Luigi Borzacchini, Department of Mathematics, University of Bari, Italy
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. 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.
The letters are unknown, whereas the syllables genus is known. (Theaetetus, 202 d10-e1)
This report is the first of a series about the origins and nature of formal and/or syntactic thinking. In this context we use "formal", in a broad sense, as manipulating signs, and "syntactic", in a more restricted sense, as manipulating strings of signs. Here "manipulating" means that signs' usage is intersubjective, i.e. all the people or, at least, a social class, to this aim recognised by all the people, agree on the manipulation rules and on the 'meaning' of the signs. Roughly speaking, syntactic thinking concerns with arithmetic and writing, formal thinking comprehends pictograms, games and icons too. Regarding geometric graphical symbols, we shall use the term "figure".
"Syntactic thinking" and "formal thinking" will be analysed most of all as a 'tool' for knowledge, hence our aim in these reports could be reformulated as an inquiry on the foundations of "knowledge representation", and, more in general, on the roots of "formal thinking". Since the beginning of Artificial Intelligence in the Sixties, there has been an increasing number of contributions in this direction, based on analytical philosophy, neuropsychology, phenomenology, cognitive psychology and so on. Our approach tries instead to define the underlying "syntactic" paradigm of the idea of knowledge representation, tracing back its features to its very beginning, in the classic Greek philosophy. In fact, we do not believe that the history of this paradigm and the relative knowledge framework has been 'linear', a sort of progressive refinement, a journey from darkness to light. We believe instead that somehow the XX century science displays, as never earlier, the deep architecture and problems of the Greek ancient foundation. And we believe that this architecture is substantially and deeply paradoxical, without any thinkable getaway. There is something Hegelian in this contradictory structure of the knowledge, in that the provisory solutions to the paradoxes are always creations, and they are the only truly creative acts in our knowledge. But there is no teleology and no dialectic reason in this creation, because the paradigm cannot be subjected to any reason, science or philosophy, being instead definitely deeply embedded instead in our unconscious roots, in the core of our individual and social never ending play. From this point of view, our aim is more Kantian in that it tries to draw the limits of the formal thinking based knowledge. Thus, this paper is about
i) the characterisation of the basic paradigm of syntactic thinking and its origin in Greek philosophy. Other forthcoming reports will be about ii) Greek mathematics and the beginning of the axiomatic-deductive method, iii) the limits of knowledge in XX century logic, mathematics and physics iv) the body/mind problem and computer science.
We place in this themes the core of the modern idea of "science". To the aim of finding its roots, we know that the 'date and place of birth' of rational thinking, at least in our 'western' meaning, are in the VI century BC, in the Greek colonies of Asia Minor ((VERNANT 1971), 343). Actually we will see that the beginning of formal thinking had to be set in the western colonies, where lived Parmenides and Pythagoras, rather than in the eastern ones. Vernant pointed out the complementarity and also maybe opposition between these two ancient 'sources' of Greek thinking: the eastern, positive and secularised, discovering the 'nature' free from Gods' interventions, and the western, more abstract and philosophical, discovering the 'thought' and 'being' problems. We could as well ascribe to the same age the birth of philosophy, of scientific thinking, and, finally, of the reason itself. In two centuries, VI and V BC, in the great area of Greek culture, human civilisation overcomes the great rivers empires cultures born after the "Neolithic revolution", creating in the IV century the bases of the modern world: the Platonic dialogues and the Greek metaphysics are the bases of our philosophy, Aristotelian logic lasted with no changes until Boole's and Frege's mathematical logic, Euclid's Elements were adopted as a school text up to few centuries ago and have founded the axiomatic-deductive structure of mathematics. So, our concern for Greek philosophy is something more than historical completeness: we are looking there for the real background of our enquiry, the origin of the modern formal thinking.
Our aim is to study this 'great' beginning most of all from a peculiar point of view: the role played by the signs. Oddly enough, signs are underevaluated from both empirical and rational approaches for the same reason: they are only 'conventional' tools to represent, respectively, perceptions or ideas. Strange indeed, for no human civilisation existed without signs and we can advocate an increasing and pervading role of signs in our civilisation, in science as well as in everyday life. Is this casual or irrelevant? Or does it mean something? The signs are so deeply inscribed in our life that we take them for obvious. Our science is formal and sign-based also in the reality it describes, from the wave function y to the DNA coding, our economy is based on symbolic money and transactions, bar-codes characterise any product, the computers pervade any aspect of modern life, and the computer-based 'virtual reality' is the new horizon of our fantasy and art. Definitely, there is nothing obvious in the world of signs and in their increasing role in our society and culture. Our lives are more and more conditioned by their lives. We are really living at the top of the realm of syntax, an empire whose beginning is the theme of this report.
As a matter of fact, we cannot say that in Egypt or Mesopotamia or Indus valley or China there were no signs: written and spoken language, arithmetic and geometry are well documented. However, we have no trace of a theory, or also of just an explicit employment, of formal and syntactic knowledge, and we shall see that from this point of view the Greek miracle represents the greatest breakthrough in the history of mankind.
