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

CHAPTER VIII. IS THERE REALLY NO DOUBT?

61. ANOTHER MAJOR CRISIS

The “evil” came from Austria. Kurt Gödel was born in Brünn (now Brno), then a Czech town in the Austro-Hungarian Empire. When that empire broke up, Gödel automatically became a Czech citizen, but he considered himself Austrian and studied at the University of Vienna. There he completed his doctorate studies and two years later, in 1931, published his famous incompleteness theorem, which we will describe below. While in Vienna, when the Nazis came to power in Germany, he was severely affected by the brutality and anti-Semitism of Nazi groups in Austria and by the murder of Moritz Schlick on the steps of the university. Schlick was a member of the university's academic staff and of the logical-philosophical Vienna Circle, of which Gödel was also a member. Although Gödel was not Jewish, he became paranoiac and remained so all his life. In the 1930s Gödel received several invitations to go to the United States, and he stayed at the Institute for Advanced Study in Princeton, alongside Einstein, who became a close friend. Gödel's yearnings for Vienna grew stronger, and despite his fears he returned to Austria. He was in Vienna in the beginning of the war as a German citizen, citizenship he had received with the German annexation of Austria. But the pressure grew and in 1940 he moved back to the United States, accepting an invitation to become a permanent member of the Institute for Advanced Study in Princeton, where he remained until his death in 1978.

Gödel's interest in the foundations of mathematics and the link between logic and the developing set theory began in the time of his doctoral studies. His doctoral thesis presented a result that merged well with the program that Hilbert envisioned. Gödel proved that with a consistent and finite list of axioms, if every system that fulfilled the axioms had a certain property, it could be proven from the axioms themselves. This was a promising step forward. What was now needed to complete Hilbert's program was “only” to prove that the Zermelo-Fraenkel system was consistent and that every property of a system that satisfies the axioms exists in every system that satisfies the axioms. Two years later, however, Gödel himself presented a result that completely negated Hilbert's program, and that was Gödel's incompleteness theorem.

The theorem showed that in every finite system of axioms (or even infinite system, if it was created by algorithmic computation) that is rich enough to include the natural numbers, there will always be theorems that cannot be proven or disproven. In other words, such a system cannot be complete!

This result was a slap in the face for all those, like Hilbert, who believed that there is nothing in mathematics that we cannot know; in other words, those who believed that the logical approach based on axioms that can be stated clearly will always result in the resolution of the question whether a particular mathematical claim is correct or not. Gödel's result also relates to systems such as the system of natural numbers constructed by Peano, the system of types developed by Whitehead and Russell, and the system of axioms of Zermelo and Fraenkel. With regard to such systems of axioms he showed that it is impossible to prove that the system does not have contradictions by using the axioms themselves. That is, if we rely only on the axioms of the system, then either one day we will discover a contradiction, or we will never know whether the system contains a contradiction. (It may be possible to prove consistency by using a broader theory, but such a theory has not yet been discovered.) The inconsistency theorem did not affect the intuitionist approach, but neither did it breathe renewed life into it. There is a strong desire to operate in an environment without potential contradictions, but not at the price of significantly limiting the range within which mathematics can develop.

Beyond the philosophical aspect and the questions relating to the foundations of mathematics, Gödel's result had a direct effect on the practice of research in mathematics. Throughout the development of mathematics, when a mathematician set out to prove a theorem, he faced two possibilities. One was that the theorem was correct, and he had to find a proof. The other was the theorem was not correct, and he had to disprove it, either by bringing a counterexample or by finding a contradiction between the theorem and other results. The incompleteness theorem gives rise to a third possibility, that the theorem cannot be proven or disproven.

Let us take as an example Fermat's last theorem, which states that for any four natural numbers X, Y, Z and n, if n is greater than 2, the sum Xⁿ + Yⁿ cannot equal Zⁿ. For more than three hundred years, mathematicians tried to prove or disprove the theorem. With the publication of Gödel's incompleteness theorem a third possibility was added: Fermat's theorem could be one of those that cannot be proven or disproven. In 1995 Andrew Wiles published a proof of the theorem, and the cloud of incompleteness around Fermat's theorem was dispelled.

