Recently Rosen (ROSEN 1987) wrote:
If I had to characterise 20-th century science I would call it the 'Age of Syntactics'. ... (a) it is supposed that any system can be thus formalised, and (b) the formalization itself, which is a purely syntactic object, from which every trace of semantics or meaning has been removed, is fully equivalent to the original system that it formalises. Thus formalization 'loses no information' present in the original system, and hence is only a reformulation, not an abstraction.
This quotation, with the related figure (Fig.1), can well characterise the "syntactic paradigm", as denoted by Rosen, which could be found elsewhere with different names, called "formal" or "neo-positivist" or "logic-positivist", but in any case recognised as the core of the modern scientific methodology.
History of science shows many examples to support Rosen's hypothesis. However, the same examples show some differences between this current paradigm, that we could call the "strong syntactic paradigm", and an earlier one, the "weak syntactic paradigm" (Fig.2). The shift between them can be set at the end of the last century, and represented by Ernst Mach and the "logic positivism". The "strong" version could be approximately identified with the one described by Rosen. In the earlier form of the paradigm the absence of "abstraction" claimed by Rosen has to be examined, and we had to consider the role played by models and mental constructs. For example at the beginning of a classic text of modern science, Hertz' Die Prinzipien der Mechanik, we read:
We build our mental images or symbols of the external objects, and we do it so that logical (denknotwendigen) consequences of the images are the images of the real (naturnotwendigen) consequences of the represented objects. As far as this assumption is accomplishable there must be sure concordances between nature and our mind (Geiste)((HERTZ 1894),1)
To discuss the figures we use Rosen's terminology: ''material system'' and ''causality'' denote the objects of the scientific enterprise, the states and their evolution, ''formal system'' and ''implication'' denote the relative theoretical representations, the signs representing states and their features, and the formulas or rules by which to represent the evolution of the system. Obviously, such representation is ''correct'' if there is a correspondence between these two worlds preserved by the 'dynamic' correspondence between the real causal evolution of the system and the formal evolution described by the implication laws. In Das Kontinuum, Weyl pointed out how the concept of natural law (Naturgesetz) was the connection between the physical concept of natural-factual dependencies (naturgegebenen Abhaengigkeiten) and the mathematical concept of algebraic-arithmetic operations (aritmetisch-algebraischen Operationen) ((WEYL 1917), 34), and in Philosophie der Mathematik und Naturwissenschaft ((WEYL 1926), 15) underlined the link between the logic implication and the causal relationship, that can be both represented by the ''neutral'' sign -- >. This connection, based on the idea of ''natural law'', is deeply embedded in modern science ever since Galilee and Descartes.
In fig.2 we introduce a 'third world' as a bridge between the ''reality'' and its ''formal representation'': a semantic model, the 'physical model' of classic physics, such as the 'planet' model of atoms, the 'wave' and the 'particle' model of light, the 'fluid' and the 'kinetic' model of heat. In this 'classic' paradigm, scientific enterprise created an analogical link between the system under analysis and earlier, already more or less formalised, pieces of (mechanical) science or 'common sense' experiences, and so fostered the formal coding of the new system. It is important to point out that such analogical link was both a tool for the learning of more complex topics and the root of the same evolution of science: as a 'pump' of possible laws or concepts for a new theory (the 'ether' was a consequence of the 'wave' model of light) and as the privileged 'topos' of the inconsistencies of the theory (so the electrons in the 'planet' model of atoms had to collapse, in contradiction with the 'common sense' stability of the matter). Such a role of the model allowed to design the so called gedanken-experimenten and the related paradoxes as valid 'moves' of the scientific 'play' (Boyd (BOYD 1979)). In the employment of 'thought' experiments we can reveal the trace of another crucial aspect of the modelling: the facility of 'locating' the ''observer'' in a fictional environment and then allowing his autonomous counterfactual reasoning.
Analogy stressed hypotheses and forecasts (a spinning electron must emit electromagnetic radiation, any wave must be relative to some elastic substance, and so on), whose possible disagreement with reality could cause crises and often foster radical restructuring of the theories. The most important contradictions were inner to the semantic world: explicitly (for example between galilean relativity, for which any speed depends on the reference framework, and Maxwell laws, for which the speed of light is the same in any inertial framework) or deduced by gedanken-experimenten (Einstein-Podolsky-Rosen argument against Bohr's interpretation of quantum mechanics). The semantic world can be employed also as an environment to get explicitly new results: Einstein's ''simultaneity'' analysis was the starting point of relativity theory, Heisenberg's analysis of the measurement process gave the ''indeterminacy'' relations, Stevin gave a gedanken-experiment 'proof' of the slope acceleration formula and Newton's and Mach's 'spinning bucket' argument ((MACH 1883), see table 1). All these are examples of a practice of the semantic reasoning by which it produced not only inconsistencies but also coherent extensions of the theory.
