NUMBERS AS SETS, LOGIC AS SETS - IS THERE REALLY NO DOUBT - The Remarkable Role of Evolution in the Making of Mathematics - Mathematics and the Real World

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

CHAPTER VIII. IS THERE REALLY NO DOUBT?

59. NUMBERS AS SETS, LOGIC AS SETS

The reliance on sets in the development of the foundations of mathematics, and in particular the redefinition of numbers on the straight line, the definition of limits and hence calculus, was in general accepted and greatly welcomed by the mathematics community. With regard to applications, it can be shown that all results previously proved geometrically were correct also on the basis of set theory. For example, the equality √2√3 = √6 can be proven by geometry but can be proven more simply using Dedekind cuts (to do this we must define and understand the product of two cuts; we will spare the reader that step). It was even more satisfying to view this from the aspect of the foundations of mathematics. Indeed, as George Boole showed, there is full parallelism between sets and the natural operations that can be performed on them on the one hand, and logical arguments on the other (Boole developed this believing that was the way to analyze probabilistic events, as mentioned above in section 41). Some examples follow.

The parallelism between sets and logical propositions can be seen when we consider a claim to mean the set of all the possibilities of satisfying it. For example, we will take the claim “it is raining” to mean the set of all the situations in which it is raining, and the claim that “the sky is blue” will refer to all the situations in which the sky is blue.

We will examine the relation between two claims, say P and Q, and compare it to the relation between two sets, say A and B. The statement “either P holds (i.e., is true) or Q holds” is equivalent to taking the union of the two sets, that is, all the elements contained in A or B. Likewise, “it is raining or the sky is blue” is equivalent to the union of the sets of situations in which it is raining and those when the sky is blue. The proposition “both P and Q hold” is equivalent to the intersection of the two sets, that is, the elements contained both in A and in B. The intersection is denoted A∩B. The claim “P does not hold” is equivalent to taking the complement of the set A, in other words, all the elements not in A. In this way, every logical argument can be stated in terms of sets. Thus, the statement “P and Q cannot both hold” can be translated into the statement that A∩B is an empty set. Such sets are also known as disjoint sets. The proposition “it is not possible that is raining and the sky is blue” can be translated into the language of sets thus: “the intersection of the ‘rain’ set of situations and the ‘blue sky’ set of situations is an empty set,” that is, they are disjoint sets. Numbers can also be based on sets, by counting the elements in the sets (this will be described shortly).

This enables us to base the whole of mathematics on sets and operations on them, including numbers and all the definitions and conclusions drawn from them, and the resulting mathematical theorems. The following is the way to establish the natural numbers from sets. The formal description of the method will be accompanied by an intuitive explanation, but we emphasize that the explanation is not part of the actual process.

We begin with the assumption that an empty set, that is, a set that has no members, exists. We select a mathematical symbol for the empty set, and the symbol normally used is Ø. The explanation is that the set Ø corresponds to the number 0. The set that corresponds to the number 1 is the set that includes the empty set, and only the empty set. We denote this set {Ø}. It is customary to write the elements of the set between the braces. The reason for considering the new set as corresponding to the number 1 is that it contains only one element, and that is the empty set. The next set will be {Ø,{Ø}}, which has two elements, the empty set, and the set that includes the empty set. This set corresponds to the number 2. The set corresponding to the number 3 is {Ø,{Ø}, {Ø,{Ø}}}, and so forth. This formulation is much more complex than simply saying 1, 2, 3, and so forth.

The sole advantage of this type of construction is that it does not use numbers at all, it is formulated entirely in terms of sets. Hence, from here we can take a further step forward and identify other sets that have a finite number of elements equivalent to those we have already constructed by matching the number of elements in those sets to those in our sets, where matching reflects counting. The next step is to define the addition of numbers by means of sets, and that too can be done by combining two sets (the union of two sets) that have no elements in common. That is how small children would act. They calculate three plus four by counting the elements in the union of a set with three members with a set with four members. The same method is used for multiplication and other operations.

