Infinity in mathematics by Manuel Valdivia

In 2001 Jaume I University in Castelló de la Plana, Spain, celebrated the 10th anniversary of its founding. To mark the occasion they awarded 14 honorary doctorates, one of these being to Manuel Valdivia. After receiving his honorary degree, he delivered the address "Infinity in mathematics" a version of which we give below:


Very Honourable Sir.
Magnificent and most excellent Sirs.
Illustrious Sirs.
Ladies and Gentlemen.

In the first place I want to thank you for the great honour that has been bestowed upon me. Second, since my work falls within the Infinite Mathematics, I will present some ideas around the mathematical infinite.

Man has always felt a special attraction for the infinite, perhaps suggested by the hypothesis of being able to move away from a point indefinitely and in a straight line, or also, in relation to time, by the possible existence of an eternal time, which will never end.

In the most famous publication of ancient Greece, the 'Elements' of Euclid, which appeared in the third century BC and which contains the mathematical knowledge discovered up until then, almost all of it obtained by the Pythagoreans, the word infinite is not used. Thus, for example, the modern-language theorem states that the set of prime numbers is infinite, is expressed in Euclid's book by saying that given any prime number there is always a prime number greater than it. At the present time this conceptual aspect gives rise to different ways that infinity can be viewed. When it is said that the set of natural numbers is infinite, it can be understood in two ways: as an ability to obtain numbers as large as we want; so, if we take the number one million, we can assure that there is a greater number than that, for example, two million. The infinite conceived in this way is called potential infinite, and it is the one used by Euclid. The second form is that used by the German mathematician Cantor, at the end of the 19th century; whole numbers, for example, are considered as a set with real and objective existence, as something that is there. The infinite understood in this other way is called the actual infinite, and has been the source of many discussions.

Before Euclid, Aristotle also avoided the use of the word infinite. Some of the reasons that contributed to this were the famous paradoxes of Zeno of Elea, which I will comment on later.

Elea was a small city of Magna Grecia founded in the 6th century BC by Greeks who came from the city of Focea in Asia Minor on the Ionian coast. These Greeks, fleeing from the Persians, came to what is now southern Italy, and there they settled. In our days it is possible to visit the ruins of Elea in Italy, that are found down the coast from Sorrento, and that now have the name of ruins of Velia, that was the name that gave the Romans to it.

Elea, in spite of being a city of little importance, was the cradle of a group of philosophers that has had a remarkable influence throughout history. The main philosopher of the Eleatic school was Parmenides who had Zeno as a disciple. It is known that in the year 450 BC, Parmenides and Zeno visited Athens as part of a diplomatic commission, promoted by their elected representatives, in order to persuade Pericles to sign a pact of alliance between the two cities. During this visit Parmenides and Zeno were able to meet with Athenian philosophers, especially with Socrates who was then twenty-five years old.

Zeno, with his paradoxes, aimed for two things. On the one hand, the defence of the philosophical ideas of Parmenides and, on the other, to make a severe criticism of certain concepts professed by Pythagoras in the previous century.

It is known that Pythagoras gave great importance to numbers, especially after having discovered the existence of a constant ratio between the lengths of the strings of a lyre and the fundamental chords of music. But Pythagoras went much further by asserting that all things were numbers. This, taken literally, makes no sense, but can be reasonably interpreted. Bertrand Russell thinks that Pythagoras considered the world as atomic, and bodies formed by molecules composed of atoms, with diverse structures. These atoms are the Pythagorean units, which can be assumed to be small spheres in the explanation that follows. Let us now imagine that we want to construct triangular figures with the Pythagorean units. We can consider a unit as a triangular figure; to form the next triangular figure three units are required to be in contact two by two; the next triangular figure has six units; and so on. Then there are numbers: 1, 3, 6, 10, etc., which correspond to the number of units needed to form the said figures. For this reason, these numbers are called triangular. If you want to form quadrangular figures, you can consider that a Pythagorean unit is a square; the next square is formed with four units; the next, with nine, etc. Thus appear the numbers 1, 4, 9, 16, etc., which are called quadrangular. By this method various geometrical figures can be obtained and, associated with them, certain numbers.

Since bodies are formed by Pythagorean units structured according to geometrical figures of various types, the corresponding numbers are associated with each body, hence Pythagoras' assertion that number is the essence of all things makes sense.

The Pythagorean line is formed by units and given that a unit has dimension, it can be said that the line is formed by points with dimension. The concept of point without dimension comes later than Pythagoras and the arguments of Zeno contributed to its acceptance by mathematicians.

With regard to time, Pythagoras considered that time was constituted of instants with duration.

On the other hand, the Pythagoreans admitted the infinite divisibility of both space and time.

Let us now turn to the paradoxes of Zeno of Elea.

