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

CHAPTER VIII. IS THERE REALLY NO DOUBT?

58. RIGOROUS DEVELOPMENT WITHOUT GEOMETRY

It was not until the nineteenth century that the mathematical community began to become involved in a reexamination of the logical basis of mathematics and the degree of rigor in its development. The words not until are used advisedly, as for thousands of years from the development of mathematics by the Greeks mathematicians relied on their intuition and were certain that it was consistent with the system of axioms determined by Euclid. This does not mean that the status of the axioms was totally ignored. There were sporadic attempts to check the compatibility of the axioms or to replace some of them with others, but these attempts focused on specific subjects, such as the discussion of the parallel-lines postulate discussed in section 27. Mathematicians in general did not consider it necessary to perform an overall examination of the foundations of mathematics.

Two interrelated factors brought about the realization that the concepts that mathematics had used for a long time and the axioms underlying mathematics should be reexamined. One was the growing number of definitions that were not sufficiently clear, definitions that even caused errors that were discovered in the use of infinitesimal calculus. Newton's formulation, specifically the use of fluxion as the basis for the description of the change of direction of a function, may have been appropriate for his own intuition, but it was thwarted many others. Leibniz's formulation, the division of infinitesimal quantities of the form was convenient to use, but it did not relate to the essence of the infinitesimal quantities, and mathematicians developed intuitions about them that often resulted in errors. Both approaches, that of Newton and that of Leibniz, and in their wake all the following developments of the theory, were based on geometry.

Then the second factor appeared, highlighting the need for renewed study of basics. The logical foundations of geometry throughout the generations were considered stable and thus not in need of examination and checking. And then, as described in section 27, questions and doubts arose at the beginning of the nineteenth century regarding the validity of the axioms that constituted the basis of geometry. The questions asked related to the absolute correctness of axioms that describe the geometry of nature, as well as to their logical completeness. For example, Euclid relates to the situation in which two points lie one on each side of a straight line. When we encounter such a description, a clear picture immediately comes into our minds of a straight line with a point on either side. But what are the two sides of the straight line? Their existence does not derive from the axioms. Imagine a long tube, and imagine a straight line drawn along its length. Does that straight line actually have two sides? Such doubts led mathematicians, with the leading French mathematician Augustine-Louis Cauchy (1789–1857) at their head, to redevelop differential and integral calculus, but this time based on the system of numbers and not on geometry.

We will not delve here into the details of the developments but will just illustrate one concept. As stated, for Newton, Leibniz, and their followers the derivative was defined as the slope of the tangent of a function at a point. A tangent and its slope are clearly defined in geometry. To define them using numbers alone, Cauchy used as a basis a precise formulation of the concept of the limit of a sequence of numbers. Archimedes had already related explicitly to the concept of a limit, but without defining it. Cauchy defined a limit as follows:

The number z is the limit of a sequence of numbers x_n if for every number ε greater than zero there is an index m such that for every index n greater than m the distance between x_n and z is smaller than ε.

Does that sound complicated? Indeed, the definition is a complicated one. We have already made the point that if a formulation has many quantifiers, and here we have at least three and the order in which they appear is also important, we cannot grasp it intuitively. If there are among our readers some who completed a course in differential calculus, they may have encountered this definition at that stage, but they will surely also remember the difficulties, or even nightmares, that this and similar definitions caused them and their colleagues.

When the concept of a limit is clear, the concept of the derivative of a function f(x) at point x₀ can be defined as follows:

The derivative is the limit of the numbers

for every sequence of numbers h_n that are different from 0 whose limit is 0.

Does that sound complicated? Indeed. Yet note, the definition is based on only numbers and is independent of geometry. The motivation for the definition, namely, the slope of the tangent, is geometric, but the definition itself does not use geometry.

