Those incompleteness theorems mean the relation of (Peano) arithmetic and (ZFC) set theory, or ph... more Those incompleteness theorems mean the relation of (Peano) arithmetic and (ZFC) set theory, or philosophically, the relation of arithmetical finiteness and actual infinity. The same is managed in the framework of set theory by the axiom of choice (respectively, by the equivalent well-ordering "theorem'). One may discuss that incompleteness form the viewpoint of set theory by the axiom of choice rather than the usual viewpoint meant in the proof of theorems. The logical corollaries from that "nonstandard" viewpoint the relation of set theory and arithmetic are demonstrated.
Много отчетливо беше всичко в тая нощ и още по-отчет ливо и ясно стана за Исая, като разбра, че н... more Много отчетливо беше всичко в тая нощ и още по-отчет ливо и ясно стана за Исая, като разбра, че него не са го правили с индиго, а един бог знае кой го е надраскал несръчно, отделно от ония, вадените с индиго.
Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two c... more Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two connotations (connotations or metaphors are inherent in literature rather than in mathematics or logics). The paradox itself refers to implication demonstrating that an intermediate implication can be always inserted in an implication therefore postponing its ultimate conclusion for the next step and those insertions can be iteratively and indefinitely added ad lib, as if ad infinitum. Both connotations clear up links due to the shared formal structure with other well-known mathematical observations: (1) the paradox of Achilles and the Turtle; (2) the transitivity of the relation of equality. Analogically to (1), one can juxtapose the paradox of the Liar (for Lewis Carroll's paradox) and that of the arrow (for "Achilles and the Turtle"), i.e. a logical paradox, on the one hand, and an aporia of motion, on the other hand, suggesting a shared formal structure of both, which can be called "ontological", on which basis "motion" studied by physics and "conclusion" studied by logic can be unified being able to bridge logic and physics philosophically in a Hegelian manner: even more, the bridge can be continued to mathematics in virtue of (2), which forces the equality (for its property of transitivity) of any two quantities to be postponed analogically ad lib and ad infinitum. The paper shows that Hilbert arithmetic underlies naturally Lewis Carroll's paradox admitting at least three interpretations linked to each other by it: mathematical, physical and logical. Thus, it can be considered as both generalization and solution of his paradox therefore naturally unifying the completeness of quantum mechanics (i.e. the absence of hidden variables) and eventual completeness of mathematics as the same and isomorphic to the completeness of propositional logic in relation to set theory as a first-order logic (in the sense of Gödel (1930)'s completeness theorems).
The paper discusses the philosophical conclusions, which the interrelation between quantum mechan... more The paper discusses the philosophical conclusions, which the interrelation between quantum mechanics and general relativity implies by quantum measure. Quantum measure is three-dimensional, both universal as the Borel measure and complete as the Lebesgue one. Its unit is a quantum bit (qubit) and can be considered as a generalization of the unit of classical information, a bit. It allows quantum mechanics to be interpreted in terms of quantum information, and all physical processes to be seen as informational in a generalized sense. This implies a fundamental connection between the physical and material, on the one hand, and the mathematical and ideal, on the other hand. Quantum measure unifies them by a common and joint informational unit. Furthermore the approach clears up philosophically how quantum mechanics and general relativity can be understood correspondingly as the holistic and temporal aspect of one and the same, the state of a quantum system, e.g. that of the universe as a whole. The key link between them is the notion of the Bekenstein bound as well as that of quantum temperature. General relativity can be interpreted as a special particular case of quantum gravity. All principles underlain by Einstein (1918) reduce the latter to the former. Consequently their generalization and therefore violation addresses directly a theory of quantum gravity. Quantum measure reinterprets newly the "Bing Bang" theories about the beginning of the universe. It measures jointly any quantum leap and smooth motion complementary to each other and thus, the jump-like initiation of anything and the corresponding continuous process of its appearance. Quantum measure unifies the "Big Bang" and the whole visible expansion of the universe as two complementary "halves" of one and the same, the set of all states of the universe as a whole. It is a scientific viewpoint to the "creation from nothing".
A few works of the famous Bulgarian writer Yordan Radichkov (1929-2004) are interpreted philosoph... more A few works of the famous Bulgarian writer Yordan Radichkov (1929-2004) are interpreted philosophically. What is investigated is the availability and inovation of well-known ideas of Western philosophy in them. The great literature refers to human life and being: thus, it shares many topics with philosophy
The paper continues the consideration of Hilbert mathematics to mathematics itself as an addition... more The paper continues the consideration of Hilbert mathematics to mathematics itself as an additional "dimension" allowing for the most difficult and fundamental problems to be attacked in a new general and universal way shareable between all of them. That dimension consists in the parameter of the "distance between finiteness and infinity", particularly able to interpret standard mathematics as a particular case, the basis of which are arithmetic, set theory and propositional logic: that is as a special "flat" case of Hilbert mathematics. The following four essential problems are considered for the idea to be elucidated: Fermat's last theorem proved by Andrew Wiles; Poincaré's conjecture proved by Grigori Perelman and the only resolved from the seven Millennium problems offered by the Clay Mathematics Institute (CMI); the four-color theorem proved "machine-likely" by enumerating all cases and the crucial software assistance; the Yang-Mills existence and mass gap problem also suggested by CMI and yet unresolved. They are intentionally chosen to belong to quite different mathematical areas (number theory, topology, mathematical physics) just to demonstrate the power of the approach able to unite and even unify them from the viewpoint of Hilbert mathematics. Also, specific ideas relevant to each of them are considered. Fermat's last theorem is shown as a Gödel insoluble statement by means of Yablo's paradox. Thus, Wiles's proof as a corollary from the modularity theorem and thus needing both arithmetic and set theory involves necessarily an inverse Grothendieck universe. On the contrary, its proof in "Fermat arithmetic" introduced by "epoché to infinity" (following the pattern of Husserl's original "epoché to reality") can be suggested by Hilbert arithmetic relevant to Hilbert mathematics, the mediation of which can be removed in the final analysis as a "Wittgenstein ladder". Poincaré's conjecture can be reinterpreted physically by Minkowski space and thus reduced to the "nonstandard homeomorphism" of a bit of information mathematically. Perelman's proof can be accordingly reinterpreted. However, it is valid in Gödel (or Gödelian) mathematics, but not in Hilbert mathematics in general, where the question of whether it holds remains open. The fourcolor theorem can be also deduced from the nonstandard homeomorphism at issue, but the available proof by enumerating a finite set of all possible cases is more general and relevant to Hilbert mathematics as well, therefore being an indirect argument in favor of the validity of Poincaré's conjecture in Hilbert mathematics. The Yang-Mills existence and mass gap problem furthermore suggests the most general viewpoint to the relation of Hilbert and Gödel mathematics justifying the qubit Hilbert space as the dual counterpart of Hilbert arithmetic in a narrow sense, in turn being inferable from Hilbert arithmetic in a wide sense. The conjecture that many if not almost all great problems in contemporary mathematics rely on (or at least relate to) the Gödel incompleteness is suggested. It implies that Hilbert mathematics is the natural medium for their discussion or eventual solutions.