Together with the above mentioned 'structured' or 'compositional' character of the three worlds, the 'correctness' requirement addresses the functional nature of the paradigm. That is, we must underline that the ''functionalism'' is a constituent part of the ''syntactic paradigm'' since its beginning (and this will be seen in more detail later). Our use of the term ''functionalism'' is homogeneous to Block's (BLOCK 1982) 'progressive' characterisation :
(1) functional analysis. In this sense of the term, functionalism is a type of explanation, and, derivatively, a research strategy, the research strategy of looking for explanations of that type. A functional explanation is one that relies on a decomposition of a system into its component parts; it explains the working of the system in terms of the capacities of the parts and the way the parts are integrated with one another. ...(2) Computation-representation functionalism. In this sense of the term, functionalism applies to an important special case of functional explanation as defined above, namely, to psychological explanation seen as akin to providing a computer program for the mind. Whatever mistery our mental life may initially seem to have is dissolved by functional analysis of mental processes to the point where they are seen to be composed of computations as mechanical as the primitive operations of a digital computer - processes so stupid that appealing to them in psychological explanation involves no hint of question-begging. The key notions of functionalism in this sense are representation and computation. Psychological states are seen as systematically representing the world via a language of thought and psychological processes are seen as computations involving these representations. (3) Metaphysical functionalism. The last functionalism ... is a theory of the nature of the mind, rather than a theory of psychological explanation. Metaphysical functionalists are concerned not with how mental states account for behaviour, but rather with what they are. The functionalist answer to ''what are mental states?'' is simply that mental states are functional states. Thus theses of metaphysical functionalism are sometimes described as functional state identity theses.
We could add a (0) Technological functionalism, not a theory but a fact: the technological achievements for which we can produce a potentially infinite number of 'equal' objects, that is objects that can be substituted in a system without changing its performance. It could be called the philosophy of spare parts. This fact appear with modern world and industry, has triumphed in this century and was almost unknown in the ancient world. The link between craftmanship and Platonic philosophy has been underlined in (CAMBIANO 1991), and we can remind Plato's Cratylus, 389 a-c, where the 'shuttle in itself' is the craftman's model of any shuttle and the 'ideal form' of the tool as well. In modern philosophy the themes of this paradigm are widespread, and shared by different schools.
Cassirer's philosophy of symbolic forms ((CASSIRER 1923)) theorised direct connections between real world and symbolic world by the concept of ''function'', so accomplishing both the idea of a ''book of the nature'' written in mathematical terms (geometric figures in Galileo, formulas after Descartes and Leibniz), and the Kantian a-priori foundation of knowledge. And neo-positivist philosophy claimed the same connection to accomplish Leibniz's idea of a reduction of reason to computation. Cassirer's philosophy is probably the thoroughest attempt of analysing the symbolic structure of modern science: the symbolic construction guarantees the forecasting function of science which no 'pure phenomenology' could allow, and it is just the refusal of any 'immediate' meaning which enhances the suitability of the sign in the measurement process. Also Weyl ((WEYL 1926), 107) underlined that connection between measurement and signs: ''the ground of the measurement processes is only the symbolic representation''. This can seem clearer reminding that equality does not exist among real objects, but is the ground of any signs manipulation, and that measurement can be not only a 'shop' activity, but also the cornerstone of an empirical science only by the possibility of repetititions of the same experiment under equal conditions and with equal experimental arrangements. Thus, the features of this paradigm are deep in our science and can then be traced back, well at the beginning of modern science and culture, but we will display its earliest roots, in the classic Greek philosophy.
Another aspect of the paradigm, which we can recognise in the ''mental models'' employment, is the role of the Self. Its appearance is a crucial feature of the ''syntactic paradigm'': it is the ''mind'' as a whole, dwelling in the mental world and there able to stay, move, act, measure, judge. Thus, the 'breakdown' of the 'semantic' world which we recognise in physics at the end of the last century, is parallel to the vanishing of the 'irreducible' role of mind in the same period : on the one hand, this process begins with Boole, continues with Frege and ends with the computer triumph, thus accomplishing the reduction of the 'rational' activity to the 'syntactic' world. On the other hand, the process appears in the reduction of the mind to the brain, thus giving birth to the modern ''neuropsychology''.
