Pythagorean school
Pythagoras founded a brotherhood of a religious, philosophical, and scientific nature with a political bias. The works, usually attributed to Pythagoras, refer not only to the legendary Pythagoras, but in general to the works of this school between 585 and 400 BC.
In his cosmological concept, Pythagoras rejected the monistic idea of the primary substance that gave birth to the entire Universe. His concept is dualistic, and in the tension between two opposing principles – limited – unlimited, odd – even, single – multiple, straight – crooked, square – oblong – he saw the reason for any development. Little interested in material elements that could give an idea of the genesis of the various components of the universe, Pythagoras, fascinated by the deep religious movement that swept Greece of the time, sought to give a global picture of the cosmos as a whole. The basis of everything he saw in the number, as evidenced by his motto: “Everything is a number”.
The most important discovery attributed to the Pythagoreans was the discovery of the irrational in the form of incommensurable straight line segments. It is possible that it was made in connection with the study of the geometric mean a: c = c: c, a quantity that interested the Pythagoreans and served as a symbol of the aristocracy. What is the geometric mean of the unit and the two, the two sacred symbols? This led to the study of the relationship of the sides and the diagonal of the square, and it was found that this ratio is not expressed by “number”, that is, what we now call rational number (integer or fraction), but only such numbers were allowed by Pythagorean arithmetic. In other words, irrational numbers were discovered when it became clear that some relationships could not be expressed using integers. This discovery marked the collapse of the Pythagorean point of view about the representability of the world with the help of integers and caused the first crisis in the history of mathematics.
Eleatics
The influence of the Eleatic School (V in. BC) on the formation of abstract scientific thought is immense. The founder of this school, Parmenides, was the first to strictly distinguish between the sensible and the intelligible, which led to the inevitable confrontation between experience and the demands of reason. that is why the Eleatics did not accept the Pythagorean doctrine, which put a number in conformity with every thing. if discrete objects can be represented by integers.
This is different in the case of continuous quantities, such as length, area, volume, and. etc., which in the general case can be interpreted as discrete sets of units, only if we assume the existence of an infinite number of very small elements of which these objects consist. In response to this last concept, Zeno of Elea (born between 495 and 480 BC) formulated four paradoxes illustrating the impossibility of infinite divisibility and any movement, if you think of space and time consisting of indivisible parts. The general goal of his arguments is to show the absurdities that come to him when they are trying to obtain continuous values from infinitely small particles taken in an infinite set.
The infinitesimal calculus originates from the Greeks’ intuitive idea of continuity, mathematical infinity and the limit, as well as from the difficulties they encountered in trying to clearly define these concepts. These three concepts were correctly defined only in the 19th century, when mathematicians wanted to systematize the achievements of their science, and they had to reconsider the foundations in order to bring a solid foundation to the mathematical building.
Numbers and geometric quantities. We have seen that the Pythagoreans likened numbers to geometric points: one to one point, some other number to a group of points forming a certain geometric figure. Each number had a discrete set of units; thus, the Pythagorean arithmetic was limited to the study of positive integers and the relations of integers that were not considered numbers.
Any continuous quantity — a line, a surface, a body — could be identified with some corresponding number — a “quantity” (length, area, volume). Just as a unit was a common measure of integers, the values had to have a common unit of measure — to be with a measure and — and each value was identified with an integer number of its constituent units. This attempt to identify integers with continuous values, to interpret the continuous in terms of the discrete, did not lead to anything and quickly failed.
The decisive role, as already mentioned, was played by the discovery of irrational numbers. In a square with side 1, the ratio of the diagonal to the side is equal; it is not expressed as the ratio of integers and, therefore, has no status at all in Pythagorean arithmetic. The side and the diagonal do not have a common unit of measurement and are called the inequality and. The mutual correspondence between magnitude and number, familiar to the Pythagoreans, turned out to be broken. If each number corresponds to a certain length, then what numbers need to be compared to incommensurable values?
Paradoxes of Zeno and the concept of infinity.
It was precisely in connection with the discovery of disparate quantities that the concept of infinity penetrated into Greek mathematics. In their search for a common unit of measurement for all quantities, the Greek geometers could consider infinitely divisible quantities, but the idea of infinity drove them into deep confusion. Even if the arguments about the infinite were successful, the Greeks in their mathematical theories always tried to circumvent and exclude it. Their difficulties in expressing the abstract notions of the infinite and continuous, opposite to the notions of the finite and discrete, clearly manifested themselves in the paradoxes of Zeno of Elea.
The arguments of Zeno were “aporia” (dead ends); they had to demonstrate that both assumptions lead to a dead end. These paradoxes are known under the name A x i lle s, C t r e l a, D i x o t o m i (division by two) and C t a d and d. They are formulated to emphasize the contradictions in the concepts of motion and time, but this is not at all an attempt to resolve such contradictions.
Aporia “Achilles and the Turtle” is opposed to the idea of the infinite divisibility of space and time. Swift-footed Achilles competes in the race with a turtle and nobly gives it a head start. Until he runs the distance separating him from the point of departure of the tortoise, the latter will crawl on; the distance between the Achilles and the tortoise has decreased, but the tortoise retains an advantage. While Achilles runs the distance separating him from the tortoise, the tortoise crawls a little more forward again, and so on. If space is infinitely divisible, Achilles can never catch up with the tortoise. This paradox is built on the difficulty of summing an infinite number of ever smaller quantities and the impossibility of intuitively imagining that this sum is equal to a finite value.
This moment becomes even more obvious in the Aporia “Dichotomy”: before passing a certain segment, the moving body must first pass half of this segment, then half half, and so on to infinity. Zeno mentally builds the series 1/2 + (1/2) 2 + (1/2) 3 + …, the sum of which is 1, but he cannot intuitively grasp the content of this concept. Modern ideas about the limit and convergence of a series allow us to state that, starting from a certain moment, the distance between Achilles and the tortoise will be less than any given number chosen arbitrarily small.
The Strela paradox is based on the assumption that space and time are composed of indivisible elements, say “points” and “moments”. At some “moment” of its flight, the arrow is in a certain “point” of space in a stationary state. Since this is true at each moment of its flight, the arrow cannot move at all.
Here is the issue of instant speed. What value should be given to the ratio x / t of the distance x to the time interval t, when the value of t becomes very small? Unable to imagine a non-zero minimum, the ancients gave it a value of zero. Now, with the help of the concept of limit, the correct answer is found immediately: the instantaneous speed is the limit of the ratio x / t with t tending to zero
Thus, all these paradoxes are connected with the concept of limit; it became the central concept of infinitesimal calculus.
The paradoxes of Zeno are known to us thanks to Aristotle, who led them in his “Physics” to criticize. He distinguishes between infinity with respect to addition and infinity with respect to division, and establishes that the continuum is infinitely divisible. Time is also infinitely divisible, and in a finite time interval one can walk an infinitely divisible distance. The paradox “Strela”, which “is a consequence of the assumption that time is made up of moments,” becomes ridiculous if one accepts that time is infinitely divisible.