The XIX century physics ends with the theory of Relativity the classical and pre-paradoxical age, and opens with the Quantum Theory the syntactic and paradoxical age. In the following we shall deal only with some crucial physical aspects, and then with the formal characterisation of quantum mechanics. The interested reader can refer to (BOHM 1951) for a technical introduction, and to (JAMMER 1974) for a complete historical and philosophical reconstruction.
A crucial physical aspect of quantum mechanics is the ''wave/particle dualism''. In the XIX century physics there was a sharp dichotomy between the ''wave'' behavior of the light and, in general, of the electromagnetic radiation, and the ''particle'' behavior of the material objects. Also other physical phenomena, as the 'heath', had been reduced to this dichotomy, which appeared 'exclusive' and 'exhaustive' of the real world physical representations.
Both models allow the representation of the temporal evolution of the system, but the ''wave'' model is continuous, and the ''particle'' model, conversely, is discrete. This is maybe the greatest difference and its effects can be illustrated by the classical ''two-slits experiment'' (fig.5). If a front of parallel waves hits a wall with two slits, beyond them there are two circular fronts centred on the slits, whose superposition produces interference phenomena, which are revealed on an intercepting surface as a succession of dark and light zones (left in the figure). If a beam of parallel electrons hits the same wall with two slits, some of them pass through the slits, maybe deflected by local interactions, and can be revealed as individual events on an intercepting surface (right in the figure). In quantum mechanics this dualism does not dichotomise the universe of the physical entities anymore, but, for any entity, its different phenomenic occurrences. Roughly speaking, both lights and objects have a 'wave' behavior in their propagation and a 'particle' behavior in their detection. So, in the two-slits experiment, single detection events on the screen, for light and electrons as well, are individual particle-like events. However, their statistical distribution shows the typically wave-like interference patterns. We could try to think that both, light and electrons, are some sort of little wave-packets, behaving like particles, but interfering as waves when in a very big number. This 'realistic' solution is impossible, for the two-slits experiment has got the same result, also if accomplished with a very low intensity, so that any individual wave-packet could not interfere with any other wave-packet. If our attempt to give an uniform, although mixed, model for light and objects, were right, in this low-intensity version of the two-slits experiment, we had to find on the screen the simple sum of two single-slit experiment detection, which is not the case. Hence, we have to stress that in quantum mechanics the dualism is not between different kinds of entities, but between different aspects of any entity. In the classical Copenhagen interpretation, more precisely, the 'wave' behavior characterises the continuous evolution of the system in absence of observation events, and the wave features allow us to compute the probability of any result of a measurement event. The 'particle' behavior describes the discontinuous behavior of the system in a measurement event, and it is not possible to find out a deeper realistic interpretation, by which to give a rational account of such dualism.
This is one of the most astonishing features of quantum mechanics: the 'autonomous' role of the observation, as a subjective event which cannot be reduced to a physical one. The very existence of a qualitatively different formalization for the observation process is something unthinkable in classic physics, where the observer can be both completely ignored and completely amalgamated, pure soul or another piece of matter as you like, in the physical representation of the event, without affecting the natural laws. In quantum mechanics, on the contrary, we have two qualitatively different evolution laws for the system: the first continuous, in absence of observation events, and the second discontinuous, accomplishing a sudden ''reduction'' of the wave-packet to one of the possible values/pure states (eigenvalues/eigenvectors) of the observable, during a measurement event. Such a behavior must be confronted with the classic description of the physical knowledge. In fact, in classic mechanics the 'knowledge' event is something not reducible to a mechanical description, as the whole history of the body/mind dualistic philosophies has shown. The odd reality of the 'knowledge' from a 'behavioural' and 'linguistic' point of view has been analysed most of all by Wittgenstein in his late works. And, also in classic physics, it is irrelevant enough to define sharply the border between the ''physical fact'' and its ''knowledge'' (in the instrument output, in the eyes of the observer, in his optical nerves, and so on), connected with the ghostly definition of the 'Self'. In addition, it is worth reminding that this 'heterogeneity' was a crucial point in the Aristotelian foundation analysed in the first report, to deal with the ''negative judgement paradox'', according to which twenty-five centuries ago Sophists taught indisputably that an 'homogeneous' being/knowledge system did not allow any kind of lie or error!
