa impressione che tutto sia facile nonostante i mille caveat nel testo. Diciamo che non credo che nessuno imparerà qualcosa sull'infinito leggendolo, ma tanto non era quello il suo scopo. ( )

Faticoso. In certi momenti,la tentazione di abbandonarlo ?� stata forte; la spinta ?¨ arrivata dall'ironia dell'Autore e dalla mia curiosit? . ( )