In relativity theory we can underline the breakdown of the special role of the inertial coordinates systems, where ''the coordinate system is the unavoidable residual of the Self vanishing (Ich-Vernichtung) in that 'geometrical-physical' world, which the Reason draws out from the facts (Gegebenen) unter the norm of Objectivity'' (Weyl, Das Kontinuum, (WEYL 1917), 72), and, in parallel to this indifference to the Self, the 'coincidence' between physical space/matter and its metric geometry, i.e. between reality and its formal embedding. Thus, the ''strong syntactic paradigm'' (fig.3) is today the dominant approach also in the area of Artificial Intelligence, as methodological characterisation of the very idea of ''knowledge representation'':
Any mechanically embodied intelligent process will be comprised of structural ingredients that a) we as external observers naturally take to represent a propositional account of the knowledge that the overall process exhibits, and b) independent of such external semantical attribution, play a formal but causal and essential role in engendering the behaviour that manifests such knowledge. (SMITH 1985)
and in the 'folklore' use of the term 'semantics' in knowledge representation literature, showed by expressions like 'to capture the semantics', which means ''to add additional syntactic conditions to constrain the data structures towards the intended meaning'', by reducing thus their set of ''models''. This 'reduction to syntax' is a strong drift in computer science: recently Kowalski (KOWALSKI 1994) explicitly claimed the uselessness of model-theoretic issues in logic and logic-based knowledge representation, and Kirsh (KIRSH 1991) underlined as model theory is not ''a theory of intentionality or a theory of meaning'', ''says nothing about the appropriateness, truth or utility of axiom sets'' and displays a complete brittleness under ''inconsistency'' (the presence of any inconsistency precludes any knowledge, also in areas thoroughly independent from the inconsistency) .<Note: Some authors support the so called 'semantic attachment' as an opposite view, i.e. the substitution of inferential procedures by computations in the 'intended' model. However, this does not a real 'opposite' view, for it is first of all a sort of ''compilation'' to get a more efficient 'procedural' solution from a substantial 'declarative' system, and second, as in the before mentioned 'semantic capturing', it is always a purely 'syntactic' technique. >
However, the most relevant aspect of the overwhelming role of the 'strong' paradigm in computer science is maybe the ''language of thought'' thesis (Fodor (FODOR 1975), but we must point out also above Block's quotation), based substantially on the idea of a 'purely syntactic' reduction of the human thought, and appearing as the most explicit background of the ''strong AI paradigm'' criticised by Searle (SEARLE 1984). This 'syntactic' reductionism took in recent years the place of the earlier 'physical' reductionism: the ''intentionality'' of human understanding advocated by Searle in the ''great debate'' against the strong AI paradigm is not so far from Einstein's critic arguments against Bohr's interpretation of quantum mechanics and the ''intuitionistic'' critique against ''formalism''. All of them tried to underline the role of the third semantic world, not as mechanistic or philosophical a-priori, but as necessary meaning presupposition for any formal or syntactic representation. Furtherly, Einstein and Searle as well used in their arguments the technique of gedanken-experimenten, and we could fairly apply, mutatis mutandis, to Searle and his critics what Jammer (JAMMER 1974) wrote about Einstein and Bohr:
...for Bohr these thought experiments were not the reason but the necessary consequence of a much more profound truth underlying the quantum mechanic description and, in particular, the uncertainty relations. Bohr consequently had the advantage that, from this point of view, he was justified in extending the chain of reasoning until he could approximately resort to the indeterminacy relations to support his thesis. Einstein on the other hand, had the advantage that if he could disprove the Heisenberg relations by a closer analysis of the mechanics of one single thought experiment, Bohr's contention of the incompatibility of a simultaneous causal and space-time descriptions of phenomena and with it his whole theory would be refuted.
Bohr's epistemology is not thoroughly accepted by the whole quantum mechanics world but is the most coherent philosophical background for the universally accepted Copenhagen interpretation. Similarly the ''strong AI hypothesis'' is not universally accepted in the AI world, but is strenuously defended as the secret core of that research program, when criticised. In our century we have been witnesses of the triumph of the gedanken-experiment 'move', with relativity and quantum mechanics theories, and its decline today with the Einstein-Bohr debate or Searle's 'Chinese room' (SEARLE 1984). I say 'decline' not for the reduced interest of such debates, but for their 'effectiveness', since the lost of the role of the semantic world makes gedanken-experimenten quite meaningless, being now more and more the only suitable 'inconsistency' the logic one. These issues will be analysed in the last reports.