discusses the option of the Gödel incompleteness statement (1931: whether "Satz VI" or "Satz X") ... more discusses the option of the Gödel incompleteness statement (1931: whether "Satz VI" or "Satz X") to be an axiom due to the pair of the axiom of induction in arithmetic and the axiom of infinity in set theory after interpreting them as logical negations to each other. The present Part II considers the previous Gödel's paper (1930) (and more precisely, the negation of "Satz VII", or "the completeness theorem") as a necessary condition for granting the Gödel incompleteness statement to be a theorem just as the statement itself, to be an axiom. Then, the "completeness paper" can be interpreted as relevant to Hilbert mathematics, according to which mathematics and reality as well as arithmetic and set theory are rather entangled or complementary rather than mathematics to obey reality able only to create models of the latter. According to that, both papers (1930; 1931) can be seen as advocating Russell's logicism or the intensional propositional logic versus both extensional arithmetic and set theory. Reconstructing history of philosophy, Aristotle's logic and doctrine can be opposed to those of Plato or the pre-Socratic schools as establishing ontology or intensionality versus extensionality. Husserl's phenomenology can be analogically realized including and particularly as philosophy of mathematics. One can identify propositional logic and set theory by virtue of Gödel's completeness theorem (1930: "Satz VII") and even both and arithmetic in the sense of the "compactness theorem" (1930: "Satz X") therefore opposing the latter to the "incompleteness paper" (1931). An approach identifying homomorphically propositional logic and set theory as the same structure of Boolean algebra, and arithmetic as the "half" of it in a rigorous construction involving information and its unit of a bit. Propositional logic and set theory are correspondingly identified as the shared zero-order logic of the class of all first-order logics and the class at issue correspondingly. Then, quantum mechanics does not need any quantum logics, but only the relation of propositional logic, set theory, arithmetic, and information: rather a change of the attitude into more mathematical, philosophical, and speculative than physical, empirical and experimental. Hilbert's epsilon calculus can be situated in the same framework of the relation of propositional logic and the class of all mathematical theories. The horizon of Part III investigating Hilbert mathematics (i.e. according to the Pythagorean viewpoint about the world as mathematical) versus Gödel mathematics (i.e. the usual understanding of mathematics as all mathematical models of the world external to it) is outlined.
The present first part about the eventual completeness of mathematics (called "Hilbert mathematic... more The present first part about the eventual completeness of mathematics (called "Hilbert mathematics") is concentrated on the Gödel incompleteness (1931) statement: if it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic, this would pioneer the pathway to Hilbert mathematics. One of the main arguments that it is an axiom consists in the direct contradiction of the axiom of induction in arithmetic and the axiom of infinity in set theory. Thus, the pair of arithmetic and set are to be similar to Euclidean and non-Euclidean geometries distinguishably only by the Fifth postulate now, i.e. after replacing it and its negation correspondingly by the axiom of finiteness (induction) versus that of finiteness being idempotent negations to each other. Indeed, the axiom of choice, as far as it is equivalent to the well-ordering "theorem", transforms any set in a well-ordering either necessarily finite according to the axiom of induction or also optionally infinite according to the axiom of infinity. So, the Gödel incompleteness statement relies on the logical contradiction of the axiom of induction and the axiom of infinity in the final analysis. Nonetheless, both can be considered as two idempotent versions of the same axiom (analogically to the Fifth postulate) and then unified after logicism and its inherent intensionality since the opposition of finiteness and infinity can be only extensional (i.e., relevant to the elements of any set rather than to the set by itself or its characteristic property being a proposition). So, the pathway for interpreting the Gödel incompleteness statement as an axiom and the originating from that assumption for "Hilbert mathematics" accepting its negation is pioneered. A much wider context relevant to realizing the Gödel incompleteness statement as a metamathematical axiom is consistently built step by step. The horizon of Hilbert mathematics is the proper subject in the third part of the paper, and a reinterpretation of Gödel's papers (1930; 1931) as an apology of logicism as the only consistent foundations of mathematics is the topic of the next second part.
The previous two parts of the paper (correspondingly, https://philpapers.org/rec/PENFLT-2 and htt... more The previous two parts of the paper (correspondingly, https://philpapers.org/rec/PENFLT-2 and https://philpapers.org/rec/PENFLT-3) demonstrate that the interpretation of Fermat's last theorem (FLT) in Hilbert arithmetic meant both in a narrow sense and in a wide sense can suggest a proof by induction in Part I and by means of the Kochen-Specker theorem in Part II. The same interpretation can serve also for a proof FLT based on Gleason's theorem and partly similar to that in Part II. The concept of (probabilistic) measure of a subspace of Hilbert space and especially its uniqueness can be unambiguously linked to that of partial algebra or incommensurability, or interpreted as a relation of the two dual branches of Hilbert arithmetic in a wide sense. The investigation of the last relation allows for FLT and Gleason's theorem to be equated in a sense, as two dual counterparts, and the former to be inferred from the latter, as well as vice versa under an additional condition relevant to the Gödel incompleteness of arithmetic to set theory. The qubit Hilbert space itself in turn can be interpreted by the unity of FLT and Gleason's theorem. The proof of such a fundamental result in number theory as FLT by means of Hilbert arithmetic in a wide sense can be generalized to an idea about "quantum number theory". It is able to research mathematically the origin of Peano arithmetic from Hilbert arithmetic by mediation of the "nonstandard bijection" and its two dual branches inherently linking it to information theory. Then, infinitesimal analysis and its revolutionary application to physics can be also re-realized in that wider context, for example, as an exploration of the way for physical quantity of time (respectively, for time derivative in any temporal process considered in physics) to appear at all. Finally, the result admits a philosophical reflection of how any hierarchy arises or changes itself only thanks to its dual and idempotent counterpart.
The book suggests a "phenomological" philosophy of science, in the sense of Husserl and... more The book suggests a "phenomological" philosophy of science, in the sense of Husserl and Heidegger. Reality is consideried as continuity. The scientific model is entangled into reality by many links in a single context rather than to redlect a certain separate part of reality studied by a scientific discipline as an "image of reality", A coherent, rather than correspondent, concept of truth is relevant to that kind of philosophy of science
The thesis is: the "periodic table" of "dark matter" is equivalent to the standard Periodic table... more The thesis is: the "periodic table" of "dark matter" is equivalent to the standard Periodic table of the visible matter once being entangled. Thus, it is to consist of all possible entangled states of the atoms of chemical elements as quantum systems. In other words, an atom of any chemical element and as a quantum system, i.e. as a wave function, should be represented as a non-orthogonal in general (i.e. entangled) subspace of the separable complex Hilbert space relevant to the system to which the atom at issue is related as a true part of it. The paper follows previous publications of mine stating that "dark matter" and "dark energy" are projections of arbitrarily entangled states on the "cognitive screen" of Einstein's "Mach's principle" in general relativity postulating that gravitational fields can be generated only by mass or energy. The "cosmological constant" introduced by Einstein additionally in 1918 is generalized to a "cosmological function" depending on space-time coordinates, and then to a "cosmological function of entanglement" being a quantum field and decomposable to two entangled subfields, correspondingly depending on space-time coordinates and energy-momentum ones. Entanglement is an additional source of gravitation and can be represented by equivalent mass and energy observable as dark ones. In fact, it violates or complements "Mach's principle", but is forced to be mapped only as mass and energy in virtue of the principle, being furthermore available implicitly in the structure of the Einstein field equation of general relativity. One can use the metaphor of Plato's "cave" about dark matter, dark energy or entanglement: the people are chained and thus can observe only the wall and shadows on it, but not what causes the shadows. So, the shadows can be described only in terms of the wall though those terms are irrelevant to the shadows by themselves: i.e. as dark matter and energy or entanglement. All possible experience of humankind is temporal: thus, the "screen of time" is what that metaphor means as depicted by the "wall of the cave". On the contrary, what causes the shadows is not temporal, nonetheless being visible only as shadows of the temporal screen. So, the "shadows of dark matter" can be observed only on the "screen" of the usual Periodic table of visible matter. Anyway, one can question what the dark "Periodic table" by itself is (i.e. not as a projection onto the visible Periodic table). What becomes visible on the "screen of time" (i.e. the non-Hermitian operators projected as Hermitian ones) can be likened as incomplete quantum calculations. The "complete calculations of the universe" as a quantum computer are all Hermitian operators and thus physical quantities, only to which the concept of the Periodic table makes sense; or in other words, the dark Periodic table by itself is relevant to non-Hermitian operators distinguishable into classes, each of which corresponds to a single subclass of Hermitian operator as any chemical element of the Periodic table can be represented.
