There is the smell of lungwort flowers Among forget-me-not In the fact that I, My distracted rigorous mind, Is the root of the Non-unit, Hiding the split point Towards what was And what will be. The pillar. V. Khlebnikov.

  INTRODUCTION

We try to analyze some logical and philosophical aspects of the quantum theory and Gödel's theorem on incompleteness in this article. It is shown, that logic must develop concepts of relativity (up to the introduction of logical "reference systems", similar to the physical reference systems) and nonpredicative notions must become foundation of the logical constructions. Model theory must more flexibly individualize elements of sets and their features. Relation between modern mathematics and philosophical concepts of Gödel and Berkley is shown.

There are some shortcomings in the article. For example, we didn't discuss some generally known aspects of Gödel's theorem, quantum phenomena. Philosophical themes are given very briefly. The theory of category is briefly touched upon too. It is desirable (but not necessary) that, despite our century of narrow specialisation, the reader get to know all these subjects. It is described in details only that, in what we managed (or it seemed to manage) to tell something new in the above mentioned questions.

 

  1. RELATION BETWEEN the SUBJECT and PREDICATE in QUANTUM THEORY

 

Superposition principle and common look of the wave function dependence Ψ on the time is important for our article. If the physical system can be in the states Ψ₁ and Ψ₂, it can be in the state αΨ₁ +βΨ₂, as well, where αΨ₁+βΨ₂, where α and β are complex constants. Dependence Ψ(t) on the time is described by the summarized equation of Schrödinger:

which is real in any of the quantum theories, which differ only in the construction specific of the H Hamilton operator. Many deep investigations was devoted Ψ-function of the semantic analysis. We should like to pay attention to one detail, which could be missed in these investigations.

The fact is, Schrödinger equation — is the fields equation, that is, a differential equation for some potential (in this case, it is a complex- symbolazed vector potential in Hilbert space). In all prequantum physical theories such equations described what philosophers call the matter, the substance, the medium. Physicists call it a field,— some essence which transfers energy and has some force effect on the matter. Strangeness of the quantum theory consists in the effect that the field equation describes only informational Ψ-function in it. Half-bilinear form (Ψ^*ĜΨ), where Ĝ,— Hermitian operator,— will be the probability density for the observed meanings of some physical value (for example, for the particle coordinate). Such nonlinear dependence of probability on linear dependent conditions (superposition principle) has some deep logical consequences.

For example, if there several alternative object states, which can't be detected by our device (that is simply by a macro-situation) then the states loose their alternativeness. The object behaves itself as if it can be in one of this states or in several states at the same time. In other words, electron is blurred as a medium all over the whole space till the device measures its coordinate, which still will be "dotted" during measurement, and will differ from other possible alternatives of the coordinate.

Indiscernibleness of unobservable alternatives goes out of the principle of superposition for the conditions and the quadratic expressions for the probabilities. Properly speaking, this proves the known two- slot mental experiment on the interference of the probability waves. There are an obstacle before the electron source whit has two slots. If to make a macro-situation such that the device fix both the slot, through which the electron has passed, and the place of the electron hit on the receiver screen, then the probability distribution will be described by the quadratic of the amplitude sum. Otherwise, when the device doesn't fix the electrons way, the probability will be the quadratic of the sum amplitudes, that is, the electron distribution will give interference picture. So the informational, logical structure (Ψ-function) is manifested physically as a field. They usually state, that we have right to discuss, which of the two alternative ways the electron has chosen, if they can't be distinguished by observation. But it isn't so, we don't speak about the results of observation, but about this ways, calculations their amplitudes summarizing. So, electron, being an empirical point, is as well "a logic field" the statements not verified empirically which status of the validity is wave process in a certain physical environment.

Ψ is a vector field, scalar quadratic function of which plays exclusively logical part. The truth measure of the statement c corresponds semantically to its "presence" degree in the given sphere of individuals and is a field intensity in the quantum theory. In usual physical fields |Ψ| ² would be energy; in the ase of quantum, it is a probability measure of the statement.

It should be noted, that empiric undeterminability of the sentence does not mean loosing the sense of this sentence at all: it reveals itself indirectly, through the observed facts. We can and discuss the non- observably electrone trajectories. It is the amplitude superposition of these ways that creates interference picture of the screen. Here lies the principal difference of the quantum correlation of uncertainties from any classical relation of the measurement mistakes. If a classical molecule in volume V puts pressure on the volume walls, consisting of local hits, the electron, localized in the some volume, will not have any definite coordinate, but distribute itself through the whole volume and, therefore,exert an even pressure on the walls (which will take some force interval, as for the impuls we have: ΔpΔx≥h/4π). This "blurring" effect we observe physically. And just it lies in the basis of the tunnel transition, the principle of the least action, as well as processes in the presence of virtual particles.

So, in the quantum theory, the density of the distribution of probability for a point particle principally is not distinguishable from the medium density, being present all over the space. It means logically some nonpredicativity: measure of the truth of the sentence about the object of some other admissible theory statement itself.

And in a classical case, the experiment also does not distinguish the point, having a coordinate, determined with the error Δx, from the medium, distributed in the interval Δx. But the classical theory takes unconstructive, inductively established idealization of the material point. The latter supposes unevidently, that density of the probability distribution of the point object is not identical with the medium distribution density.

And what is more, both classical and constructive logic takes analogical idealization when proclaims a consistency tautology in the kind: A&(~A)=0. When following the concept of the truth of the quantum theory, it is wrong in the case, when the sentence А is unestablishable by some effective means. Perhaps, we must consider some "reference systems" in the quantum logic, which a formed by a definite choice of the disjunctive system of the events.

For better understanding of the difference between the quantum and classical situation, we shall give one obvious example. Suppose, on a flat surface a heap of sand is poured out so that mass density over the surface is distributed; for instance, normally. And suppose, there is one black grain of sand in this heap. Then the relative (i.e., fixed) weight of the sand over a unit of the area will present the density of some medium, and the density of probability to find the black grain of sand over the given unit of the area. And nevertheless, the medium presents one thing in the classical case, and the information is quite different (probability to find the black grain of sand). But in case of the quantum the situation is quite different. If the black grain of sand is somewhere in the depth of the heap, then it cannot be observed directly and is does not have to be a grain: it will behave itself as a medium, distributed according to the probability. And the whole heap will take blackish tint, thickening to the top. But if we begin to mix the sand, all this blackness will be concentrated in one point, when the grain of sand will come out and will become observable.

