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.


-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.


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.




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

zip folder

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 dellAula Magna al prof. Aldo COSSU

21, maggio, 2007. Prolusione




Luigi Borzacchini's PUBLISHWARE

last update: January, 14th, 2013

Luigi Borzacchini, Dip. di Matematica, Università di Bari, via Orabona, 70125 BARI, ITALY/ email: gibi@dm.uniba.it