Secrets of Infinity

Kvant 1970, No. 3 p. 7, 9, 11, 12, 13

Rough automatic draft, low quality: expect mistakes and awkward wording. It will be replaced by a better translation. Original →

If necessary, each of the parts could be divided again into parts. But if you continue this division, then sooner or later a moment will come when the parts turn out to be so small that it becomes practically impossible to divide them further. In order to express such small “individual” quantities, the images of a “speck of dust”, “poppy seed”, etc. have long been used. But if a poppy seed can still be divided into parts, albeit with difficulty, then can a speck of dust fluttering in a ray of sunlight be divided into parts? Everyday experience has not given an answer to this question, and we have to rely only on reasoning and conduct a purely speculative analysis.

One of the additional answers to this question before us belonged to someone who lived in the 5th century BC. e. Greek philosopher Anaxagoras. He read:

“Among small quantities there is no smallest, but the decrease is continuous, or the existing cannot cease to exist.”

He also said:

“All things (at the beginning of miracles) were together, they are infinitely numerous and infinitely small.”

So, according to Anaxagoras, the whole state is made up of an infinite number of infinite small parts. In this case, of course, by infinitesimal we meant not a variable tending to zero, as is now customary in mathematics, but a point without dimensions.

But this contradicted two basic axims generally accepted in Greek mathematics. Greek scientists divided all quantities into extended and non-extended. At the same time, they believed that:

The sum of an infinitely large number of any, even very small, extended quantities must necessarily be infinitely large.
The sum of any, albeit infinitely large, number of unextended quantities is always equal to zero and can never be equal to some predetermined extended quantity.

It turned out that the infinitesimals, from which, according to Anaxagoras, the world was composed, could not be either extended or non-extended, that is, they could not exist.

Other possibilities were also put forward against the idea of infinite divisibility. Anaxagoras Plutarch wrote:

“If the division of two quantities into parts can continue indefinitely, then there is no reason to consider one quantity larger than the other, and the very nature of inequality is destroyed.”

The meaning of this objection is that after dividing the quantities will turn out to be composed of the same (infinite) number of identical (infinitely small) quantities.

STILL WORLD

Please, the strongest arguments against the idea of infinite divisibility were put forward by the philosopher Zeno of Elea, who lived in Italy in the middle of the 5th century BC. e. He “showed” that the assumption of infinite division leads… to the denial of the possibility of movement. Zeno dressed his reasoning in scales in a living, paradoxical form. So, in one of them he “proves” that the fast-long Achilles will never finish the slow turtle - after all, first he must reach point A, from where the turtle crawled, then accurately

a l and higher surfaces (having a minimum thickness). For example, a cone, according to Democritus, consisted of circles with minimal thickness, and a pyramid consisted of the same polygons. He imagined the ball to consist of pyramidal needles, the tops of which are located in the center of the ball. Aristotle wrote:

“According to Democritus, the ball also cuts, since it is also angular.”

This approach allows Democritus to solve many problems that geometers faced. So, he was the first to calculate the volume of a cone and a pyramid, he was the first to calculate the volume of a ball and to calculate the area of a sphere. This is apparently how he dealt with the pyramid.

First he proved the theorem that two pyramids with equal bases and equal heights have the same volume. Here there were no additions for him - pyramids with the same height consisted, according to Democritus, of an equal number of corresponding triangles, and their volumes were the sum of the “volumes” of the triangles (after all, according to Democritus, triangles were material and had a minimum thickness). Since the areas of the bases of the pyramids were the same, the “volumes” of the corresponding triangles in both pyramids coincided, and therefore the volumes of the pyramids coincided. And then he divided the triangular prism into three pyramids, each having the same height and equal bases (see figure). In this way, Democritus “proves” that the volume of a pyramid is equal to one third of the volume of a prism having the same height and the same base. And the equality of the volumes of a pyramid and a cone, having the same height and the same base area, was proven quite simply - they consisted of the same number of figures of the same “volume”.

It is also clear how Democritus approached the problem of the volume of a sphere. Since the ball consists of needle-pyramids, then the volume of the ball is equal to the sum of the volumes of these pyramids. And he already knew how to find the volume of each pyramid. Summarizing these volumes, he comes to the answer: “The volume of a sphere is equal to one-third of the product of the radius of the sphere and the surface area of the sphere.” He was unable to calculate this area.