Resuming we an make following conclusions:

  1) Wave-particle duality.

  Wave-particle duality is not a problem by itself. It is only the consequence of a deeper dualism between the medium and the predicate, the material and ideal. There is no paradox in the unity of a wave and corpuscular descriptions at all. That is why physics so persistently searched for electromagnetic ether in the nature at the beginning of the century. Any medium, discrete or continuous, having the simplest elastic force, is a carrier of wave processes with we have in acoustics, for example. And vise versa, with the help of wave packets we can model any classical corpuscles.

2) Dualism of the subject and predicate.

Real paradox is another dualizm: what was density of probability of some event under one condition, becomes a physical substance under the others. And these conditions lie in the way the outer macrosituation distinguishes the range of disjunctive microconditions of the object. The same Ψ-vector describes both field properties of the object, and informational, logical properties of sentences about the object. In the end, this state of the things is caused by the fact that empirically the object is distinguished by us only by the presence or absence of some properties in it, these properties being manifested in the nature neither worse nor better than the objects possessing these properties. "To be an object" is also one of the properties. It is just logically that set elements from predicates modelled on this set. As we can see, the nature is built somehow differently. And that is why strict division of the world into elements of individual sphere and the predicates outwardly applied to the sequences of these elements is a shortcoming of the modern formal logic. A conclusion can be made that future development of logic must go by means of more delicate study of model processes and more flexible solution of the individuation problem. Namely, the relation (х ∈ М) must be determined effectively by some individuation means. The same is in the relation of the truth of А (х₁,...х_n)=1 the sentence А to the sequence (х_i)| х_i∈M,i=1,...,n; of the set elements М.

As for set theory itself, plurality of molecules in a glass of water, undoubtfully makes one physical object and has a physical reality. Which cannot be so simply said, for example, about boolean set of this set. The nature, apparently, tends "to turn on and of" notorious "function of collecting ∈ ". In any case, in quantum theory, only those sets of microconditions are real, which are alternatively distinguished by a macrosituation.

Now, let us pass on the effectiveness problems. The first attempt to substantiate mathematics by empiric realities is constructivism. All constructivism appeals to intuitive clearness come only to feasibility of empiric processes of calculation. And only empiric practice makes these processes clear intuitively. The quantum theory shows, that empiric means do not come only to the calculation processes. There many processes their effectiveness being groundlessly supposed self-evident by the constructivism. We have already pointed out violation of tautology A&(~A) =0, which is true both in boolean and heiting logic. Analogical is the case with the self-obvious effectiveness of the reflexive property of the equation: Х=Х in the constructive and classical theories. Ist inner contradictoriness was notices from ancient times. To experimentally determine Х=Х, it is necessary to take the same twice for the comparison with itself, which is impossible. The modern method of inscriptions consideration (of unlimited reserve of copies of the given object or symbol) simply diverts from the problem considering it being solved a priori. However, nontriviality of individuation of the quantum objects does not require effective determination of reflexivity. For the same reasons it is impossible to consider admissible idealisation: "a point in a continuum", or in general: "the set element" if this concept is defined independently both from set, and from internal structure of the element. These of idealisation cease to work already at least because of basic integrity of the quantum description: Ψ- function sets a condition of all Universe, instead of set of the allocated objects. Complementarity principle points to the fact, that effective means are relative, comprise some equitable "reference systems", much more difficult, than system of bases in linear space. This relativity of the logical theory is still to be studied. It is thought that Brauer tried in intuitionizm to come to deeper ideas, rather than constructive "inspection" of constructions of the Cantor...

3) Quanta and probability.

Despite unavoidable error of measurement, classical theory ascribes to the physical quantity point meaning in the interval of this error. This idealization is the main difference of classical physics from the quantum one. The idealization is based on induction we can always construct a more precise device and repeat the measurement. The nonrepeatability of the measurement act in the quantum theory prohibits this induction. But the medium, distributed in the error interval of one measurement, is indistinguishable from the point, which could be fixed in this interval by some other, ideal measurement. It is because we have only one measurement. And the thing, which in some other measurement would be a point, an element of the set of values of physical quantity, is an interval now, a subset. To be more precise not even a subset, but measure of the truth of some predicate. Nature does not know any strict selfidentical elements in a new way, depending on distinguishing of their features in some basic set ("reference system" of the device). Let some physical quantity be determined through the set of values of another quantity now. Then point localization of one of them will lead to the loosing of localization of another one. And "the value" of this another quantity will become some set of ist common values, that is,— predicate in the sphere, which it runs trough. Such is the case with coordinates and impulses, as you know, the speed values, representing derivatives, are determined through the open intervals of coordinate values (let us remember the Fourier transform, used for the derivation of relationship of uncertainties). Hence principles uncertainty of the quantum description come out.

Here one should pay attention to the fact, that physics could face the violation of the idealization of the point values of the quantities not obligatory in microcosm simply because of the necessity of very precise measurements. The examined idealization itself has not physical, but logic and statistical meaning.

That is why it is necessary to specify logical foundations of mathematical statistics to understand quantum phenomena. Theory of probability is applied in statistics in some nonpredicative manner, striving to ground its axioms empirically, end the same time postulating their empirical truth beforehand.

So that "quantum" phenomena may arise in statistics not only because of the high measurements precision, but also by some other reasons, for example, when increasing the selection volume (however, the last follows from the most quantum theory: the rubber ball thrown in a concrete wall, will necessarily make through it tunnel transition if to make astronomically a great number of throws. However not all here so is simple. Both classical, and the quantum physics describe the same world, and the quantum theory does not deny any classical laws, and opposite, only follows them. Actually all their distinctions purely statistical: to the events unequivocally occurring or impossible in the classic, distributions of probabilities answer. And the quantum description always needs classical verification. Differently, it is necessary to Newton and Einstein's physics, without rejecting the laws, and at all without paying attention to the microcosm phenomena, all the same somehow to explain abnormal behaviour of a ball). Not classical physics, but modern theory of probability is the approximate description of nature. It is necessary to regard only concrete meaning of the Plank constant to the microcosm phenomena specificity. When probability density is undistinguishable from the medium density, then even simple notion of the object, will be described by the wave process, and it is naturally to expect, that we'll come to some field equation of Schrödinger type.