We will repeat what we have already mentioned several times. These structures were developed purely and simply to show that mathematics can be based on sets and operations on sets, thus establishing a geometry-free logical basis for mathematics. Nobody thought that this would yield a better intuitive understanding. Some mathematicians opposed this type of development on principle. Leopold Kronecker (1823–1891), for example, is quoted as saying, “God made natural numbers; all else is the work of man.” In other words, there is no need to justify the existence of the natural numbers. Poincaré also saw no need for such structures. Most mathematicians of the period, however, accepted these developments enthusiastically.

The reliance on sets resulted in renewed interest in the concept of infinity. The sets used in defining the natural numbers include a finite number of members, but the sets required to form more complex foundations, such as irrational numbers, contain an infinite number of elements. The question arises naturally, is it possible to implement the arithmetic operations that are translated into logical claims on sets with infinite numbers of elements? As noted previously, evolution did not equip the human brain with the tools to develop intuition about the concept of infinity. Throughout the thousands of years of the development of mathematics in the Babylonian and Assyrian Empires and in Egypt, the concept of infinity was not considered. The term infinity itself was mentioned in the context of very large quantities or large numbers that could not be counted, meaning that it was difficult or impossible to count to such high numbers. Reference to God was also sometimes a reference to the infinite, meaning that God's wisdom and strength were so great that they could not be described. The Greeks were the first to relate to the mathematical infinity, for instance, counting that goes on and on, or lines that did not end. Underlying the interest in the infinite were the questions of whether the world had always existed and if it would exist forever. The Greek's solution to the inability to analyze infinite sets was to make a distinction between potential infinity and an actual infinite collection, a distinction based on Aristotle's methodology. They simply did not think of an infinite collection as a legitimate mathematical entity eligible for consideration. Potential, not actual, infinity—for example, constantly increasing collections of numbers, or finite lines that keep lengthening, or a world that will exist for a constantly increasing length of time—the Greeks considered to be collections of finite sets.

This doctrine of the Greeks persisted, seemingly, but as most mathematical developments were not based on axioms, mathematicians did not hesitate to use the concept of infinity even in the non-potential sense. For example, they referred to infinite lines in researching plane geometry, and thereby even reformulated the axioms, ignoring the distinction between potential and “ordinary” infinity. For many generations there was no re-discussion of the concept of infinity itself, apart from a contribution by Galileo. He noted that although there are “more” natural numbers than squares of natural numbers, there was a one-to-one relation between the natural numbers and their squares, thus:

1, 2, 3, 4,…

1, 4, 9, 16,…

This was also the correspondence that Galileo found between time and the distance that a body falls, as we have seen, and apparently that was the research that brought him to consider infinity. However, this discovery of Galileo's did not go beyond the statement that infinity has strange properties, a saying that was not accompanied by further study. Now, with the increasing dependence on infinite sets as a basis of mathematics, the time had come to explore that strangeness. That step was taken by Cantor.

Georg Cantor was born in St. Petersburg, Russia, in 1845 to a Christian family of merchants and musicians, a family apparently with Jewish roots. When Georg was eleven, the family moved to Germany, where he excelled in his studies. After studying at the University of Zurich, he returned to Germany and completed his doctorate at the University of Berlin. He studied under Leopold Kronecker and Karl Weierstrass. These two were bitter opponents, and their rivalry also held implications for Cantor himself. On completing his studies he hoped to obtain a post in Berlin or in another major city in Germany, but his path was blocked, apparently by Kronecker. Cantor settled for a post in the less-prestigious University of Halle, Germany, about a hundred miles (160 kilometers) from Berlin, and there he developed the mathematics of infinity. Kronecker was strongly opposed to this new mathematics and, among other things, also blocked Cantor's attempts to publish his papers in professional journals. These rejections, both regarding the posts he had hoped to be appointed to and of his findings, had a lasting effect on the young Cantor and apparently contributed to his mental crises. Cantor spent a large part of his time in the sanatorium in Halle, where he died in 1918. Nevertheless, he saw his theory accepted by the mathematics community, even with the logical difficulties and paradoxes it brought to the foundations of mathematics.

We will now give a short description of Cantor's theory. Its starting point is the same analysis of Galileo's that we spoke of above. Cantor suggested that we agree to state that both sets have “the same number” of elements if the elements of one can be matched one-to-one (bijection) with the elements of the second. Thus the set of natural numbers and the set of their squares have the same number of elements. Similarly, he showed that the set of rational numbers and the set of natural numbers have the same number of elements, although the rational numbers are spread tightly all along the real number line, while the natural numbers have empty spaces between them.