We can also observe as this 'border fuzziness' is mirrored in one of the most ancient problem of logic: the odd behavior of the 'if...then...' constructs. In ancient and modern logic we distinguish between two different interpretations. First, as a ''connective'' =>, which 'statically' builds complex propositions from simpler ones, and whose truth-value is ''false'' iff the antecedent is ''true'' and the consequence is ''false''. This interpretation is thoroughly wrong in a naive natural language usage: in fact it gives 'true' value to any sentence with a antecedent as ''if the moon is a block of cheese, then ...'' or with a conclusion as ''...then two is even''. Second, as the ''deduction'' symbol |- , expressing the logic proof of a formula from some hypotheses by some inference rules. This 'dynamic' interpretation tries to formalise the normal mathematical proof process. Well, a theorem (deduction theorem) of logic asserts that, given a set of formulas A1,..., An, B, whatever dichotomy C/D of the set of Ai we choose,
A1,..., An |- B iff |- A1 and ... and An => B iff C |- D => B
expressing, the same 'fuzziness' about the choice of the border between hypothetical 'facts' and their consequence. We underline, once more, the analogy, as different instances of the 'asking' process, between physical measurement process and logic inference. I think that the real problem with the quantum mechanics measurement theory can be understood by referring to the ''two-slits experiment''. There, the real oddity is placed in the difficulty in homogeneously locating the measurement act in a mental image of the physical event. We see the particle behavior on the screen, and we cannot avoid to imagine the particle flying before the impact through one of the slits, 'ignoring' the possible existence of the second slit. That is, the discontinuity of the 'knowledge' event in classic physics is 'saved' by its continuity in the 'mental' model. To deal with this 'mental' problem, most of the quantum mechanics theorists (as Bohm (BOHM 1951)) insisted on the 'wholeness' of the world, as a way out from the necessity of a reasonable and functional mental representation (the 'physical reality') of the real world. Again, the real problem is the shift from the old ''weak syntactic paradigm'' to the new ''strong'' one.
This kind of complementarities between the wave and the particle behavior of the physical entities, between the 'spontaneous' and the 'measurement caused' evolution laws, is a constant feature of quantum mechanics, entailing also a set of complementarity relations between pairs of physical magnitudes, as position/momentum or time/energy, expressed in Heisenberg's ''indeterminacy relations''. According to these relations, the product of the errors in measuring for the same system in the same time two complementary magnitudes cannot be less than a quantity approximately given by the Plank constant. Also the continuous/discrete description of the system appears to be not 'ontological', that is linked to the specific entity to be described, but simply implied by the respectively absence/presence of boundary conditions. So, unbounded positions entail a continuous representation, whereas bounded magnitudes admit a discrete spectrum of eigenvalues.
To outline the ''semiotic triangle'' representation for quantum mechanics we introduce the 'richer' diagram shown in fig.6. Here, as told at the beginning of the report and as in similar figures in the following, the round-corners squares represent the subjective questioning in the paradigm, and we hypothesise the commutative property for the diagram. In fact, if we want to describe the working of the paradigm, we must warrant that the real evolution of any phenomenon will be faithfully mirrored in its formal representation. For example if we want a faithful representation of the motion of a comet in the sky by our 'celestial mechanics' theory, we must develop our computations in the theory, whereas the comet runs in the sky. However, at the end we need to find the comet in the point forecasted by our equations. In this model this need is warranted by the ''commuting'' of the inner rectangle. In fig.6 this means that the real measurement in a classic equipment of an ''observable'' must coincide, for the ''correspondence principle'', with the value of the observable following the corresponding ''wave-packet reduction''. The necessity of a 'classic' observation counterpart, beyond the formal theory, has been pointed out by Bohm:'' ...the present form of Quantum Theory implies that the world cannot be put into a one-to-one correspondence with any conceivable kind of precisely defined mathematical quantities, and that a complete theory will always require concepts that are more general than that of analysis into precisely defined elements.'' ((BOHM 1951). 622)
Since the beginning of quantum mechanics there has been a sharp debate about the nature of the 'indeterminacy' relations: simply an 'epistemological' limit of our knowledge acquisition or a sort of 'ontological' berkeleyan temporary lack of existence for non-observed physical quantities? It is noteworthy that this alternative can be found as well in modern interpretations of Protagoras' relativism (''the man is the measure of all things''):''...(a) that all properties perceived by anybody coexist in a physical object but some are perceived by one man, others by another, or (b) that the perceptible properties have no independent existence in the object, but come to be as they are perceived, and for the percipient.((GUTHRIE 1962),III, 184)''
In the standard Bohr interpretation, complementarity becomes a general principle, substituting the realistic and materialistic philosophy of classic physics. This could not be without effect on the physicists community in the thirties. The most important criticism came from Einstein who claimed the ''incompleteness'' of Quantum Mechanics in a renowned paper, containing the so called ''Einstein-Podolski-Rosen experiment'', maybe the last great Gedanken-experiment of modern physics. Normally, it is said that Einstein's 'incompleteness' must not be mistaken with the logic ''incompleteness''. However, I believe that maybe there is more than a linguistic coincidence, as we shall see in the following. We refer to Jammer's reconstruction (JAMMER 1974) for more details, and here we give a short synthesis of the argument.