A case study of quantum mechanics is investigated in the framework of the philosophical oppositio... more A case study of quantum mechanics is investigated in the framework of the philosophical opposition "mathematical model-reality". All classical science obeys the postulate about the fundamental difference of model and reality, and thus distinguishing epistemology from ontology fundamentally. The theorems about the absence of hidden variables in quantum mechanics imply for it to be "complete" (versus Einstein's opinion). That consistent completeness (unlike arithmetic to set theory in the foundations of mathematics in Gödel's opinion) can be interpreted furthermore as the coincidence of model and reality. The paper discusses the option and fact of that coincidence it its base: the fundamental postulate formulated by Niels Bohr about what quantum mechanics studies (unlike all classical science). Quantum mechanics involves and develops further both identification and disjunctive distinction of the global space of the apparatus and the local space of the investigated quantum entity as complementary to each other. This results into the analogical complementarity of model and reality in quantum mechanics. The apparatus turns out to be both absolutely "transparent" and identically coinciding simultaneously with the reflected quantum reality. Thus, the coincidence of model and reality is postulated as necessary condition for cognition in quantum mechanics by Bohr's postulate and further, embodied in its formalism of the separable complex Hilbert space, in turn, implying the theorems of the absence of hidden variables (or the equivalent to them "conservation of energy conservation" in quantum mechanics). What the apparatus and measured entity exchange cannot be energy (for the different exponents of energy), but quantum information (as a certain, unambiguously determined wave function) therefore a generalized law of conservation, from which the conservation of energy conservation is a corollary. Particularly, the local and global space (rigorously justified in the Standard model) share the complementarity isomorphic to that of model and reality in the foundation of quantum mechanics. On that background, one can think of the troubles of "quantum gravity" as fundamental, direct corollaries from the postulates of quantum mechanics. Gravity can be defined only as a relation or by a pair of non-orthogonal separable complex Hilbert space attachable whether to two "parts" or to a whole and its parts. On the contrary, all the three fundamental interactions in the Standard model are "flat" and only "properties": they need only a single separable complex Hilbert space to be defined.
Toluene conversions over metal-zeolitic catalysts at 450/sup 0/C and 1 atm in hydrogen flow proce... more Toluene conversions over metal-zeolitic catalysts at 450/sup 0/C and 1 atm in hydrogen flow proceeded via methyl-group disproportionation to xylenes and trimethylbenzenes in parallel with dealkylation to benzene. The conversion level, which did not exceed 3-4% over a CaY zeolite prepared by ion exchanging a NaY form with 2.4:1 SiO/sub 2/-Al/sub 2/O/sub 3/ ratio (R) to 85% Ca(+2), increased to 40% for CaMgY, 42% for CaHY, and 33% for rare-earth-containing Y zeolites (REY) and to 45-52% by introducing small amounts of nickel (0.4% for REY, 2.4% for CaY, and 8% for NaY) or platinum (0.5% for CaY or CaMgY) into the zeolites. By contrast, catalysts with nonacidic supports, such as 1.5% NiCaA or 0.5% Pt/Al/sub 2/O/sub 3/, and a mechanical mixture of NiO with CaY showed low activities and selectivities for toluene disproportionation. The selectivities, which depended both on the metalic and acidic functions of the catalysts, were very low for A and X zeolites, but exceeded 80-85% for REY and synthetic CaNi faujasites with R greater than 1.75:1. Increasing R of the faujasites from 1.3:1 to 2.4:1 also increased the conversion from approx. 7 to 50%, because of the improved acidic properties.
Can the so-ca\\led first incompleteness theorem refer to itself? Many or maybe even all the parad... more Can the so-ca\\led first incompleteness theorem refer to itself? Many or maybe even all the paradoxes in mathematics are connected with some kind of self-reference. Gбdel built his proof on the ground of self-reference: а statement which claims its unprovabllity. So, he demonstrated that undecidaЬle propositions exist in any enough rich axiomatics (i.e. such one which contains Peano arithmetic in some sense). What about the decidabllity of the very first incompleteness theorem? We can display that it fulfills its conditions. That's why it can Ье applied to itself, proving that it is an undecidaЬle statement. It seems to Ье а too strange kind of proposition: its validity implies its undecidabllity. If the validity of а statement implies its untruth, then it is either untruth (reductio ad absurdum) or an antinomy (if also its negation implies its validity). А theory that contains а contradiction implies any statement. Appearing of а proposition, whose validity implies its undecidabllity, is due to the statement that claims its unprovability. Obviously, it is а proposition of self-referential type. Ву Gбdel's words, it is correlative with Richard's or liar paradox, or even with any other semantic or mathematical one. What is the cost, if а proposition of that special kind is used in а proof? ln our opinion, the price is analogous to «applying» of а contradictory in а theory: any statement turns out to Ье undecidaЬ!e. Ifthe first incompleteness theorem is an undecidaЬ!e theorem, then it is impossiЬle to prove that the very completeness of Peano arithmetic is also an tmdecidaЬle statement (the second incompleteness theorem). Hilbert's program for ап arithmetical self-foundation of matheшatics is partly rehabllitated: only partly, because it is not decidaЬ!e and true, but undecidaЬle; that's wby both it and its negation шау Ье accepted as true, however not siшultaneously true. The first incompleteness theoreш gains the statute of axiom of а very special, semi-philosophical kind: it divides mathematics as whole into two parts: either Godel шathematics or Нilbert matheшatics. Нilbert's program of self-foundation ofmatheшatic is valid only as to the latter
Those incompleteness theorems mean the relation of (Peano) arithmetic and (ZFC) set theory, or ph... more Those incompleteness theorems mean the relation of (Peano) arithmetic and (ZFC) set theory, or philosophically, the relation of arithmetical finiteness and actual infinity. The same is managed in the framework of set theory by the axiom of choice (respectively, by the equivalent well-ordering "theorem'). One may discuss that incompleteness form the viewpoint of set theory by the axiom of choice rather than the usual viewpoint meant in the proof of theorems. The logical corollaries from that "nonstandard" viewpoint the relation of set theory and arithmetic are demonstrated.
Много отчетливо беше всичко в тая нощ и още по-отчет ливо и ясно стана за Исая, като разбра, че н... more Много отчетливо беше всичко в тая нощ и още по-отчет ливо и ясно стана за Исая, като разбра, че него не са го правили с индиго, а един бог знае кой го е надраскал несръчно, отделно от ония, вадените с индиго.
Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two c... more Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two connotations (connotations or metaphors are inherent in literature rather than in mathematics or logics). The paradox itself refers to implication demonstrating that an intermediate implication can be always inserted in an implication therefore postponing its ultimate conclusion for the next step and those insertions can be iteratively and indefinitely added ad lib, as if ad infinitum. Both connotations clear up links due to the shared formal structure with other well-known mathematical observations: (1) the paradox of Achilles and the Turtle; (2) the transitivity of the relation of equality. Analogically to (1), one can juxtapose the paradox of the Liar (for Lewis Carroll's paradox) and that of the arrow (for "Achilles and the Turtle"), i.e. a logical paradox, on the one hand, and an aporia of motion, on the other hand, suggesting a shared formal structure of both, which can be called "ontological", on which basis "motion" studied by physics and "conclusion" studied by logic can be unified being able to bridge logic and physics philosophically in a Hegelian manner: even more, the bridge can be continued to mathematics in virtue of (2), which forces the equality (for its property of transitivity) of any two quantities to be postponed analogically ad lib and ad infinitum. The paper shows that Hilbert arithmetic underlies naturally Lewis Carroll's paradox admitting at least three interpretations linked to each other by it: mathematical, physical and logical. Thus, it can be considered as both generalization and solution of his paradox therefore naturally unifying the completeness of quantum mechanics (i.e. the absence of hidden variables) and eventual completeness of mathematics as the same and isomorphic to the completeness of propositional logic in relation to set theory as a first-order logic (in the sense of Gödel (1930)'s completeness theorems).
The paper discusses the philosophical conclusions, which the interrelation between quantum mechan... more The paper discusses the philosophical conclusions, which the interrelation between quantum mechanics and general relativity implies by quantum measure. Quantum measure is three-dimensional, both universal as the Borel measure and complete as the Lebesgue one. Its unit is a quantum bit (qubit) and can be considered as a generalization of the unit of classical information, a bit. It allows quantum mechanics to be interpreted in terms of quantum information, and all physical processes to be seen as informational in a generalized sense. This implies a fundamental connection between the physical and material, on the one hand, and the mathematical and ideal, on the other hand. Quantum measure unifies them by a common and joint informational unit. Furthermore the approach clears up philosophically how quantum mechanics and general relativity can be understood correspondingly as the holistic and temporal aspect of one and the same, the state of a quantum system, e.g. that of the universe as a whole. The key link between them is the notion of the Bekenstein bound as well as that of quantum temperature. General relativity can be interpreted as a special particular case of quantum gravity. All principles underlain by Einstein (1918) reduce the latter to the former. Consequently their generalization and therefore violation addresses directly a theory of quantum gravity. Quantum measure reinterprets newly the "Bing Bang" theories about the beginning of the universe. It measures jointly any quantum leap and smooth motion complementary to each other and thus, the jump-like initiation of anything and the corresponding continuous process of its appearance. Quantum measure unifies the "Big Bang" and the whole visible expansion of the universe as two complementary "halves" of one and the same, the set of all states of the universe as a whole. It is a scientific viewpoint to the "creation from nothing".
A few works of the famous Bulgarian writer Yordan Radichkov (1929-2004) are interpreted philosoph... more A few works of the famous Bulgarian writer Yordan Radichkov (1929-2004) are interpreted philosophically. What is investigated is the availability and inovation of well-known ideas of Western philosophy in them. The great literature refers to human life and being: thus, it shares many topics with philosophy
The paper continues the consideration of Hilbert mathematics to mathematics itself as an addition... more The paper continues the consideration of Hilbert mathematics to mathematics itself as an additional "dimension" allowing for the most difficult and fundamental problems to be attacked in a new general and universal way shareable between all of them. That dimension consists in the parameter of the "distance between finiteness and infinity", particularly able to interpret standard mathematics as a particular case, the basis of which are arithmetic, set theory and propositional logic: that is as a special "flat" case of Hilbert mathematics. The following four essential problems are considered for the idea to be elucidated: Fermat's last theorem proved by Andrew Wiles; Poincaré's conjecture proved by Grigori Perelman and the only resolved from the seven Millennium problems offered by the Clay Mathematics Institute (CMI); the four-color theorem proved "machine-likely" by enumerating all cases and the crucial software assistance; the Yang-Mills existence and mass gap problem also suggested by CMI and yet unresolved. They are intentionally chosen to belong to quite different mathematical areas (number theory, topology, mathematical physics) just to demonstrate the power of the approach able to unite and even unify them from the viewpoint of Hilbert mathematics. Also, specific ideas relevant to each of them are considered. Fermat's last theorem is shown as a Gödel insoluble statement by means of Yablo's paradox. Thus, Wiles's proof as a corollary from the modularity theorem and thus needing both arithmetic and set theory involves necessarily an inverse Grothendieck universe. On the contrary, its proof in "Fermat arithmetic" introduced by "epoché to infinity" (following the pattern of Husserl's original "epoché to reality") can be suggested by Hilbert arithmetic relevant to Hilbert mathematics, the mediation of which can be removed in the final analysis as a "Wittgenstein ladder". Poincaré's conjecture can be reinterpreted physically by Minkowski space and thus reduced to the "nonstandard homeomorphism" of a bit of information mathematically. Perelman's proof can be accordingly reinterpreted. However, it is valid in Gödel (or Gödelian) mathematics, but not in Hilbert mathematics in general, where the question of whether it holds remains open. The fourcolor theorem can be also deduced from the nonstandard homeomorphism at issue, but the available proof by enumerating a finite set of all possible cases is more general and relevant to Hilbert mathematics as well, therefore being an indirect argument in favor of the validity of Poincaré's conjecture in Hilbert mathematics. The Yang-Mills existence and mass gap problem furthermore suggests the most general viewpoint to the relation of Hilbert and Gödel mathematics justifying the qubit Hilbert space as the dual counterpart of Hilbert arithmetic in a narrow sense, in turn being inferable from Hilbert arithmetic in a wide sense. The conjecture that many if not almost all great problems in contemporary mathematics rely on (or at least relate to) the Gödel incompleteness is suggested. It implies that Hilbert mathematics is the natural medium for their discussion or eventual solutions.
discusses the option of the Gödel incompleteness statement (1931: whether "Satz VI" or "Satz X") ... more discusses the option of the Gödel incompleteness statement (1931: whether "Satz VI" or "Satz X") to be an axiom due to the pair of the axiom of induction in arithmetic and the axiom of infinity in set theory after interpreting them as logical negations to each other. The present Part II considers the previous Gödel's paper (1930) (and more precisely, the negation of "Satz VII", or "the completeness theorem") as a necessary condition for granting the Gödel incompleteness statement to be a theorem just as the statement itself, to be an axiom. Then, the "completeness paper" can be interpreted as relevant to Hilbert mathematics, according to which mathematics and reality as well as arithmetic and set theory are rather entangled or complementary rather than mathematics to obey reality able only to create models of the latter. According to that, both papers (1930; 1931) can be seen as advocating Russell's logicism or the intensional propositional logic versus both extensional arithmetic and set theory. Reconstructing history of philosophy, Aristotle's logic and doctrine can be opposed to those of Plato or the pre-Socratic schools as establishing ontology or intensionality versus extensionality. Husserl's phenomenology can be analogically realized including and particularly as philosophy of mathematics. One can identify propositional logic and set theory by virtue of Gödel's completeness theorem (1930: "Satz VII") and even both and arithmetic in the sense of the "compactness theorem" (1930: "Satz X") therefore opposing the latter to the "incompleteness paper" (1931). An approach identifying homomorphically propositional logic and set theory as the same structure of Boolean algebra, and arithmetic as the "half" of it in a rigorous construction involving information and its unit of a bit. Propositional logic and set theory are correspondingly identified as the shared zero-order logic of the class of all first-order logics and the class at issue correspondingly. Then, quantum mechanics does not need any quantum logics, but only the relation of propositional logic, set theory, arithmetic, and information: rather a change of the attitude into more mathematical, philosophical, and speculative than physical, empirical and experimental. Hilbert's epsilon calculus can be situated in the same framework of the relation of propositional logic and the class of all mathematical theories. The horizon of Part III investigating Hilbert mathematics (i.e. according to the Pythagorean viewpoint about the world as mathematical) versus Gödel mathematics (i.e. the usual understanding of mathematics as all mathematical models of the world external to it) is outlined.
