Fallibilist a priori. Thought of the Day 127.0

Figure-1-Peirce's-ten-classes-of-sign-represented-in-terms-of-three-core-functions-and

Kant’s ‘transcendental subject’ is pragmatized in this notion in Peirce, transcending any delimitation of reason to the human mind: the ‘anybody’ is operational and refers to anything which is able to undertake reasoning’s formal procedures. In the same way, Kant’s synthetic a priori notion is pragmatized in Peirce’s account:

Kant declares that the question of his great work is ‘How are synthetical judgments a priori possible?’ By a priori he means universal; by synthetical, experiential (i.e., relating to experience, not necessarily derived wholly from experience). The true question for him should have been, ‘How are universal propositions relating to experience to be justified?’ But let me not be understood to speak with anything less than profound and almost unparalleled admiration for that wonderful achievement, that indispensable stepping-stone of philosophy. (The Essential Peirce Selected Philosophical Writings)

Synthetic a priori is interpreted as experiential and universal, or, to put it another way, observational and general – thus Peirce’s rationalism in demanding rational relations is connected to his scholastic realism posing the existence of real universals.

But we do not make a diagram simply to represent the relation of killer to killed, though it would not be impossible to represent this relation in a Graph-Instance; and the reason why we do not is that there is little or nothing in that relation that is rationally comprehensible. It is known as a fact, and that is all. I believe I may venture to affirm that an intelligible relation, that is, a relation of thought, is created only by the act of representing it. I do not mean to say that if we should some day find out the metaphysical nature of the relation of killing, that intelligible relation would thereby be created. [ ] No, for the intelligible relation has been signified, though not read by man, since the first killing was done, if not long before. (The New Elements of Mathematics)

Peirce’s pragmatizing Kant enables him to escape the threatening subjectivism: rational relations are inherent in the universe and are not our inventions, but we must know (some of) them in order to think. The relation of killer to killed, is not, however, given our present knowledge, one of those rational relations, even if we might later become able to produce a rational diagram of aspects of it. Yet, such a relation is, as Peirce says, a mere fact. On the other hand, rational relations are – even if inherent in the universe – not only facts. Their extension is rather that of mathematics as such, which can be seen from the fact that the rational relations are what make necessary reasoning possible – at the same time as Peirce subscribes to his father’s mathematics definition: Mathematics is the science that draws necessary conclusions – with Peirce’s addendum that these conclusions are always hypothetical. This conforms to Kant’s idea that the result of synthetic a priori judgments comprised mathematics as well as the sciences built on applied mathematics. Thus, in constructing diagrams, we have all the possible relations in mathematics (which is inexhaustible, following Gödel’s 1931 incompleteness theorem) at our disposal. Moreover, the idea that we might later learn about the rational relations involved in killing entails a historical, fallibilist rendering of the a priori notion. Unlike the case in Kant, the a priori is thus removed from a privileged connection to the knowing subject and its transcendental faculties. Thus, Peirce rather anticipates a fallibilist notion of the a priori.

Advertisement

Tarski, Wittgenstein and Undecidable Sentences in Affine Relation to Gödel’s. Thought of the Day 65.0

 maxresdefault

I imagine someone asking my advice; he says: “I have constructed a proposition (I will use ‘P’ to designate it) in Russell’s symbolism, and by means of certain definitions and transformations it can be so interpreted that it says: ‘P is not provable in Russell’s system.’ Must I not say that this proposition on the one hand is true, and on the other hand is unprovable? For suppose it were false; then it is true that it is provable. And that surely cannot be! And if it is proved, then it is proved that it is not provable. Thus it can only be true, but unprovable.” — Wittgenstein

Any language of such a set, say Peano Arithmetic PA (or Russell and Whitehead’s Principia Mathematica, or ZFC), expresses – in a finite, unambiguous, and communicable manner – relations between concepts that are external to the language PA (or to Principia, or to ZFC). Each such language is, thus, essentially two-valued, since a relation either holds or does not hold externally (relative to the language).

Further, a selected, finite, number of primitive formal assertions about a finite set of selected primitive relations of, say, PA are defined as axiomatically PA-provable; all other assertions about relations that can be effectively defined in terms of the primitive relations are termed as PA-provable if, and only if, there is a finite sequence of assertions of PA, each of which is either a primitive assertion, or which can effectively be determined in a finite number of steps as an immediate consequence of any two assertions preceding it in the sequence by a finite set of rules of consequence.

