Impossibility (book)

John D. Barrow Impossibility (1998) There are several kinds of impossibility, but they fall into three groups. There is the practical impossibility, reflecting some limits to the resources we (or any other creature) can command. Then the nature of the Universe itself sets limits on the possible. And all logical systems above a certain level of complexity exhibit impossibilities.
     An example of practical impossibility is the solution of problems that would take more computing time than the lifetime of the Universe; another is travelling beyond the solar system. Whether the Universe has a beginning or not is an example of a question we cannot answer because, although we can specify what we should need to know in order to settle the question, we cannot get the necessary knowledge. An example of a logical impossibility is expressed in Godel's theorem, which states that any axiomatic system at least as complex as arithmetic contains statements whose truth or falsehood cannot be determined
     A more interesting example is Arrow's Impossibility theorem: as the number of candidates for office increases, the probability that there will be no majority winner. approaches certainty. What this means in practice is that whoever wins, most people wanted someone else. The result can be generalised to any situation with multiple, mutually independent choices . It also applies to sporting events. Where several teams compete for a championship, there is surprisingly large possibility that the winner can be (and often has been) beaten by one or more of the losers. With 8 teams, the odds of this happening are 1 in 3.
     Barrow is a somewhat turgid writer. There are irritating typographical errors throughout the book, mostly of the wrong-word variety; an effect of reliance on spell checkers. The book is heavy going in places. I have read similar discussions elsewhere, and so didn't get hopelessly lost, but anyone who hasn't at least a senior high school understanding of physics, logic, mathematics, and other disciplines will probably have trouble following some of Barrow's arguments. Nevertheless, it's worth reading, if only to disabuse one of the notion that all things are possible. Barrow's most subtle point is this: that impossibilities, the limits of action and knowledge, tell us more about the nature of our Universe than the possibilities do. *** (1999)

Update 2012: if quantum computers do become a reality, then the range of solvable problems will enlarge by many orders of magnitude. Then question then become which of these problems are worth solving, which may be impossible to answer without solving the problem.

Update 2019: Minor correcctions in style and spelling.