The present first part about the eventual completeness of mathematics (called "Hilbert mathematic... more The present first part about the eventual completeness of mathematics (called "Hilbert mathematics") is concentrated on the Gödel incompleteness (1931) statement: if it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic, this would pioneer the pathway to Hilbert mathematics. One of the main arguments that it is an axiom consists in the direct contradiction of the axiom of induction in arithmetic and the axiom of infinity in set theory. Thus, the pair of arithmetic and set are to be similar to Euclidean and non-Euclidean geometries distinguishably only by the Fifth postulate now, i.e. after replacing it and its negation correspondingly by the axiom of finiteness (induction) versus that of finiteness being idempotent negations to each other. Indeed, the axiom of choice, as far as it is equivalent to the well-ordering "theorem", transforms any set in a well-ordering either necessarily finite according to the axiom of induction or also optionally infinite according to the axiom of infinity. So, the Gödel incompleteness statement relies on the logical contradiction of the axiom of induction and the axiom of infinity in the final analysis. Nonetheless, both can be considered as two idempotent versions of the same axiom (analogically to the Fifth postulate) and then unified after logicism and its inherent intensionality since the opposition of finiteness and infinity can be only extensional (i.e., relevant to the elements of any set rather than to the set by itself or its characteristic property being a proposition). So, the pathway for interpreting the Gödel incompleteness statement as an axiom and the originating from that assumption for "Hilbert mathematics" accepting its negation is pioneered. A much wider context relevant to realizing the Gödel incompleteness statement as a metamathematical axiom is consistently built step by step. The horizon of Hilbert mathematics is the proper subject in the third part of the paper, and a reinterpretation of Gödel's papers (1930; 1931) as an apology of logicism as the only consistent foundations of mathematics is the topic of the next second part.
The previous two parts of the paper (correspondingly, https://philpapers.org/rec/PENFLT-2 and htt... more The previous two parts of the paper (correspondingly, https://philpapers.org/rec/PENFLT-2 and https://philpapers.org/rec/PENFLT-3) demonstrate that the interpretation of Fermat's last theorem (FLT) in Hilbert arithmetic meant both in a narrow sense and in a wide sense can suggest a proof by induction in Part I and by means of the Kochen-Specker theorem in Part II. The same interpretation can serve also for a proof FLT based on Gleason's theorem and partly similar to that in Part II. The concept of (probabilistic) measure of a subspace of Hilbert space and especially its uniqueness can be unambiguously linked to that of partial algebra or incommensurability, or interpreted as a relation of the two dual branches of Hilbert arithmetic in a wide sense. The investigation of the last relation allows for FLT and Gleason's theorem to be equated in a sense, as two dual counterparts, and the former to be inferred from the latter, as well as vice versa under an additional condition relevant to the Gödel incompleteness of arithmetic to set theory. The qubit Hilbert space itself in turn can be interpreted by the unity of FLT and Gleason's theorem. The proof of such a fundamental result in number theory as FLT by means of Hilbert arithmetic in a wide sense can be generalized to an idea about "quantum number theory". It is able to research mathematically the origin of Peano arithmetic from Hilbert arithmetic by mediation of the "nonstandard bijection" and its two dual branches inherently linking it to information theory. Then, infinitesimal analysis and its revolutionary application to physics can be also re-realized in that wider context, for example, as an exploration of the way for physical quantity of time (respectively, for time derivative in any temporal process considered in physics) to appear at all. Finally, the result admits a philosophical reflection of how any hierarchy arises or changes itself only thanks to its dual and idempotent counterpart.
The book suggests a "phenomological" philosophy of science, in the sense of Husserl and... more The book suggests a "phenomological" philosophy of science, in the sense of Husserl and Heidegger. Reality is consideried as continuity. The scientific model is entangled into reality by many links in a single context rather than to redlect a certain separate part of reality studied by a scientific discipline as an "image of reality", A coherent, rather than correspondent, concept of truth is relevant to that kind of philosophy of science
The thesis is: the "periodic table" of "dark matter" is equivalent to the standard Periodic table... more The thesis is: the "periodic table" of "dark matter" is equivalent to the standard Periodic table of the visible matter once being entangled. Thus, it is to consist of all possible entangled states of the atoms of chemical elements as quantum systems. In other words, an atom of any chemical element and as a quantum system, i.e. as a wave function, should be represented as a non-orthogonal in general (i.e. entangled) subspace of the separable complex Hilbert space relevant to the system to which the atom at issue is related as a true part of it. The paper follows previous publications of mine stating that "dark matter" and "dark energy" are projections of arbitrarily entangled states on the "cognitive screen" of Einstein's "Mach's principle" in general relativity postulating that gravitational fields can be generated only by mass or energy. The "cosmological constant" introduced by Einstein additionally in 1918 is generalized to a "cosmological function" depending on space-time coordinates, and then to a "cosmological function of entanglement" being a quantum field and decomposable to two entangled subfields, correspondingly depending on space-time coordinates and energy-momentum ones. Entanglement is an additional source of gravitation and can be represented by equivalent mass and energy observable as dark ones. In fact, it violates or complements "Mach's principle", but is forced to be mapped only as mass and energy in virtue of the principle, being furthermore available implicitly in the structure of the Einstein field equation of general relativity. One can use the metaphor of Plato's "cave" about dark matter, dark energy or entanglement: the people are chained and thus can observe only the wall and shadows on it, but not what causes the shadows. So, the shadows can be described only in terms of the wall though those terms are irrelevant to the shadows by themselves: i.e. as dark matter and energy or entanglement. All possible experience of humankind is temporal: thus, the "screen of time" is what that metaphor means as depicted by the "wall of the cave". On the contrary, what causes the shadows is not temporal, nonetheless being visible only as shadows of the temporal screen. So, the "shadows of dark matter" can be observed only on the "screen" of the usual Periodic table of visible matter. Anyway, one can question what the dark "Periodic table" by itself is (i.e. not as a projection onto the visible Periodic table). What becomes visible on the "screen of time" (i.e. the non-Hermitian operators projected as Hermitian ones) can be likened as incomplete quantum calculations. The "complete calculations of the universe" as a quantum computer are all Hermitian operators and thus physical quantities, only to which the concept of the Periodic table makes sense; or in other words, the dark Periodic table by itself is relevant to non-Hermitian operators distinguishable into classes, each of which corresponds to a single subclass of Hermitian operator as any chemical element of the Periodic table can be represented.
A case study of quantum mechanics is investigated in the framework of the philosophical oppositio... more A case study of quantum mechanics is investigated in the framework of the philosophical opposition "mathematical model-reality". All classical science obeys the postulate about the fundamental difference of model and reality, and thus distinguishing epistemology from ontology fundamentally. The theorems about the absence of hidden variables in quantum mechanics imply for it to be "complete" (versus Einstein's opinion). That consistent completeness (unlike arithmetic to set theory in the foundations of mathematics in Gödel's opinion) can be interpreted furthermore as the coincidence of model and reality. The paper discusses the option and fact of that coincidence it its base: the fundamental postulate formulated by Niels Bohr about what quantum mechanics studies (unlike all classical science). Quantum mechanics involves and develops further both identification and disjunctive distinction of the global space of the apparatus and the local space of the investigated quantum entity as complementary to each other. This results into the analogical complementarity of model and reality in quantum mechanics. The apparatus turns out to be both absolutely "transparent" and identically coinciding simultaneously with the reflected quantum reality. Thus, the coincidence of model and reality is postulated as necessary condition for cognition in quantum mechanics by Bohr's postulate and further, embodied in its formalism of the separable complex Hilbert space, in turn, implying the theorems of the absence of hidden variables (or the equivalent to them "conservation of energy conservation" in quantum mechanics). What the apparatus and measured entity exchange cannot be energy (for the different exponents of energy), but quantum information (as a certain, unambiguously determined wave function) therefore a generalized law of conservation, from which the conservation of energy conservation is a corollary. Particularly, the local and global space (rigorously justified in the Standard model) share the complementarity isomorphic to that of model and reality in the foundation of quantum mechanics. On that background, one can think of the troubles of "quantum gravity" as fundamental, direct corollaries from the postulates of quantum mechanics. Gravity can be defined only as a relation or by a pair of non-orthogonal separable complex Hilbert space attachable whether to two "parts" or to a whole and its parts. On the contrary, all the three fundamental interactions in the Standard model are "flat" and only "properties": they need only a single separable complex Hilbert space to be defined.