The philosophical dimensions of this emerges if we take M as the standard, arithmetical, interpretation of PA, where:

(a)  the set of non-negative integers is the domain,

(b)  the integer 0 is the interpretation of the symbol “0” of PA,

(c)  the successor operation (addition of 1) is the interpretation of the “ ‘ ” function,

(d)  ordinary addition and multiplication are the interpretations of “+” and “.“,

(e) the interpretation of the predicate letter “=” is the equality relation.

Now, post-Gödel, the standard interpretation of classical theory seems to be that:

(f) PA can, indeed, be interpreted in M;

(g) assertions in M are decidable by Tarski’s definitions of satisfiability and truth;

(h) Tarskian truth and satisfiability are, however, not effectively verifiable in M.

Tarski made clear his indebtedness to Gödel’s methods,

We owe the method used here to Gödel who employed it for other purposes in his recently published work Gödel. This exceedingly important and interesting article is not directly connected with the theme of our work it deals with strictly methodological problems the consistency and completeness of deductive systems, nevertheless we shall be able to use the methods and in part also the results of Gödel’s investigations for our purpose.

On the other hand Tarski strongly emphasized the fact that his results were obtained independently, even though Tarski’s theorem on the undefinability of truth implies the existence of undecidable sentences, and hence Gödel’s first incompleteness theorem. Shifting gears here, how far was the Wittgensteinian quote really close to Gödel’s? However, the question, implicit in Wittgenstein’s argument regarding the possibility of a semantic contradiction in Gödel’s reasoning, then arises: How can we assert that a PA-assertion (whether such an assertion is PA-provable or not) is true under interpretation in M, so long as such truth remains effectively unverifiable in M? Since the issue is not resolved unambiguously by Gödel in his paper (nor, apparently, by subsequent standard interpretations of his formal reasoning and conclusions), Wittgenstein’s quote can be taken to argue that, although we may validly draw various conclusions from Gödel’s formal reasoning and conclusions, the existence of a true or false assertion of M cannot be amongst them.

Algorithmic Subfield Representation of the Depth of Descent Tree

1-up

A finite field K admits a sparse medium subfield representation if

– it has a subfield of q2 elements for a prime power q, i.e. K is isomorphic to Fq2k with k ≥ 1;

– there exist two polynomials h0 and h1 over Fq2 of small degree, such that h1Xq − h0 has a degree k irreducible factor.

We shall assume that all the fields under consideration admit a sparse medium subfield representation. Furthermore, we also assume that the degrees of the polynomials h0 and h1 are uniformly bounded by a constant δ. Any finite field of the form Fq2k with k ≤ q + 2 admits a sparse medium subfield representation with polynomials h0 and h1 of degree at most 2.

In a field in sparse medium subfield representation, elements will always be represented as polynomials of degree less than k with coefficients in Fq2. When we talk about the discrete logarithm of such an element, we implicitly assume that a basis for this discrete logarithm has been chosen, and that we work in a subgroup whose order has no small irreducible factor to limit ourselves to this case.

Proposition: Let K = Fq2k be a finite field that admits a sparse medium subfield representation. Under the heuristics, there exists an algorithm whose complexity is polynomial in q and k and which can be used for the following two tasks.

1. Given an element of K represented by a polynomial P ∈ Fq2[X] with 2 ≤ deg P ≤ k − 1, the algorithm returns an expression of log P (X ) as a linear combination of at most O(kq2) logarithms logPi(X) with degPi ≤ ⌈1/2 degP⌉ and of log h1(X).

2. The algorithm returns the logarithm of h1(X) and the logarithms of all the elements of K of the form X + a, for a in Fq2.

Let P(X) be an element of K for which we want to compute the discrete logarithm. Here P is a polynomial of degree at most k − 1 and with coefficients in Fq2. We start by applying the algorithm of the above Proposition to P. We obtain a relation of the form

log P = e0 log h1 + ei log Pi,

where the sum has at most κq2k terms for a constant κ and the Pi’s have degree at most ⌈1/2 degP⌉. Then, we apply recursively the algorithm to the Pi’s, thus creating a descent procedure where at each step, a given element P is expressed as a product of elements, whose degree is at most half the degree of P (rounded up) and the arity of the descent tree is in O(q2k). At the end of the process, the logarithm of P is expressed as a linear combination of the logarithms of h1 and of the linear polynomials, for which the logarithms are computed with the algorithm in the above Proposition in its second form.

