THE SOPHIST.

GENESIS OF FORMAL THINKING IN GREEK PHILOSOPHY AND MATHEMATICS.

e-mail: gibi@pascal.dm.uniba.it. http://www.dm.uniba.it/psiche

MAN KNOWS BY SIGNS. God knows directly the world because he is the Creator, whereas man ahs not got this possibility, and then his knowledge can just be result of syntactic manipulation. This awareness appears in its sharpness at the heart of Leibniz' ars combinatoria, but can be revealed throughout all the history of the european civilization, from its Greek dawn to modern science, until today pervasiveness of the computer, the syntactic machine.

It is something so evident to be a crucial ingredient of common sense. A book of physics is full of "formulas", such that F = G m M / R², i.e. simple 'traces' of ink on a piece of paper, that are however believed "natural laws". Obviously nobody believes those scribbles sufficient to cause the movement of planets and stars: actually they 'stand for', 'represent' a 'something', a 'natural law' indeed, that we can not see or describe otherwise than by signs, and, we guess, that 'something' plays a causal role in the sky.

Such syntactic representation links two worlds: the real world (semantics) and the signs world (syntax). However: what do we mean here with sign? Sustantially a 'trace' (written on paper, carved on a rock, impressed on clay), chosen in a finite set, an alphabet, of traces. In addition our 'signs' have something abstract and ideal, since we consider them identically and infinitely reproducible and perfectly distinguishable: they are indeed the one beings of which we can actually assert absolute equality and diversity. On the other side they have even something material and solid, as far as we consider them technically producible and movable.

From the point of view of the representation function, signs are considered conventional as to the form intersubjective as to the meaning and the rule-driven usage. They are our numerical, algebraic, logic signs, and even the alphabetic ones, as far as natural language can be employed as a communication protocol of facts.

Linguistics often sets this "syntactic" representation against an "iconic" one, where the oblects of the real world are represented by their more or less stylized 'images' (for example men's and women's silhouettes on the toilet door). Such a representation seems simpler, common in the earliest civilizations and writings, and, in its most elementary instances as a natural and adaptive representation, maybe achievement even of living beings we usually do not ascribe intelligence to.

Signs and their syntactic representation are the ground of what we call formal thinking. Something which is a crucial ingredient of our life, so that it appears thoroughly plain. But is it really so?

We can imagine an astronomer looking at Venus at midnight near the top of the mountains, asking: where will it be tomorrow at midnight (fig. 1)? He could be a Babylonian or a Copernican or a modern astronomer, he will make the same thing: he will represent the position of the planet by its astronomical coordinates, and this is the 'static' side of representation (it concerns the realm of being, heir of Leibniz' "characteristica universalis"). Then he will make some computations (by either tables and abacus, or paper and pencil, or computer), and finally he will interpret the result as a position in the sky. If his astronomic theory is 'correct' he will find Venus in the computed position (we should say mathematically that the diagram 'commutes'), and this is the 'dynamical' side of representation (it deals with becoming, change, heir of Leibniz' "calculus ratiocinator"). ‘Representation’ and ‘interpretation’ are the coding/decoding relationship between our real world and a world of signs in which we embed our knowledge.

These two worlds, syntax and semantics, appear thoroughly heterogeneous and thus we must first face the problem of its 'working', and we must recognize that it does work very well according to its unbroken sequence of triumphs (first and foremost the achievements of mathematical physics in the last four centuries), not only as simple empirical 'recipe' of facts recording, but even as foundation of the very idea of truth. Probably so far the most 'convincing' answer is Leibniz' thesis of a preestablished harmony between mind and body, established by God ever since the beginning of time. Or even the 'neokantian' thesis which advocates the 'apriori' nature of this syntactic paradigm.

A second problem is to explain how Man could 'discover' or 'produce' this representation paradigm. Heterogeneity between the two worlds makes hard to give it an analogical or empirical root, evolved by forms of iconic representation, and it does not seem possible to 'formally' deduce 'formal thinking'.

However, the alphabetic signs are the heirs of an evolution whose start can be recognized in Neolithic Age and then developed in the great eastern civilizations with the literacy. We can imagine the 'ancestors' of our Chaldean astronomer to single out the equinox with the dawn of the sun on a hill, and then, in Egypt or at Stonehenge, place stones in circle, associate the equinox to a monolite and transmit orally this knowledge. Graffiti and pictures marked the world, rocks and caves were the web of a tissue whose weft were Man's symbols, unique and irrepeatable 'emblems', not yet signs.

But, after few thousands years, we find, both in Middle East and China, something astonishing: a scribe, a palace and temple intellectual, crouched in front of a stone layer or a clay tablet or a thin paper produced from some retted and pressed vegetal, with a chisel or brush or pen in his hand: it is record, sign, picto-logographic writing. And after a few more thousands years we see him in the same position, in front of a stone or wood or sand-filled stone, while moving little pebbles and rods: it is computing, draughts and chess, divination. Ultimately literacy is going to become the ground of the cultural reproduction in the 'polis' and it will be literature, law, philosophy and science.

Ever since its beginning the "world of signs" is a world, autonomous and splitted from the 'real' world, a world made up with infinitely reproducible and manageable in accordance with rules: from abacus to computer keyboard it is a fully detailed world that can be held and managed on one's knees. And the history of the realm of signs is a history of 'shadow areas' where ruptures can be recognized that a linear evolution can not bridge.

The intriguing thesis that the shift from oral to literate cultural reproduction can deeply restructure the mind is after all something understandable even from a purely 'computational' point of view. For 'mind' in an oral culture must be structured as a finite state automaton, because it can remember only by cerebral states which are finite in number. Instead a mind which can read/write/move on a potentally infinite world must be structured as a Turing machine

This rupture is hardly understandable and this shadow area on the border of the realm of signs is scarcely superable, if we simply guess an evolutionary process either from the chimpanzees' behaviour or from a practical knowledge transmitted by gestures and sounds or from a neural network model of mind: how can a "Turing machine" emerge from a "finite state automaton"? And how can "unique and irreproducible symbols" become "infinitely reproducible signs"?

Leaving the most philosophical aspects of these questions, we must try to better outline the problems inner to the very idea of "syntactic representation": hence our attention must be placed over another shadow area, connected to the genesis of formal thinking, when writing, in its alphabetic form, becomes the ground of a (relatively) mass culture, in Greek classic civilization, and the main medium of the whole linguistic practice.

First, let us analyze the double characterization of the syntactic representation, both conventional and intersubjective. Intersubjectivity would be better satisfied by an iconic than by a syntactic representation, for the latter, insofar conventional, is substantially subjective: writing on the toilet's door 'men' and 'women' is a more subjective representation than drawing silhouettes (for example the words are not understood by a not-english-speaker. The two aspects of the 'sign' are strangely contradictory: a large massive spreading of literacy paradoxically undermines its intersubjectivity.

But compare the smooth, continuous, regular motion of the planet in the sky with the shaking, discrete, irregular operations accomplished by pebbles, pencil, microchips: in two thousand years nothing is changed, knowledge representation seems completely heterogeneous to the represented reality and bound to discreteness.