Toluene conversions over metal-zeolitic catalysts at 450/sup 0/C and 1 atm in hydrogen flow proce... more Toluene conversions over metal-zeolitic catalysts at 450/sup 0/C and 1 atm in hydrogen flow proceeded via methyl-group disproportionation to xylenes and trimethylbenzenes in parallel with dealkylation to benzene. The conversion level, which did not exceed 3-4% over a CaY zeolite prepared by ion exchanging a NaY form with 2.4:1 SiO/sub 2/-Al/sub 2/O/sub 3/ ratio (R) to 85% Ca(+2), increased to 40% for CaMgY, 42% for CaHY, and 33% for rare-earth-containing Y zeolites (REY) and to 45-52% by introducing small amounts of nickel (0.4% for REY, 2.4% for CaY, and 8% for NaY) or platinum (0.5% for CaY or CaMgY) into the zeolites. By contrast, catalysts with nonacidic supports, such as 1.5% NiCaA or 0.5% Pt/Al/sub 2/O/sub 3/, and a mechanical mixture of NiO with CaY showed low activities and selectivities for toluene disproportionation. The selectivities, which depended both on the metalic and acidic functions of the catalysts, were very low for A and X zeolites, but exceeded 80-85% for REY and synthetic CaNi faujasites with R greater than 1.75:1. Increasing R of the faujasites from 1.3:1 to 2.4:1 also increased the conversion from approx. 7 to 50%, because of the improved acidic properties.
Can the so-ca\\led first incompleteness theorem refer to itself? Many or maybe even all the parad... more Can the so-ca\\led first incompleteness theorem refer to itself? Many or maybe even all the paradoxes in mathematics are connected with some kind of self-reference. Gбdel built his proof on the ground of self-reference: а statement which claims its unprovabllity. So, he demonstrated that undecidaЬle propositions exist in any enough rich axiomatics (i.e. such one which contains Peano arithmetic in some sense). What about the decidabllity of the very first incompleteness theorem? We can display that it fulfills its conditions. That's why it can Ье applied to itself, proving that it is an undecidaЬle statement. It seems to Ье а too strange kind of proposition: its validity implies its undecidabllity. If the validity of а statement implies its untruth, then it is either untruth (reductio ad absurdum) or an antinomy (if also its negation implies its validity). А theory that contains а contradiction implies any statement. Appearing of а proposition, whose validity implies its undecidabllity, is due to the statement that claims its unprovability. Obviously, it is а proposition of self-referential type. Ву Gбdel's words, it is correlative with Richard's or liar paradox, or even with any other semantic or mathematical one. What is the cost, if а proposition of that special kind is used in а proof? ln our opinion, the price is analogous to «applying» of а contradictory in а theory: any statement turns out to Ье undecidaЬ!e. Ifthe first incompleteness theorem is an undecidaЬ!e theorem, then it is impossiЬle to prove that the very completeness of Peano arithmetic is also an tmdecidaЬle statement (the second incompleteness theorem). Hilbert's program for ап arithmetical self-foundation of matheшatics is partly rehabllitated: only partly, because it is not decidaЬ!e and true, but undecidaЬle; that's wby both it and its negation шау Ье accepted as true, however not siшultaneously true. The first incompleteness theoreш gains the statute of axiom of а very special, semi-philosophical kind: it divides mathematics as whole into two parts: either Godel шathematics or Нilbert matheшatics. Нilbert's program of self-foundation ofmatheшatic is valid only as to the latter
The presentation for the paper and talk of the same title at: Nordic Network for Philosophy of sc... more The presentation for the paper and talk of the same title at: Nordic Network for Philosophy of sciences, Pärnu, Eesti April 22-23, 2016 (NNPS2016)
“Aristotle in Phenomelogy”, Fort Wayne, IN, USA
April 23-24, 2016
Heidegger called Aristotle’s P... more “Aristotle in Phenomelogy”, Fort Wayne, IN, USA April 23-24, 2016
Heidegger called Aristotle’s Physics “the secret, never sufficiently rethought base book of Western Philosophy”. This can explain the choice of Heidegger to comment namely it
Moscow, VII International Congress "Modes of Thinking, Ways of Speaking“
The School of Philosophy... more Moscow, VII International Congress "Modes of Thinking, Ways of Speaking“ The School of Philosophy of the Faculty of Humanities (National Research University Higher School of Economics) 30 April 2016 The “improper interpretation” of an infinite set-theory structure founds the “proper interpretation” and thus that structure self-founds itself as the one interpretation of it can found the other
The CMI Millennium "P vs NP Problem" can be resolved e.g. if one shows at least one counterexampl... more The CMI Millennium "P vs NP Problem" can be resolved e.g. if one shows at least one counterexample to the conjecture. A certain class of problems being such counterexamples will be formulated. This implies the rejection of the hypothesis for any conditions satisfying the formulation of the problem. Thus, the solution "P is differnt from NP" of the problem in general is proved. The class of counterexamples can be interpreted as any quantum superposition of any finite set of quantum states. The Kochen-Specker theorem is involved. Any fundamentally random choice among a finite set of alternatives belong to "NP' but not to "P". The conjecture that the set complement of "P" to "NP" can be described by that kind of choice exhaustively is formulated https://arxiv.org/abs/2005.01412?fbclid=IwAR2tLJPQ-lmfhX3mkPQlV7_SLmHVTSmtA4dTFxDD9OMwNRbkAKtS61fd8KU
A proof of Fermat's last theorem is demonstrated. It is very brief, simple, elementary, and absol... more A proof of Fermat's last theorem is demonstrated. It is very brief, simple, elementary, and absolutely arithmetical. The necessary premises for the proof are only: the three definitive properties of the relation of equality (identity, symmetry, and transitivity), modus tollens, axiom of induction, the proof of Fermat's last theorem in the case of í µí±í µí± = 3 as well as the premises necessary for the formulation of the theorem itself. It involves a modification of Fermat's approach of infinite descent. The infinite descent is linked to induction starting from í µí±í µí± = 3 by modus tollens. An inductive series of modus tollens is constructed. The proof of the series by induction is equivalent to Fermat's last theorem. As far as Fermat had been proved the theorem for í µí±í µí± = 4, one can suggest that the proof for í µí±í µí± ≥ 4 was accessible to him.