The first refers to the story of Achilles and the tortoise. Achilles competes in a race with the tortoise giving him an advantage at the start. The argument is that Achilles can never reach the tortoise. The race begins and while Achilles reaches the starting point of the tortoise, it will have advanced some distance. When Achilles reaches the new position of the tortoise, it will have advanced a little more. Thus, every time Achilles reaches a new position, the tortoise moves away. Since these positions are points that, according to Pythagoras, have dimension, and there are an infinite number of them, Achilles will be forced to travel an infinite distance and, therefore, the movement will never end: Achilles will never reach the tortoise.

The second paradox, called the race course, states that no one can traverse the race course from one end to the other, as it has to reach the midpoint. To move from the starting position to the midpoint, you have to go to the midpoint of the first half and so on. Therefore, it has to pass through an infinite number of midpoints. Therefore, if it is admitted that the points have dimension, one can never even begin to move.

In these paradoxes, Zeno does not prove that Pythagoras was wrong, but it does show that the Pythagorean theory of unity and the infinite divisibility of the line can not be accepted at the same time, since the two are incompatible.

Mathematicians, in view of Zeno's arguments, opted for the choice that was most convenient, admitting that a line segment can be divided indefinitely and assuming that the geometric points lack dimension. These geometrical points are those that appear in the 'Elements' of Euclid. I must say that the acceptance of points without dimension was not easy, because in the world that surrounds us it does not seem to exist; however, human mathematics has been constructed with this hypothesis. In this respect, one can place oneself in the Platonic position and consider these points as belonging to the world of ideas, either admitting that they are thoughts of God, or adopting a more modest posture and stand next to Aristotle by assuming them mere abstractions of material points.

Mathematicians, in addition to accepting the concept of dimensionless point, were somehow influenced by Zeno to rigorously establish the concept of boundary and introduce convergent series.

There are two more paradoxes of Zeno: that of the arrow shot at a target and that of three athletes, one of whom is sitting in the stands and the other two run in a circular track in the opposite direction and with the same speed. In these paradoxes the fact that time is formed by moments with duration is challenged, and they explain that the theory of Pythagorean units continues to be contradictory even if it is admitted that a segment of a line is formed by a finite number of units and, as a consequence, that it can not be subdivided infinitely. As mathematicians we have chosen points without dimension and these two paradoxes do not concern us.

Turning now to the concept of actual infinity, I will say that this was not accepted as such until the nineteenth century.

Galileo, in the early seventeenth century, realized that there are as many natural numbers as perfect squares. To do this, he assigned 1 to 12, to 2 to 22, to 3 to 32, and in general, to n to n2. In this way he established a one-to-one correspondence between natural numbers and all perfect square numbers. But the perfect squares form a part of the natural numbers so Galileo thought that the correspondence built by him was against the principle that says that the whole is greater than the part, hence he did not accept that natural numbers constituted a set. This correspondence has long been called the paradox of Galileo.

It was in the nineteenth century that a famous mathematician, Bernard Bolzano, illuminated the concept of actual infinity.

Bolzano's father was an Italian from northern Italy, an antique merchant who, married to a German, lived in Prague. From this marriage was born Bolzano in 1781. Following his father's wishes he made an ecclesiastical career, and was ordained in 1805. From 1805 to 1820 he taught theology at the University of Prague.

For his interventions in favour of the national independence of the Czech people and against the Austrian monarchy, he was prevented from teaching, assigning him a petty pension with which he could barely live. He went to Techbuz where he dedicated himself, with some of his friends, to mathematics and philosophy. In 1841 he returned to Prague, where he lived until his death, which occurred in 1848.

Bolzano, a year before his death, wrote a book entitled 'Paradoxes of the Infinite', which is fundamentally philosophical in character. Bolzano states in this work that most of the paradoxical statements found in the domain of mathematics are theorems that directly contain the concept of the infinite, or rely on it. Bolzano, for this reason, attempts to classify the notion of infinity. He shows that the set of mathematical propositions is infinite and presupposes clearly that integers form an infinite set. He puts in a one-to-one correspondence a set of real numbers as a subset of itself, but instead of using this fact to dismiss the concept of an infinite set, as Galileo did, he made it serve as a definition. This was later rediscovered by the German mathematician Dedekind who formulated the following definition: a set is infinite when it can be put in one-to-one correspondence with proper subset of itself.

Later, Cantor, in the late nineteenth century, developed his theory of infinite sets, partly setting out transfinite arithmetic.

I will make it clear here that in modern mathematics the infinite is fundamental, and so in the axioms of set theory, that of Zermelo-Fraenkel-Skolem, the existence of infinite sets is guaranteed by axiom number five.

Let me now comment on one of the most elusive concepts of mathematics: the infinitesimals.

A very clear example of the handling of infinitesimals is the determination of the area of the circle as obtained in the fifteenth century by Nicholas of Cusa, who was a cardinal in the church and a mathematician. Reason as follows: the circumference of a circle is a regular polygon of infinitely many infinitesimal sides. Each side of this polygon is the base of an isosceles triangle whose opposite vertex is the centre of the circle and whose height is the radius; the corresponding infinitesimal area is half the base times the height. Nicolas de Cusa concludes by asserting that by summing all the areas of these infinite triangles it turns out that the area of the circle is the product of the length of the semicircle by the radius.

