Is God a Mathematician_ - Mario Livio [89]
Zero is a number.
The successor to any number is also a number.
No two numbers have the same successor.
The problem was that while Peano’s axiomatic system could indeed reproduce the known laws of arithmetic (when additional definitions had been introduced), there was nothing about it that uniquely identified the natural numbers.
The next step was taken by Bertrand Russell. Russell maintained that Frege’s original idea—that of deriving arithmetic from logic—was still the right way to go. In response to this tall order, Russell produced, together with Alfred North Whitehead (figure 50), an incredible logical masterpiece—the landmark three-volume Principia Mathematica. With the possible exception of Aristotle’s Organon, this has probably been the most influential work in the history of logic (figure 51 shows the title page of the first edition).
Figure 50
In the Principia, Russell and Whitehead defended the view that mathematics was basically an elaboration of the laws of logic, with no clear demarcation between them. To achieve a self-consistent description, however, they still had to somehow bring the antinomies or paradoxes (additional ones to Russell’s paradox had been discovered) under control. This required some skillful logical juggling. Russell argued that those paradoxes arose only because of a “vicious circle” in which one was defining entities in terms of a class of objects that in itself contained the defined entity. In Russell’s words: “If I say ‘Napoleon had all the qualities that make a great general,’ I must define ‘qualities’ in such a way that it will not include what I am now saying, i.e. ‘having all the qualities that make a great general’ must not be itself a quality in the sense supposed.”
To avoid the paradox, Russell proposed a theory of types, in which a class (or set) belongs to a higher logical type than that to which its members belong. For instance, all the individual players of the Dallas Cowboys football team would be of type 0. The Dallas Cowboys team itself, which is a class of players, would be of type 1. The National Football League, which is a class of teams, would be of type 2; a collection of leagues (if one existed) would be of type 3, and so on. In this scheme, the mere notion of “a class that is a member of itself” is neither true nor false, but simply meaningless. Consequently, paradoxes of the kind of Russell’s paradox are never encountered.
Figure 51
There is no question that the Principia was a monumental achievement in logic, but it could hardly be regarded as the long-sought-for foundation of mathematics. Russell’s theory of types was viewed by many as a somewhat artificial remedy to the problem of paradoxes—one that, in addition, produced disturbingly complex ramifications. For instance, rational numbers (e.g., simple fractions) turned out to be of a higher type than the natural numbers. To avoid some of these complications, Russell and Whitehead introduced an additional axiom, known as the axiom of reducibility, which in itself generated serious controversy and mistrust.
More elegant ways to eliminate the paradoxes were eventually suggested by the mathematicians Ernst Zermelo and Abraham Fraenkel. They, in fact, managed to self-consistently axiomatize set theory and to reproduce most of the set-theoretical results. This seemed, on the face of it, to be at least a partial fulfillment of the Platonists’ dream. If set theory and logic were truly two faces of the same coin, then a solid foundation of set theory implied a solid foundation of logic. If, in addition, much of mathematics indeed followed from logic, then this gave mathematics some sort of objective certainty, which could also perhaps be harnessed to explain the effectiveness of mathematics. Unfortunately, the Platonists couldn’t celebrate for very long, because they were about to be hit by a bad