As for the empiric shortcomings of modern statistics, consecutively carried out principle of the probability approach should consider metaprobabilities, that is probability measures. All situations with the beforehand unknown laws of distribution, lead to such an approach. And that is why we empirically begin to speak about the probabilities, that we do not know them. And frequency of some event in the consecutive theory must give information about the distribution laws not the first, but higher metalevels. But it is the subject for another work.

Let us examine one more famous mental experiment in conclusion. It is the experiment of Einstein- Podolsky-Rosen. That teleportation of properties by which have now learnt to make experimentally.

Two photons flying in the opposite sides, are created as a result of annihilation of an electron and a positron. Far as they fly away to the moment of their being observed, determination of polarisation of one of the photons will cause exactly this polarization of the other, which is required for preservation of the moment of momentum of the whole system. It turns out, that some physically important information can spread at any speed, exceeding that the speed of light. There is no contradiction here, a luminous spot on the screen of a long enough kinescope can also move at any high speed. The mental experiment simply affirms that not all physical phenomena are caused by spreading of the signal carrying energy. That is, purely phase characteristics of Ψ- function describe the observed physical effects. But Ψ-function is such a "truth measure", phase changes of which describe qualitative peculiarity of the sentences, their interrelation and dependance on the time. The world, weaved by the movable light spot, has quite different informational characteristics, which classical physics supposes. In a classical world, the information exists only by Shannon, as a measure of nonuniform material substance: some medium or energy-carrying signal. But in the world, weaved by the light spot, we have information as a self-sufficient ideal essence, from which purely theologically, as by the drawing or plan, the matter is constructed. The Einstein-Podosky-Rosen experiment, the explanation of the least action principle through the wave packets, the tunnel transition, combination of undistinguishable alternatives, virtual effects, all this says that the real world has exactly such, teleological features, but not always and not in everything.

This has some deep philosophical and logical consequences. We have tried to prove, that understanding of the quantum phenomena should be looked for in logic. Namely, it is required to study the relations between the individual sphere and the sphere of predicates more flexible and in more details, nontrivially decide the individuation problem. Individual of elements must be effectively defined by the available means, forming something like reference systems. It requires some reconstruction and the probability theory as the base.

 

  2. A LITTLE of PHILOSOPHY

 

Foundations of the mathematical logic, especially in the semantic sphere and in the model theory are inseparable from some global philosophical questions, which we'll examine briefly.

1) Material and Ideal.

This question is directly connected with our main subject: logical relation between objects and their properties.

Classical philosophy paid much attention to the material and ideal. However these notions themselves have not reached a sufficient strictness level. Under ideal philosophers understand intuitive manner of psychical processes, which begin in the sensation and finish somewhere on the level of thinking and ideas. Philosophy represents ideal generally, turning off the pure human qualities and perceiving similar phenomena in the whole nature. As for the material, here the initial image was the aspect sensation, which goes out of the limits of psychical and belongs to the outer world. Nevertheless, classical philosophy left us two clear-cut aspects of the material and the ideal.

Firstly. The ideal has no other essence except reflection, modelling, homomorphism. The material is the reflected itself, thus appearing as something external to the ideal. These results are formulated most fully by Marx and express some functional properties of the material and the ideal. We deliberately digress from the marxism specificity and do not state after Marx, that "material exists outside and independently of the ideal", and that "material is primary, ideal is secondary".

Secondly. The ideal is in the role of a predicate, a property and has ist foundation in itself in this sense. Material is a subject of a predicate, a carrier of properties, the main of which (axioms) are externally applied to the matter. And in this sense the material has a foundation out of itself in something else. This aspect is most exactly formed by Hegel. And again, we turn off the rest of the specific properties, which Hegel ascribes to the idea and the matter, and we stop at these logical properties.

Now let us pay our attention to the fact, that physics and mathematics brought about much new to this question and more strictly studied both the first, and the second aspects. The point is, that in modern science both these aspects characterize the ideal only as an information, and the material — as the thing, which models, reflects the given information. Attention should be paid to the fact, that the information here is understood in a wide sense, as some measure on properties, expressing their qualitative differences and quantitative characteristics of their truth. But the logic has not developed yet such an information concept. Shannon's definition is only a particular case purely quantitative, comparison of the information of qualitatively uniform sentences, having real values of the truth in the [0,1] interval. So, the philosophical notion of the ideal corresponds to the sentences, their probabilities, their measures of truth in different logics, as well as to the values depending on them. All the above mentioned arguments of the quantum theory consider the relationship between the material and the ideal (the medium and the probability). As we see, the nature is build so that these points are the sides of the same medal. The material and the ideal are unstable notions relativated in some reference systems, determining and completing each other. In the quantum theory, the only concept describes them the Ψ- function. So, the logical problem of the subject and the predicate has a direct philosophical accent.

2) What is Dialectics?

Now this word seems to be somehow incomprehensible, which is interpreted almost like "sophistry", i.e. toing with paradoxes, as ungrounded attempts to draw strict conclusions in nonformal and even nonformalizable situations.

In the past philosophers often argued, whether the laws of logic are objective or not. But now it is almost evident, that the laws of logic are just the laws of logic are just the laws of our thinking and nothing more. And these laws are universal for the whole world just because they are the laws of making language constructs. And they are objective so far as they describe some part of work of our brain.

That why many different logics can be constructed which will describe the same theoretical content. The logic collides with reality in verification and modelling processes only, which are empirical in the long run.

But the reality itself has not to obey any logical laws at all. Moreover, everything shows that the reality is not logical altogether. For example, it is impossibility simultaneous completeness and noncontradictoriness of the formal arithmetic and empirical completeness and feasibility of all real count processes. And our theoretical cognition itself, turning to the experiment, is forced to give old formal theories up and create new, more precise ones. And, judging by everything, this process is infinite. That means that no formal theory is capable of being empirically complete. I.e., the world have no formal model. In other words, it is contradictory logically! But it is contradictory logically only. These formal contradictions are not only solved in their content, but, possibly, do not even arise.