Second, let us compare in the above astronomical example the real moon motion during the night is continuous, smooth and uniform, with the syntactic manipulation of the astronomers, with the motion of the tokens on the abacus, with the scrabbles of the arithmetical computations, with the state transitions in a computer. All these are discrete processes and display a chaotic behaviour thoroughly not comparable with the regular moon motion. This continuous/discrete opposition entalis many problems for 'representation'. Syntactic representation requires 'elementary' symbols, iconic representation doesn't; the parts of of the syntactic representation are not representations of parts of the object (which part of Socrates does the 'o' of Socrates represent?), whereas this happens for the iconic representation (the hand of Socrates' image is the image of Socrates' hand); the relation is not 1-1: a predicate concerns many objects and an object satisfies many predicates.

Third, the correspondence is plain for empirical terms (as 'house' or 'yellow'), clumsier for abstract terms (as 'beauty' or 'mind') or theoretical constructs (as 'electric charge' or 'wave function'), and there are words for which the correspondence is impossible, as 'truth', 'being', 'negation'. In the 'fact' represented by the proposition "there is nothing red" we find not only nothing corresponding to 'nothing' or 'there is', but also nothing corresponding to 'red'. Then, there are mathematical terms, as numbers or relations ('equal' or 'more/less'), which do not allow 'analytic' correspondences , but only 'olistic' ones: in "there are two stars" or "the planets have equal magntiude" 'two' or 'equal' can not be interpreted as any single object, but only linked to the whole fact.

These aspects are well known theoretical questions in our century, from Wittgenstein [1] to the debate about neural networks and the subsymbolic knowledge vs. traditional Artificial Intelligence and symbolic knowledge. However, these themes appear in their most radical and perspicuous form in the platonic dialogues, most of all in Cratylus, Theaetetus, Sophist.

In the middle, there are more than two thousand years of reduced or no interest for these problems. Why?

I believe that the reason of this reduced interest for the troubles of the syntactic representation has been the role played by the great Platonic achievement, the world of forms, that for centuries has been, together with the world of signs and the world of objects, the third corner of the semiotic triangle (fig.2). Forms were 'amphibious' entities, half 'visual', 'iconic', as according to their etimology, half 'syntactical', as according to a tradition stemming from Plato and prevalent since Leibniz and overwhelming since Hilbert. This double face made "forms" and "mind" into a 'buffer state', a 'bridge' between reality and signs, between semantics and syntax: you could say that the signs 'stood for' the corresponding forms and these were in turn connected to reality, or vice versa the signs connected the forms to the images of things.

The form, the idea, was the ground of the Aristotelism and of the Middle Ages thought, and survived in Modern Science as the geometrical and mechanical model, losing its role in our century under the blows of neo-positivism, of anti-mentalism, and, most of all today, of Artificial Intelligence. This sunset of the mind as autonomous world reduces the semiotic triangle to the simple correspondence, established in modern physics, metamathematics and computer science, between syntax and semantics, tout court known as representation or (by its inverted relationship) as interpretation.

The most likely solution to these representation problems in modern science is to consider the link between syntactic laws and natural phenomena based on the reduction of experiment to measurement activity. However, 'measurement' is nothing else that 'signs perception' on laboratory instruments, and hence experimental agreement is just signs agreement between syntactic manipulation of algebraic natural laws and signs perception on tools in special places called "laboratories", which are the most artificial and less natural things in the world. What about 'nature'? I do not know.

The process of "mind demolition" therefore can not avoid the reestablishment of the dilemmas on which the Platonic effort was centred. For example today, facing the 'natural' and 'adaptive to the simple visual peception' representation supplied by the neural networks, we address the problem of how can a 'symbolic', 'syntactic' representation emerge from that 'subsymbolic' representation.

Two knowledge metaphors compare with each other in these AI problems: the first 'mimesis', centred on the syntactic and formal representation, can be called the "metaphor of the writer", the second one, grounded on the iconic and natural representation, can be called the "metaphor of the painter" (here we do not thematize this second mimesis, see Borzacchini [2]).

These double metaphors can be found explicitly in many Plato's dialogues:

I think the soul at such a time is like a book…. Memory unites with the senses, and they and the feelings which are connected with them seem to me almost to write words in our souls; and when the feeling in question writes the truth, true opinions and true statements are produced in us; but when the writer within us writes falsehoods, the resulting opinions and statements are the opposite of true…. Then accept also the presence of another workman in our souls at such a time…. A painter, who draws (grafei) in our souls pictures (eikonaV) to illustrate the words which the writer has written…. When a man receives from sight or some other sense the opinions and utterances (legomenwn) of the moment and afterwards beholds in his own mind the images of those opinions and utterances…. And the images of the true opinions are true, and those of the false are false? (Philebus 38e-39b)

We must face a double problem: on the one hand justify the syntactic nature of human knowledge and intelligence, on the other explain the "emergence" of this form of knowledge from simpler iconic forms. It is very hard to say something new about these foundational questions, but it can be useful to analyze, or at least describe, the coming of formal thinking and its 'questions' on the european civilization scene.

At the periphery of european civilization we find areas where we lose the tracks of formal thinking. Piaget's 'baby' gets in the reach of formal thinking slowly from 6 to 11 years old, and in a quite mysterious way, even though school seems to play therein a crucial role; Levi-Strauss' Bororo [3] and Luria's Uzbek [4] appear outside; Levi-Strauss points out that it is not 'illogicality' but just 'concrete logic' instead of 'formal logic', and Luria displays the rising of formal thinking after mass school attendance. We can say almost nothing about Egyptians and Babylonians, but we begin to know something about Chinese civilization even on the scientific side; it has been a great civilization, even from a scientific and technological side, if not superior to the western one until three centuries ago; however there is a strong evidence that it kept out of essential aspects of "formal thinking" out of the western influence.

By and large, I believe that formal thinking was born just once. And this emergence can be recognized approximately in the V century B.C. somewhere in Greece: sometimes between Homer and Aristotle. Probably the first traces are in Pythagoras and Parmenides, then a complex development in Plato and then a final form in Aristotle, solid enough to last for two thousand years. Renaissance, from Galilee to Leibniz, marks a restucturing of formal thinking, whose final form can be recognized in our century, from Hilbert to Turing and the Computer Science.

We can assign to Parmenides the 1.0 release of formal thinking, to Aristotle the 1.2, so we can look at the 1.1 as a not well defined release, developed by the Sophists and Plato, a 'prototype' where we can see formal thinking 'in statu nascendi', and where we can find most hints for an enquiry about its never ending problems.

Why this V century emergence? These are the centuries that see the complete establishment of a money-based economy, of the 'polis' and its school, with the relative mass diffusion of an alphabetic writing, the emergence of a lay intellectual class, and the breakdown of the enlarged family, with the related social spreading of the colonization process. In this sort of 'melting pot' we want to underline, according to our signs centred analysis, the pervasive role of the alphabetic technology. Many authors stressed the crucial role of the linguistic forms and of the communication medium in the very form of a civilization, from von Humboldt to the socalled 'Whorf-Sapir hypothesis', till, in more recent times, Havelock, Mac Luhan, Goody, Ong.

Even alphabet was born just once: all known alphabets seem to stem, directly or indirectly, from a proto-alphabet, born between Sinai and Phoenicia in the II millennium B.C., and then perfectioned by the Greeks.