To underline the differences between weak and strong paradigm we must observe that: i) the ''strong syntactic paradigm'' removes the semantic model leaving just the formal representation. It reflects substantially the Machian and neo-positivist critique against ''mechanism'', unconscious metaphysical postulates and philosophy. We can say that the aim of the whole methodological debate at the beginning of this century was the suppression of the semantic world, paving this way the road for a 'radical' quantum mechanics foundation, based on a purely symbolic procedure and a complete experimental arrangement, as claimed by Bohr. ii) in the ''weak paradigm'' all the terms of the formal alphabet have a semantic counterpart, and hence 'must' be interpreted in the material system, whereas in the ''strong paradigm'' the non-observable properties are uninterpreted terms and the calculus is then only partially interpreted. For example, the wave function y in quantum mechanics is not observable and then uninterpreted in the Copenhagen interpretation, and the ''working parameters'' in the software are 'hidden' to the I/O functions. Besides, good programming practice stresses such ''information hiding''. iii) in the ''strong paradigm'' we have to distinguish between 'spontaneous' and 'driven' evolution of the system, between 'objective description' and 'subjective questioning' of the theory: the former aspects are represented by causality and implication, as in the ''Schroedinger equation'', the latter ones require an explicit reference to an external 'subject', as in the ''wave-packet reduction''. iv) some room for semantic models can also be given in the ''strong paradigm'' for learning or for 'unconscious' aspects of the ''discovery context''. However, this means to create a substantial difference between the 'asemantic' structure of science and its 'semantic' process of production and reproduction. The gap between science and lebenswelt is one of the price to be paid for the strong syntactic paradigm. v) the mental world becomes a piece of the real world too, and enter the range of the formal representation: this substantially is the background of the ''Artificial Intelligence strong hypothesis''.
The ''three-worlds'' representation we have analysed recalls a classic scheme of our scientific and philosophical culture, sometimes called ''semiotic triangle'', and whose origin can be traced back to Aristotle (fig.4). Its modern version can be found in Frege, but it is a 'topos' of modern culture 'turn' toward the linguistic fact, as pointed out also in Weyl((WEYL 1971), 20):
As in a central point we can find here the greatest philosophical problems: the relation between 'objectual reality'(Sachverhalt) - 'thinking' (Gedanke) - 'expression' (Aussage), where any attempt to find certainty in some Realismus of pure being wrecked.
I consider such diagram, with its functional inner architecture (we had to call it the ''syntactic-functional paradigm''), the 'red thread' we have to follow in our analysis. The first world is the reality register whose functional structure is deeply embedded both in our science and in our technology and industry; the second world is the signs or language register, whose compositional structure characterises both formal (logic and mathematics) and natural languages; the third world is the less well-defined, consists of ideas and concepts, maybe in 'visual' forms, and is the mental images or ideas register. If we give an 'objective' role to the ideal world, in Frege's terminology the sides from the language world to the mental and real worlds are respectively the Sinn (intension, meaning) and Bedeutung (extension, reference) of a given 'word'. The sides represent, if we consider the scientific or formal 'version' of the triangle, homomorphism relations, at least as regards the connection with the reality world, because the correspondence between elementary objects extends univocally to a correspondence between complex objects, according to the ''compositionality'' of the Tarskian extensional semantics. It is noteworthy that the compositional and extensional character of meaning for formal languages is substantially rejected for natural languages by most of linguists, who rather would underline intensional and holistic characters of semantics, following a tradition as old as Wilhelm von Humboldt's (von HUMBOLDT 1988):
...language never represents the objects, but always the concepts that the mind has spontaneously formed from them in producing language. (von HUMBOLDT,84) From the first elements onwards the production of language is a synthetic procedure, and that in the truest sense of the word, where synthesis creates something that does not lie per se in any of the conjoined parts. ... The complete synthesis we are talking of does not proceed from details, but from the whole composition and form of the language. (von HUMBOLDT, 88-89) Also Frege underlines this difference: in natural languages to analyse the meaning of a term is useless, but we have directly the meaning of the proposition including the term; in his Begriffschrift instead, any elementary symbol must be given a rigorously unique meaning. This thesis is to compare with Tarski's remark about the ''universality'' of colloquial (natural) languages: It would not be in harmony with the spirit of this language if in some other language a word occurred whivh could be not translated into it; it would be claimed that if we can speak meaningfully about anything at all, we can also speak about it in colloquial language. (164).
Obviously, this preference for the intensional side does not allow an easy application of the ''strong syntactic paradigm'' to natural language studies. Nonetheless, the ''syntactic paradigm'' remains as a sort of 'logo' of our culture, with different instances for different aspects, scientific (fig.5, left) and linguistic (fig.5, right). In the right figure dotted lines represent 'fuzzier' connections, to account for these special features of natural language and common sense reasoning. The triangles could share the objects/facts vertex, and represent the steady modern shape of the cleavage between 'common sense' and 'science'. Such a distinction will be revealed as a constant in the human topography of knowledge. In the syntactic paradigm it can be seen to arise in Parmenides and since then it will be an undisputed feature of the paradigm.