This report concerns most of all with Plato: the reason is that there we find the struggle to overcome the great paradoxes connected to the beginning of the 'realm of syntax', from Parmenides and Zeno to the Sophists' "negative judgement paradox" and to the mathematical rupture of the "incommensurability". In Plato these paradoxes are linked to the growing role of the two faces of the world of signs: mathematics, with the beginning of the abstract idea of "number" and the new axiomatic-deductive method that found in the Platonic Academia their cradle, and writing, whose alphabetic form in those times in Athens became the overwhelming technique both for social life and culture. And Plato was completely inside this passage concerning language and knowledge, as witnessed by his well-known words in the dialogue Phaedrus (PLATO 1964):
At the Egyptian city of Naucratis there was a famous old God whose name was Theuth; the bird which is called the Ibis is sacred to him, and he was the inventor of many arts, such as arithmetic and calculation and geometry and astronomy as well as draughts and dice, but his great discovery was the use of letters. Now in those days (the God) Thamus was the king of the whole country of Egypt; and he dwelt in that great city of Upper Egypt which the Hellenes call Egyptian Thebes, and the god himself is called by them Ammon. To him came Theuth and showed his inventions, desiring that the others Egyptians might be allowed to have the benefit of them; he enumerated them, and Thamus inquired about their several uses, and praised some of them and censured others, as he approved or disapproved of them. It would take a long time to repeat all that Thamus said to Theuth in praise or blame of the various arts. But when they came to letters, Theuth said: O king, here is a study which will make the Egyptians wiser and give them better memories; it is a specific both for the memory and for the wit. Thamus replied: O most ingenious Theuth, the parent or inventor of an art is not always the best judge of the utility or inutility of his own inventions to the users of them. And in this instance, you who are the father of letters, from a paternal love of your own children have been led to attribute to them a quality which they cannot have; for this discovery of yours will create forgetfulness in the learners' souls, because they will not use their memories; they will trust to the external written characters and not remember of themselves. And so the specific which you have discovered is an aid not to memory but to reminiscence. As for wisdom, it is reputation, not the reality, that you have to offer to those who learn from you; they will have heard many things and yet received no teaching; they will appear to be omniscient and will generally know nothing; they will be tiresome company, having acquired not wisdom, but the show of wisdom. (Phaedrus, 274c-275b)
In these words, we get the feeling of the changes the alphabetic writing was introducing in the Greek world and the paradox of the fear of Plato, one of the greatest writer of mankind though he was, to deal with that epoch change. The writing waxed board became the basic metaphor in Plato (the 'block of wax') and Aristotle (the 'tabula rasa') for the mind before knowledge, and the alphabetic writing their basic metaphor for scientific knowledge. We can confront Plato's text with a myth about the 'gift of writing' in China, cited in (FAZZIOLI 1986):
Fu Xi the first of the Five Emperors of the legendary period. ...he is credited with the invention of rope, fishing- and hunting-nets, musical instrument and the eight trigrams. He taught man how to use fire to cook food and how to raise and tend livestock, becoming the protective deity of nomadic life... In a commentary to the Book of Changes (I Ching) ...: ''When Fu Xi governed everything under the sky, he looked upward and admired the splendid designs in the heavens, and looking down he observed the structure of the earth. He noted the elegance of the shapes of birds and animals and the balanced varieties of their territories. He studied his own body and the distant realities and afterwards invented the eight trigrams in order to be able to reveal the transformations of nature and understand the essence of things.'' (12-13)
In the older environment, based on hunting and nomadic breeding, the writing, connected to the magic understanding and control of the external world, mirrors the nature (Fu Xi precedes Shen Nong, discoverer of agriculture, in the list of the mythic 'discoverer-emperors'), whereas in the Platonic version of the Egyptian myth, in a full agricultural and commercial society and with (implicit in Plato's experience) reference to the new alphabetic technology, the writing is the gift of a lawgiver, is connected to mathematics and has to substitute memory to foster human knowledge.
Recently, many authors have more and more underlined the deep cultural changes caused by the birth and the social spreading of alphabetic writing ((GOODY 1986),(HAVELOCK 1978),(ONG 1982)). Without accepting any strict cause-effect connection (criticised by (THOMAS 1992)), this report however claims the creative cultural role of that revolution, both for philosophy and for mathematics. In addition, we shall consider the Chinese mathematics as counterevidence of a different development in a non-alphabetic cultural environment, which nevertheless achieved the highest cultural levels, also in mathematics and technology.
V century was an age of great dialectic struggles witnessed by Plato's writings and based on subtle discussions about being and not-being. Schooltexts of philosophy tend to consider them not very important, almost absurd riddles, and most of the modern science tends to conceal this Sophist-Platonic crisis as completely overtaken by the Aristotelian foundation, so that "sophist" today is synonymous of 'intellectually useless', purely verbal and meaningless. I think that the problems and paradoxes Plato dealt with, have been not solved then once forever, but only 'removed' by Plato and Aristotle, condemned to sudden returns, and in our century again clearly visible in the antinomical arguments of logic, theoretical computer science and physics. This 'eternal recurrence' will be the topic of the third and fourth reports.
Often in these reports we shall refer to the similarity between two authors: Parmenides and Wittgenstein. They were the philosophers who most of all tried to institute a thorough correspondence between reality and language, 'natural language' for the former, 'formal logic' for the latter. They were the most coherent thinkers of more diffused tendencies, 'rationalism' and 'logic-positivism' respectively. Both of them lived at the dawn of a season of paradoxes, and at the end of their life both founded a new form of 'scepticism': for Parmenides we can remind Zeno's paradoxes, the sceptical arguments in the Platonic dialogue Parmenides and Sophists' teaching, and for Wittgenstein we can remind his late works and the methodological debate in the 'thirties'. This similarity displays the core of the 'never-ending' paradoxical framework of the 'realm of syntax': the relation between being and sign.
Hence we know where to look for our enquiry, but, before beginning with classic Greek philosophy, we have to make more precise the questions about ''formal thinking'' and the ''syntactic paradigm''. Then we are going to analyse the features of negative and being ideas in the Greek culture. The central sections are devoted to the evolution of the paradigm and the related paradoxes in Parmenides, Sophists, Plato and Aristotle. In the conclusion we try to outline a synthesis aimed to a better focusing of both the paradoxes and the paradigm for the concerns of the following reports. In the appendix we give a short outline of negative and being usage in archaic Chinese.