We approach the formal contradictoriness of any real content for purely formal reasons as well. Let us again turn to Gödel's theorem of noncompleteness. Any formal theory, rich enough in empirical content, includes the arithmetics. Hence, being noncontradictory, this theory cannot be complete. Whereas in reality this theory can have a complete model (arithmetic, for instance). Hence follows contradictoriness of the reality, as well as the necessity of constant turning of the formal theory to its empirical model for solving statements which are unestablishable formally.

Thus, the real world is contradictory formally, but this is not the problem of the world, it is our problem. Because the formal logic is the law of our thinking. All this is a direct consequence of the statements being unsolvable formally, always solvable by their content but not vice verso. And here follows, that the substantial object is simply nonformalizable, it cannot exist formally. But in the logic (and not only in logic), incapability to exist is a synonym of contradictoriness. But the substantial objects exist really. There is difference between actual and logical existence. So that in reality everything is contradictory logically in the nature. That is why we have to model noncontradictory the contradictory, reconciling ourselves to idealizations, relativity of our truths and groundlessness of our induction methods. And the cognition is just constant pulling on of a straight-jacket of the logic on the evident absurdness of the reality.

Therefore, the dialectics is a logical contradictoriness which is realized logically through a noncontradictory method. An excellent illustration to it is an example from the quantum theory — identification of nonobservable alternatives. In fact, all the science consists of such examples. And a point in the continuum is just an attempt to connect full totality, lack of individuality of the medium with "elementness", strict individualization of discrete objects. Thus, the logically contradictory world is projected noncontradictorily in our logic. It is just a dialectical contradictoriness. It is a cognition.

3) Relativity.

The problem of relativity was put forward by Berkley. It is very important, that relativity of the truth consisted for him just in the presence of effective means of determining it. And this relativity was total: every object was created only through the correlations of sensations. "To exist is to be apprehended".

Only physics was included in further development of the relativity. So reference systems appeared. So excessive anthropomorphism was excluded and even by "interaction". Thus, "exist" became the synonym for "being perceived and perceive".

The relativity itself has a purely dialectic nature. So far as the reality is contradictory, А as well must be true besides the truth of some sentence (~A). One of the methods of logically noncontradictory modelling of this situation is the truth А in the reference system and the truth in (~A) in another, the order reference system finitizing the world, "carving" a finite part out of it and choosing effectively not any, but a particular negation, i.e. sentence В such, that A&В=0.

As for the concrete character of the relativity, the modern science gives here two models: the Einsteinian and the quantum ones. These models differ in essence, though in both cases it is a question of transformations of basis of linear space. The reference systems of the theory of relativity are built on the passive act of observation, which leads to co-existence of different reference systems, exchanging the information, and to the possibility to change our system rather at will. But in the quantum theory, Ψ-function always describes the situation as a whole, it is always the only and unique at the given moment of time. The observation act is replaced by the interaction act and there is no changing the choice of the complete system of the really measurable values. There is no observer from the other reference system who would measure impulses when I am measuring the coordinate. The quantum observer is always one as well as an instrument which he has chose for measuring characteristics of a particular particle in a given condition.

Thus, Einstein's relativity says of different sides of the real. The quantum relativity treat of different variations of the possible with complete absoluteness of choice from these possibilities. The choice is only one and once being made it cannot be changed. As we see, the differences have a modal character. But in both cases, the relativity is modelled mathematically by transforming the bases of some space. In the Einsteinian case, it is a metric space. In the quantum case, we have different presentations of the state vector in Gilbert's space.

And the identification of nonobservable alternatives gets a very interesting formulation: "The class of possibilities not realized by a given reality, presents itself as the only and indivisible reality. And vice verso, any reality is the class of possibilities which are not realized by it." Just this happens with Ψ-function of the electron when it "gropes" ist real way with superlight phase speed to fulfil the principle of the least action.

And again, we see, that the cause of the quantum phenomena lies not in physics, but in the logic and the theory of probabilities. But the modern logic lags behind these ideas, it does not model the relativity of sentences. It has happened, because the effectiveness problem has been developed by the mathematics without any relation to the relativity problem. Thus, different models of the given predicate only conjuctively add to its properties a new list chosen from a strictly fixed set of properties coinciding with the initial ones. The concept of relative solvability developed by the constructivizm, is notable for analogical scholasticism. The relativity of modern physics requires that the predicate model does not regulate trivially strict relation of this predicate with the elements of the individual sphere. And the value of the truth itself must also be relativated.

4) Hegel.

Now let us remember Hegel's dialectics. In the basis of the Universe he puts the idea, the notion which is identical to the God in absolute. The world of phenomena is presented as not quite adequate embodiment of the idea, almost as a parody on it. Thus the matter is determined negatively as a negation of the notion and, at the same time, the matter has only the notion as the basis. So the matter - this is what has cause of something else. The notion also has its foundation only "in itself and for itself" - no more characteristic properties of this dialectical pair has. And the direct self-truth of the axiom cannot, in fact, be self-validity of the notion, for is empty and conceals in reality the basis of the axiom in the belief or in practical needs. In the same way the notion creates itself and deduce through the self-denial in its other existence in the matter.

Two properties of Hegel's dialectics are striking the eye. The first one is the relativity. All this expressions of the kind "in itself", "for itself", "for the other" are simply the indicators of different reference systems. But with Hegel, the interrelation of these systems is rather peculiar. They are unstable and flow over one into another constantly. Just in this flowing consists both the process of meditation over the nature of the notion and the evolution of the notion itself. "In this aspect, it looks so that requires quite different aspect where it looks quite different..." is a typical turn of Hegel's thought. It should be noted that the nature behaves so as well, performing constantly according to Schrödinger's equation unitary transformation of the Ψ-function.

And the second, the main characteristic of the notion is nonpredicativity self-applicability. And Hegel, as it makes Russel in the 20-th century, arrives at the self-negation of such a notion. To avoid misunderstanding, later on we shall use inverted commas to distinguish Hegel's notion from the notion in general.

