## 6509 - Mental approach of Archimedes

### N. Lygeros

Translation: Paola Vagioni

If the opus of Archimedes is less known, it is due to the absence of material according to the philologists of society. We do not have the equivalent schema with Aristotle. The big difference is that Archimedes was actually engaged in many activities and a multidimensional personality. He was the Socrates of mathematics, while Aristotle was the Protagoras of science. The reality of the first is our world, while the world of the second is not our reality. Moreover there is another big difference, which is not theoretical but experimental. Aristotle is not puzzled, he provides solutions, while Archimedes puzzles everybody with his unorthodox solutions. The discovery of the approach of the circle was done by none other but Archimedes, because where everyone was looking at the circle he was seeing the number π. Even in his calculations with the 96sided polygons, he made use of two fractional approaches of the number √3 without explaining them anywhere. In his research the mental schema is solid, without explanations. He does not give all the details. The guidelines suffice. He does not attempt to convince the others, because he functions with the notion of proof. Therefore why demonstrating. Whoever is capable of following him, may he become his disciple in order to learn his methodology. Only in this way does he also approach integration. It is interesting too that he does not regard it in relation to differentiation, he does not examine the notion of the function but only as a trait of a mechanical procedure. He functions in this framework solely as a geometer. He studies knowing his limits and for this reason he discovers equivalences with the problem of the squaring of the circle without falling into the trap, like many, of the solution. Moreover he reveals via his proof the value of the spiral too, which carries his name ever since. He is capable of seeing new structures where others were having difficulties with the finite number of the Platonic solids. He does not look at the 5, but he sees all 13 of them. As for the sphere and the cylinder, he is the first to reveal the simplicity of their relation. He explains in this way the contribution of the *Theory of Relations* in advanced mathematics, on which Caratheodory will operate mentally via *Theory of Somas*. Complex entities can have simple relations and these via the process of elimination can even define them in an efficient way. He succeeded in promoting even the elements of linearity inside the sphere of cyclicity. He achieved this with the introduction of the barycenter via the three dimensional space, in order to elegantly solve the problem of the semi-disc. It took centuries for humanity to understand the value of his opus, even in more technical areas of mathematics like combinatorics and only the Non-standard analysis theory managed to structurally invent the notion of the infinitesimal, which is so important in various integrations. Society may compare him to Aristotle due to philology, but in reality they are incomparable, as it proven by the creation of the polyspaston. It will still take many more men and a purely mental approach in order for everyone to see the difference that makes the difference, via thought and not reasoning.