Luigi Borzacchini Department of Mathematics, University of Bari, Italy
Our science, with its modernity and technology, shows in the ruptures of its paradoxes and limits of the knowledge its very ancient roots, deeply embedded in the hearth of old Greek thought. The syntactic paradigm was the result of the Platonic and Aristotelian foundation, and developed in its 'strong' form at the beginning of 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 deepest structure of the link between human knowledge and reality.
Die objective Welt ist schlechtin, sie geschieht nicht. Objective world simply is, it does not happen.([WEYL, 1926 #221], 83)
This report concerns with those deep characters of modern science, most of all logic and physics, through which we can read in transparency the traces of ancient paradoxes and problems concerning being, negation, infinity and truth. These themes in the ancient Greek philosophy and mathematics were the core of the Aristotelian foundation against presocratic philosophers and their celebrated (Zenonian, Sophist, Megaric) paradoxes, and their analysis was developed in the first two reports. It is not easy to justify now a jump of two thousand years in an historical reconstruction, but I think that the years between Aristotle and modern science have been characterized by a relatively well-established framework, centred on the Aristotelian version of the syntactic paradigm, introduced and discussed in the first report. It would be false to say that in-between there were no antinomies (we can remember for example the Kantian ''antinomies of pure reason'' or the medieval ''insolubilia''), deeply embedded in theological and philosophical medieval thinking, but I think that they did not play that foundational, both philosophic and scientific, role they played in Greek culture and that they are playing in modern science. We will underline furthermore, at the end of this report, that the paradoxes we are dealing with do not concern with semantic conceptual antinomies (about God for example), but with contradictions arising from the simplest linguistic primitives (being, truth, not, etc.). Maybe the distinction is not very sharp: in fact, it is not to ignore the relevance of the theological debate for the development of the modern philosophy and science, and it is not to ignore as well the formal aspects of the theological paradoxes. So, for example, the birth of the modern idea of ''actual infinite'' can be probably ascribed to the development of the idea of God, and the idea of God was influenced by the idea of infinite developed in the late Greek philosophy: the link is thoroughly explicit from Descartes to Cantor, whose first aspect of actual infinite was the absolute to be found in Deo, in God. In addition, by the idea of God the mental world becomes the world of the objective knowledge. However, this slow and long development eventually gave rise, in the XVII century, to a substantially only 'positive' thinking, without paradoxes, and finally brought the science to its modern triumphs until, within the XX century, a new wave of antinomical arguments suddenly appeared. This time they seemed deeply rooted and hardly disposable after some conceptual clarification. Since the beginning these antinomical arguments showed their similarities with the ancient Greek ones. And it is possible, to show the diffused awareness of this heritage, to list many modern thinkers, philosophers, logicians and physicists, who found the reference for present logic and quantum-mechanic antinomies directly in those old examples.
What about these paradoxes? We could hope to solve them with some 'adhoc' clever device. But the 'liar' paradox seems to stand up easily against any attempt of solution, the 'hidden variable' theories do not seem able to deal with the quantum mechanics puzzles, and Goedels incompleteness arguments have in themselves the stated impossibility of simple adhoc solutions. We could think that these questions are not actually relevant. In fact, physicists employ quantum mechanics without losing their substantial realism, mathematicians use formalised theories without being upset by their incompleteness, and people argue about truth and utter self-referential sentences without any troubles. This sounds reasonable. Unless we do not believe that these antinomies are the tip of an iceberg, whose size in our culture is far wider than the simple paradoxes utterance. This belief is our starting point.
These reports claim such a wider hidden extent of the paradoxical nature of our thinking as due to an unanalysed role and weight of the sign in our culture, with effects reaching any aspects of our western civilisation. I stress the role of the signs and their syntax, as something irreducible both to ideas and experience. Both idealism and materialism always underestimated the role of signs. It is rare to find a different approach. However, there exists a 'red thread', from Leibniz to Cassirer and Wittgenstein, in which it is possible to recognise the attempt to define a major role for the signs systems.
We regard understanding as the essential thing, and signs as something unessential. But in that case, why have the signs at all? If you think that it is only so as to make ourselves understood by others, then you are very likely looking on the signs as a drug which is to produce in other people the same condition as my own.([WITTGENSTEIN, 1974 #165],I.2)
Also Nietsche in his Froeliche Wissenschaft remarked that signs determine things. Signs are so common in our life that we give them for sure, trivial things. But for thousands years signs played a very poor role (or no role at all) in human life, so there is nothing natural or obvious in their emergence. This is the problem: Why the signs? And, why are the signs becoming more and more important in our civilisation? And, why is the computer, the 'syntactic' machine, becoming today so relevant in our society and culture? I think that the autonomous role of signs has become more and more relevant in the history of mankind, so much to characterise the contemporary age as the realm of syntax. And, understanding the deep role of the signs systems is crucial to understand the inner architecture of our culture.Also the first book of Marx' Capital can be read as the analysis of the development of the ''syntactic paradigm'' in political economics. Beginning with the truly 'semantic' exchange, through the 'syntactic' money, based on the idea of a ''general equivalence'', until the thorough autonomy of the value-increasing process of the capital. In more recent times, the increasingly financial nature and the progressive tranfiguration of the 'money' as a simple 'sign' in a formal language, let us realize that there is nothing casual in today's centrality of the computer, the 'syntactic' machine, in modern economics and society as well.
This role is somehow outlined in the syntactic paradigm. In this report our aim is to show the paradoxical nature of this paradigm. After a short outline of its features, we analyse its occurrences in logic and quantum mechanics. Then, we try to find the common features of the corresponding occurrences of antinomical arguments, and to give an interpretation of this paradoxical nature. The connection of this enquiry with the themes of Artificial Intelligence and the mind/body problem, will be the object of another report.
| 1. The syntactic paradigm.
|| 2. Wittgenstein and Parmenides.
|| 3. XX century paradoxes.
|| 4. Logic and mathematics in the syntactic paradigm.
|| 5. Quantum Mechanics.
|| 6. The ''never-ending paradox''.