The infinite endless: a route between cognitive

philosophy, logic, mathematics and art .

• Epimenide Paradox (if all the Cretan are false and a Cretan pronounced the sentence "All Cretans are false" ... is this the truth or falsehood? It creates an endless circle in which the truth leads to falsehood and vice versa ). - Other examples of propositions that trigger an endless process of self ( "This sentence contains an error"). - Reading of the L. Carroll, What the tortoise said to Achilles (summary, the impossibility of principle to give demonstration demonstrations, that regression "described by Carroll).

A) activity started. Identification of the concept

Listen to this…

• - Listening to the canon of Bach, contained in the music (dedicated to King Frederick II), entitled "Canon for Tonos, the so-called Canon Eternally Ascending. In it Bach, from the tone of C minor, modulating the issue develops in such a way that ends in D minor, and without the listener will notice a thing. This process is repeated for six times, that is, until you return to the original tone and the piece closes permanently. However the interesting in all this is that, given the structure "potentially uninterruptible", the fee could continue to rise indefinitely, thus demonstrating that despite the apparent impression very clear conclusion, in reality the infinite ascend is truncated so artificial. Bach, however, noted in the margin: "May the Glory of the King ascends ascend as modulation."

- An object placed between two parallel

mirrors is played in an endless drain on the

same mirrors.

- The mind as your self: the theory of knowledge

= knowledge of knowledge of knowledge of knowledge…

Vision and comment of the picture of M.C. Escher

• - Anassimandro and Apeiron: the unlimited as Archè of things. • -1 Pythagoreans and the discovery dell'infinito as immeasurable (the

diagonal of the square). • - Parmenides: the infinite as "always" of being. • - Zeno and the denial of the motion from the infinite divisibility of space

geometric (refutation of Zeno through geometric progressions). • - Empedocle and the endless cycle of the interplay of cosmic Neikos and

Philia. • - Aristotle: denial of the possibility of founding sillogisticamente (deductive)

the syllogism: regression indefinitely. • - Advances medieval concept of infinity as equipotent (G. of Ockham: "It is

not inconsistent that the party is not equal to or less than its all because this happens every time a part of everything is infinite).

• - Cantor: equipotent and infinite combinations.

B) Synthetic thematic development path with texts / authors / problems

• "Some classes are elements of themselves, others are not: the class of all classes is a class, the class of non-teapots is not a teapot.

• 'Now consider the class of all classes that are not elements of themselves. If it is something of herself, then is not an element of itself. If it is not, it is. "

• B. Russell • "Usually the sets are not elements of themselves [...]. From this point of

view most of the sets can be considered as "routine". However there are some groups to "self-ingestion, which contain themselves as elements, for example the set of all sets, the set of all things except Joan of Arc, and so forth. Clearly, all together or is routine, or to self-ingestion and no set can be either one than the other. Now, nothing prevents us from inventing all OA: the set of all sets of ordinary administration. At first glance, OA may seem rather ordinary administration of an invention, but this view should be reviewed as soon as we ask the question: "OA itself is a set of routine or set to auto-ingestion?". The answer is: "OA is neither a set of routine or a set of self-ingestion, because each of these alternatives leads to the paradox." Try to believe. "

D. Hofstadter

C) Anthology of Texts

From Zero to infinity• PART 1:They are two sides of the same coin - Seife writes in Zero, the story of

a dangerous idea- multiplying zero for whatever amount you get zero, multiplying infinity for any amount Infinity is obtained. The division by zero offers infinite, the infinite division offers zero. Add a zero to the number unchanged, add an infinite number to leave the infinite as such.; We have emphasized offers ;presents" because there is the difficulty; dividing by zero or infinity stopped classic mathematics. And the verb ; in fact, mathematically, does not mean anything. We wanted a leap in quality, a genuine revolution to solve these problems. The first thoughts on this probably come from India. We have already talked about the views of Mahavira (we are in the century AD) for which divide a number by zero, leaves the number unchanged, with a confusion between zero and nothing. More interesting is the opinion of Bhaskara (eleventh century AD), for which 3 / 0 equals infinity. At the same time, however, says that (3 / 0) x 0 = 3, showes a lot of confusion. Confusion remains over the centuries that followed, with the maths experts always embarrassed when dealing ;infinite zero for the inconsistencies that inevitably resulted from it. Still; in Euler Algebra is of 1770, we find 1 / 0 = ¥ and immediately after 2 / 0 = ¥.

• Part 2: A century had passed since when Leibniz and Newton had invented the new, powerful tool to the problems of infinitesimal and ;infinite : the infinitesimal. Calculastia Mathematically, however, neither Leibniz or Newton had explained and justified the division by zero. The infinitesimal calculation required an act of faith, but in the practical worked. Bishop Berkeley was right, a stubborn opponent of Newton who wrote: If we raise the veil and look under there, we discover empliness, dark and somewhat confusing, to say, if I am not mistaken, certain contradictions and even impossible. He was right to demand more rigor, but was wrong to believe the theory wrong. We thought the mathematicians, in later years, to clarify and define the infinitesimal calculation. Would have gone almost two hundred years before they could Cauchy and Weierstrass to build a rigorous mathematical theory on which to base the mathematical analysis, by defining the concept of; limit. The development of this differential calculus also explains the huge success of the method in various applications - Keith Devlin writes in the language of mathematics – despite it depended on the methods of reasoning whide were not fully understood. People knew what to do, even if he did not know why it worked. Many students in the course of analysis still have similar experience. It is a pity is that the mathematical knowledge of students stop at the seventeenth century. And perhaps not even there, because if they had awareness of the problems faced by Newton and Leibniz, would not have difficulty in understanding the differential calculus