Mathematics and the Real World: The Remarkable Role of Evolution in the Making of Mathematics (2014)

CHAPTER VIII. IS THERE REALLY NO DOUBT?

60. A MAJOR CRISIS

With the exception of a relatively small group of mathematicians, the mathematics community welcomed the reliance on set theory with open arms. Gottlob Frege (1848–1925), a leading German mathematician at the beginning of the twentieth century, decided to put in writing the set theory basis of mathematics. He published the first tome and was in the latter stages of the second when the crisis erupted. Bertrand Russell wrote him a dramatic letter and set out the famous paradox he had discovered in Frege's text, a paradox that led Frege to halt the publication of the second part of his book. For a while Frege tried to correct the theory he had developed but eventually decided to abandon the project he had undertaken. Bertrand Russell was then a young mathematician whose astuteness had already earned him a reputation in Britain. He was later acknowledged as one of the founders of analytic philosophy and was known for his radical social and political views. He was a pacifist, a conscientious objector in the First World War, and bitterly critical of totalitarian regimes throughout the world. In 1950 he was awarded the Nobel Prize in Literature for his writings in which he championed humanitarian ideals and freedom of thought and for his famous bestseller A History of Western Philosophy.

Russell's paradox was in fact a variation on a paradox known already in the time of the Greeks, the liar's paradox. A person says of himself, “I am a liar.” Can we believe him? If his statement is untrue, he is telling the truth, so he is a liar. If it is true, then he is a liar, so we cannot believe him and he not a liar. We have a paradox. Likewise with regard to sets, claimed Russell; define a set such that it contains all sets that do not include themselves as one of the members of the set. Does such a set include itself as one of its elements? If not, it is itself a member of the set and therefore includes itself. If it includes itself, then by the definition of the set it is not one of its elements. A paradox.

Russell's paradox could have been solved in the same way as the Greeks solved the paradox of the liar and similar paradoxes. The solution is to determine that a statement in natural language relating to itself is not acceptable, and it is not legitimate to analyze it by mathematical means. This rule can also be adopted with regard to sets. Just as the set of all sets was excluded from being a set that can be analyzed mathematically, it can be decided that a set whose definition relates to itself is not a “legitimate” set, and the set that Russell used in the paradox was such a set. However, Russell's paradox raised a more fundamental problem, one that even the Greeks were unaware of. We will examine again one of the basic laws of inference, the law of excluded middle.

For every claim P, either P holds or P does not hold.

As this rule of inference relates to itself, among other things, it is not acceptable! The removal of this rule of inference from the area of mathematics is too great a blow for the mathematics community to bear. One reason is that the method of proof via contradiction is based on this rule. Abolishing all the claims that were based on this system means going right back to square one, including, for example, placing a question mark by the discovery in Pythagoras's time that the number √2 is not rational (see section 7) and also doubting a large part of mathematics developed since then. It was clear that this proposed method of solving the paradox would not work, so there was an urgent need to get to the source of the problem and to reexamine the foundations of mathematics.

The attempts to reconstruct the foundations of mathematics focused on three main avenues.

The first was proposed by Bertrand Russell himself, together with his colleague the well-known English mathematician and philosopher Alfred North Whitehead (1861–1947). They realized that elimination of sets or logical claims would not yield the desired results. Instead, they decided to define what is a permitted system of logic, and they started with a very delicate classification of permitted logical claims, constructing the permitted structure of logic “from below.” They called the permitted bases “types,” and constructed a theory of types from which all mathematics could be developed. Whitehead and Russell began to write their theory and published the first parts in a weighty tome, from which it would eventually be possible to develop mathematics. Its title was Principia Mathematica. Their monumental project was never completed, because the approach was too complex. For example, the proof of the equality 1 + 1 = 2 did not appear before page 362. Clearly such a system could not play a future role in a vibrant mathematics.

A second approach, called intuitionism, was developed by a group of mathematicians led by the Dutchman Luitzen Brouwer (1881–1966). The mathematics that this approach allowed was limited to concrete constructive operations. For example, if you wish to show that a geometric body that has certain properties exists, you must point to it directly. Inferring that the body exists from indirect evidence is not acceptable as proof. In particular, according to this approach the method of proof by contradiction is not acceptable. Brouwer and his colleagues managed to reconstruct a large part of mathematics in accordance with their approach, but the awkwardness in mathematical practice deriving from that method, together with the need to relinquish many results in the existing mathematics, led to the non-adoption of the method by the mathematics community. Hilbert himself was strongly opposed to intuitionism, repeating and emphasizing that proof by contradiction is at the core of mathematics.