In non-alphabetic languages writing can be read, but oral language remains substantially autonomous, so that chinese writing is the same for the different Chinese dialects and learned people who spoke different languages shared the same literature, and the earliest writing tokens in Egypt and Mesopotamia concern topics which are stranger to the oral and common language employment, such as victories of the Pharaohs or the accounting of the mesopotamian temple-palace.

Alphabet restuctures language as one entity with two different expressive media, spoken and written, and this is the necessary condition to shift from oral cultures, where mass cultural reproduction is grounded on a memory-based oral transmission, to literate cultures, where instead writing dominates the cultural reproduction. Alphabet was born when "what is said" began to overlap with "what is written".

This happened foremost in Greece by a relatively 'mass' diffusion of literacy in the polis' school, not based on music and gymnastic anymore, but on signs manipulation, i.e. reading, writing and computing. This process is the background of the genesis of Greek polis and democracy.

Shifting the focus to the signs, we can remark that in an oral culture the 'speech' is the whole language considered as the core of the social structure, including the mass culture constrained into memory and oral tradition, whereas writing is bounded to recording and accounting, legacy of a little intellectual priesthood, functionally linked to government and religious activities. Signs in non-alphabetic languages preserve the trace of their semantics, even though the common 'phonetic loan', by which a concept is written by a symbol which denotes a different concept, but of analogous pronounce, shows that even pictographic and ideographic languages are logographic: the link between orality and literacy can not be avoided if language must express more than simple concrete facts..

Alphabet instead creates an indissoluble bond between oral social communication and the "realm of signs", and extends the employment of writing from its earlier administrative and religious goals to politics, laws, mass culture. Language, that was in the oral civilizations something 'continuous' (Levi-Strauss' [3] Nambikwara tries to 'write' by drawing a continuous wavy line, Homer's, Hesiod's and even Plato's language metaphors employ terms concerning something that 'flows'), becomes 'discrete' (just in Aristotle oral language, by the fwnh, 'voice', is considered a discrete magnitude).

This overlapping or written signs and oral speeches restructures the whole culture, and ground the very idea of representation, where the earlier 'linguistic-pragmatic social event' breaks down in two autonomous worlds, a "world of signs", language, conventional, interpreted by meanings and rules, that can be 'written' and 'spoken' and that must 'reflect' a 'deaf and dumb' "world of things", reality, that let itself 'be represented'.

In the half-light of "formal thinking", at the periphery of our cultural world, sign was instead symbol, bearer of an efficacious and reality-building power, something that has got nothing conventional and whose interpretation is not meaning or rule, but wisdom: in classic Chinese thought, in magic, cabalistic and hermetic European traditions, and even in Greece from Pythagoras to Neo-Platonism, symbol is an integral and efficacious part of reality and wisdom. Here we grasp an essential aspect of "formal thinking" and "syntactic representation" in our scientific tradition: language and reality are two thoroughly "autonomous and symmetric" universes. Language, even though completely lacking in 'efficacious' functions on reality, can however thoroughly "represent" it, and this turns language into the first and basic scientific tool and the 'place' of the very truth; reality, totally free from magic-symbolic influences and indipendent from the knowing subject' actions, can be represented, and is composed only of objects.

In classic Greek culture the whole intelletual class and function is deeply affected from the more and more relevant role plaued by writing. We hear the echo of this epoch-making change in Plato's words, (Phaedrus, 274b-275c), the myth of writing's (even mathematics' and combinatorial games') from the egyptian God Theuth. This God shows to king Thamus his inventions and claims their benefits:

The story goes that Thamus said many things to Theuth in praise or blame of the various arts, which it would take too long to repeat; but when they came to the letters, "This invention, O king," said Theuth, "will make the Egyptians wiser and will improve their memories (mnhmonikwterouV); for it is an elixir of memory (mnhmhV) and wisdom that I have discovered." But Thamus replied, "Most ingenious Theuth, one man has the ability to beget arts, but the ability to judge of their usefulness or harmfulness to their users belongs to another; [275a] and now you, who are the father of letters, have been led by your affection to ascribe to them a power the opposite of that which they really possess. For this invention will produce forgetfulness (lhJhn) in the minds of those who learn to use it, because they will not practice their memory. Their trust in writing, produced by external characters which are no part of themselves, will discourage the use of their own memory within them. You have invented an elixir not of memory (mnhmhV), but of reminding (upomnhmhwV); and you offer your pupils the appearance of wisdom (doxan), not true wisdom (alhJeian): for they will read many things without instruction and will therefore seem to know many things, when they are for the most part ignorant and hard to get along with, since they are not wise, but only appear wise.

The other side of the "world of signs" is mathematics, another great gift of the egyptian God. Pythagoreanism reveals indeed how deep was the relation between mathematics and earliest Greek philosophy.

Even in Greek mathematics we can recognize the transition from an 'iconic' to a 'syntactic' mimesis. An iconic earlier mimesis can be recognized in the Pythagorean connection between mathematics and reality based on 'figured' (triangular, squared, gnomons, etc.) numbers, till Euritos' constructions, where 250 was man's definition because he was able to build an human image by 250 pebbles ([5], 45,3), and even the etymology of the geometric terms shows the earlier 'visual' grounds of Greek geometry ('proof', 'theorem' are diagramma, Jewrhma, apodeixiV, all etymolically 'visual' terms); on the other side numerical regularities necessarily entail, in an 'iconic representation', an immanent presence of number in nature.

However it is in the syntactical Euclidean Geometry and Aristotelean axiomatic-deductive method ('axiom', 'postulate', are axiwma, aithma, stoiceion, all 'dialectical' or 'alphabetical' terms) the landing place of the evolution of Greek mathematics and the starting place of its role in the history of modern science. We do not try to describe this great construction, but we'll limit ourselves just to the elementary aspects of the concept of number and numerical sign.

In the ancient Near and Far East civilizations numbers were both words in the oral languages and written signs. These seem to have been built according to the general architectural scheme of pre-alphabetic writing, such as icons of the computing tools fingers, hands, pebbles, circles, rods, etc., with even likely traces of phonetic loans. As words numbers were always "numbers of…", say adjectives or determinatives, and even in Greek 1, 2, 3, 4 were declined according to gender and case, and from fialh, 'cup', fialithV , 'cup number', could be derived. Number had therefore a semantic-computational aspect on the one hand, the nature of a cardinal attribute of a set on the other. In Plato ariJmoV means often simply 'set' and it is considered an essential attribute of everything exists.

In Greek mathematics there is something peculiar: there are two numbers' notational systems. The first, said "erodianic", looks like the Latin system, with vertical strokes for the units, signs given by the first letter of the word for 5, 10, etc., and the remaining numbers given by juxtaposition. The semantic and cardinal nature appears in the existence of 'mixed' signs, made with numbers and measure units, so that '5 talents' is represented by a P, first letter of pente, wiyh a T, first letter of talanton inside.

The second system has got no non-Greek analogous ones, and it is built straightforwardly from the Greek alphabet with three more phoenicians letters to get 27 signs. The first nine denote the first nine numbers, the next nine the tens from 10 to 90, the last nine the hundreds from 100 to 900. The other numbers were built by juxtaposition and apices were added for numbers greater than 999. Historians did not place a great emphasis to this system because they often deemed it technically inferior to the earlier Middle East systems: embarassing situation for the numerical system which expressed the great Greek mathematics!