 

  3. NON-PREDICATIVE CONCEPTS in the CONTEMPORARY LOGIC.

 

We are having an interesting situation. The formal logic can model Hegel's ideas nonpredicatively only. At the same time, Russel has driven out of the logic all nonpredicative statements. But it is not quite so. As we have mentioned, the logic presents the contradictory noncontradictorily. Russel's paradox (if: Х≡{α|~(α ∈ α)}, then: (Υ ∈ X) ↔ ~(Υ ∈ Х), or nominal variant: "In the village lives only one barber; he shaves all the men who do not shave themselves. Who, then shave it?" — Where we can see, it's not about the specifics of the theory of sets, and in our thinking in general) is "solved" by simple refusal to contemplate such sets. For instance, by introducing one's own classes which do not represent sets. Further, if the above mentioned classes are considered within the limits of the theory of modes, the nonpredicative concept is still present in the theory not manifestly, being "diffused" all over the hierarchy of modes. But even without turning so theory of modes, we come to the concept of a universal set (the set closed relative to all set forming operations) together with the axiom of belonging of any set to a universal one. In this case, this universal set will be one of the models of the theory of sets as a whole. That is, if the logic refuses to consider the nonpredicative object "set of all set" directly, it all the same introduces its more strict model, a universal set. That is why one should not insist that such philosophical generalizations as "the World as a whole" are logically groundless. The universal set is just one of the formal models of the Universe. How much this model is adequate to the prototype — or can the Universe be considered either as an open or closed system — are different questions, but the fact is, that the theory of sets has not refused the concept: "set of all set", but just proved that this concept should be modelled not trivially, i.e. as a universal set.

The same with the other absolute concepts: the logic does not forbid their formal models at all, but models them not directly, not trivially. On the contrary, these general concepts can be interpreted scientifically only in a formal and logical way. Remember, for instance Hegel's definition: the infinity is something self- identical due to its self-difference. As a matter of fact, it coincides with the "concept" as being "self-affirmed through its self-denial" (we are again free to quote Hegel). When formalizing such an infinity trivially, we come down to the absurd: (Х=Х)↔(Х≠Х). But replacing here equality by the equivalence, Cantor arrives at the following: "the set Х is infinite if there exists such a one-to-one immersion f: X→X, that f(X)≠X." And it is clear now, that Hegel correctly defined the infinity for all that. And if we take a closer look at the situation, it will be clear that the empirically set equality is always a coincidence of a limited class of properties, i.e. some effective equivalence. In this case, one should not simple substitute the classical definition: (Х=Х)&[(X=Y)→(A(X)→A(Y))] (where А is any statement) for (Х=Y)≡[A(X)&A(Y)] (where А is an element of a effective limited class of statements). As has been mentioned above, an uneffectiveness of a reflexive character should be somehow shown. As we see, the problem of infinity is inseparable from the problem of individulization.

But with all this coincidence of the concepts of the infinity according to Hegel and Cantor, it should not be forgotten that Cantor's definition is only a formal image of the substantial concept of Hegel, and the image far from perfection at that, for it is created by the classical mathematics with a trivially clear understanding of an element and a function as a plurality of put in order pairs of elements with a formula of having a single meaning of values. But for Hegel's approach all those are accidental, not justified by anything factors. Meanwhile, the modern mathematics have only Cantor's definition of the infinity. At the same time, Russel's paradox and troubles with the axiom of choice illustrate an incompleteness of this definition.

The next, and perhaps the only step in formalizing the infinity of the "concept" was made by Robinson in this theory of the nonstandard mathematics. The basis here is in the fact that the formal theory having a countable model has a model of any infinite power. What if the first model is a subset of the second one? What if the theorems of the first model result from the theorems of the second one due to the restriction of the sphere of action of quantifiers and the sphere of changing variables? By using the theorem of compactness, ultrafilters and directed relations, such a pair of models can be built for any formal theory. And then the following situation arises. The theorems true in the standard sphere (individual sphere of the first model) can get broken in the expanded set which includes nonstandard elements as well. But the same theorems are still true in the expanded set, only their standard presentation is broken. Here we have a substantial nonpredicativity which is expressed quite predicatively and noncontradictorily formally. The theory is modelled so, that is theorems have the "truth which is realized through the moment of falsity". Of course, formally it looks quite different. But here the content is the same as in the "concept". As to the reality, the infinity of the field mass of the particle (where the mass and in fact appears to be measurable, but an infinite number of) is more natural to explain by nonstandard methods than by renormalizing ones. Analogous is the matter which zero probabilities of point events forming are continuum.

But here as well we have but an incomplete model of the "concept". Making the infinity according to Cantor nonstandard, we still cannot eliminate the aforementioned shortcomings. For instance, the standardness predicate remains here something external for the theory, having relation to its model only. But the "concept" requires the modelling process (being embodiment of an idea into the matter according to Hegel) to be something self-depended but be included organically into the construction of the logical language and theory.

Another instance of real action of nonpredicative structures in the mathematics is the recursive function apparatus. Here we also avoid any direct application of the nonpredicativity through some variation of the term in the right and left parts of the equation: f(n+1)=g[f(n)], where the natural numder n is changed to (n+1). As a matter of fact, it should be mentioned that the nonpredicativity in modern mathematics always causes much trouble even if it does not lead to a contradiction. The reasons will become clear, if we consider, for example, the functional equation: f(х)=g[f(х)], where х is real or complex. The situation is typical for the mathematical analysis. The equation presents itself as a determination of a function f(х). But to acquire the full information of it, one should solve the equation, that is, find the expression of f(х) through other functions and exclude nonpredicativity. Such striving for determinizm and irrelativity of the cognition says that the mathematics is still too classical, still lags behind the latest parts of the physics. We try to show, that the mathematical logic to come should have principles of additionality, relativity, etc. of its own.

Gödel has given the best result of applying nonpredicative methods in constructing the proof of his theorems. But it is the theme of a special subject.

 

  4. Gödel INCOMPLETENESS THEOREM.

 

Gödel had proved nonpredicativity of the arithmetic concealed in the metatheory. The essence of the incompleteness theory consists just in possibility to code numerically the logical symbols in the sentences of numerals and, using the recursive functions,— the logical relations between these symbols. But this time, the content relates to the metalogic. The fact, that this coding gives the possibility to formulate the liar paradox (one of the modifications of Russell's paradox) is just the consequence of provide by Gödel nonpredicativity.

