First crisis in the foundations of mathematics

Since secondary school we believe in strictness and steadfastness of mathematics. Therefore it is full surprise for us that in process of its development mathematics was subjected to crisis's. Moreover it is surprise for us that starting since the beginning of the 20th century modern mathematics is in the state of huge crisis and modern mathematicians do not see the way out from this crisis.

At this page of our Museum we will tell about first crisis in the mathematics history. However, to understand this page of our Museum we should strain all our mathematical and even philosophical skills and knowledge. And if this page is very complicated for you we can admit it and move then.

So, why the first crisis in foundations of mathematics aroused? To answer this question we should remember that the main philosophical Pythagorean's doctrine was reduced to assertion: "Everything is a number", that is, all things in the world can be expressed through natural numbers and their ratios.

The early Pythagorean mathematics was based on the so-called "commensurability principle". According to this principle any two geometric values Q and V have some common measure, i.e. both values are divisible by it. Thus, their ratio can be expressed as the ratio of the relative prime numbers m and n:


Considering the ratio of the diagonal and the side of the square denoted by (Fig. 1) the Pythagoreans encountered the contradiction.

Incommensurable line segments
Figure 1. Incommensurable line segments.

Indeed, suppose where m and n are relative prime numbers. Then m2 = 2n2. Hence it follows from here that the number m2 is even as the number m. Since the numbers m and n are the relative prime numbers, the number n is odd according to (1). However, if m is even, the number m2 is divisible by 4, and hence, n2 is even. Thus, n is also even. But n cannot be even and odd simultaneously! This contradiction shows that the premise of the commensurability of the diagonal and the side of the square is wrong and therefore the ratio is an irrational number.

The discovery of the incommensurability shocked the Pythagoreans and caused the first crisis in the foundations of mathematics because the latter refuted the main philosophical doctrine of the Pythagoreans. The discovery of irrational numbers generated the complicated mathematical notion, which had not direct connection with human experience.

According to the legend, Pythagor committed the "hecatomb", i.e. sacrificed one hundred oxen to Gods. The discovery was worthy such sacrifice because it became the "turning point" in mathematics development. It ruined the former system created by the Pythagoreans (the "commensurability principle") and generated a lot of new and celebrated theories.

The importance of the discovery may be compared with the discovery of the non-Euclidean geometry in the 19-th century or the theory of relativity at the beginning of the 20-th century. Along with these theories the problem of incommensurable line segments was well-known among educated people. Plato and Aristotle are known to discuss the problem of "incommensurability".

Eudoxus-Archimede and Cantor Axioms

To overcome the first crisis in mathematics the famous geometer Eudoxus developed his "exhaustion" method and created a new theory of values. The essence of the "exhaustion" method can be explained using the following practical example. If we have the "barrel of bier" and the "beer mug" then this "barrel of bier" will be "exhausted" sooner or later even the "barrel of bier" would be enormous and the "beer mug" would be very small.

Eudoxus's theory of incommensurability (see "Euclidian Elements", book 5) can be viewed as one of the greatest achievements of mathematics throughout its history and coincides in general with the modern theory of irrational numbers suggested by Dedekind in 1872.

The geometric values measurement theory dating back to the incommensurable line segments is based on the group of the so-called "continuity axioms", which comprises both the axiom of Eudoxus-Archimede and the Cantor axiom or the axiom of Dedekind.

Eudoxus-Archimede's axiom (the "measurement" axiom): For anyone of two line segments A and B (Fig. 2) one can find the positive integer of n, so that

nB > A.(2)

Eudoxus-Archimede axiom

Cantor axiom
Figure 2. Eudoxus-Archimede axiom.Figure 3. Cantor axiom.

Cantor's axiom (of the "contracting" line segments): If there is the infinite sequence of the "enclosed" line segment A0B0, A1B1, A2B2, ... , AnBn, ... (Fig. 3), i.e. each line segment is the part of the preceding one, there exists at least one intersection point C common for all the line segments.

The principal result of the measurement theory of geometric values is the proof of the existence and uniqueness of the solution q for the "basic measurement equation":

Q = qV,(3)

where V is the unit of measurement; Q is the measurable value and q is the result of measurement.

It is difficult to imagine that the setting up of the "continuity axioms" and the creation of the mathematical measurement theory was the result of more than a 2000-year's period in the development of mathematics. The "continuity axioms" and "basic measurement equation" (3) comprise a number of great mathematical ideas influencing on formation and development of different branches of mathematics.

It is necessary to note that the "measurement" axiom expressed by (2) is a reflection of the Eudoxus' "exhaustion" method in modern mathematics. The axiom generalizes the thousand-year's mankind experience in measuring distances, areas and time intervals. It is the brief presentation of the easiest algorithm of measurement of the line segment A by the line segment B lesser than A. This algorithm consists of the successive applying of B to A and counting up the number of B' s put on A. And it is called the counting algorithm.

The "counting algorithm" is the basis of various fundamental notions of arithmetic and the number theory, viz. of the notion of the natural number (n' = n + 1), the prime and composite numbers, and also of the notion of multiplication, division, etc. In this connection the Euclidean definition of prime (the "first") and composite numbers ("the first number is measured by only 1", "the composite number is measured by some number") is of great interest.

The "measurement" axiom generates the "division theorem", which plays a fundamental part in number theory. The "theory of divisibility" and the "comparison theory" are based on the "division theorem".

It is necessary to note that the same subject of the classical number theory, which "studies the common theorems about the natural numbers 1, 2, 3, ... of the traditional arithmetic", is derived from the "counting algorithm" generating both the natural numbers and all the theories connected with them.