The next question was, do all infinite sets have the same number of elements? Here Cantor made a surprising discovery. He proved that the set of real numbers and the set of rational numbers do not have the same number of elements. Every rational number can be represented by a point on the straight line, but there is no such match for all the real numbers. From that he derived that the latter set had “more members.” He denoted the “number” of members in the set of natural numbers as aleph-null, denoted images, and sets with a number of elements like the set of natural numbers he called countable infinite sets. Cantor called the indication of the size of the set its power, or cardinality. Other powers of sets greater than that of the natural numbers he denoted images, images and so on. It is not clear why Cantor chose to use the Hebrew letter aleph for this mathematical notation, and some relate that choice to his family's possible Jewish roots. Others note that as the Bible, including Hebrew writing, was studied by religious Christian groups in Germany, Cantor was familiar with Hebrew, and that was why he chose the first letter in the Hebrew alphabet, aleph. He denoted the power of the set of the real numbers C, the first letter of the Latin word continuum. Cantor went on to prove that the power of C was the same as that of the set of subsets of the natural numbers. Therefore, just as the number 2n measures the number of subsets of the set with n elements (we recommend the reader to check this), Cantor denoted the power of C by 2images.

Cantor also developed the arithmetic of powers. For example, it can be seen that the equality images + images = images holds when the sum is defined in the same way as we defined the sum of numbers by means of sets. That is, the sum of powers is the power of the union of disjoint sets that have the corresponding powers. And indeed, both even numbers and odd numbers are of power images, and that is also the power of their union, that is, all the natural numbers. Cantor proved generally that the collection of subsets of a non-empty set is of a greater power than that of the set itself, thus ensuring that, as with the numbers themselves, the possible powers of sets can increase boundlessly.

During the development of this elegant theory, some troubling questions arose. For example, is there a power greater than that of the natural numbers and smaller than that of the continuum? In mathematical notation the question is, does C = images? This question greatly perturbed Cantor himself and generations of mathematician who came after him until the answer was found in 1964, as we will describe below. Another question was, what happens to the set of all sets? It was always clear that there was no natural number that was the largest of all numbers. With any number, 1 can be added to it, and a larger number will be obtained. When examining sets, however, the question can be asked, what is the power of the set whose elements are all the sets in the world? Such a property would ensure that the power of that set is the greatest possible. On the other hand, Cantor proved that for every set that is not empty, and hence also the set of all sets, the power of the collection of its subsets is greater than the power of the set itself. We seem to have arrived at a contradiction. It was solved by the decision that not all sets are “legitimate” from the aspect of the new mathematics. Thus, the set of all sets, although we have called it a set, is not a set in the sense that we can apply the new mathematics to it, just as the arithmetic of the natural numbers cannot be applied to infinity.

That was the situation toward the end of the nineteenth century and at the beginning of the twentieth. Natural numbers were defined by sets. Then, as we have seen, the positive rational numbers could be defined, followed by the negative rational numbers (we spared the reader the detailed description of that stage). Next, the irrational numbers could be defined by means of Dedekind cuts, which were themselves defined as sets of power images and were thus permitted infinite sets. From there it was possible to continue to derivatives, integrals, and the rest of mathematics. Operations in logic, and in particular the laws of inference, could also be explained in terms of sets. At that time it seemed that the solid foundations of mathematics had been found that would replace the shaky foundations set by the Greeks.