Thus, the arithmetic is nonpredicative in the sense that the sentences of numerals are expressible by these numerals and metasentences — by the sentences. As we see, it is more fine nonpredicativity than that considered by Russell. Russell's nonpredicativity is its consequence. But the character of "derivation" this consequence goes beyond the boundaries of the formalism, in which we construct the arithmetic, and makes us to study the whole class of other formalisms.

What sentences are unsolvable in the arithmetic? The only ones for which there exists such a system of Gödel's enumeration in which they turn out to be the liar paradox (becoming nonpredicative according to Russell).

This enumeration class is virtually the whole class of formal languages which are isomorphic to that on which the arithmetic is built. The proof of this isomorphism is the main point of Gödel. In that case, if in one of this languages the sentence interpreted as the liar paradox, it will be unsolvable. An impression is created that in the boundaries of the classical mathematics Gödel uses methods radically different from formal and constructivistic ones. The statement is regarded here not as something rigidly formulated in the only formal language, but as a pattern of statements, as a function of the class of isomorphic languages. As a result, it is proved that this function has some properties which are invariant to the class as a whole. The consequance of this is unsolvability of the statements with nonpredicative interpretation in one of the languages. Moreover, this class of languages contains obligatorily the elements nonpredicative in the sense that they are built just of the "letters" of the individual sphere which models the theory. The fact, that this property is fulfiled for any theory in the boundaries of which the arithmetic can be modelled (and this is just the property of any theory we are interested in) points out clearly to the following general conclusions.

First, Gödel's theorems describe the correlations of the "concept" and its material embodiment according to Hegel's philosophy. It is precisely the ideal structure (the formal theory) that is reflected in this material structure which it describes (numerals, individual sphere) and, owing to this,reflects itself. Here we have a self-denying autoreflexive nature of the "concept". Thus, the infinite set together with the induction and recursion principles presents a formal model of the "concept". As Gödel has proved, it is not complete and enlargeable. Its noncontradictoriness is not to be determined practically. As to the further refinement of the model, the global character of the "concept" requires all the power limitations to be removed; besides, as we have mentioned above, the nonstandard variation of its formalism looks more preferably. That is why the proper class of all ordinals of the nonstandard set theory with the limitation axiom is to be considered as the most complete formalization of the "concept". The limitation axiom is equivalents to the fact that any set is obtained from 'Ø' by taking a Boolean value and uniting it is transfinite number of times. It corresponds to the "concept" in Hegel's philosophy. We have not discussed yet the truth of this philosophy, an adequate formalization of the "concept" has been considered. And this formalization, as mentioned above, is still far from being perfect.

Second, in the modern logic, Gödel's theorem pops up suddenly and accidentally, like a trick. For in the arithmetic itself the absence of the incompleteness is evident. This incompleteness puts forWard much more problems than solves them. Therefore, Gödel's theorem shows that the modern logic somehow exhausted itself and is required to be revised. And it is the way of proofing the theorem that shows the direction of the further development. The process of constructing a formal language (as well as the logic and theories in this language) cannot be separated from the process of its modelling. A structure uniting to faces: the predicate and the individual in itself should be built. The formalism as well as its model should loose their stiff unambiguity. The main apparatus must enclose some class of isomorphic languages and models. Having taken any of aspects of this structure, we should have obtained a formalism model or a model formalization.

It is difficult to say more concrete now about this future logic. But the fact mentioned earlier can serve as a confirmation of just this tendency of the development: the mathematics discerns and registers the individuals through the feasibility of some predicates. At the same time, the predicates themselves are defined in the long run as functions from the individual sphere into the Boolean (or Heyting) algebra as well, i.e. only through their own models. Therefore, the predicate cannot be considered separately from the individual. The problem of their correlation cannot be considered as solved a priori, as it makes the modern logic. All the problems arise just with this correlation. Boolean (as well as intuitionistic) algebra of "pure" statements is complete, solvable and noncontradictory. The problems arise just in the calculus of predicates in modelling the first-order theory. And the nature itself, as was shown for the quantum theory, does not correlate the individuals and predicates in a trivially external way. Therefore, nonpredicative methods acquire a special meaning. And despite the impossibility to use them directly, just they will serve the basis for the development of the logic and the key to understanding Hegel's dialectics.

 

  5. FORMALISM and CONSTRUCTIVISM.

 

Thus, the indeterminate and complementarity principles of the quantum theory arise, possible, from the same source that Gödel's incompleteness in the mathematics. In the quantum theory these principles arise almost like an establishment of empirical facts, and in the mathematics — as a sudden and not quite pleasant surprise. Thus, to find their deeper grounds is the matter of future. But may be here a constructive trend in the mathematics could help?

But despite all its theoretical and practical significance, the constructivism does not solve the global logic problems. The modern formal systems are not able to substantiate themselves. There can be formulated unsolvable questions and paradoxes arise in them: the concepts contradictory by themselves turn contradictory in this formalism. The paradoxes are eliminated by accumulating new and new, rather artificial axioms. The trouble is that these systems do not cover the deep reasons of origin of a paradox. The algorithmical method is not free from these problems as well. But the constructive solvability problem is not solvable. That means that there will always exist problems the solvability of which cannot be cleared by the algorithm theory. It is not simply the impossibility to state the truth, it is just meta-undecidability. And if such a problem is just not solvable owing to some concealed reasons, the algorithmical method will never find it out.

But the constructivism could not really get rid of such problems. These two trends in mathematics have begun with violent discussions and now they are trying to come to mutual understanding. And they do come to it. Any formal language has a strict algorithm of its construction of initial symbols. Introductions topological methods makes it possible to contract pseudoboolean algebra in Boolean algebra, and vice verso. The constructivism becomes a logic of solving problems, and the formalism — by the logic of mathematical deduction. Both approaches have become, as a result, equivalent and treat the same subject in two different aspects. Probably Brauer himself tried to find something quite different. Let us examine these problems more closely. The constructivism is considered to drive the infinity out of the mathematics. But it is not quite so. The potential infinity is a nonobvious applying of the actual one. As we have seen, the analogical situation originates in the set theory. Proper classes are introduced to avoid Russel's paradox. But having refused to consider nonpredicative concepts, we force them into the sphere of the metatheory, and then we are not immune against surprises either accepted (of the universal set kind), or not quite accepted (incompleteness theorem kind).