In fact, the first law of conservation (that of mass) was found in chemistry and generalized to ... more In fact, the first law of conservation (that of mass) was found in chemistry and generalized to the conservation of energy in physics by means of Einstein's famous "E=mc^2". Energy conservation is implied by the principle of least action from a variational viewpoint as in Emmy Noether's theorems (1918): any chemical change in a conservative (i.e. "closed") system can be accomplished only in the way conserving its total energy. Bohr's innovation to found Mendeleev's periodic table by quantum mechanics implies a certain generalization referring to the quantum leaps as if accomplished in all possible trajectories (according to Feynman's interpretation) and therefore generalizing the principle of least action and needing a certain generalization of energy conservation as to any quantum change. The transition from the first to the second theorem of Emmy Noether represents well the necessary generalization: its chemical meaning is the generalization of any chemical reaction to be accomplished as if any possible course of time rather than in the standard evenly running time (and equivalent to energy conservation according to the first theorem). The problem : If any quantum change is accomplished in all possible "variations (i.e. "violations) of energy conservation" (by different probabilities), what (if any) is conserved? An answer : quantum information is what is conserved. Indeed, it can be particularly defined as the counterpart (e.g. in the sense of Emmy Noether's theorems) to the physical quantity of action (e.g. as energy is the counterpart of time in them). It is valid in any course of time rather than in the evenly running one. That generalization implies a generalization of the periodic table including any continuous and smooth transformation between two chemical elements.
A principle, according to which any scientific theory can be mathematized, is investigated. That ... more A principle, according to which any scientific theory can be mathematized, is investigated. That theory is presupposed to be a consistent text, which can be exhaustedly represented by a certain mathematical structure constructively. In thus used, the term " theory " includes all hypotheses as yet unconfirmed as already rejected. The investigation of the sketch of a possible proof of the principle demonstrates that it should be accepted rather a metamathematical axiom about the relation of mathematics and reality. Its investigation needs philosophical means. Husserl's phenomenology is what is used, and then the conception of " bracketing reality " is modelled to generalize Peano arithmetic in its relation to set theory in the foundation of mathematics. The obtained model is equivalent to the generalization of Peano arithmetic by means of replacing the axiom of induction with that of transfinite induction. Accepting or rejecting the principle, two kinds of mathematics appear differing from each other by its relation to reality. Accepting the principle, mathematics has to include reality within itself in a kind of Pythagoreanism. These two kinds are called in paper correspondingly Hilbert mathematics and Gödel mathematics. The sketch of the proof of the principle demonstrates that the generalization of Peano arithmetic as above can be interpreted as a model of Hilbert mathematics into Gödel mathematics therefore showing that the former is not less consistent than the latter, and the principle is an independent axiom. An information interpretation of Hilbert mathematics is involved. It is a kind of ontology of information. Thus the problem which of the two mathematics is more relevant to our being (rather than reality for reality is external only to Gödel mathematics) is discussed. An information interpretation of the Schrödinger equation is involved to illustrate the above problem.
Philosophy of quantum information. Einstein and Gödel. Sofia: IPhR-BAS, 2009
Поредицата от 7 книги, посветени на философията на квантовата информация, започва с възникването ... more Поредицата от 7 книги, посветени на философията на квантовата информация, започва с възникването на дисциплината в тясна близост с квантовата механика и дебата около нейните основи. Предпочита се нова мета-математическа интерпретация и се обсъждат wзаимоотношенията със съществуващите. Философска и онтологическа проекция е предлаганото видоизменено, а именно „дуалистично питагорейство”. Като символ е използван „Принстънският дух” и приятелството между Айнщайн и Гьодел. Набедената непълнота на квантовата механика и доказаната непълнота на аритметиката са преплетени и „сдвоени”, така че взаимни отблясъци осветяват аритметиката и математиката с онтологичен пламък, но и квантовата механика и информация – с философска фундаменталност и способност да обосновава. Неразрешими твърдения ли са самите т. нар. теореми на Гьодел за непълнотата, ако те се отнесат към самите себе си? Може ли парадоксът на Скулем да се използва за обобщаване на Айнщайновия „принцип на относителността” (1918) от дифеоморфизми и за дискретни морфизми? Как следва да се тълкуват явленията на сдвояване (entanglement), квантовият компютър и квантовата информация аритметически и логически? Книгата е предназначена за научни работници в областта на физиката, математиката и философията, за докторанти и студенти, за всеки, който се интересува от този съвсем нов отрасъл на знанието.
S&T Foresight Workshop “A quest for an interface between information and action”, 2021
The thesis is: the “periodic table” of “dark matter” is equivalent to the standard periodic table... more The thesis is: the “periodic table” of “dark matter” is equivalent to the standard periodic table of the visible matter being entangled. Thus, it is to consist of all possible entangled states of the atoms of chemical elements as quantum systems. In other words, an atom of any chemical element and as a quantum system, i.e. as a wave function, should be represented as a non-orthogonal in general (i.e. entangled) subspace of the separable complex Hilbert space relevant to the system to which the atom at issue is related as a true part of it. The paper follows previous publications of mine stating that “dark matter” and “dark energy” are projections of arbitrarily entangled states on the cognitive “screen” of Einstein’s “Mach’s principle” in general relativity postulating that gravitational field can be generated only by mass or energy.
Space and Time: an Interdisciplinary approach, 2019
Background and prehistory:
The French mathematician Henri Poincaré offered a statement known as “... more Background and prehistory: The French mathematician Henri Poincaré offered a statement known as “Poincaré’s conjecture” without a proof. It states that any 4-dimensional ball is equivalent to 3-dimensional Euclidean space topologically: a continuous mapping exists so that it maps the former ball into the latter space one-to-one. At first glance, it seems to be too paradoxical for the following mismatches: the former is 4-dimensional and as if “closed” unlike the latter, 3-dimensional and as if “open” according to common sense. So, any mapping seemed to be necessarily discrete to be able to overcome those mismatches, and being discrete impies for the conjecture to be false. Anyway, nobody managed neither to prove nor to reject rigorously the conjecture about one century. It was included even in the Millennium Prize Problems by the Clay Mathematics Institute. It was proved by Grigory Perelman in 2003 using the concept of information. Physical interpretation in terms of special relativity: One may notice that the 4-ball is almost equivalent topologically to the “imaginary domain” of Minkowski space in the following sense of “almost”: that “half” of Minkowski space is equivalent topologically to the unfolding of a 4-ball. Then, the conjecture means the topological equivalence of the physical 3-space and its model in special relativity. In turn, that topological equivalence means their equivalence as to causality physically. So, Perelman has proved the adequacy of Minkowski space as a model of the physical 3-dimensional space rigorously. Of course, all experiments confirm the same empirically, but not mathematically as he did. An idea of another proof of the conjecture based on that physical interpretation: Topologically seen, the problem turns out to be reformulated so: one needs a proof of the topological equivalence of a 4-ball and its unfolding by 3-balls (what the “half” of Minkowski space is, topologically). If one adds a complementary, second unfolding to link both ends of the first unfolding, the problem would be resolved: 4-ball would be equivalent to two 3-spaces topologically. Two 3-spaces are equivalent to a single one as follows: one divides a 3-space into two parts by a certain plane (that plane does not belong to any of them). Any part is equivalent topologically to a 3-space for any open neighborhood is transformed into an open one by the mapping of each part (excluding the boundary of the plane) into the complete 3-space. That idea is linked to the original proof of Perelman by the concept of information. It means that any bit of information interpreted physically conserves causality. In other words, the choice of any of both states of a bit (e.g. designated as “0” and “1” recorded in a cell) does not violate causality (the cell, either “0” or “1”, or both “0” and “1” are equivalent to each other topologically and to a 3-space).
