Is God a Mathematician? (26 page)

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 case of déjà vu.

The Non-Euclidean Crisis All Over Again?

In 1908, the German mathematician Ernst Zermelo (1871–1953) followed a path very similar to that originally paved by Euclid around 300 BC. Euclid formulated a few unproved but supposedly self-evident postulates about points and lines and then constructed geometry on the basis of those axioms. Zermelo—who discovered Russell’s paradox independently as early as 1900—proposed a way to build set theory on a corresponding axiomatic foundation. Russell’s paradox was bypassed in this theory by a careful choice of construction principles that eliminated contradictory ideas such as “the set of all sets.” Zermelo’s scheme was further augmented in 1922 by the Israeli mathematician Abraham Fraenkel (1891–1965) to form what has become known as the Zermelo-Fraenkel set theory (other important changes were added by John von Neumann in 1925). Things would have been nearly perfect (consistency was yet to be demonstrated) were it not for some nagging suspicions. There was one axiom—the
axiom of choice
—that just like Euclid’s famous “fifth” was causing mathematicians serious heartburn. Put simply, the axiom of choice states: If
X
is a collection (set) of nonempty sets, then we can choose a single member from each and every set in
X
to form a new set
Y.
You can easily check that this statement is true if the collection
X
is not infinite. For instance, if we have one hundred boxes, each one containing at least
one marble, we can easily choose one marble from each box to form a new set
Y
that contains one hundred marbles. In such a case, we do not need a special axiom; we can actually prove that a choice is possible. The statement is true even for infinite collections
X,
as long as we can precisely specify how we make the choice. Imagine, for instance, an infinite collection of nonempty sets of natural numbers. The members of this collection might be sets such as {2, 6, 7}, {1, 0}, {346, 5, 11, 1257}, {all the natural numbers between 381 and 10,457}, and so on. In every set of natural numbers, there is always one member that is the smallest. Our choice could therefore be uniquely described this way: “From each set we choose the smallest element.” In this case again the need for the axiom of choice can be dodged. The problem arises for infinite collections in those instances in which we cannot define the choice. Under such circumstances the choice process never ends, and the existence of a set consisting of precisely one element from each of the members of the collection
X
becomes a matter of faith.

From its inception, the axiom of choice generated considerable controversy among mathematicians. The fact that the axiom asserts the existence of certain mathematical objects (e.g., choices), without actually providing any tangible example of one, has drawn fire, especially from adherents to the school of thought known as
constructivism
(which was philosophically related to
intuitionism
). The constructivists argued that anything that exists should also be explicitly constructible. Other mathematicians also tended to avoid the axiom of choice and only used the other axioms in the Zermelo-Fraenkel set theory.

Due to the perceived drawbacks of the axiom of choice, mathematicians started to wonder whether the axiom could either be proved using the other axioms or refuted by them. The history of Euclid’s fifth axiom was literally repeating itself. A partial answer was finally given in the late 1930s. Kurt Gödel (1906–78), one of the most influential logicians of all time, proved that the axiom of choice and another famous conjecture due to the founder of set theory, Georg Cantor, known as the
continuum hypothesis
, were both consistent with the other Zermelo-Fraenkel axioms. That is, neither of the two hypotheses could be refuted using the other standard set
theory axioms. Additional proofs in 1963 by the American mathematician Paul Cohen (1934–2007, who sadly passed away during the time I was writing this book) established the complete independence of the axiom of choice and the continuum hypothesis. In other words, the axiom of choice can neither be proved nor refuted from the other axioms of set theory. Similarly, the continuum hypothesis can neither be proved nor refuted from the same collection of axioms, even if one includes the axiom of choice.

This development had dramatic philosophical consequences. As in the case of the non-Euclidean geometries in the nineteenth century, there wasn’t just one definitive set theory, but rather at least four! One could make different assumptions about infinite sets and end up with mutually exclusive set theories. For instance, one could assume that both the axiom of choice and the continuum hypothesis hold true and obtain one version, or that both do not hold, and obtain an entirely different theory. Similarly, assuming the validity of one of the two axioms and the negation of the other would have led to yet two other set theories.

This was the non-Euclidean crisis revisited, only worse. The fundamental role of set theory as the potential basis for the whole of mathematics made the problem for the Platonists much more acute. If indeed one could formulate many set theories simply by choosing a different collection of axioms, didn’t this argue for mathematics being nothing but a human invention? The formalists’ victory looked virtually assured.

An Incomplete Truth

While Frege was very much concerned with the meaning of axioms, the main proponent of formalism, the great German mathematician David Hilbert (figure 52), advocated complete avoidance of any interpretation of mathematical formulae. Hilbert was not interested in questions such as whether mathematics could be derived from logical notions. Rather, to him, mathematics proper consisted simply of a collection of meaningless formulae—structured patterns composed of arbitrary symbols. The job of guaranteeing the foundations of mathematics was assigned by Hilbert to a new discipline, one he referred to as “metamathematics.” That is, metamathematics was concerned with using the very methods of mathematical analysis to prove that the entire process invoked by the formal system, of deriving theorems from axioms by following strict rules of inference, was consistent. Put differently, Hilbert thought that he could prove mathematically that mathematics works. In his words:

Figure 52

Hilbert’s program sacrificed meaning to secure the foundations. Consequently, to his formalist followers, mathematics was indeed just a game, but their aim was to rigorously prove it to be a fully consistent game. With all the developments in axiomatization, the realization of this formalist “proof-theoretic” dream appeared to be just around the corner.

Not all were convinced, however, that the path taken by Hilbert was the right one. Ludwig Wittgenstein (1889–1951), considered by some to be the greatest philosopher of the twentieth century, regarded Hilbert’s efforts with metamathematics as a waste of time. “We cannot lay down a rule for the application of another rule,” he argued. In other words, Wittgenstein did not believe that the understanding of one “game” could depend on the construction of another: “If I am unclear about the nature of mathematics, no proof can help me.” Still, no one was expecting the lightning that was about to strike. With one blow, the twenty-four-year-old Kurt Gödel would drive a stake right through the heart of formalism.

Kurt Gödel (Figure 53) was born on April 28, 1906, in the Moravian city later known by the Czech name of Brno. At the time, the city was part of the Austro-Hungarian Empire, and Gödel grew up in a German-speaking family. His father, Rudolf Gödel, managed a textile factory and his mother, Marianne Gödel, took care that the young Kurt got a broad education in mathematics, history, languages, and religion. During his teen years, Gödel developed an interest in mathematics and philosophy, and at age eighteen he entered the University of Vienna, where his attention turned primarily to mathematical logic. He was particularly fascinated by Russell and Whitehead’s
Principia Mathematica
and by Hilbert’s program, and chose for the topic of his dissertation the problem of
completeness
. The goal of this investigation was basically to determine whether the formal approach
advocated by Hilbert was sufficient to produce all the true statements of mathematics. Gödel was awarded his doctorate in 1930, and just one year later he published his
incompleteness theorems,
which sent shock waves through the mathematical and philosophical worlds.

Figure 53

In pure mathematical language, the two theorems sounded rather technical, and not particularly exciting:

1. Any consistent formal system
S
within which a certain amount of elementary arithmetic can be carried out is incomplete with regard to statements of elementary arithmetic: there are such statements which can neither be proved nor disproved in
S.