Einstein-Podolski-Rosen's necessary condition for ''completeness'' is : < every element of the physical reality must have a counterpart in the physical theory>, while a sufficient condition for ''physical reality'' is: <if, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity>. Beyond these two 'explicit' conditions, many authors underlined the presence in Einstein-Podolski-Rosen's paper of other 'implicit' hypotheses, among which a ''locality'' assumption, according to which between two systems no longer interacting, anything done to one of them cannot cause any change to the other one. Then, the authors outline an experiment which, in their opinion, proves the ''incompleteness'' of quantum mechanics. A simplified version of the experiment is not difficult to sketch: let us suppose to have two particles scattered in opposite directions, and an observable A (for example, the ''spin'' component along the x axis) whose knowledge for the first particle, for some conservation principle, allows us to know its value for the second particle. The same could be said for another observable B, non-commuting with A (for example, the ''spin'' component along y axis). The measurement of A for the first allows us to predict the value of the same observable for the second, which then must correspond to an element of the physical reality. The same could be done for B. Hence there must be in the theoretical description of the second system the two corresponding elements, for the second system cannot 'know' which observable has been measured on the first system. But this is impossible, for the second element, not disturbed by any measurement event and no more interacting with the first, cannot have exact values of A and B simultaneously. We do not try to give a more complete version, for the core of the problem was already given in the formalistic reduction of the theory, and the paradox is only an example of the conceptual framework of the current quantum mechanics Copenhagen interpretation. We have to remind that this is a 'paradox' and not a 'theorem', and it is useful only to illustrate some aspects of the theory. So in fig.8 we refer to the paradox, but the 'syntactic paradigm' scheme of the theory (fig.6) does not refer only to the paradox. Bohm's conclusions are that:'' ...the paradoxical results obtained by ERP... will not be obtained if one avoids making their implicit assumptions ... that the world can correctly be analysed into elements of reality, each of which is a counterpart of precisely defined mathematical quantity appearing in a complete theory. ... we assume that the one-to-one correspondence between mathematical theory and well-defined 'elements of reality' exists only at the classical level of accuracy ''((BOHM 1951),619)
We can synthetically represent the development of the reasoning in the ''syntactic paradigm'' framework as shown in the following figure.
The 'critical' concepts concern 'locality' and 'individuality' principle in the syntactic reduction of our representation of the world. And these concepts are crucial ingredients of the ''strong syntactic paradigm''. As with the formalistic mathematics, we here encounter a halt to the strong paradigm. The answer of Bohr was aimed to stress the peculiar role of measurement in quantum mechanics: '' ...the procedure of measurement has an essential influence on the conditions on which the very definition of the physical quantities in question rests. Since these conditions must be considered as an inherent element of any phenomenon to which the term 'physical reality' can be unambiguously applied, the conclusion of the above mentioned authors will not appear to be justified.''(N. Bohr, quoted in(JAMMER 1974)) The role of measurement is really a crucial point in the Copenhagen interpretation.