A more balanced evaluation reveals however pros and cons. As to our analysis we must first and foremost underline the nature of the signs. The erodianic system reflects the employment of the abacus even in that simple sums and subtractions could be obtained from the signs by an easy addition and deletion of signs: PII and II is PIIII. This does not happen in the alphabetic system, where z and b is J. The semantic-computational base is left. On the other hand this numeric system grows on the 'ordered' alphabet's structure and is learnt in an unique approach with the alphabetic signs manipulation: "number" thus obtains a new ordinal and purely syntactic character. The relevance of this connection is enhanced by that the greek word for 'letters' of the alphabet is stoiceia, etimologically linked to the image of 'elements ordered in a row', that we translate today with 'elements' and will be employed, since the Platonic age, to denote the basic elements of reality and the proof principles as well (such as in the title of Euclid's masterpiece), to replace earlier terms denoting 'principles', arcai, of reality which showed a clear biological origin (rizwmata, ‘roots’ in Empedocles, spermata, ‘seeds’ in Anaxagoras). And Euclid will employ letters to represent points and segments, Aristotle to denote terms of the syllogism, Diophantus to create the first algebraic notations.

Greek mathematics will employ most of all the alphabetic numeric system, whereas the erodianic one will be employed first and foremost for weights and money: this double system lets us realize that the sharp Platonic opposition between philosophers' and shopkeepers' mathematics had deep roots in Greek mathematics and was not only a Pythagorean mysticism.

What the real role of this emergence of a specifically 'ordinal' idea of number? Piaget showed how far this idea affects the 'mature' idea of number, and even in modern mathematics the 'ordinal' aspect is not less important of the 'cardinal' one. We must remark that in the V century B.C. the concept of 'infinite' gains its sharply quantitative character, by addition in Archytas and by division in Anaxagoras and Eudoxus: in both cases it is a 'potential' infinite, thus linked to the 'ordinal' idea of number.

The 'infinite' is an idea, as far as we know, completely lacking in Egyptian, Babylonian and Chinese mathematics: it is the greatest achievement of Greek mathematical thought. A side effect of this idea of number is the terrible idiosyncrasy of Greek thought toward zero, that in the semantical-computational systems, such as the babylonian, could find some occurrence, but that was thoroughly rejected by Greek mathematics. On the other hand a child grasps the numerical infinite when he realizes that there is no limit in counting.

In addition, modern mathematics, from Cantor to Frege and to computer science, with aritmetization of syntax, mathematical logic and programming languages, revealed the importance of the connection between numerical system and alphabet most of all in the idea that integer numbers allows a coding of any set of finitely expressible elements in any denumerable alphabet, and vice versa that integer numbers can be coded in any finite alphabet.

Maybe the earliest expression of the idea of Maybe the earliest expression of the idea of "syntactic representation" can be ascribed to Simonides (end VI- beginning V century B.C.). According to Michele Psello he said that "word is the image (eikon) of reality". The term 'image' reminds us the visual, 'iconic', mimesis. Simonides is a 'modern' poet, who says "poetry is skeaking painting, painting is silent poetry". It could seem today quite trivial, but at that time poets and painters were sharply different social characters. The latter were just craftsmen, often unknown, bearers of a simply technical knowledge, episthmh, inside an 'iconic' mimesis, whereas the former in a oral culture were the prominent intellectual class in the social structure, whose mnemonic skills were the tool of reproduction, a 'linguistic' mimesis, of the mass culture.

Poet was the instrument of Muses, who were the 'trustees' of Truth (alhJeia), of the myths-embedded truth who gave an account of the genesis and nature of cosmos, and of the more positive truth, whose ground was given by the myths, and that ruled the everyday productive and social life. Hesiod in his verses reminded the former in his Theogony, the latter in his Works and Days. This way the constellations have got the names of heroes and remind us their myths, but are at the same time the places of the working and social time organization. Hence Homer was the dominant figure of the cultural panorama even at Plato's time and he asked the Muses too, to remember Achilles' and Odysseus' deeds.

For Simonides instead memory is 'mnemotechnic', he writes and in addition reforms the alphabet, and his style is thoroughly lay, actually venal, if Aristotle's reference in Rhetoric is true, that he was asked for little money a poetry to celebrate some prize-winning mules, and he refused maintaining that the mules were scarcely a poetic theme; however, when the pay increased, he stroke up the verses: "You, daughters of horses whose hooves run as fast as the wind". But he was most of all a great poet, who wrote for the heroes of Thermopilae: "Go tell the Spartans, thou that passest by, that here, obedient to their laws, we lie": this way the dead and their graves speak by writing, with the style of the personal appeal, i.e. with the purest style of orality.

At dawn of the modern world writing outlines a new relationship between Truth and Knowledge, between 'narrated truth' and 'seen reality'; a relationship no longer centred on myth and memory, but on the linguistic aptitude to 'syntactically represent' a world of objects and facts. But myths' world and everyday life world show an opposite behaviour with respect to negation and becoming.

Myths and their 'memorized and narrated' truth were by their very nature descriptively genetic and positive, even though often actually incoherent or absurd, like the dreams. Everyday language described instead a pragmatically coherent reality, but subject to becoming and then linguistically always threatened by contradiction. To mediate between Truth and Knowledge language had two ways: either accepting language in its 'natural' form as an infinite already given land to be explored, thus reducing Falsity to something linguistically impossible and Truth to something compatible with contradiction, or looking at language as something to be reconstructed to be able to 'fix' reality deleting its contradictory aspects. Both ways had a price to be paid.

The former was Heraclitus' way and its price was the "paradox of becoming": if everything is becoming and if knowledge must be linguistically steady, then it is impossible to find an expressible knowledge. With the words of the platonic Theaetetus, reflecting Protagorean theses largely common among the Sophists:

Second solution was advocated by Parmenides and its price was the "negative judgement paradox": if an assertion matches with a fact that is, then a refutation matches with something that is not, but a sentence concerning something that is not, is about nothing and hence is impossible.

The european and his science are the result of this second way, a choice which consisted in the settlement of formal thinking, with the following appearance of not-being paradoxes. Plato in the Sophist (235d-236c) tries to define the "sophist", and in this search he sees him concealed inside the double 'mimesis' we spoke about, between the 'icastic' (eikastikh), concerning the exact 'copy', and the 'imaginative' (fantastikh), concerning the 'appearance'. If the former looks 'pictoric', the latter shows 'linguistic' characters:

for the matter of appearing (fainesJai) and seeming (dokein), but not being, and of saying things, but not true ones--all this is now and always has been very perplexing. You see, it is extremely difficult to understand how a man is to say or think that falsehood really exists and in saying this not be involved in contradiction…..This statement involves the bold assumption that not-being exists, for otherwise falsehood could not come into existence. (Sophist 236e-237a)

There are two linguistic phenomena that heavily affect the relevance and the evolution of the "negative judgement paradox" and give it the 'depth' it could not get today: first, the verb 'being', eimi, that from an earlier 'dynamic' indo-european value, i.e. 'appearing', (in tenses different from present and future, eimi could be substituted by gignomai, 'to be born, to happen, to become'), gets a new 'static' meaning, 'being fixed'.

We must underline the link between this static-locative function and the static value we ascribe to the ideas of 'predicate', 'truth assertion', 'fact' and 'existence'. A link that in the ancient Greek philosophy is a rigid constraint for the analysis and the very expression of becoming, and that on the other hand turns the permanence of being the necessary condition for knowledge (Kahn [6], 415).