There are, nevertheless, 'holes' in the link between reality and signs world: we must take into account that there are sentences or formulas not corresponding to real facts, for example ''green colourless ideas sleep furiously'' or ''all the unicorns are blue-eyed'', and only some of them can have a corresponding mental image, and hence a meaning (for example the second sentence, but not the first, in the above examples). With the words of Weyl (Das Kontinuum, (WEYL 1917),
1): Only a meaningful (sinnvoll) sentence accomplishes a judgement (Urteil), only a true judgement a fact (Sachverhalt); a fact however simply 'is' (besteht). Maybe meaningless sentences can appear only in 'linguistic' (sprachlich), never in 'factual' (sachlich), thoughts; any case here it is a great danger from the language, that it allows meaningless combinations of words (Wort-Symbole)...that from a formal-grammatic point of view (Hinsicht) look like real (echt) uttered judgements.
Hence, roughly speaking, we can say that any fact has a mental image and any mental image has a sentence, but not vice versa. < Note: We do not make any difference between ''sentences'' and ''propositions'', a distinction believed crucial by many authors, as Kneale and Kneale(KNEALE 1962), for this distinction is not relevant before stoic logic. Anyhow the Greek philosophic usage is linked to the idea of 'utterance' (logos) but with an 'onthologic' flavour due to the fact that Greek philosophy is 'without self', a 'third person' ontology (see Khan (KAHN 1973)). >
The false facts are ruled out by the Aristotelian-Tarskian definition of ''truth as correspondence'' between reality and sentences: a proposition is true if and only if the fact it describes occurs. But if we accept to structure the signs world with the first order predicate calculus, we have that predicates, which in 'this world' interpretation (i.e. such that the extension of the predicative letter 'red' is the set of the real red things) have an empty extension (for example unicorn), imply, in 'this world', any other predicate, hence ''all unicorns are blue-eyed'' is true, for it is satisfied in any interpretation in which there are no unicorns. Hence, the reality world is made by a finite set of real true facts and an infinite set of 'invisible' true facts. That is, we could create infinite true sentences with empty-extension predicates, corresponding to mental images, and infinite 'unthinkable' true facts, if we create also new 'meaningless' words as new empty-extension predicates. Russell-Quine's theory gets rid of these 'pathologic' facts by a suitable reshaping of the sentences. However, this solution is 'ad hoc' and is not generally accepted. This sort of 'size' hierarchy between the worlds is an 'invariant' of the paradigm at least since Gorgias. This is a version of the old empty-extension paradox, linked to the extensional character of tarskian semantics but already known to Duns Scoto (XIII century), as linked to the connective nature of the implication and hence really linked to the compositional character logic imposes to its formulas. But, at the same time, also the opposite inclusion relationship is tenable, for words and propositions are particular 'objects'. This ambiguity grounds crucial 'moves' of the formal thinking, as the ''metareasoning'' and the ''self-reference'', and will be found again in the following. In addition, it is the core of the paradoxes: ''this universality of everyday language which is the primary source of all semantical antinomies'' (164)
The ''strong syntactic paradigm'' is quite young, it is a product of this century, the ''weak syntactic paradigm'' is older, at least as the beginning of Renaissance science. But a form of this paradigm is still older, traceable back to Aristotle, strongly linked with the birth of the axiomatic-deductive method. Here, with ''axiomatic-deductive'', I mean something more than the simple proof procedure starting from axioms: a general framework instead, whose components are the association of ''concepts'' to ''names'' and their categorical establishment, formal inference, truth as correspondence between propositions and reality, non-contradiction principle and reductio-ad-absurdum proof technique, substitution principle, compositional semantics. This bundle of themes suddenly appears in the world at the beginning of the formal thinking, with the classic Greek philosophy, and, together, with the first appearance of the theoretical paradoxes. They appear in Greece at the sunset of the classic ''mythological paradigm'', in which the real knowledge was divine, and the Muses, daughters of Mnemosines, the ''memory'', reminded it to the mortals. Such knowledge furthermore supplied the everyday life of mankind with the cultural framework: names and rationales of natural beings and phenomena, grounds of rites and customs, proverbial and popular wisdom. The making of a new non-mythological paradigm, but suitable to ground everyday life as well, was the true problem of Greek philosophy.