Enquiries about the
genesis of formal thinking and about syntactic knowledge representation.
by Luigi Borzacchini (Dept. of Mathematics, University of Bari, Italy)
J.P.Vernant
reminds us that in Francois' vase (VI B.C.) we can see all the olympian Gods
attending a great party for the wedding of Peleus and Thetis. All of them are
on profile, according to the classic style, but Dyonisus, depicted instead from the
front . Gorgon and
Satyrs were depicted in the same way, and in many vases it is possible to see
two great eyes, credibly Dyonisus' or Gorgon's. J.P. Vernant connects these
figures to theater's 'masks', the Latin 'personae', the 'subject' on the stage
of the earliest tragedies. It is the dawn of the 'conscience' and the 'self'.
In that age almost
everything begins. Science and reason, axiomatic-deductive method and logic,
modern mathematics and model-based astronomy, philosophy, theater and history.
Codified written laws, democracy and elections, modern school and alphabetic
writing, money-based economy. Conscience and soul, the Self and the mind.
Paradoxes and formal
thinking, signs-based and soul-centred knowledge are the arguments of the
following reports centred on that age and also on our century, because today we
face the full unfolding of that genesis and the trace of a possible
metamorphosis.
Our enquiry deals with the
establishment of the realm of syntax, from Parmenides to our computer
age, with the connected formal paradoxes and with the genesis of the
theoretical concepts by the astonishing power of the negative.
-A
WITTGENSTEINIAN MYTH-
-BEING
and SIGN I: Syntactic paradigm and negative judgement paradox.
Knowledge
representation and formal thinking have 'ancient roots', deeply embedded in our
modern culture. This core can be revealed in the classic Greek philosophy and
mathematics, where we can find the ancestors of the paradoxes and the limits of
knowledge concerning 'being' and 'negation', which we can discover in modern
logic, physics and computer science.
I believe that to deal with
such limits we have to go back to their roots and analyse their inner
structure, linked to the genesis of the syntactic paradigm. This report,
the first of a series, circumscribes the field of our enquiry in the Greek
philosophy, as well as analyses the first instances of the above paradoxes,
from Parmenides to Aristotle, through the debate between the Sophists and
Plato. We stress in this process the role played by the alphabetic writing
revolution, and, as counterevidence of a great civilisation which never
produced a formal thinking as the western science, we analyse the same issues
in classic Chinese culture.
-BEING
and SIGN II: Axiomatic-deductive method and infinite in Greek mathematics.
Knowledge
representation and formal thinking have 'ancient roots', deeply embedded in our
modern culture. This has been analyzed in the first report, in the Platonic and
Aristotelian foundation of the syntactic paradigm, against Sophists' negative
judgement paradox and beyond the Parmenidean 'being' philosophy.
In this report we extend
the analysis to the evolution of Greek mathematics, dealing with
incommensurability's discovery and Zeno's paradoxes, from Pythagoras till the
Aristotelian foundation of the axiomatic-deductive method and the
continuous/discrete idea of infinite. The core of this analysis is in Euclid's Elements
where we can acknowledge the passage from an ancient non-syntactic proof-theory
to the axiomatic-deductive method. Also in this report the Chinese archaic
mathematics is employed as counterevidence for our hypotheses.
-BEING
and SIGN III: The ancient soul of the new science
Our science,
with its modernity and technology, shows in the ruptures of its paradoxes and
limits of formal knowledge its very ancient 'roots', deeply embedded in the
hearth of old Greek culture. The syntactic paradigm was the result of
the Platonic and Aristotelian foundation, and developped its 'strong' form at
the beginning of the XX century science.
In this framework modern
science achieved its greatest breakthroughs: quantum mechanics, formal logic
and computer science. All of them, however, can not avoid the occurrences of
the never ending paradox connected to the syntactic paradigm. Below the
surface of the antinomical form, we can maybe reveal the deep 'preestablished
disharmony' of the link between human knowledge and reality.
-BEING
and SIGN IV: The syntactic Self
This report
analyses the features of the syntactic paradigm in the field of Artificial
Intelligence, centred on the functionalism term, and the role played by
the idea of 'negation' and 'existence' in the contemporary idea of formal
representation. We show the substantially unified reductionism of the mind
we encounter today under the labels of materialism, behaviorism, connectionism,
functionalism, after the breakdown of the autonomous mental world
due to the 'strong' version of the paradigm. We show that also in the area of
Artificial Intelligence we can recognise such a paradoxical character, in the never
ending paradox form. Some remarks on the idea of paradigm developped
in our reports, compared with the kuhnian one, and about the mind/body
problem conclude this paper.
References.
On
certainty. Genesis and metamorphoses of formal thinking.
Formal
thinking appears in the Greek world as connected to a wide reshaping of human
mind, linked to the alphabetic writing technology, to the new money-based
economy, to the polis democracy. It can be characterised as a syntactic
paradigm by well defined features, and thus its evolution can be analysed
till our century. Phylogenetical evolution is also somehow reflected in
ontogenetical cognitive development, such as described by Piaget. Today these
features are widely debated in Metamathematics, Phylosophy and Artificial
Intelligence as well. We can recognise in Wittgenstein's late books a seminal
work toward a new metamorphosis of the paradigm. We set out a phenomenological
analysis of the mathematical certainty as trace for the development of
such evolution.
The Sophist. Genesis of Formal Thinking in greek Philosophy and Mathematics. Eng. Transl. Of "Il Sofista. Genesi del pensiero formale nella filosofia e matematica greca" (Matematica e Cultura 2, Springer Verlag Italia, 1999)
There have
been different releases of Formal Thinking since its genesis in the
pre-socratic Greek culture. We analyze the first releases, in particular the
release 1.1 due to the Sophists and Plato: it was just a work in progress, only
a prototype for the great Aristotelean release 1.2, which lasted for almost two
thousand years.
MUSIC AND MATHEMATICS
Our aim is
to employ computers and mathematics to experience ancient and not-well-tempered
music.
We can
yield at the moment a paper about the research program and a zip folder
containing a Mathematica notebook about ancient Greek music and
mathematics by which you can play ancient Greek music, choosing genus, mode,
tuning.
A Mathematica notebook about ancient Greek Music and
Mathematics
The Mathematica
notebook can be downloaded
In the
folder you can choose between MacIntosh and Windows-PC.
In addition
you can choose between English or Italian version.
In the
subfolder Music you find AIFF and MP3 files containing Pythagoreans and well-tempered
performances of ancient Greek musics.
Please,
read carefully section 5.1 of the notebook for correct installation.
Intitolazione dell’Aula Magna al prof. Aldo COSSU
21, maggio, 2007. Prolusione
IL TEMPO DELLA SCIENZA SENZA TEMPO
Luigi
Borzacchini's PUBLISHWARE
last update: July, 13-th, 2007
Luigi Borzacchini, Dip. di Matematica, Università di Bari, via Orabona, 70125 BARI, ITALY/ email: gibi@dm.uniba.it