This theme became part of that popular diffusion of the philosophical debate, witnessed by Aristophanes' comedies. Thus, at the beginning of the V century B.C. Epicharmus ([5], 23), father of the doric comedy, turned the "paradox of becoming" in a 'common place' of comic effects, as for the debtor who refused the acknowledgement of his debt because, in a heraclitean mood, he was no longer 'the same'.

It is clear that the Greek archaic culture was unable to 'thematize' the 'ambiguity' of being itself, so that "not being something" could not avoid to be a "not being". Such disambiguation of that semantic field, and even of more other terms of the philosophical lexicon, was accomplished in the V book of Aristotle's Metaphysics. There are different opinions about the disambiguation of existential, copulative and identifying function of the verb even in a relatively 'modern' text as Plato's Sophist.

To understand how relevant was the 'restructuring' of the semantic field of 'being', it is sufficient to consider the different functions of the verb in logic and set theory:

- predicate-argument relationship, p(a) (a is p),

- membership, a Î A (a is a A),

- inclusion A Í B (the A are B),

- identity, a = b (a is b),

- existenzial quantification, $ x (there is a x),

A not disambiguated employment of 'being' would entail the merging of such notations, and then make impossible modern formalized mathematics!

The second linguistic aspect concerns the visual origin of the Greek verb of knowledge: idein, eidoV, noein, Jewria, apodeixiV. All these verbs are related to 'vision' and progressively become expressions of a theoretical knowldege. It is hardly necessary to underline that this visual background turns "thinking what is not" into "seeing what is not", and entails the devastating character of the paradox.

Is that which is assumed in common speech possible at all, and can any human being hold an opinion which is not (tiV anJrwpwn to mh on doxasei), … And is the same sort of thing possible in any other field?…Yes, that a man sees something, but sees nothing (tiV ora men ti ora de ouden). (Theaetetus 188e)

Hence Parmenides is the true father of formal thinking, and to him we can ascribe the 1.0 release:

And the imperative of finding something fixed in the flux of becoming as necessary condition for a linguistically expressible knowledge entails the sudden appearance of the "negative judgement paradox":

The solution or rather the simple 'removal' of the paradox is to be ascribed to Aristotle, whose 1.2 release of formal thinking will last for two thousand years, till the birth of Modern Science. The aristotelean solution was based on two great innovations: on the one hand the soul with his forms built a bridge long enough to connect the 'world of signs' to the 'real world', looking for a distinction between subjective, such as 'saying' or 'thinking', and objective, such as 'seeing' or 'being', functions; it is a move that we can already read in Plato: "…the soul is to know, the being to be known (thn men yukhn gignwskein, thn d ousian gignwskesJai)…" (Sophista, 248 d1), and in Theaetetus the soul, and not the eyes, can 'perceive' concepts such as being, equal, and the other numerical concepts.

The second innovation was the solution of the opposition between a continuous reality and a discrete language, by the introduction of the ideas of meaning and 'syntactic' truth, with the following possibility of disambiguating verbs like 'being' in a new philosophical lexicon. Before Aristotle and after Parmenides there are the Sophists, Socrates and Plato who worked on a 1.1 release who was never something more than a 'work in progress', focused on the solution of the formal thinking paradoxes, first and foremost that "negative judgement paradox" that Plato in Sophist 238 d2 defines as "the greatest and the first quandary" (twn aporiwn h megisth kai prwth) and that set the basis for the arisotelean 'solution'.

Sophists' paradoxes were neither foolish nor symptoms of ignorance or intellectual incorrectness, but were a true expression of formal thinking in statu nascendi. It is easy in Plato and Aristotle to find them and verify the connection between them and the above outlined liguistic aspects. Thus, the 'visual' nature of the verbs of knowledge was almost a common place and drives Plato to make the young mathematician Theaeteus assent, after Protagoras: "… Perception, you say, is knowledge… Yes." (Theaetetus 151e), and Socrates remark: " Perception, then, is always of that which exists and, since it is knowledge, cannot be false." (Theaetetus 152c), echoing the words og Antiphon: "What is, is always seen and known, what is not, is neither seen nor known"; and the negative judgement paradox affects the very possibility of knowledge acquisition:

Syntactic knowledge, differently from iconic knowledge, must deal with 'negation', 'false', 'error'. Looking for a characterization of 'wisdom', swfrosunh, Socrates says:

But, it this science were 'iconic' it would be:

There are even the sophisms which appear connected to the 'binary relationships': "this dog is a father, this dog is yours, hence this dog is your father" (Sophistici Elenchi 179 a34). It is relevant how far paradoxes connected to the ambiguity of 'being' and to the binary relationships displays a straightforward mathematical aspect. For example, in the Sophistici Elenchi Aristotle writes "two is double of one, and not double of three, hence two is double and not double" (167 a 29), "Five is two and three, hence five is odd and even " (166 a 33). And Plato:

He finds the same troubles in the idea of 'representation' we pointed out at the beginning of this essay. I think these problems were the core of the whole Platonic enterprise, at least from the epistemological view, and in the following we are going to analyze three of them: the natural/conventional opposition for language, the negative judgement paradox even in mathematics, and the role of number in building dialectics.

Cratylus is the dialogue where Plato deals with the problem of 'language'. There he is the 'referee' in the debate between Hermogenes and Cratylus: the former maintains the 'conventional', the latter the 'natural' nature of language.

According to Hermogenes, for the 'conventional' character of the relationship between language and reality, there can not be either error or falsity, but everything said must be substantially true, echoing thus Protagorean themes. According to Cratylus instead, there must be a 'natural' relationship, an almost 'pictoric' mimesis betwen reality and language, so that false words do not represent anything, are not words but just 'noise': maintaining Heraclitean theses Cratylus denies the presence of 'falsity' in language, what is uttered, if does not 'reflect' reality, is just noise.

At the beginning Plato inclines toward Cratylus' solution, but suddenly it appears a problem we already stressed: applying language to reality, both experienced as 'flowing, continuous' entities, turns the very idea of truth into a sort of iconic matching of the sentence with respect to reality, where the "image of the part" coincides with the "part of the image", and then: "And is the true speech thoroughly true and its parts not true? Oh, no, true even the parts" (Cratylus, 385c). Then the names too must be either 'true-matching' or simple 'noise'. And the truth-matching of a name cam be ascertained looking at its etymology, i.e 'destructuring' it in simpler words: even the name of Hermogenese does not match with its bearer!

By the way the Platonic etimologies in Cratylus, even if ill founded, reveal interesting aspects of the earliest usage of some terms: this way episthmh is said stemming from pragmasin epomenhV "following moving things" (412a), and it shows that even in Plato this term displayed its technical and practical origin. AlhJeia, is said stemming from ale Jeia "divine ceaseless wandering", losing any trace of the privative a, revealing its mythological root (421a). Onoma from on ou zhtema estin "being on which there is research" (421b), reflecting the link between language and reality characterizing the parmenidean idea of 'syntactic representation'.