The question of rightfulness of applying the infinity concept turns only in the following: how much is the induction principle rightful in the empiric practice? Just by spreading the generalization of the observed in many experiments on all other experiments, one comes to the ideas of infinity, running the risk of erring and coming to a contradiction. And he do err by making the first inductive conclusions more accurate with the help of subsequent ones which are also inductive. But there is no cognition without the induction at all. Without induction we could not have designated the classes of objects by words, we could not have united a number of sensations of a certain magnitude, hardness, form, and weight with a word "stone". And this induction erroneous by definition (a conclusion based on a finite number of cases is groundless for all the infinite class of cases) is the main tool of the theoretical cognition. The induction itself represents this cognition without being able to substantiate itself. It is possible, that just in the practice of induction lies the cause of the self-negation properties of Hegel's "concept". But the main thing for us now is that the induction is simply unavoidable for thinking. The infinity is unavoidable for thinking as well. By denying the actual infinity, the constructivists act unconstructively, for they do not have arguments in favour of its rejection, as well as its maintaining. The only argument remains, that the infinity is not perceivable empirically. But many other things are not perceived empirically as well, such simple notions as "stone" (or "number"), for instance. Just simply being a notion, it has purely inductive nature, being linked with the experience but indirectly. If very strict refine the concept of account, the account of the empirical process will only unconscious sorting items. The concept of being impossible without experience, a purely inductive, out-experienced entity. At the time, it is well demonstrated Berkeley and Kant.

We have turned now to the sphere of empirical facts not accidentally. As we have mentioned, the empirical counting process presents the foundation of the constructivism. In essence, it is an experimental construction of the same classical mathematics, and the formalism follows the way fully equivalent to the constructive one, because the formalism builds its constructions of discretely discernible symbols using certain combinatorial rules. But the world perceived is more complex than the counting processes or the collection of discretely discernible elements. Observing smooth movement of pointers of his instruments, the physicist draws an inductive conclusion of a dense everywhere degree of order of values of the measured quantity. Nevertheless, he has to graduate these values by the discrete marks of the natural series to draw maximum of information out of the position of the pointers.

All this presents idealization of our intellect and our reference systems. But if the formal logic is occupied with noncontradictory interpretation of the contradictions, the intuitionism one is engaged in constructive interpretation of the nonconstructive world.

 

  6. The CONTENT and the FORM.

 

In the paragraph 5 a subject of infinity was regarded. As to, the real World, we should admit its infinity at least due to its logical contradictoriness and principal nonpredicativity. But our knowledge is always finite, for it is always expressed through a finite number of symbols of the formal language. And it just turns out, that we cognize the infinite by using the finite means, somehow reducing, limiting the infinity in our imagination. We think, the logical "reference system" within the limits of which we imagine the future logic, will just reflect and limit differently the infinity of the World with its finite means.

Speaking of the infinity of the World and of the finiteness of the knowledge, as a matter of fact, we speak of the substantial and the formal. Because under the content, the formal theory semantics mean always not obviously its real empirical model. The term "content" has simply no other meaning. The formal arithmetic studies the properties of numbers expressible by the formal and logical method. The substantial arithmetic is occupied with real count processes. Being preoccupied with the form exclusively, the mathematics applies in the end for the content to the empirical practice only. But just in the end, because there is a relativity of terminological and modal character. On the whole, the group theory can be called the form, and such properties as commutativity, cyclicity, etc., attributed to the content. But this will be the content which can be formalized, i.e. the formal one. It is done so in the mathematics by modelling a theory in the individual sphere of another one. For all that, the "true", absolute content rests, nevertheless on the empirical reality. The algorithmical language is of value only in that one can write an algorithm down here and now, on this sheet of paper and then apply it to the really written symbols.

Thus, the "true" content is a complete model of the formal theory. This model can be said much about, but it exists only in reality as a concrete object, virtually being without formal existance. But the abstract, formal object exists as a language structure. Gödel's theorem just proves inexhaustibility of the content by the form. Thus, the content is an infinite form gradation coming true really but not formally. The form presents an infinite plurality of contents embraced by the formal theory, infinite as much as to be realized formally.

All the terms of any language are formal in the sense that they designate an abstract "object in general" of the kind "stone", "house", "variable", etc. But the real house is rich in content, having an infinite plurality of properties, and any its axiomatic description will be incomplete. On any concrete projects of architects and builders an infinite number of houses, of copies of their formal plan can be constructed. Hence, for any formal means some rich in content objects are indiscernible and even become apparent as coinciding (according to the quantum theory), although they are alternative.

Consequently, the problem of the object individualization is far from trivial the empirical point of view, as supposed by the set theory relative to the elements or the algorithm theory relative to the totality of the inscriptions, representing similar objects. Besides, when studying modelling and semantic problems, the mathematics inaviteble turns to the empirical practice which requires revision of the initial mathematical postulates.

 

  7. METAMATHEMATICS.

 

Despite the fact, that the modern mathematics understand the truth of statements rather absolutely, some relativity moment is still inevitable. Therefore, any theory or logic imply metatheory and metalogic nonmanifestly. A formal language cannot be built on nothing. A metalanguage is necessary for this. And the number of these prefixes: "meta-", in fact, should be built up infinitely. So that operating on the finite objects only, the constructivism simply ignores the infinite number of the metalevels of its determinations. This sequence is usually cut off artificially, trying to build the theory so that the metastatements were as trivial as possible; then all the necessary metainformation is reduced to the thesis that the letters of the language are discernible and there is an unlimited stock of inscriptions for any letter.

But Gödel's theorem has destroyed this seeming simplicity as well. Figuratively speaking, a "short circuit" between the individual sphere and the sphere of statements of them is created. This nonpredicativity pierces all the infinite sequence formed by the prefix "meta-". All the more, it is necessary to construct the formalism so, that all metalevels and logical reference systems were taken into consideration in the foundations of the mathematics itself.