We are left with the complexity analysis of the descent process. Each internal node of the descent tree corresponds to one application of the algorithm of the above Proposition, therefore each internal node has a cost which is bounded by a polynomial in q and k. The total cost of the descent is therefore bounded by the number of nodes in the descent tree times a polynomial in q and k. The depth of the descent tree is in O(log k). The number of nodes of the tree is then less than or equal to its arity raised to the power of its depth, which is (q2k)O(log k). Since any polynomial in q and k is absorbed in the O() notation in the exponent, we obtain the following result.

Let K = Fq2k be a finite field that admits a sparse medium subfield representation. Assuming the same heuristics as in the above Proposition, any discrete logarithm in K can be computed in a time bounded by

max(q, k)O(log k)

Mappings, Manifolds and Kantian Abstract Properties of Synthesis

sol-lewitt

An inverse system is a collection of sets which are connected by mappings. We start off with the definitions before relating these to abstract properties of synthesis.

Definition: A directed set is a set T together with an ordering relation ≤ such that

(1) ≤ is a partial order, i.e. transitive, reflexive, anti-symmetric

(2) ≤ is directed, i.e. for any s, t ∈ T there is r ∈ T with s, t ≤ r

Definition: An inverse system indexed by T is a set D = {Ds|s ∈ T} together with a family of mappings F = {hst|s ≥ t, hst : Ds → Dt}. The mappings in F must satisfy the coherence requirement that if s ≥ t ≥ r, htr ◦ hst = hsr.

Interpretation of the index set: The index set represents some abstract properties of synthesis. The ‘synthesis of apprehension in intuition’ proceeds by a ’running through and holding together of the manifold’ and is thus a process that takes place in time. We may now think of an index s ∈ T as an interval of time available for the process of ’running through and holding together’. More formally, s can be taken to be a set of instants or events, ordered by a ‘precedes’ relation; the relation t ≤ s then stands for: t is a substructure of s. It is immediate that on this interpretation ≤ is a partial order. The directedness is related to what Kant called ‘the formal unity of the consciousness in the synthesis of the manifold of representations’ or ‘the necessary unity of self-consciousness, thus also of the synthesis of the manifold, through a common function of the mind for combining it in one representation’ – the requirement that ‘for any s, t ∈ T there is r ∈ T with s, t ≤ r’ creates the formal conditions for combining the syntheses executed during s and t in one representation, coded by r.

Interpretation of the Ds and the mappings hst : Ds → Dt. An object in Ds can thought of as a possible ‘indeterminate object of empirical intuition’ synthesised in the interval s. If s ≥ t, the mapping hst : Ds → Dt expresses a consistency requirement: if d ∈ Ds represents an indeterminate object of empirical intuition synthesised in interval s, so that a particular manifold of features can be ‘run through and held together’ during s, some indeterminate object of empirical intuition must already be synthesisable by ‘running through and holding together’ in interval t, e.g. by combining a subset of the features characaterising d. This interpretation justifies the coherence condition s ≥ t ≥ r, htr ◦ hst = hsr: the synthesis obtained from first restricting the interval available for ‘running through and holding together’ to interval t, and then to interval r should not differ from the synthesis obtained by restricting to r directly.

We do not put any further requirements on the mappings hst : Ds → Dt, such as surjectivity or injectivity. Some indeterminate object of experience in Dt may have disappeared in Ds: more time for ‘running through and holding together’ may actually yield fewer features that can be combined. Thus we do not require the mappings to be surjective. It may also happen that an indeterminate object of experience in Dt corresponds to two or more of such objects in Ds, as when a building viewed from afar upon closer inspection turns out to be composed of two spatially separated buildings; thus the mappings need not be injective.