In addition, the truth-matching of a word must be reducible to the truth-matching of its letters. But this approach enters in a deep troubles if applied to the single letters, where it faces the opposition between a real continuous, without last natural finite 'elements', and a linguistic discrete, based on finite conventional alphabetic elements. The solution, that appears in Plato and becomes explicit in Aristotle, will be centred on the distinction between truth and meaning, both seen as threshold-properties, being the proposition the threshold for truth (it is impossible to speak about the truth of 'parts', i.e. subject, verbs, etc., of the proposition), and being the word the threshold for the meaning (it is impossible to speak of the meaning of 'parts', i.e. the single letters, of a word).

The discrete nature of language allows also the birth of the idea of "syntax" and the evolution of the idea of "truth" from the earlier idea of simple "truth-matching" with reality, which will become the Aristotelean idea of truth as correspondence between language and reality, toward the new idea of analytcal-syntactical truth, agreement between a property, expressed in the verb or in the predicate, and an individual, expressed in the subject, that will be developed in Plato's Sophist, will play a key role in the meJexiV, the 'participation' between objects and forms, crucial ingredient of his "theory of forms", and will appear again, in modern times, most of all in Leibniz.

In the Sophist the analysis of the "negative judgement paradox" is at the center of Plato's interest, and a first form of solution of the paradox appears based on reading not being as 'different' and not 'opposite' to being. More generally, a complete disambiguation of the verb 'being' begins its adventure in philosophy.

We point out the explicit link between the paradox and the troubles Greeks faced when thinking of the number 'zero', and their connection with the structure of Greek language:

but seriously, one of his pupils were asked to consider and answer the question "To what is the designation 'not-being' (to mh on) to be applied?" how do we think he would reply to his questioner, and how would he apply the term, for what purpose, and to what object?….But this is clear, anyhow, that the term "not-being" cannot be applied to any being (to on).….And if not to being, then it could not properly be applied to something (to ti), either…..And you will agree that "something" (ti) or "some" in the singular is the sign of one, in the dual (tine) of two, and in the plural (tine) of many….And he who says not something, must quite necessarily say absolutely nothing….Then we cannot even concede that such a person speaks, but says nothing? We must even declare that he who undertakes to say "not-being" does not speak at all? (Sophist 237c-237e)

We have singular, dual and plural, and there is no 'grammatical number' characterizng the 'absence': indo-european flexion affects the expressibility of not being as 'multiplicity', and thus its impossibility stems from the very idea of representation by natural language, and this in turn involves the concept of number, in its classic Greek form, "number of …", essential attribute of everything which 'is':

How deep this question will be in the following centuries can be remarked even in Lorenzo Fibonacci's Liber Abaci, where he introduces the new indo-arab numerical system with the "novem figure Indorum" (nine indian figures) and the "hoc signum 0" (this sign zero), and in Leibniz' Dissertatio de arte combinatoria, where on a set of 4 elements 15 'complexions' (subsets) can be defined.

In modern mathematics "number" becomes a 'relative measure', with the 0 as a divide between positive and negative, whereas 'discrete' and 'continuous' are fused in the decimal representation, so that every integer is a peculiar real number, and any real number can be represented by an infinite sequence of figures or as a limit of rational numbers.

In Greek mathematics instead integers and fractions are two opposite numerical worlds, often with a differenr numerical base, 5 'to count' and 12 'to divide', and 1 plays the role of a 'seed' of the opposition, whereas 0 has simply no reason to exist at all. Only Renaissance Science will overcome the idea of becoming as a contradiction, turning becoming in a form of being, rest in a form of motion and zero a special number, and not the very negation of the idea of number as multiplicity.

The troubles in connecting the continuous structure of reality with the discrete one of language was clear most of all in the problems of the elements, stoiceia, of their 'knowledge' and 'function' in a hierarchical-functional structure of science. These themes can be found most of all in Plato's Theaetetus, where Socrates/Plato tells what he calls a 'dream':

…I in turn used to imagine that I heard certain persons say that the primary elements (prwta stoiceia) of which we and all else are composed admit of no rational explanation (logon); for each alone by itself can only be named, and no qualification can be added, neither that it is nor that it is not, for that would at once be adding to it existence or non-existence, whereas we must add nothing to it, if we are to speak of that itself alone. Indeed, not even "itself" or "that" or "each" or "alone" or "this" or anything else of the sort, of which there are many, must be added; for these are prevalent terms which are added to all things indiscriminately and are different from the things to which they are added; but if it were possible to explain an element, and it admitted of a rational explanation of its own, it would have to be explained apart from everything else. But in fact none of the primal elements can be expressed by reason; they can only be named, for they have only a name; but the things composed of these are themselves complex, and so their names are complex and form a rational explanation; for the combination of names is the essence of reasoning. Thus the elements are not objects of reason or of knowledge (aloga kai agnwsta), but only of perception, whereas the combinations of them (sullabaV) are objects of knowledge and expression and true opinion. (Theaetetus 201e-202b)

This is the problem of the primitives of a representation language. If trhe meaning of the complex term can be elicited by way of 'composition' from the meaning of the primitives, the meaning of these latter must be somehow built-in, i.e. not expressible and not computable in the language. In logic, algebra and programming languages as well this means the existence of syntactic structures (respectively "logic constants", "algebraic operations and predicates" and "elementary instructions") whose meaning can not be 'interpreted' and whose working is "given" (by rules, axioms or compilers).

The first, 'naive', idea of representation that we find in Parmenides' release 1.0 shows problems which are the core of the Platonic dialogues, and in such problems we recognize the special role of the mathematical entities. The connection between the concept of number and the structuring of the Platonic dialectics was at the center of a well known enquiry of Jakob Klein [7] in 1934. This was a deep insight, even if the strategic role of 'number' was read in a Husserlian mood as a conquest paid with the lack of the 'natural' (i.e. 'cardinal' in the husserlian approach) idea of number ([7], 99). We are going to show instead as such 'lack' was, more than a 'price', the conquest of the idea of ordinal number and the necessary condition for the beginning of formal thinking and of the same modern mathematics.

The reference now must be to the Platonic world of forms, built to connect reality and language. We underlined above that this view modeled the forms as amphibious between an 'iconic' mimesis, a kind of "images", and a 'syntactic' mimesis, substantially sort of "words". Hence in Theaetetus, the "dialectic knowledge", dianoia, word stemming from a verb of 'visual' nature, noew, becomes a logoV , the "dialogue of the soul with itself". In the world of forms the idea of good 'enlightens' like the sun, and this is a 'pictorial' metaphor, but in Phaedo (73d-75d) we discover that the relationship equal (isoV), which we know syntactic in nature, seems to play a special role: it is the ideal touchstone of that relationshpi of similarity (omoioV) connecting the real objects which 'partake' in the same form: it is a sort of meta-form. To this extent meJexiV is based on the one hand on an 'iconic' characterization of similarity, on the other hand on a 'syntactic' characterization of equality. This ambiguity is necessary to overcome the problems of the "syntactic representation" we underlined at the beginning of this paper. To make it even sufficient Plato had to look for a solution of the connected paradoxes.

In order to deal with 'truth' and 'being', the "forms" must be immune from change and contradiction, but, in order to deal with the knowledge of reality, they must account for becoming and contradiction. "Something contrary (enantia pragma) must be generated from a contrary thing, but a contrary in itself (auto to enantion) can never become the contrary of itself" (Phaedo 103b).