Goldbach's conjecture, formulated by Christian Goldbach in 1742, is also simple: Every even number is the sum of two prime numbers. Despite the simplicity of the formulation, still today no one has proven or disproven the conjecture, and here too nobody knows whether it is one of the theorems that Gödel's theorem applies to.

Now for a third hypothesis, Georg Cantor's continuum hypothesis, that is, the question referred to in section 59: Does C = ? Cantor invested much effort in trying to prove or disprove this claim, using the naive approach to set theory, the approach in which Russell found the contradiction. Following the formulation of the Zermelo-Fraenkel axioms, the question became: Is the equality C = correct or incorrect in the context of those axioms?

Another axiom also played a major role in the attempts to answer the question, and that is the axiom of choice, a highly intuitive claim when referring to a world about which human beings developed intuition. It says that given a collection of non-empty sets, a new set can be formed by selecting one member from each set in the collection. Such a selection is simple if the collection of sets is finite, but we have already seen that the concept of infinity can be deceptive. The axiom of choice was indeed recognized as an axiom, and the question of what would happen if it was added to the Zermelo-Fraenkel axioms remained unanswered.

Gödel himself contributed to the answer to this question and proved that if the Zermelo-Fraenkel system is consistent, that is, it does not contain contradictions (which has not been proven), adding the axiom of choice to the system will leave it consistent. Then Gödel added a slightly confusing finding: if the Zermelo-Fraenkel system is consistent, even if we add the axiom of choice to it, it will be impossible to prove that C = . Prior to the incompleteness theorem, this result would have ended the search for the truth: if it is impossible to prove the correctness of a particular claim, and if everything can be proven or disproven, then the claim is incorrect. In the world after the incompleteness theorem, however, there is another possibility, that the theorem is one of those the truth of which is impossible to resolve.

In 1964 Paul Cohen (1934–2007), a young mathematician from Stanford University, showed that the continuum hypothesis was in that category. Moreover, he showed that if the Zermelo-Fraenkel system is consistent, the claim C = or its negation can be added to the system without affecting its consistency. The question of whether the Zermelo-Fraenkel system is consistent, meaning, does not contain contradictions, is still unanswered today.

Research along these avenues, that is, the attempts to find a system of logic without contradictions on which mathematics can be based without doubts or uncertainties, continues. On the one hand, research into Gödel's theorem shows that the phenomenon is very general. For instance, Gödel's proof of the incompleteness theorem was based on the liar's paradox, that is, on a claim that relates to itself. Since then proofs have been found that do not use claims that relate to themselves so that we cannot dispose of incompleteness merely by not allowing claims that relate to themselves. On the other hand, the various attempts to find ever-more complete systems sometimes yielded strange results. For example, as the axiom of choice seems intuitive, we can define other axioms, no less intuitive, that contradict the axiom of choice but whose status regarding the basic axioms of set theory is the same as that of the axiom of choice. That is to say, even if we add them to the Zermelo-Fraenkel system we will not find a contradiction (provided, of course, that the Zermelo-Fraenkel system is consistent).

So what is the correct mathematics? Does mathematics that is correct beyond all doubt exist? The answer is clear: we do not know. At this stage there is room for faith. Some believe that in the world, the physical world and the Platonic ideal world, there is a correct mathematics, but we just have not found it yet. In that mathematics, for example, Goldbach's conjecture is definitely correct or incorrect. Similarly, in that mathematics, either the equality C = is fulfilled or it is not. Others believe that there is no such absolute truth. Mathematics rests on the axioms that define it, and different systems of axioms yield different types of mathematics, possibly even contradictory ones, and we must learn to live with that dichotomy. But most mathematicians simply do not care. The many years of involvement in mathematics have resulted in faith that there is no basic contradiction in mathematics, and we can continue to work. If the logicians manage to arrive at absolute mathematical truth, whether via axioms or some other way, all the better. But even if not, even without any logical “proof beyond doubt,” the mathematics we produce is correct beyond any doubt.