The interaction of the directedness of the index set and the mappings hst is of some interest. If r ≥ s, t there are mappings hrs : Dr → Ds and hrt : Ds → Dt. Each ‘indeterminate object of empirical intuition’ in d ∈ Dr can be seen as a synthesis of such objects hrs(d) ∈ Ds and hrt(d) ∈ Dt. For example, the ‘manifold of a house’ can be viewed as synthesised from a ‘manifold of the front’ and a ‘manifold of the back’. The operation just described has some of the characteristics of the synthesis of reproduction in imagination: the fact that the front of the house can be unified with the back to produce a coherent object presupposes that the front can be reproduced as it is while we are staring at the back. The mappings hrs : Dr → Ds and hrt : Ds → Dt capture the idea that d ∈ Dr arises from reproductions of hrs(d) and hrt(d) in r.

Rhizomatic Topology and Global Politics. A Flirtatious Relationship.

 

rhizome

Deleuze and Guattari see concepts as rhizomes, biological entities endowed with unique properties. They see concepts as spatially representable, where the representation contains principles of connection and heterogeneity: any point of a rhizome must be connected to any other. Deleuze and Guattari list the possible benefits of spatial representation of concepts, including the ability to represent complex multiplicity, the potential to free a concept from foundationalism, and the ability to show both breadth and depth. In this view, geometric interpretations move away from the insidious understanding of the world in terms of dualisms, dichotomies, and lines, to understand conceptual relations in terms of space and shapes. The ontology of concepts is thus, in their view, appropriately geometric, a multiplicity defined not by its elements, nor by a center of unification and comprehension and instead measured by its dimensionality and its heterogeneity. The conceptual multiplicity, is already composed of heterogeneous terms in symbiosis, and is continually transforming itself such that it is possible to follow, and map, not only the relationships between ideas but how they change over time. In fact, the authors claim that there are further benefits to geometric interpretations of understanding concepts which are unavailable in other frames of reference. They outline the unique contribution of geometric models to the understanding of contingent structure:

Principle of cartography and decalcomania: a rhizome is not amenable to any structural or generative model. It is a stranger to any idea of genetic axis or deep structure. A genetic axis is like an objective pivotal unity upon which successive stages are organized; deep structure is more like a base sequence that can be broken down into immediate constituents, while the unity of the product passes into another, transformational and subjective, dimension. (Deleuze and Guattari)

The word that Deleuze and Guattari use for ‘multiplicities’ can also be translated to the topological term ‘manifold.’ If we thought about their multiplicities as manifolds, there are a virtually unlimited number of things one could come to know, in geometric terms, about (and with) our object of study, abstractly speaking. Among those unlimited things we could learn are properties of groups (homological, cohomological, and homeomorphic), complex directionality (maps, morphisms, isomorphisms, and orientability), dimensionality (codimensionality, structure, embeddedness), partiality (differentiation, commutativity, simultaneity), and shifting representation (factorization, ideal classes, reciprocity). Each of these functions allows for a different, creative, and potentially critical representation of global political concepts, events, groupings, and relationships. This is how concepts are to be looked at: as manifolds. With such a dimensional understanding of concept-formation, it is possible to deal with complex interactions of like entities, and interactions of unlike entities. Critical theorists have emphasized the importance of such complexity in representation a number of times, speaking about it in terms compatible with mathematical methods if not mathematically. For example, Foucault’s declaration that: practicing criticism is a matter of making facile gestures difficult both reflects and is reflected in many critical theorists projects of revealing the complexity in (apparently simple) concepts deployed both in global politics.  This leads to a shift in the concept of danger as well, where danger is not an objective condition but “an effect of interpretation”. Critical thinking about how-possible questions reveals a complexity to the concept of the state which is often overlooked in traditional analyses, sending a wave of added complexity through other concepts as well. This work seeking complexity serves one of the major underlying functions of critical theorizing: finding invisible injustices in (modernist, linear, structuralist) givens in the operation and analysis of global politics.

In a geometric sense, this complexity could be thought about as multidimensional mapping. In theoretical geometry, the process of mapping conceptual spaces is not primarily empirical, but for the purpose of representing and reading the relationships between information, including identification, similarity, differentiation, and distance. The reason for defining topological spaces in math, the essence of the definition, is that there is no absolute scale for describing the distance or relation between certain points, yet it makes sense to say that an (infinite) sequence of points approaches some other (but again, no way to describe how quickly or from what direction one might be approaching). This seemingly weak relationship, which is defined purely ‘locally’, i.e., in a small locale around each point, is often surprisingly powerful: using only the relationship of approaching parts, one can distinguish between, say, a balloon, a sheet of paper, a circle, and a dot.