Theaetetus is where Plato faces the problem of error. In 'iconic' representations 'error' can be a difference between "perception" and "memory", between what I see and what I remember, imprinted in a memory structured as a 'block of wax': I see Socrates but I recognize the figure of Theaeteus.

However, how it is possible adding five and seven finding eleven nstead of twelve? We are wrong even though numbers are perfectly memorized and there are no perceptions. If the whole knowledge is positive, only perception and memory, and not being is just difference, how does error appear when it is impossible to speak about a difference between perception and memory or between memories? To 'grasp' the wrong memory is an error on its own. At the end you have to recognize in the subject and his memory, beside knowledge, even not-knowledge (anepisthmosunaV), and this is again the problem of the 'negative'.

Once again it is 'number' the place where the iconic mimesis displays its weakness in grounding a theory of knowledge, and it is error, not-being, negation, paradoxes the point where the Platonic release halts on the way toward the syntactic mimesis. We already underlined that the Sophists' paradoxes could be distinguished in two schemes: the first connetced to the relationships many-predicated-forms in one-object or one-predicated-form in many-objects which involves the meJexiV; the second linked to the binary relatioships, among which the numeric ones: 'equal', 'different', 'greater'.

The analysis of such paradoxes reveal the peculiar role played by the mathematical concepts: once again the starting point is the difference between a purely 'visual', 'analytic' and structurally always 'positive' knowledge and a 'holistic' and 'paradoxical' one.

if you can discern that some reports of our perceptions do not provoke thought to reconsideration because the judgement of them by sensation seems adequate, while others always invite the intellect to reflection because the sensation yields nothing that can be trusted. …The experiences that do not provoke thought (ta ou parakalounta) are those that do not at the same time issue in a contradictory perception. Those that do have that effect I set down as provocatives, when the perception no more manifests one thing than its contrary, alike whether its impact comes from nearby or afar. …. Naturally, then, it is in such cases as these that the soul first summons to its aid the calculating reason and tries to consider whether each of the things reported to it is one or two. And if it appears to be two, each of the two is a distinct unit. If, then, each is one and both two, the very meaning of 'two' is that the soul will conceive them as distinct. For if they were not separable, it would not have been thinking of two, but of one. Sight too saw the great and the small, we say, not separated but confounded. And for the clarification of this, the intelligence is compelled to contemplate the great and small, not thus confounded but as distinct entities, in the opposite way from sensation. And is it not in some such experience as this that the question first occurs to us, what in the world, then, is the great and the small? By all means. And this is the origin of the designation "intelligible" for the one, and "visible" for the other. This, then, is just what I was trying to explain a little while ago when I said that some things are provocative of thought and some are not, defining as provocative things that impinge upon the senses together with their opposites, while those that do not I said do not tend to awaken reflection. (Respubblica 523b-524c)

In the vision of a finger there can not be any contradiction, neither in far objects or pictures, simply because you can not see the negative of something. The canonic place of 'negative' and then of 'contradiction' is in the relationship and hence in the judgement of a comparison: to be at the same time 'great' and 'little', 'hard' and 'soft', etc.. Syntactic representation is sharply different from the iconic one, first and foremost in its being the place of birth of the reason facing negation. And hence contradiction is the cradle of thinking.

But: what the role of number?

"To which class, then, do you think number and the one belong?" "I cannot conceive," he said. "Well, reason it out from what has already been said. For, if unity is adequately seen by itself (auto kaJ''auto) or apprehended by some other sensation, it would not tend to draw the mind to the apprehension of essence, as we were explaining in the case of the finger. But if some contradiction is always seen coincidentally with it, so that it no more appears to be one than the opposite, there would forthwith be need of something to judge between them, and it would compel the soul to be at a loss and to inquire, by arousing thought in itself, and to ask, whatever then is the one as such (auto to en), and thus the study of unity will be one of the studies that guide and convert the soul to the contemplation of true being." "But surely," he said, "the visual perception of it does especially involve this. For we see the same thing at once as one and as an indefinite plurality. " "Then if this is true of the one," I said, "the same holds of all number, does it not?" "Of course." "But, further, reckoning and the science of arithmetic are wholly concerned with number." "They are, indeed." "And the qualities of number appear to lead to the apprehension of truth." "Beyond anything," he said. "…."It is befitting, then, that this branch of learning should be prescribed by our law and that we should induce those who are to share the highest functions of state to enter upon that study of calculation and take hold of it, not as amateurs, but to follow it up until they attain to the contemplation of the nature of number, by pure thought, not for the purpose of buying and selling, as if they were preparing to be merchants or hucksters, but for the uses of war and for facilitating the conversion of the soul itself from the world of generation to essence and truth…. that it strongly directs the soul upward and compels it to discourse about pure numbers (autwn twn ariJmwn), never acquiescing if anyone proffers to it in the discussion numbers attached to visible and tangible bodies. For you are doubtless aware that experts in this study, if anyone attempts to cut up the 'one' in argument, laugh at him and refuse to allow it; but if you mince it up, they multiply, always on guard lest the one should appear to be not one but a multiplicity of parts. Suppose now, someone were to ask them, 'My good friends, what numbers are these you are talking about, in which the one is such as you postulate, each unity equal to every other without the slightest difference and admitting no division into parts?' What do you think would be their answer?" "This, I think--that they are speaking of units which can only be conceived by thought, and which it is not possible to deal with in any other way." "You see, then, my friend," said I, "that this branch of study really seems to be indispensable for us, since it plainly compels the soul to employ pure thought (auth th nohsei) with a view to truth itself (Respubblica 524d-526a)

The numbers must face the problem of contradiction, and they have its key. At the same time only the 'soul' can know them, because it is the subject of the syntactic representation, to the same extent that the eye can know not numerical and not contradictory aspects because subject of the iconic representation. The one/many paradoxes are the key to enter in the dialectic of forms and the heart of the theoretical arithmetic as well.

On the other hand a static meJexiV, where 2 is even just for its partaking in the idea of evenness, would give an 'iconic' representation of reality:

Sure, no contradiction. However not able to account for becoming, and then for reality.

We must be able to deal with the very nature of the mathematical entities, through arithmetic, at the same time ideal and necessary to account for the contradictory expression of reality:

This power of giving to the world of forms its dynamical features characterizes the mathematical concepts, so that many 'monads' together, say 'five', create a numerical concept, the number 'five', that was not already included in the earlier idea of monad: the holistic nature of the concept of number is essential to overcome the 'analytical' nature of the simple "iconic representation". With the words of Klein [7]:

In the world of forms the "numers in themselves", ariJmoi autoi, are not just 'ideas', existing in some iperuranious world, as in a certain 'platonism' so common among mathematicians; they are instead the real ground of formal thinking, its deep structure. It is clear, in the VI book of Respubblica, that the mathematical method is just a 'static' version of the dialectic method, because it can not avoid the presupposition of those 'principles' which are instead the goal of the latter. At the same time the mathematical objects appear as 'images' with respect to the world of forms: here is the problem of the "ideal numbers" (ariJmoi eidhtikoi), beyond the mathematical and the 'material' ones, whose ordinal character appears in their genetic nature (born from One and the 'infinite dyad', aoristoV duaV), that will be objects of different interpretations by Plato's followers in the (Xenocrates, Speusippus) and of the relative Aristotelean criticism.

