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11 Δεκ 2013 (πριν από 2 χρόνια και 10 μήνες)

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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

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


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

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

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