The third avenue pursued was the one accepted by mathematicians as a whole. The idea of basing the structure on sets remained, but like Whitehead and Russell's approach of constructing logic from the foundations, instead of declaring which sets are not permissible and risking future encounters with other paradoxes, here the construction is “from the very core.” We start with permitted sets and, via specific building axioms, we show which sets can be formed from those already in existence. Only sets that can be constructed via the axioms are “legitimate” from the aspect of mathematical analysis. It was Ernst Zermelo, of whom we wrote with regard to his contribution to game theory, who presented these axioms, and they were completed later by Abraham Halevi Fraenkel.

Zermelo was a German mathematician who studied in Berlin, worked for some years in Zurich, returned to Germany, to the University of Freiburg, but resigned in 1936 in protest against the Nazi regime's treatment of the Jews. After World War II he was reinstated to his honorary professorship in Freiburg. Abraham Halevi Fraenkel was also born in Germany, where he published his work on the foundations of set theory and reached the position of professor. He was an active Zionist, and in 1929 he immigrated to pre-State Israel (then Palestine), joined the Hebrew University of Jerusalem, and worked there for the rest of his life.

The system of axioms developed by Zermelo and completed by Fraenkel were named the Zermelo-Fraenkel axioms. The axioms themselves are too technical to be of interest to the general public, and we will not present them here, but once they were put forward and tried on a range of problems, it seemed that hope had been reawakened that mathematics could be based on set theory. In addition to the set-theory axioms, other specific systems of axioms were also being examined, for example, the axioms of the natural numbers developed by the Italian mathematician Giuseppe Peano (1858–1932). Those axioms are quite simple and incorporate some self-evident statements, such as: the rule that the number 1 exists, the rule stating every natural number is followed by a number that is larger than it by 1, the rules of how to add and multiply and how to use induction, which is actually an independent axiom. The system is simple, and its purpose is to show that mathematics can be based on simple axioms, and those axioms can also be translated into terms related to sets. We mentioned above that Euclid's geometric axioms also underwent a reexamination that was led by Hilbert himself. It seemed that the new versions are free from lack of clarity and the errors of the mathematics of Euclid and his followers.

Alongside the efforts to improve the axiom systems, emphasis was also placed on understanding the properties that a system of axioms should have in order to make it acceptable. As mentioned, for the Greeks, axioms reflected absolute unassailable truths. The more modern approach permitted alternative systems of axioms and even systems that contradict each other. It was therefore important for all of that to clarify what is expected of a system of axioms itself for it to be accepted as reliable. Here are two basic requirements:

Consistency: Mathematical inferences from the use of axioms should not result in contradictions; in other words, mathematical conclusions deriving from the axioms should not contradict each other.

Completeness: Every mathematical claim about the system that the axioms describe can be proven or disproven by using the axioms themselves.

The consistency requirement is self-evident. In many day-to-day events the human reaction to an encounter with logical contradictions is not one of great upheaval because we are constantly exposed to situations that we do not examine in depth or even to apparent contradictions that we are prepared to accept as part of our daily lives. Mathematics, however, cannot allow itself internal contradictions. That is, a mathematical system cannot allow a conclusion that is both true and untrue.

The completeness requirement is more intricate. The basic idea is that when we formulate a hypothesis, let us say about numbers, the system of axioms that describes the numbers must be rich enough for us to be able to decide whether the hypothesis is or is not correct within the system of axioms. If that is not so, there could be, say, number systems that contradict each other that satisfy the system of axioms and we will not know which is correct and which is not. This does not mean that a system that is incomplete is useless. The completeness property, however, ensures that in principle we can conclude whether a claim is correct without having to resort to additional axioms.

When the Zermelo-Fraenkel system of axioms was formulated and the first successful steps were taken to corroborate its consistency and completeness, the mathematics community was spiritually uplifted. The axioms seemed reasonable and were formulated meticulously. Although their consistency and completeness have not been completely proven, the intuition and care with which the axioms were constructed appears flawless, and the first steps to establish their consistency and completeness are promising. Hilbert himself published the grand program: a full formulation of the system of axioms for mathematics that would be consistent and complete. He declared in a lecture in 1930 on the occasion of his retirement

We have to know, and we will know.

And that is what is inscribed, in German, on his tombstone.