The point should be made, and it will be repeated in connection with other developments, that the purpose of this rigorous development was not to give a better understanding of the concepts. We will go further and say that for a better understanding of the concepts they should be illustrated by a geometric drawing. The incentive behind the development was similar to that driving the Greeks, an attempt to prevent errors deriving from geometric illusions.

Basing infinitesimal calculus on numbers avoided the need to rely directly on geometric axioms, but it did not avoid indirect dependence on geometry, because the definition of the numbers was itself geometric. An example we have quoted previously is the definition of irrational numbers such as √2, defined as the length of the diagonal of a square with sides of length 1. With this realization, attempts began to provide a non-geometric base for irrational numbers themselves. Two of Germany's leading mathematicians at that time, Karl Weierstrass, mentioned in the previous section, and Bernhard Bolzano (1781–1848) based the concept of an irrational number on the limits of numbers. Thus, √2 would be defined as the limit of positive rational numbers, say r_n, that satisfy the requirement that the series (r_n)² itself has a limit, and that is the whole number 2. Later, the German mathematician Richard Dedekind (1831–1916) proposed a different definition of irrational numbers. His definition uses what is named after him, that is, Dedekind cuts, and is the definition that is taught still today in mathematics classes in universities.

The number √2, for example, is defined as the pair of sets of rational numbers, say (R₁, R₂), where R₁ is the set of rational numbers in which each one is smaller than a rational number the square of which is smaller than 2, and R₂ is the set of rational numbers in which each one is larger than a rational positive number the square of which is greater than 2. Other irrational numbers are similarly defined as pairs of sets of rational numbers.

To the reader who has not personally experienced this type of definition in the past, this will seem strange. A single number, whose meaning is clear and has been clear for thousands of years, is now defined as a pair of sets of rational numbers. Yet this is the price to be paid for the aspiration to avoid geometry. The definition of irrational numbers by means of Dedekind cuts was not intended to make it any easier to understand what an irrational number is. No one thinks he is clarifying what an irrational number is by presenting it as a pair of sets of numbers. The geometric definition is in fact simpler and more comprehensible. The reason for this development was to avoid using geometric language, even if by so doing it greatly complicated the concepts.

Defining irrational numbers without resorting to geometry did not entirely remove geometry from the picture, as rational numbers, sets of which are used to define irrational numbers, are also defined geometrically, based on the plane axioms. Again, the need arose to define the rational numbers without geometry. I will now present a development that I learned in my first class in university. It will be relevant to the last part of this book, but the details can be omitted without losing the main message.

Let us agree that we know what the natural numbers are, that is, 1, 2, 3, and so on, and we also know how to add and multiply the natural numbers. We now define the positive rational numbers.

First, we look at pairs of natural numbers (a, b) (the explanation, which helps us understand but which may not be used in the definition, is that (a, b) is the rational number ). We define equivalent pairs thus: the pair (a, b) is equivalent to the pair (c, d) if ad = bc (according to our explanation, the equivalence does ensure the equality = , in other words, that it is the same rational number). Once we have understood what equivalence means, we can define a positive rational number as follows: a positive rational number is a collection of pairs, with the numbers in each pair equivalent to each other. In addition, addition and multiplication of the rational numbers must also be defined. We define the addition of (a, b) and (c, d) as the collection of pairs equivalent to (ad + cb, bd), and their product as the collection of pairs equivalent to (ac, bd) (we suggest that the reader check the operations in light of the interpretation).

These definitions reflect what we understand by the term rational numbers, and they are entirely free of dependence on geometry. It should be repeated and stressed that it is difficult to understand the definitions without relating to the intuitive grasp of rational numbers, and the only reason for this somewhat strange presentation of the definitions of quantities that we all essentially understand, is the desire to avoid reliance on geometry. Thus, as we have seen, irrational numbers, and hence the real number line, can be defined without resort to geometry.

Note that these definitions, and others that we have not included here, use the natural numbers and also the concept of a set. The sets were used in the definition of rational numbers, via equivalence of sets, and also in the definition of irrational numbers, via Dedekind cuts.