Students intuitively feeling that they are being tortured in vain by nonsenses about some non-existent numbers, suddenly have a question: “What is a number?” They never come to mind that not a single professor could not answer this question even if he has reread everything that is in mathematics. One of them even could not bear the mocking hints and had published a whole book called “What is a number?” [13, 29]. In it, he has written so many whatnots that students have very well understood – such a question it’s better not to ask.

Pic. 26. Francis Viète

Meanwhile, scientists continued to move science forward, not bothering with such trifles as the essence of the notion of number. So, they created a whole bunch of new algebras taking advantage of the fact that there were no obstacles along the way. But they were not a continuation of what was a real one, the founder of which was the first royal mathematician François Viète served as an advisor to the court of the French king Henry III. But if these new algebras are special, then their terminology and bases are also special.

So, little by little in the science began to form a particular bird language understandable only to the authors of these most innovative developments. It even reached the point where mathematical societies creating a science only for themselves began to appear and in addition to this, the newest numbers appeared out of nothing: “hypercomplex”, “quaternions”, “octonions” etc. But the impression from the new achievements sometimes was spoiled from the same mare tail,19 which from somewhere appeared again. Getting this tail in the face is not very pleasant, but this is already the costs of a profession. In an effort to get away from such costs, a brilliant way out of the difficulties with the definition the essence the notion of a number was found. Scientists have finally grasped that it needs to be derived from other simpler notions, for example, such as the notion “set”. Everything turned out so simple, a set is that what is a lot. Well, is it not clear? However, it was found out again that one cannot do without empty set and in this case, it would see like nothing, and the question again arises, so what is a set number or not?

Georg Cantor has developed his theory of sets, which other mathematicians such as, for example, Henri Poincaré, called all sorts of bad words and did not want to admit at all. But suddenly unexpected for everyone the respectable "Royal Society of London" (the English Academy of Sciences) in 1904 decided to award Cantor with its medal. So, it turns out that here is the point, where the fates of science are decided!20

Pic. 27. Georg Cantor

And everything would be fine, but suddenly another trouble struck again. Out of nowhere in this very theory of sets insurmountable contradictions began to appear, which are also described in great detail in Singh’s book. In the scientific community everyone immediately was alarmed and began to think about how to solve this problem. But it has rested as on the wall and in no way did not want to be solved. Everyone was somehow depressed, but then they yet cheered up again.

It was so happened because now David Hilbert himself got down to it, the great mathematician that first solved the very difficult Waring problem, which has a direct relationship to the FLT. 21 It is also curious that Hilbert repeated Euler's experiment apparently inspired by the FLT problem. It seems that at some point Euler began to have doubts that the FLT is generally provable and he assumed the equation a