Though, even without taking into account Gödel's theorem, the formal logic has to touch upon the metalevels, because any logical operation can be interpreted only as a metastatement semantically. And this is just the deep basis for constructive criticism of the classical implication and negation. Namely: (А→B) is a metastatement of the relation of the truth values of the А and В statements. That is why the constructivists insist that without a real derivation of В from А, the implication is simply a logically groundless formal trick of Boolean algebra. Analogically, the quantum theory calls the classical operations (А&В) and (АvВ) is questions as well.

Apropos, in in physics not only the quantum theory contains nonpredicativity of objects and their properties. The same is in the general relativity theory. Being a relation between material objects, space-time itself becomes a material objects possessing even nonzero stress- energy tensor (a pseudotensor, more accurately, which is immaterial in this case). The space-time can possess a curvature of its own, not induced by the matter but influencing manifestly the matter behavior. So that her as well obtain nonpredicativeness.

But the real nonpredicativeness of the nature means that everything becoming apparent in our formalism as metalogic, must have an analogue in the physical reality as well. All property sequence of any metalevels should be realized somehow physically. It is just formally, that we can come to the agreement not to consider this sequence. But in reality it must be real.

In this connection, it is necessary to return to the problem of correlation between the material and ideal. Such properties as "to be material" and "to be ideal" are deeply relative. The matter is that which exists outside and independently of perceiving it reference system (let us venture to exclude anthropomorphism out of Marx's definition). I.e., this is what is reflected in the given reference system but has its cause in the other system. Therefore, the matter always exists "outside itself". Here we come close to Hegel's ideas. According to Marx, the ideal is defined as a reflection of the material in the material. But Marx does not take into account the reference system. Where is it, this ideal? It is somewhere, where it is present as a reflection, i.e. in the third object perceiving both the reflecting and the reflected. But then, the 4-th object, etc. is necessary for the existence of the ideal. For the metareflection itself must be ideal. But all this unlimited "meta"-sequence can become something whole (from a whole reference system) in the self-reflection only. So it turns out, that the ideal exists in itself and for itself, its nature is in autoreflection; it is something that perceives itself and is perceived by itself. And it is only in itself, that the ideal is ideal. The information, the reflected image, is not in a mirror or in a reflected object but in the reflection itself taken from the side of its self- definition. If some physical and mathematical abstractions are excluded, one should admit the nonlinearity of all existing in the nature reflections, interactions. But the nonlinearity is a synonym of self-action. Material objects exist through interaction only. Therefore, the matter always creates the information, the ideal. And vice versa. The ideal existing only in itself, simply does not exist for anything another. But the ideal possessing the reality for something else (differing from it), just materializes. The matter that exists (perceives and is perceived) outwardly and in the outward (outside itself). In this sense, the matter is limited, finite. And the ideal is infinite just in the sense of autoreflection (let us remember Cantor).

And now let us turn to the mathematical aspect of the subject. We say that the set Y contains an information of the set Х, if there is a structural homomorphism f: X→Y. In reality, the information is all three (X,Y,f), and in Y it is contained but relatively. Actually, the information is contained in the Galois correspondence G(f)[X,Y], describing this structure,— that forces us to consider the mappings the booleans: 2^X and 2^Y, etc. But looses all its sense without its metatheory, etc. We have an infinite sequence Booleans. Objective realisation of reflexion demands objective realisation the Booleans of all orders. In nature, this metastructure present in complete form. The only way to define the location of the information exactly is the requirement of Gödel's incompleteness (X,Y,f). The statement of this structure should be encoded in itself, and it must be an integral part of the structure and not an amusing incident arising post factum of its construction. We suppose, the logic must develop just in this direction.

 

  8. THEORY of CATEGORIES.

 

Let us again formulate briefly the requirements to the future development of the mathematics and logic which we have tried to ground here. Virtually, it is one requirement: total relativity brought to the reference system level. In this case, the following relativity moments are the key ones: 1) self-relativity meaning nonpredicativity of the formalism in essence, 2) relativity of the truth, 3) relativity of effective means, 4) relativity of the individualization of cognition objects.

But we should like to present this future logic more concretely. From our point of view, the theory of categories is a forerunner of the new mathematics, the first attempt to build it. Why?

The theory of categories is an abstraction of purely functional relations between objects. It considers not sets and reflections, but their purely algebraical properties. Hence nontrivial view of the individualization of elements.

In fact, abstraction of categories embraces not only sets and reflections, but also a class of any mathematical structures and their homomorphisms. The central structure of the theory become not the objects with their elements, but the homomorphisms of the objects — arrows. This leads to strengthening of the relativity moment and to loss of stiff individualization of the object typical to the set theory and algorithm theory. Any arrow is a representative of a class of the arrows isomorphic to it, formed by its multiplication by the isoarrows. The category isomorphic arrows are indiscernible by their main properties. But all this is not as simple as that. The main definitions of the category theories are constructed so that selection of an arrow of the class isomorphic to it depends on the analogical selection from other classes. For instance, the diagram limit is unique to within isomorphism (but there is the only arrow letting to pass through it a given cone over the diagram). Thus, the individualization of certain arrows determines strictly the individualization of others. It happens because in the category theory, the algebraical approach (associative composition law) is combined with the combinatory geometrical one (partiality of composition law; presence of the left and right units for each arrow). It is interesting as well, that the category approach allows to formulate most completely nonstandard methods, in the sheaf theory. But the concept of the limit, topos, sheaf comes to different particular cases of one concept — of the category conjugacy. The conjugation is the most accurate model of Hegel's "concept". Substantion of this statement would have required a too detailed semantic analysis of Hegel's works and exceeded the bounds of this article. Therefore, we ask the reader, if he is interested in it, to solve the problem of truth of our statement on his own. We stress only, that the apparatus of arrows and functors allows to express formally the semantics of the philosophical term "reflection" which has a purely informational nature. For instance, such formations as category К↓а all arrows of the category К in some its object а, allow to say of peculiar "reference systems" in the logic. If К is a topos, then the К↓а is topos as well and just as the К, has the subobject classifier and the logic of its own.

Thus, using language of categories, one can create working models of our main requirements to the logic: nonpredicativity, relativity flexibility of the individualization. But it is impossible to maintain that these are good models. And the category theory itself is equivalent in many respects to the set theory. But deeper principles which can be developed in future, stand probably behind the construction of the category theory.

 

 

Vladimir Zhuravlev