This reasoning, which no mathematician accepts, even if the conclusion is correct, is typical of the use of infinitesimals.

Archimedes, a mathematician of the third century BC, uses infinitesimals with a heuristic character and determines the area of the parabolic segment, but convinced of the non-existence of infinitesimals, invents the method of exhaustion and thus gives a rigorous test of the formula for the area that he had obtained.

Bothered by his contemporaries' criticisms of his use of infinitesimals, he used to say that the heart intervened to facilitate the work.

Newton called any function of time which gives the position of a moving body its "fluent" and the function giving the velocity he called the "fluxion". The determination of each of these in terms of the other is the essence of the infinitesimal calculus invented independently by Newton and Leibniz in the 1660-1670 decade. Although the foundations of this part of the mathematics are rigorously explained by both authors, at the time of setting up the calculus there appeared confusion due to the use of the infinitesimals. Newton strove to avoid them, but there were steps in which he could not. On the other hand, Leibniz claimed that one could reason with infinitesimals without any error, as if they existed.

The criticism of infinitesimals made by the Irishman Berkeley, the philosopher who denied the existence of matter and who became bishop in the year 1734, is famous and brilliant. It is said that an English astronomer, Edmund Halley, Newton's friend who financed the publication of the 'Principia', tried to convince an acquaintance of Berkeley of how inconceivable Christian doctrine was. The bishop reacted by stating that Newton's fluxions were as confusing and precarious as the darkest points of theology, and, in a publication entitled 'The Analyst', he made a destructive and accurate critique of infinitesimals.

In spite of Berkeley's objections, the first book of infinitesimal calculus, published in 1696 and authored by the Marquis de l'Hôpital, uses the infinitesimal without any limitation. At the beginning of the book he states that two quantities that differ by an infinitesimal number are equal and each curve is considered to be formed by an infinite number of rectilinear segments of infinitely small lengths.

It was necessary to wait until the nineteenth century, exactly in 1872, when Weierstrass rigorously established the concept of limit by the method called epsilon-delta. At this moment, infinitesimals were expelled from mathematics.

It might be believed that with the advent of rigour the problems were completely settled, but one sees that physicists and engineers have not resigned themselves to this fact and continue to use infinitesimals to this day, undoubtedly encouraged by the fact that although the method is obscure, results are correct. This situation forces us to continue research on this subject, and thanks to mathematical logic, which is the great intellectual monument of our century, we achieve results that would never have been suspected. Abraham Robinson in 1964, with the creation of non-standard mathematical analysis, managed to recover in a completely rigorous way the infinitesimal methods in the differential and integral calculus in the manner of the classics, as used by Newton, Leibniz, Euler, Bernouilli, etc.

I will now give you an intuitive idea of how this is achieved. The Norwegian logician Skolem realized in studying the axioms of the natural numbers, the so-called Peano axioms, that these admit models with strange objects not contemplated in ordinary arithmetic. This was Robinson's starting point. We now consider the set R of the real numbers which we call the real universe, and the following infinite number of true statements relative to R.

1. There are numbers greater than 0 and less than 1/2.

2. There are numbers greater than 0 and less than 1/3.

3. There are numbers greater than 0 and less than 1/4.

4. There are numbers greater than 0 and less than 1/5.

And so on.

It is obvious that if you take a finite number of these statements, they are true at the same time. We now apply a theorem of the Russian mathematical logician Malcev, the so-called compactness theorem, which allows us to state that if we have a universe U and a collection of true statements such that each finite subset of them is true, then there exists a universe V containing U so that all the statements in the collection V are true simultaneously. Then it can be ensured that there exists a universe S containing the real universe in which all the statements quoted above are true simultaneously, that is to say, there are elements in S greater than 0 and smaller simultaneously than 1/2, 1/3, 1/4, 1/5, etc., that is, less than any positive real number. These are the infinitesimals. From here the theory is developed.

The infinitesimal calculus, differential equations and even the calculus of probabilities and mathematical statistics are presented in a very simple way in non-standard analysis, but in return the meanings are not as clear as in classical mathematical analysis.

I would like to say here that mathematics is probably the best man-made intellectual work, but it is not perfect, even as we have seen, imperfections have contributed to its development. Sometimes errors have played a fundamental role in it, and so it is that, logically, mathematics is subject to the human condition. Absolute perfection is not possible in this world. This is well expressed by the great Bengali poet Rabindranaz Tagore, who says the following: If you shut your door to all errors truth will be shut out.

Finally, I think that mathematics is much more than its applications. In this connection, I quote from a letter from the German mathematician Jacobi in the last century to the French mathematician Legendre. Jacobi said: a theorem on number theory is as important as a question about the world system, because the fundamental purpose of science is to honour the human spirit.

Nothing else. Thank you very much for the attention you have given me.


JOC/EFR August 2017

The URL of this page is:
http://www-history.mcs.st-andrews.ac.uk/Extras/Valdivia_Infinity.html