The essentially new feature in the analysis of quantum phenomena is, however, the introduction of a fundamental distinction between the measuring apparatus and the objects under investigation. This is a direct consequence of the necessity of accounting for the functions of the measuring elements in purely classical terms ... the unambiguous account of proper quantum phenomena must, in principle, include a description of all relevant features of the experimental arrangement. ((BOHR 1958) 310-311)
It looks impossible to define without ambiguity the border between the observed system and the observer, although the measurement event must be describable in 'classic' terms, whereas the observed system must be described in 'quantum-mechanical' terms. The ''correspondence'' between them assures the ''commutativity'' of the rectangle in fig.6 and hence the consistency of the theory. This is a crucial limit in the pure syntactic paradigm application to our knowledge of the world. However, the term 'incompleteness' used by Einstein-Podolsky-Rosen is not far from the general use we introduced at the beginning of this report: it underlines a structural impossibility of syntactically reproducing the whole semantic system. The connection will be enhanced in the following section. Now we can synthesise the EPR argument in terms of a 'gap' in the commutativity of the diagram in fig.6. And we underline that this incompleteness and the antinomical aspects of the 'measurement theory' in quantum mechanics appear, exactly as in the logic limitative theorems, as the price to be paid for the consistency of the representation:
In fact, we can argue that the knowledge by measurement of the x-component of the spin for the first particle had, being accomplished in a classic arrangement, to allow us to compute, by classic arguments and correspondence principle, the knowledge of the same component of the second particle, which, in turn, had then to be also computed by the 'other', quantum-mechanical, way in the rectangle, for it commutes. But this is not the case according to quantum mechanics principle. If it were necessary to give all parts of the world a completely quantum mechanical description, a person trying to apply quantum theory to the process of observation would be faced with an insoluble paradox. This would be so because he would then have to regard himself as something connected inseparably with the rest of the world. On the other hand the very idea of making an observation implies that what is observed is totally distinct, from the person observing it. This paradox is avoided by taking note of the fact that all real observations are, in their last stages, classically describable.((BOHM 1951) 584-585)For Bohr the absence of a formal theory of measurement did not indicate any imperfection or incompleteness of his epistemological analysis of quantum mechanics, but was rather required for reasons of consistency (472) In his view classical concepts, representing the ultimately immediate data of common experience, are in the last resort not formalizable, for nay formal elaboration becomes physically meaningful only if it is interpreted in terms of classical concepts. Bohr's insistence on the logical (though not physical) necessity of drawing a sharp distinction between object and measuring instrument can therefore never be replaced by any formal treatment.(473) (JAMMER 1974)
There is a sharp difference between Relativity and Quantum Theory, despite their flourishing ages are almost the same, and the philosophical aspects of such difference are only reflected in the Bohr-Einstein debate, but concern their thoroughly different relation with the central themes of the syntactic paradigm. Relativity's 'subject' is 'Cartesian', sharply distinct from the world, observed in a space-time Cartesian framework. Measurement is based on 'clear and distinct' ideas and observations, and the physical magnitudes, though 'operative' they be, are also the counterpart of ideas and concepts, whose center is a well-defined idea of reality. Quantum Theory's 'subject' is 'phenomenological', part of a whole arrangement including also measurement and physical systems, with clear distinction of roles but without sharp borders between the different parts. Consequently the very idea of 'reality' and, more in general, models and ideas play no role at all.
The same difference, in my opinion, can be revealed between geometry and arithmetic at the end of the XIX century. We must remind that since Plato and Euclid the true core of the philosophical aspects of Mathematics had been the Geometry, with Arithmetic in a secondary role: Descartes and the geometrical, ''rational'', translation of mechanics, the geometrical intuition of the ''real number'', the Kantian a-priori had been the cornerstones of the connection between philosophy and mathematics. At the same time, and this is not casual, Physics and Mathematics, during the entire modern age, had been substantially paradox-free, at least rid of those paradoxes grounded on 'being' and 'negation' we are analysing in these reports. The actual infinite was a source of paradoxes and for that reason had been ignored in mathematics, but the very idea of 'continuity' could not avoid to present again the actual infinite under the idea of 'point', unattainable as a numerical concept before Dedekind.
The turning point at the end of the last century shifted the attention on logic and arithmetic: axiomatization, arithmetization of analysis, logicism, formalism and intuitionism as well, endeavoured to ground all mathematical areas on a discrete world of numbers or other logical signs, and, again not casually, met a new season of paradoxes. This way also geometry lost its basic role, as linked to the being and founded on evident concepts. Probably the role of the ''axioms'', from evident truths to manipulable strings of signs, can depict that 'turn'.