In this context another important new development took place in understanding the concept of an axiom. For the Greeks, axioms expressed agreed properties of entities in nature. As such, the axioms related to familiar concepts, such as points and straight lines, which the mathematicians defined rigorously. The problem is that the definition itself uses concepts that have not themselves been defined. Here the question reappears: To what extent are the defined entities self-explanatory? The answer, on which there was a consensus in the nineteenth century, was that the axioms relate also to abstract entities, which do not need to pertain to anything recognized and defined but just need to be denoted, say, by letters, x, y, A, B, and so on. When we want to apply abstract mathematics, we must ascribe to undefined quantities a reference to known entities. If the explanation conforms to the axioms, the mathematics to be developed in line with the axioms will indeed describe reality. Unlike with the Greeks, however, in the mathematics of the nineteenth century, the elements dealt with by the axioms could have nothing to do with nature or with other uses. Speaking of elements that at the outset have no explanation, the English mathematician-philosopher Bertrand Russell (1872–1970) described a mathematician as a person who does not know what he is talking about, nor does he care whether what he is saying is true. Not knowing what we are talking about refers to entities that mathematics deals with that from the outset have no explanation or use. Not knowing whether what we are saying is true refers to truth for a particular purpose, for instance, in nature. In other words, the mathematician can engage in mathematics without any explanation and without being at all interested in an explanation of the entities he is analyzing. Despite Russell's humoristic note, and despite the agreement that axioms relate to abstract entities that are not represented in nature, I do not know mathematicians who can discuss and analyze what can be derived from the axioms without having in mind some sort of representation of a system that fulfills the axioms, except in limited and extreme cases. As we have seen, the human brain is incapable of relating intuitively to logical systems that are completely abstract.

The attitude of the leading mathematicians of the time, at the turn of the century (nineteenth to twentieth), to the basing of mathematics on set theory is interesting. Of particular interest is the reaction of the most famous mathematicians of that period, the German David Hilbert, and the Frenchman Henri Poincaré.

Hilbert (1862–1943) was born in and studied in Königsberg, Prussia (today the Russian city Kaliningrad), then moved to Göttingen, where he remained until the end of his life. In his lifetime he saw regimes in Europe change several times, and he died while the Nazis were in power. He was not one of their supporters, and after 1933 he tried to help persecuted Jewish mathematicians and physicists, even though at that time he was no longer at his prime. A leading figure in the Nazi regime turned to Hilbert in the course of an official dinner and said, “Herr Hilbert, at last we have rid ourselves of the Jews who contaminated German mathematics.” Hilbert's reply was, “Yes, sir, but since the Jews have left, mathematics has ceased to exist in Germany.” He made many contributions to mathematics in various spheres. He was responsible for fundamental developments, in particular the abstraction of concepts and methods, and was interested in logic and the foundations of mathematics. He made a major impact on mathematics worldwide. When he was invited to deliver the keynote lecture at the Second International Congress of Mathematicians in 1900, instead of presenting his own achievements, as usually happened at such congresses, Hilbert chose to present a list of unsolved problems in mathematics, which he predicted would become the central mathematical problems of the twentieth century. Those problems did indeed feature in research in mathematics throughout the twentieth century; some were solved relatively quickly, and others are as yet unsolved and await solutions in the twenty-first century.

Henri Poincaré, who featured in our discussion of the events that led to the development of the theory of relativity, actually came from the field of engineering. He studied in the École des Mines (a mining, or engineering, school), which was and remains a highly prestigious school in France. His abilities were discovered when he was very young, and he was elected to the French Academy, taught at the Sorbonne, and was possibly the most influential French mathematician of his time. One of his nonacademic activities was engaging in the defense of Alfred Dreyfus at the appeal stage of the famous trial. He and his colleagues, the mathematicians Paul Appel and Jean Gaston Darboux, examined the evidence and declared in a written report submitted to the court that probability theory showed that the charges did not hold up under serious scientific examination. In his private life Poincaré exhibited great courage in other situations too. In mathematics he was active in mathematical physics and dynamics. His career surged when in the context of a competition announced by the king of Sweden he promoted the understanding of the three-body problem, that is, the dynamics of three or more bodies in space, for instance, the Sun and the planets. In his research he discovered and characterized the dynamic behavior that results in what is studied today in the context of chaos theory.

Hilbert enthusiastically adopted the new developments in understanding the axioms and their connection to logic. He himself drew up a series of geometry axioms, which refined Euclid's axioms, and succeeded in showing that they do not depend on unreliable intuition, nor do they contain internal contradictions. Poincaré also adopted the dependence on logic with enthusiasm and actually said that logic is the material that disinfects mathematics by inhibiting errors. Nevertheless, with regard to the use of sets as the basis of mathematics, opinions were divided. Hilbert declared that set theory is the crowning achievement of man's creativity. Poincaré is reputed to have declared that set theory is a sickness that mathematics will be cured of. We will see the effects of this dispute on the status of set theory when we discuss matters related to the teaching of mathematics, in the last chapter of the book.