To each delineated concept, one should distinguish and associate a topological space, in a (necessarily) non-explicit yet definite manner. Whenever one has a relationship between concepts (here we think of the primary relationship as being that of constitution, but not restrictively, we ‘specify’ a function (or inclusion, or relation) between the topological spaces associated to the concepts). In these terms, a conceptual space is in essence a multidimensional space in which the dimensions represent qualities or features of that which is being represented. Such an approach can be leveraged for thinking about conceptual components, dimensionality, and structure. In these terms, dimensions can be thought of as properties or qualities, each with their own (often-multidimensional) properties or qualities. A key goal of the modeling of conceptual space being representation means that a key (mathematical and theoretical) goal of concept space mapping is

associationism, where associations between different kinds of information elements carry the main burden of representation. (Conceptual_Spaces_as_a_Framework_for_Knowledge_Representation)

To this end,

objects in conceptual space are represented by points, in each domain, that characterize their dimensional values. A concept geometry for conceptual spaces

These dimensional values can be arranged in relation to each other, as Gardenfors explains that

distances represent degrees of similarity between objects represented in space and therefore conceptual spaces are “suitable for representing different kinds of similarity relation. Concept

These similarity relationships can be explored across ideas of a concept and across contexts, but also over time, since “with the aid of a topological structure, we can speak about continuity, e.g., a continuous change” a possibility which can be found only in treating concepts as topological structures and not in linguistic descriptions or set theoretic representations.

Universal Inclusion of the Void. Thought of the Day 38.0

entering-the-void

The universal inclusion of the void means that the intersection between two sets whatsoever is comparable with the void set. That is to say, there is no multiple that does not include within it some part of the “inconsistency” that it structures. The diversity of multiplicity can exhibit multiple modes of articulation, but as multiples, they have nothing to do with one another, they are two absolutely heterogeneous presentations, and this is why this relation – of non-relation – can only be thought under the signifier of being (of the void), which indicates that the multiples in question have nothing in common apart from being multiples. The universal inclusion of the void thus guarantees the consistency of the infinite multiplicities immanent to its presentation. That is to say, it underlines the universal distribution of the ontological structure seized at the point of the axiom of the void set. The void does not merely constitute a consistency at a local point but also organises, from this point of difference, a universal structure that legislates on the structure of all sets, the universe of consistent multiplicity.

This final step, the carrying over of the void seized as a local point of the presentation of the unpresentable, to a global field of sets provides us with the universal point of difference, applicable equally to any number of sets, that guarantees the universal consistency of ontological presentation. In one sense, the universal inclusion of the void demonstrates that, as a unit of presentation, the void anchors the set theoretical universe by its universal inclusion. As such, every presentation in ontological thought is situated in this elementary seizure of ontological difference. The void is that which “fills” ontological or set theoretical presentation. It is what makes common the universe of sets. It is in this sense that the “substance” or constitution of ontology is the void. At the same stroke, however, the universal inclusion of the void also concerns the consistency of set theory in a logical sense.

The universal inclusion of the void provides an important synthesis of the consistency of presentation. What is presented is necessarily consistent but its consistency gives way to two distinct senses. Consistency can refer to its own “substance,” its immanent presentation. Distinct presentations constitute different presentations principally because “what” they present are different. Ontology’s particularity is its presentation of the void. On the other hand, a political site might present certain elements just as a scientific procedure might present yet others. The other sense of consistency is tied to presentation as such, the consistency of presentation in its generality. When one speaks loosely about the “world” being consistent, where natural laws are verifiable against a background of regularity, it is this consistency that is invoked and not the elements that constitute the particularity of their presentation. This sense of consistency, occurring across presentations would certainly take us beyond the particularity of ontology. That is to say, ontological presentation presents a species of this consistency. However, the possibility of multiple approaches does not exclude an ontological treatment of this consistency.

Causality

Quantum_Computer

Causation is a form of event generation. To present an explicit definition of causation requires introducing some ontological concepts to formally characterize what is understood by ‘event’.

The concept of individual is the basic primitive concept of any ontological theory. Individuals associate themselves with other individuals to yield new individuals. It follows that they satisfy a calculus, and that they are rigorously characterized only through the laws of such a calculus. These laws are set with the aim of reproducing the way real things associate. Specifically, it is postulated that every individual is an element of a set s in such a way that the structure S = ⟨s, ◦, ◻⟩ is a commutative monoid of idempotents. This is a simple additive semi-group with neutral element.