For the Greeks the axioms were self-evident, and the postulate reasonable. Subsequent philosophers took different views. For Kant and his school, geometry dealt with rather a vague property called 'externality' and it was believed that the human mind could not construct a rationale universe for itself without starting with certain assumptions. The axioms came to be considered as necessities for rational thought. The opposite opinion was that they were facts experimentally determined. Neither of these views is held widely today. No one can experiment with geometrical figures which, as defined, are pure abstractions with no physical substratum.... the modern view that the raw material of mathematics is a small number of undefined objects, point, class, order, number, etc. and that the assumptions which we make about them are matters of pure logic. Besides the undefined objects we have certain relations such as equal, greater than, between, part of, which are also undefined, except for their logical interrelations. The axioms are definite assumptions which we make about these things. They are arbitrary in that we are free to make any assumptions we please according to our taste or inclination, or the particular ends in view. If we are to set up a system of axioms for a particular sort of geometry, two qualities are essential, and two desirable. The essential qualities are that: 1) They should be consistent. 2) They should include all of the assumptions necessary for the purpose in hand. The desirable qualities are: 3) They should be independent of one another and include nothing unnecessary. 4) The mathematical system built on them should be interesting rather than trivial. (COOLIDGE, 84-85)
We must remember that, at the beginning of the foundational debate, there was no idea of a 'formalist' employment of the axioms to give the 'meaning' of the primitives, but an effort to find an 'essential' characterization of both. This is explicit, for example, in Frege (FREGE 1971) and Riemann (RIEMANN 1854). The epistolar debate between Frege, Hilbert and Korselt (FREGE 1971) elucidate that Frege employed the word ''axiom'' in the old euclidean sense: we have to grasp before the sense of the concepts and of the relative properties, and then to write down such properties as axioms. Also a proof, reduced to an inference-chain of psudo-propositions, does not contain a thought. A definition must be constructed out of primitive elements and ''is a constituent of the system of a science''. Primitive elements are indefinable, and for mutual understanding among investigators we must employ some ''explications'', which have ''no place in the system of a science'' and are useless for a researcher ''who pursued research only by himself''. (''On the foundations of Geometry'' in Kluge (FREGE 1884)).Thus, Frege eagerly criticizes Hilbert' formalism and Riemann begins his 'Ueber die Hypothesen, welche der Geometrie zugrunde liegen', looking for the relationship between the 'nominal' definitions (Nominaldefinitionen) of the fundamental concepts and their 'essential' determination (wesentlichen Bestimmungen) described by the axioms. Nevertheless their efforts must find a solid base to stand upon: the new symbolic logic for Frege's arithmetic, the analysis of the 'infinitesimal' line-element for Riemann's geometry. The difference is also sharper comparing Riemann with Dedekind's 'Was sind und was sollen die Zahlen'. (DEDEKIND 1888) Whereas Riemann's foundation was based on analysing the hidden hypotheses of our concept of space (infinitesimal line elements as homogeneous second degree function of the components, homogeneity and isotropy of space), Dedekind's foundation begins trying the same way to elucidate the idea of number, but then analyses also the grounds of our general thinking practice (to establish relations between entities, to represent something by something else, to create sets). Once again we can underline the different evolution of the two fields: geometry is going to tie with physics, displaying its original link with the being, whereas the border between arithmetic and logic is going to become more and more not revealable, core of the signs world, and subjected to the paradoxical aspects of the syntactic paradigm. But this evolution 'replays' that steady connection, nature with continuity, culture with discreteness, we underlined in the first report as ancient at least as the mythological paradigm. However, as regards the passage from the ''weak'' to the ''strong'' form of the syntactic paradigm, the sharpest traces can be found, beyond the anti-metaphysical wave in physics outlined in the first report, maybe in the debate between Frege and the formalists (Hilbert, Korselt, Thomae) about the formal foundations of geometry and arithmetic (FREGE 1971)
Frege's logic is a language, both suitable to represent the contents and to allow the inference of mathematical reasoning, whereas Hilbert's formalism is a methodology to rebuild the whole scientific enterprise. Frege's description of the connection between definitions and axioms remind the classic Aristotelian framework: there are primitive elements which are indefinable, and their understanding, in the social and didactic scientific practice but not in the system of a science, can be fostered by 'explications'. On this ground we can give definitions and axioms, which must always 'have thoughts'. For Frege it is absurd to reduce the definition to the formal structure of the axioms: '' If Mr.Korselt means that an arrangement of chess pieces expresses a thought merely because of the rules of chess, the I question this and shall continue to question it until I am presented with this thought as expressed in German.''(94) Also the inference cannot be a purely formal, 'thoughtless', process. Thus, arithmetic can be grounded as a piece of logic, whereas other sciences, geometry included, require peculiar, 'thoughtful', axioms.