Agustín Rayo - On the Brink of Paradox: Highlights from the Intersection of Philosophy and Mathematics (The MIT Press)

Galileo Galileis 1638 masterpiece, Dialogues Concerning Two New Sciences, has a wonderful discussion of infinity, which includes an argument for the conclusion that the attributes larger, smaller, and equal have no place [] in considering infinite quantities (pp. 7780).

Galileo asks us to consider the question of whether there are more squares (12, 22, 32, 42, ) or more roots (1, 2, 3, 4, ). One might think that there are more roots on the grounds that, whereas every square is a root, not every root is a square. But one might also think that there are just as many roots as squares, since there is a bijection between the roots and the squares. (As Galileo puts it, Every square has its own root and every root has its own square, while no square has more than one root and no root has more than one square p. 78.)

We are left with a paradox. And it is a paradox of the most interesting sort. A boring paradox is a paradox that leads nowhere. It is due to a superficial mistake and is no more than a nuisance. An interesting paradox, on the other hand, is a paradox that reveals a genuine problem in our understanding of its subject matter. The most interesting paradoxes of all are those that reveal a problem interesting enough to lead to the development of an improved theory.

Such is the case of Galileos infinitary paradox. In 1874, almost two and a half centuries after Two New Sciences, Georg Cantor published an article that describes a rigorous methodology for comparing the sizes of infinite sets. Cantors methodology yields an answer to Galileos paradox: it entails that the roots are just as many as the squares. It also delivers the arresting conclusion that there are different sizes of infinity. Cantors work was the turning point in our understanding of infinity. By treading on the brink of paradox, he replaced a muddled pre-theoretic notion of infinite size with a rigorous and fruitful notion, which has become one of the essential tools of contemporary mathematics.

In part I of this book, Ill tell you about Cantors revolution (chapters 1 and 2) and about certain aspects of infinity that are yet to be tamed (chapter 3). In part II Ill turn to decision theory and consider two different paradoxes that have helped deepen our understanding of the subject. The first is the so-called Grandfather Paradox, in which someone attempts to travel back in time to kill his or her grandfather before the grandfather has any children (chapter 4); the second is Newcombs Problem, in which probabilistic dependence and causal dependence come apart (chapter 5). Well then focus on probability theory and measure theory more generally (chapters 6 and 7). Well consider ways in which these theories lead to bizarre results, and explore the surrounding philosophical terrain. Our discussion will culminate in a proof of the Banach-Tarski Theorem, which states that it is possible to decompose a ball into a finite number of pieces and then reassemble the pieces (without changing their size or shape) so as to get two balls, each the same size as the original (chapter 8).