Is God a Mathematician?

by Mario Livio

  Around the same time that Frege was developing his logicist program, the Italian mathematician and logician Giuseppe Peano was attempting a somewhat different approach. Peano wanted to base arithmetic on an axiomatic foundation. Consequently, his starting point was the formulation of a concise and simple set of axioms. For instance, his first three axioms read:

  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 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.

&n
bsp; 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

  My investigations in the new grounding of mathematics have as their goal nothing less than this: to eliminate, once and for all, the general doubt about the reliability of mathematical inference…Everything that previously made up mathematics is to be rigorously formalized, so that mathematics proper or mathematics in the strict sense becomes a stock of formulae…In addition to this formalized mathematics proper, we have a mathematics that is to some extent new: a metamathematics that is necessary for securing mathematics, and in which—in contrast to the purely formal modes of inference in mathematics proper—one applies contextual inference, but only to prove the consistency of the axioms…Thus the development of mathematical science as a whole takes place in two ways that constantly alternate: on the one hand we derive provable formulae from the axioms by formal inference; on the other, we adjoin new axioms and prove their consistency by contextual inference.

  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.

  2. For any consistent formal system S within which a certain amount of elementary arithmetic can be carried out, the consistency of S cannot be proved in S itself.

  The words may appear to be benign, but the implications for the formalists’ program were far-reaching. Put somewhat simplistically, the incompleteness theorems proved that Hilbert’s formalist program was essentially doomed from the start. Gödel showed that any formal system that is powerful enough to be of any interest is inherently either incomplete or inconsistent. That is, in the best case, there will always be assertions that the formal system can neither prove nor disprove. In the worst, the system would yield contradictions. Since it is always the case that for any statement T, either T or not-T has to be true, the fact that a finite formal system can neither prove nor disprove certain assertions means that true statements will always exist that are not provable within the system. In other words, Gödel demonstrated that no formal system composed of a finite set of axioms and rules of inference can ever capture the entire body of truths of mathematics. The most one can hope for is that the commonly accepted axiomatizations are only incomplete and not inconsistent.

  Gödel himself believed that an independent, Platonic notion of mathematical truth did exist. In an article published in 1947 he wrote:

  But, despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception.

  By an ironic twist of fate, just as the formalists were getting ready for their victory march, Kurt Gödel—an avowed Platonist—came and rained on the parade of the formalist program.

  The famous mathematician John von Neumann (1903–57), who was lecturing on Hilbert’s work at the time, canceled the rest of his planned course and devoted the remaining time to Gödel’s findings.

  Gödel the man was every bit as complex as his theorems. In 1940, he and his wife Adele fled Nazi Austria so he could take up a position at the Institute for Advanced Study in Princeton, New Jersey. There he became a good friend and walking partner of Albert Einstein. When Gödel applied for naturalization as an American citizen in 1948, it was Einstein who, together with Princeton University mathematician and economist Oskar Morgenstern (1902–77), accompanied Gödel to his interview at the Immigration and Naturalization Service office. The events surrounding this interview are generally known, but they are so revealing about Gödel’s personality that I will give them now in full, precisely as they were recorded from memory by Oskar Morgenstern on September 13, 1971. I am grateful to Ms. Dorothy Morgenstern Thomas, Morgenstern’s widow, and to the Inst
itute for Advanced Study for providing me with a copy of the document:

  It was in 1946 that Gödel was to become an American citizen. He asked me to be his witness and as the other witness, he proposed Albert Einstein who also gladly consented. Einstein and I occasionally met and were full of anticipation as to what would happen during this time prior to the naturalization proceedings themselves and even during those.

  Gödel whom I have seen of course time and again in the months before this event began to go in a thorough manner to prepare himself properly. Since he is a very thorough man, he started informing himself about the history of the settlement of North America by human beings. That led gradually to the study of the History of American Indians, their various tribes, etc. He called me many times on the phone to get literature which he diligently perused. There were many questions raised gradually and of course many doubts brought forth as to whether these histories really were correct and what peculiar circumstances were revealed in them. From that, Gödel gradually over the next weeks proceeded to study American history, concentrating in particular on matters of constitutional law. That also led him into the study of Princeton, and he wanted to know from me in particular where the borderline was between the borough and the township. I tried to explain that all this was totally unnecessary, of course, but with no avail. He persisted in finding out all the facts he wanted to know about and so I provided him with the proper information, also about Princeton. Then he wanted to know how the Borough Council was elected, the Township Council, and who the Mayor was, and how the Township Council functioned. He thought he might be asked about such matters. If he were to show that he did not know the town in which he lived, it would make a bad impression.