ROUGH PYRAMID

Democritus added remarkable results in determining the volumes of geometric bodies. But most of his contemporaries responded with his methods. The ideas of Democritus were too different from the results of direct experience. After all, even dividing a segment in half was a very difficult gap in Democritus’s geometrics. A segment consisting of an odd number of weekly debts could not be divided in half: the average weekly debt went to one of the halves, and this half became larger than the other. It was impossible to separate

Aristotle’s Vatel, medieval scholastic Thomas Bradvarin, “all scientific systems are true only insofar as they are not based on the assumption that a continuous state is made of weekly ones.” And back in the 17th century, the Parlament of Paris threatened with the death penalty all those who opposed the opinions of Aristotle. And in part - to all supporters of atomism.

EXHAUSTION METHOD

So, mathematicians had to respond to both Democritus’ method of weeks and infinite processes - Zeno’s criticism undermined any confidence in the results obtained by using the concept of infinity. As a result, a difficult situation arose - the methods of Democritus, although through loose calculations, provided the correct formulas for calculating areas and volumes; now it was necessary to develop a new procedure for calculating geometric quantities, in which not a word would be said about infinity, or infinitesimals, or weeks. This procedure was created in the 4th century. BC e. Euclid’s predecessor was the mathematician Eudoxus, who developed the “method of exhaustion” (or, otherwise, “exhaustion”), in which many see the ancestor of the modern method of limits. This method was widely used by Euclid in his Elements.

For example, here is how Euclid proves that the area of two circles is the same as the square of their diameters:

$\frac{S_1}{S_2} = \frac{d_1^2}{d_2^2}.$

This may not be true. Then $\frac{d_1^2}{d_2^2} = \frac{S_1}{S},$ where $S$ is either less than or greater than $S_2$ .

Let us first assume that $S < S_2$ . Let us inscribe a square in a circle with diameter $d_2$ and construct regular inscribed polygons, successively doubling the number of sides. It is easy to prove that after a few steps the difference between the area of the circle and the area $P_2$ of the inscribed polygon

became than $S_2 - S$ , that is, $S_2 - P_2 < S_2 - S$ , and therefore $P_2 > S$ . Now let’s inscribe in a circle with diameter $d_1$ a polygon similar to that inscribed in a circle with diameter $d_2$ ; Let’s denote the area of this polygon by $P_2$ . Since the areas of similar regular polygons are related as the squares of the diameters of circumscribed circles, then

$\frac{P_1}{P_2} = \frac{d_1^2}{d_2^2}.$

But we admitted that $\frac{d_1^2}{d_2^2} = \frac{S_1}{S},$ and therefore

$\frac{P_1}{P_2} = \frac{S_1}{S}$

or, due to the properties of proportions,

$\frac{P_1}{S_1} = \frac{P_2}{S}.$

But $P_1 < S_1$ (the area of the inscribed polygon is less than the area of the circle), and $P_2$ , as just shown, is greater than $S$ . So, the left side of the equality (*) is less than one, and the right side is greater than one, which is absurd. This means that the inequality $S < S_2$ is impossible.

In the same way, it is proved that the inequality $S > S_2$ is impossible (do the proof yourself!).

So, the assumptions $S < S_2$ , $S > S_2$ are determined and remain the only possible one: $S = S_2$ , that is, $\frac{d_1^2}{d_2^2} = \frac{S_1}{S_2}.$

But if for a circle the result could be easily guessed, and the calculations were not too complicated, then calculating the volume of a cone or the parameters of the source was a complex and confusing matter. However, it was still possible to cope with the cone and the parameter - after all, the answer was known at the time of Democritus, and Eudoxus only had to clarify Democritus’s messages, replacing the division of parameters into weekly given triangles with differences according to the source method.

But when it was necessary to look for the volumes of figures from which Democritus did not deal

had, for example, a vertical ellipsoid or a vertical parabolide, the situation was completely bad - it was unknown for what value to prove that the volume could not be greater or less than this value. In other words, the mode calculus method serves only to prove already found results, but not to discover new ones.

AND LAZINESS CAN BE USEFUL

However, mathematicians who lived after Eudoxus were not too eager to discover new formulas. The center of scientific thought at this time moved from Greece to Egypt, where, under the tutelage of the Ptolemies in the Alexandrian Museum, scientists studied the sky, systematized previously accumulated mathematical knowledge, wrote books in honor of the high patrons of science, and copied ancient manuscripts. It must be said that the Alexandrian scientists did not so little, only their work was more of an epigone character. They developed what was previously available rather than selling new paths.