Even the nature of continuum, the more-less paradoxes, can be overcome by the 'denoting' on the continuum and its flowing guaranteed by the 'number', true prototype of the 'sign'. In Philebus we read the events before the gift of the alphabet from the God Theuth: voice is continuous, one and infinite at the same time:

Sound, which passes out through the mouth of each and all of us, is one, and yet again it is infinite in number (apeiroV au plhJei). And one of us is no wiser than the other merely for knowing that it is infinite or that it is one; but that which makes each of us a grammarian is the knowledge of the number and nature of sounds. And it is this same knowledge which makes the musician. Sound is one in the art of music also, so far as that art is concerned. And we may say that there are two sounds, low and high, and a third, which is the intermediate, may we not? But knowledge of these facts would not suffice to make you a musician, although ignorance of them would make you, if I may say so, quite worthless in respect to music. But, my friend, when you have grasped the number and quality of the intervals of the voice in respect to high and low pitch, and the limits of the intervals, and all the combinations derived from them, which the men of former times discovered and handed down to us, their successors, with the traditional name of harmonies, and also the corresponding effects in the movements of the body, which they say are measured by numbers and must be called rhythms and measures--and they say that we must also understand that every one and many should be considered in this way-- when you have thus grasped the facts, you have become a musician, and when by considering it in this way you have obtained a grasp of any other unity of all those which exist, you have become wise in respect to that unity. But the infinite number of individuals and the infinite number in each of them makes you in every instance indefinite in thought and of no account and not to be considered among the wise, so long as you have never fixed your eye upon any definite number in anything….I will, when I have said a little more on just this subject. For if a person begins with some unity or other, he must, as I was saying, not turn immediately to infinity, but to some definite number; now just so, conversely, when he has to take the infinite first, he must not turn immediately to the one, but must think of some number which possesses in each case some plurality, and must end by passing from all to one. Let us revert to the letters of the alphabet to illustrate this…..When some one, whether god or godlike man,--there is an Egyptian story that his name was Theuth--observed that sound was infinite, he was the first to notice that the vowel sounds in that infinity were not one, but many, and again that there were other elements which were not vowels but did have a sonant quality, and that these also had a definite number; and he distinguished a third kind of letters (grammatwn) which we now call mutes. Then he divided the mutes until he distinguished each individual one, and he treated the vowels and semivowels in the same way, until he knew the number of them and gave to each and all the name of letters (stoiceion). Perceiving, however, that none of us could learn any one of them alone by itself without learning them all, and considering that this was a common bond which made them in a way all one, he assigned to them all a single science and called it grammar. (Philebus, 17b-18c)

Noteworthy here the opposition of two terms both translated as 'letters': grammata are the letters as 'traces', cut, engraved, stoiceia are the letters as 'signs', which denote and distinhguish considered as a whole, in the alphabet. These characterize the vocal flow as a language. We must underline the similarity between this 'denoting and distinguishing' and the diairesiV, the progressive dicotomy of the conceptual categories, that Plato employs for example in the Sophist to define the 'sophist'.

That voice, fwnh, oral speech, that in Plato is a continuous magnitude, made discrete by the creative act of Theuth to create the world of signs, will become in Aristotle (Categoriae 4b 30) tout court discrete and the sign will become a social convention, and this will mark, by the written medium, the complete interiorization of the alphabet as model and paradigm of theoretical knowledge, so that stoiceion is 'element' of the alphabet, of the axiomatic system and of the same reality as well, last 'element' in the three worlds of the semiotic triangle.

And this original ability of number to denote and single out in the reality, to turn the traces into signs, is the secret of creation and its human knowledge:

Here Plato faces the trace of a not 'iconic', but, say, model-based, Pythagorean tradition, developed during their, first anf foremost Archytas', studies about musical harmony, astronomy and proportions. The dialogue where this trace is most clear is Timeus, where we read the first statement of a 'preestablished harmony' between the 'sky orbits' and the analogous ones in the human mind (47 b-c), necessary condition for a knowledge of the 'laws' that rule nature.

This deep 'complicity' between arithmetic and dialectics is going to vanish in the following evolution of Greek philosophy: Aristotle's criticism (for example in the XIII book of Metaphysics) will be centred most of all on this 'confusion', for example between "one" as number and as logical item, and will lead to the reduction of dialectics to a logic 'organon', and to the reduction of arithmetic to a special science of discrete quantity, with the following return of the earlier 'cardinal' character of the "mathematical number", i.e. as determined multiplicity.

This connection between signs in formal thinking and numbers, with the following beginning of a ars combinatoria, will appear again in Leibniz and will become the ground of modern logic and mathematics: from Gauss to Hilbert and Gödel the connection between logic and arithmetic and then the arithmetization of syntax are cornerstones of formal thinking.

However, this connection is usually considered just as a complex 'coding', a genial 'trick'. Arithmetic had to behave as a 'disguised logic' or as a axiomatized mathematical theory: anyway reducible to or 'other' from the pure logic. But Gödel [8]wrote:

The 'miracle' is in that arithmetic, and computability, is not one of the many formal languages, but the "mother" of all the syntactic representations, 'consubstantial' to the very possibility of a 'formal' thinking. Something like this has already been stressed in the debate about the foundations of mathematics: intuitionism underlined the 'genetic' role of arithmetic and ordinal numbers (for rxample: the "induction principle") with respect to formal logic. It is useful to underline that this thesis in our analysis is not the consequence of a philosophy of the a-prioris of the transcendental subject, but is the result of a creation of classic Greek culture, and, most of all, the origin of the european culture itself.

Hence Arithmetic is the source of that preestablished harmony between reality and language that we can not not believe after almost four centuries of astonishing achievements, but we must even say that, neither tendentially, syntactic representation can thoroughly mirror reality, become someway iconic. And this because it is marked in its basic principles with a preestablished disharmony, that is even its hidden evolutive principle.

It plays the role of source of never ending paradoxes well recognizable ever since the beginning of formal thinking. Negation, truth and being ground an antinomical argument, from the "negative judgement paradox" (impossibility of asserting falsity), through the "liar paradox" (contradictory nature of self-asserting falsity), to set-theoretical paradoxes and to Gödel's and Tarski's limitative theorems.

There is no 'ignorabimus', but an essential and changing 'incompleteness', never-ending and changing 'breakdown' in our capacity of 'saying' the world, that makes our scientific work not the patient reconstruction of an image of reality, but a wonderful adventure not only 'outside', but even 'inside' us.

REFERENCES

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[2] BORZACCHINI, L. "Light as a metaphor of knowledge. A preestablished disharmony.". in: PETRILLI, S. and PONZIO, A. edts. Semiotica, special issue. In way of publishing (1999)

[3] LEVI-STRAUSS, C. Le cru et le cuit. Paris, Libr. Plon., 1964

[4] LURIA, A. R. La storia sociale dei processi cognitivi. Firenze, Giunti-Barbera, 1976. (orig. russian ed.: 1974)

[5] DIELS, H. and KRANTZ, W., Ed. Die fragmente der Vorsokratiker. Zurich, Weidmannsche Verlag, 1903-1966.

[6] KAHN, C. H. "The verb 'be' in ancient Greek." Foundations of language, suppl. series, 16 (1973)

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[8] GÖDEL, K. "Remarks before the Princeton bicentennial conference on problems in mathematics." in: Klibanski edt. Contemporary philosophy. Firenze, La Nuova Italia, 1968