In the structure S, s is the set of all individuals, the element ◻ ∈ s is a fiction called the null individual, and the binary operation ◦ is the association of individuals. Although S is a mathematical entity, the elements of s are not, with the only exception of ◻, which is a fiction introduced to form a calculus. The association of any element of s with ◻ yields the same element. The following definitions characterize the composition of individuals.

1. x ∈ s is composed ⇔ (∃ y, z) s (x = y ◦ z)
2. x ∈ s is simple ⇔ ∼ (∃ y, z) s (x = y ◦ z)
3. x ⊂ y ⇔ x ◦ y = y (x is part of y ⇔ x ◦ y = y)
4. Comp(x) ≡ {y ∈ s|y ⊂ x} is the composition of x.

Real things are distinguished from abstract individuals because they have a number of properties in addition to their capability of association. These properties can be intrinsic (Pi) or relational (Pr). The intrinsic properties are inherent and they are represented by predicates or unary applications, whereas relational properties depend upon more than a single thing and are represented by n-ary predicates, with n ≥ 1. Examples of intrinsic properties are electric charge and rest mass, whereas velocity of macroscopic bodies and volume are relational properties.

An individual with its properties make up a thing X : X =< x, P(x) >

Here P(x) is the collection of properties of the individual x. A material thing is an individual with concrete properties, i.e. properties that can change in some respect.

The state of a thing X is a set of functions S(X) from a domain of reference M (a set that can be enumerable or nondenumerable) to the set of properties PX. Every function in S(X) represents a property in PX. The set of the physically accessible states of a thing X is the lawful state space of X : SL(X). The state of a thing is represented by a point in SL(X). A change of a thing is an ordered pair of states. Only changing things can be material. Abstract things cannot change since they have only one state (their properties are fixed by definition).

A legal statement is a restriction upon the state functions of a given class of things. A natural law is a property of a class of material things represented by an empirically corroborated legal statement.

The ontological history h(X) of a thing X is a subset of SL(X) defined by h(X) = {⟨t, F(t)⟩|t ∈ M}

where t is an element of some auxiliary set M, and F are the functions that represent the properties of X.

If a thing is affected by other things we can introduce the following definition:

h(Y/X ) : “history of the thing Y in presence of the thing X”.

Let h(X) and h(Y) be the histories of the things X and Y, respectively. Then

h(Y/X) = {⟨t,H(t)⟩|t ∈ M},

where H≠ F is the total state function of Y as affected by the existence of X, and F is the total state function of X in the absence of Y. The history of Y in presence of X is different from the history of Y without X .

We can now introduce the notion of action:

X ▷ Y : “X acts on Y”

X ▷ Y =def h(Y/X) ≠ h(Y)

An event is a change of a thing X, i.e. an ordered pair of states:

(s1, s2) ∈ EL(X) = SL(X) × SL(X)

The space EL(X) is called the event space of X.

Causality is a relation between events, i.e. a relation between changes of states of concrete things. It is not a relation between things. Only the related concept of ‘action’ is a relation between things. Specifically,

C'(x): “an event in a thing x is caused by some unspecified event exxi“.

C'(x) =def (∃ exxi) [exxi ∈ EL(X) ⇔ xi ▷ x.

C(x, y): “an event in a thing x is caused by an event in a thing y”.

C(x, y) =def (∃ exy) [exy ∈ EL(x) ⇔ y ▷ x

In the above definitions, the notation exy indicates in the superscript the thing x to whose event space belongs the event e, whereas the subscript denotes the thing that acted triggering the event. The implicit arguments of both C’ and C are events, not things. Causation is a form of event generation. The crucial point is that a given event in the lawful event space EL(x) is caused by an action of a thing y iff the event happens only conditionally to the action, i.e., it would not be the case of exy without an action of y upon x. Time does not appear in this definition, allowing causal relations in space-time without a global time orientability or even instantaneous and non-local causation. If causation is non-local under some circumstances, e.g. when a quantum system is prepared in a specific state of polarization or spin, quantum entanglement poses no problem to realism and determinism. The quantum theory describes an aspect of a reality that is ontologically determined and with non-local relations. Under any circumstances the postulates of Special Relativity are violated, since no physical system ever crosses the barrier of the speed of light.