One of the few exceptions in the quiet family of Alexandrian scientists was a relative of the Syracusan king, Archimedes, who came to study from distant Syracuse. Unlike most scientists of that time, he was interested not only in high theory, but also in the practical applications of science, invented a screw pump, which was then used for many centuries, made mechanical toys, and later, when his hometown was besieged by the Romans, built military machines. It is clear that Archimedes became interested in mechanics, the balance of levers, the floating of bodies, etc. And mechanics has been separated from pure science since the time of Plato and was considered something akin to craft. Therefore, mechanics did not carefully monitor the rigor of reasoning, and mechanics widely used atomism in their research, splitting the wedges into atoms-wedges having a common vertex with the whole wedge, water outlets - into a number of “atomic” circles of small thickness, etc. *).

And in many of Archimedes’ works he encountered a curious combination of the method of exudation with reasoning drawn from mechanics. He learned to apply the concepts of the center of gravity and leverage to the solution of geometric problems and in this way obtained new results that were inaccessible to the “pure mathematics” of that time**). And since the results of mechanics were ultimately based on atomistic reasoning, then in fact Archimedes in these works relied on atomism. But obviously he didn’t write anything about it. The good mathematical tone of the time seemed to require that scientists present their results without referring to the anathematized and den of Democritus. Or maybe Archimedes had not yet read Democritus - this is indicated by some details of his works.

From the great works of Archimedes, which reached the 16th-17th centuries, mathematicians could not find out how he agreed, what results he needed to prove. They only had to follow the bends of Archimedes’ thoughts, only at the very end becoming convinced of his correctness. But in some passages quoted by Heron and other scholars, there was mention of some message from Archimedes to Eratovthenes. In this message, Archimedes wrote “about reasoning based on the method, but which is not yet proof for the text of Archimedes’ works. But to be sure of this, it was necessary to have

*) It is curious that even now engineers and physicists prefer to talk about elementary planes, infinitely thin layers, etc., without resorting to the precise, but somewhat complicated language that mathematicians used.
**) See about this in V. F. Kagan’s book “Archimedes”.

in your hands the full text of the sequel. As often happens, a happy accident helped.

In 1906, a private assistant professor at St. Petersburg University, Popadopulogo-Keramevs, found some Bogoslav treatise in the library of one of the non-Rusalem monasteries. Since parchment was very expensive in the Middle Ages, they usually took ancient books, washed or erased pagan texts from them, and wrote the life of some holy martyr. The hand that interested Popadopul was also of the same origin. The assistant professor was very weak in mathematics and was not too interested in the remnants of the erased text (fortunately, the monk who wrote the treatise was lazy and only erased the text, and did not erase it). He gave only a small excerpt from the ancient manuscript. But for the famous Danish source of mathematics, Heiberg, this excerpt was enough to establish that the monk washed away the text of Archimedes. Among the many works read by Heiberg was the last one by Eratovfenu.

In this message, Archimedes pays tribute to the merits of Democritus, saying that he was the first to establish the theorems on the volume of a cone and a pramid, without, however, giving them proper proof. And the method with which Archimedes conducted a preliminary study of the problem turned out to be close to the methods of Democritus. He decomposes cylinders, cones and balls into extremely thin circles, proves the position he needs for one of them, notes that the conclusion must be true for the rest, and finally says: “Since the whole body is made up of such circles and is completely filled with them, the statement is true for the whole body.” But this coincides almost word for word with the reasoning of the atomists!

Archimedes well understood the value of his method and hoped that with the help of this method “one of the current or future researchers… will find other theorems that have not yet occurred to me.”

Unfortunately, the posture of Archimedes turned out to be unknown to precisely those mathematicians who needed it most - the creators of the calculus of infinite malch, Kepler, Galil, Cavalieri. But they, using individual hints, reconstructed Archimedes’ train of thought.

The story of how atomistic ideas were revived and how modern mathematical analysis was created will appear on pages «Kvant».

LITERATURE

Vander Waerden, Awakening Science. Fizmatgiz, 1959.
M. Ya. Vygodsky, Arithmetic and algebra in the ancient world. Science, 1967.
I. A. Geiberg, Natural science and mathematics in classical antiquity.
V.F. Kagan, Arrifmet. Gostekhidat, 1951.
S. Ya. Lurie. The theory of infinite malch among the ancient atomists. Publishing house of the USSR Academy of Sciences, 1935.
S. Ya. Lurie, Arrifmet. Publishing house of the USSR Academy of Sciences, 1935.