The power of the square of opposition has been proved during millennia. It supplies logic by the ... more The power of the square of opposition has been proved during millennia. It supplies logic by the ontological language of infinity for describing anything... 6th WORLD CONGRESS ON THE SQUARE OF OPPOSITION http://www.square-of-opposition.org/square2018.html
Abstr act. The paper addresses Leon Hen.kin's proposition as a " light house", which can elucidat... more Abstr act. The paper addresses Leon Hen.kin's proposition as a " light house", which can elucidate a vast territory of knowledge uniformly: logic, set theory, information t heory, and quantum mechanics: Two st rategies to infinity are equally relevant for it is as universal and t hus complete as open and thus incomplete. Henkin's, Godel's, Robert Jeroslow's, and Hartley Rogers' proposition are reformulated so that both completeness and incompleteness t o be unified and t hus reduced as a joint property of infinity and of all infinite sets. However, only Henkin's proposition equivalent to an internal posit ion t o infinity is consistent. This can be retraced back to set theory and its axioms, where that of choice is a key. Quantum mechanics is forced to int roduce infinity implicitly by Hilbert space, on which is founded its formalism. One can demonst rate that some essential properties of quantum information, entanglement, and quantum comput er originate direct ly from infinit y once it is involved in quant um mechanics. Thus, these phenomena can be elucidated as both complete and incomplete, after which choice is the border bet ween them. A special kind of invariance to t he axiom of choice shared by quantum mechanics is discussed to be involved that border between the completeness and incompleteness of infinity in a consistent way. The so-called paradox of Albert Einstein, Boris Podolsky, and Nathan Rosen is interpreted ent irely in the same terms only of set theory. Quantum computer can demonstrat e especially clearly t he privilege of t he internal position, or " observer'' , or " user" to infinity implied by Henkin's proposition as the only consistent ones as to infinity. An essential area of cont emporary knowledge may be synt hesized from a single viewpoint. Both completeness and incompleteness are well distinguishable as to finite-ness: Completeness supposes that any operat ions defined over any finite sets do
The state of nothing passes spontaneously (by itself)
into the state of being
❖ This represents t... more The state of nothing passes spontaneously (by itself) into the state of being ❖ This represents the “creation”. The transition of nothing into being is mathematically necessary ❖ The choice (which can be interpreted philosophically as “free will”) appears necessary in mathematical reasons ❖ The choice generates asymmetry, which is the beginning of time and thus, of the physical word ❖ Information is the quantity of choices and linked to time intimately
1. Quantum information can be discussed as the counterpart of action.
2. Quantum information is w... more 1. Quantum information can be discussed as the counterpart of action. 2. Quantum information is what is conserved, action is what is changed. 3. The gap between mathematical models and physical reality, needing truth as adequacy to be overcome, is substituted by the openness of choice. 4. That openness in turn can be interpreted as the openness of the present as a different concept of truth recollecting Heidegger’s one as “unhiddeness”. 5. Quantum information as what is conserved can be thought philoso[hically as the conservation of that openness.
What is the form by which the past is represented in the present as far as the proper quality of ... more What is the form by which the past is represented in the present as far as the proper quality of the past cannot be conserved literally in the present?
The outlined approach allows a common philosophical viewpoint to the physical world, language and... more The outlined approach allows a common philosophical viewpoint to the physical world, language and some mathematical structures therefore calling for the universe to be understood as a joint physical, linguistic and mathematical universum, in which physical motion and metaphor are one and the same rather than only similar in a sense.
Hilbert space underlying quantum mechanics and pseudo-Riemannian space underlying general relativ... more Hilbert space underlying quantum mechanics and pseudo-Riemannian space underlying general relativity share a common base of quantum information. Hilbert space can be interpreted as the free variable of quantum information, and any point in it, being equivalent to a wave function (and thus, to a state of a quantum system), as a value of that variable of quantum information. In turn, pseudo-Riemannian space can be interpreted as the interaction of two or more quantities of quantum information and thus, as two or more entangled quantum systems. Consequently, one can distinguish local physical interactions describable by a single Hilbert space (or by any factorizable tensor product of such ones) and non-local physical interactions describable only by means by that Hilbert space, which cannot be factorized as any tensor product of the Hilbert spaces, by means of which one can describe the interacting quantum subsystems separately. Any interaction, which can be exhaustedly described in a single Hilbert space, such as the weak, strong, and electromagnetic one, is local in terms of quantum information. Any interaction, which cannot be described thus, is nonlocal in terms of quantum information. Any interaction, which is exhaustedly describable by pseudo-Riemannian space, such as gravity, is nonlocal in this sense. Consequently all known physical interaction can be described by a single geometrical base interpreting it in terms of quantum information.
THE SECOND WORLD CONGRESS ON ANALOGY, POZNAŃ, MAY 24-26, 2017
(The Venue: Sala Lubrańskiego (Lubr... more THE SECOND WORLD CONGRESS ON ANALOGY, POZNAŃ, MAY 24-26, 2017 (The Venue: Sala Lubrańskiego (Lubrański’s Hall at the Collegium Minus), Adam Mickiewicz University, Address: ul. Wieniawskiego 1) The presentation: 24 May, 15:30
“Formal ontology” is introduced first to programing languages in different ways. The most relevan... more “Formal ontology” is introduced first to programing languages in different ways. The most relevant one as to philosophy is as a generalization of “nth-order logic” and “nth-level language” for n=0. Then, the “zero-level language” is a theoretical reflection on the naïve attitude to the world: the “things and words” coincide by themselves. That approach corresponds directly to the philosophical phenomenology of Husserl or fundamental ontology of Heidegger. Ontology as the 0-level language may be researched as a formal ontology
A possible interpretation of the equivalence of the Schrödinger equation and the Einstein field e... more A possible interpretation of the equivalence of the Schrödinger equation and the Einstein field equation
« Le cours de linguistique générale 1916-2016 »
« Arbtrariness of the sign », Suitzerland, Gene... more « Le cours de linguistique générale 1916-2016 » « Arbtrariness of the sign », Suitzerland, Geneva, University of Geneva, 10-12 January 2017: 11 January, 14:40-15:10
“The Square of Opposition”, 5th World Congress Rapanui (Easter Island), Chile, 10-15, November ... more “The Square of Opposition”, 5th World Congress Rapanui (Easter Island), Chile, 10-15, November 2016 http://www.square-of-opposition.org/Rapanui2016.html
“The Real of Reality”, An International Conference in Philosophy and Film
Karlsruhe, Germany, 2-... more “The Real of Reality”, An International Conference in Philosophy and Film Karlsruhe, Germany, 2-6 November 2016 Zentrum für Kunst und Medientechnologie (ZKM, Kube, 3 Nov, 11:20-12:50)
The world (wide) web generates a new “estate”, to which a huge part of mankind belongs. Unlike th... more The world (wide) web generates a new “estate”, to which a huge part of mankind belongs. Unlike the other “estates”, any human being even a child of own free will might joint it or not. That amorphous, “liquid”, non-professional, and virtual “estate” is the real “power” in the world, the most powerful one. The world web as a power overcomes the classical three powers and also the so-called fourth one, media. The trends in favor of it are even stronger than its real power and influence in the contemporary world.
My presentation at the conference “The Historical responsibility: from the myths of the past to ... more My presentation at the conference “The Historical responsibility: from the myths of the past to the strategies of future” September 22-23, 2016 (22 Sep, 12:20 – 14:00) Yekaterinburg, Russia (Lenin av. 51,Ural Federal University, Room 314)
My presentation at:
2nd Annual Conference of The European Network of Japanese Philosophy (ENOJP)
... more My presentation at: 2nd Annual Conference of The European Network of Japanese Philosophy (ENOJP) Université libre de Bruxelles 2016, December 7-10 8 December 15:30 (room 3)
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Papers by Vasil Penchev
April 22-23, 2016
(NNPS2016)
April 23-24, 2016
Heidegger called Aristotle’s Physics “the secret, never sufficiently rethought base book of Western Philosophy”.
This can explain the choice of Heidegger to comment namely it
The School of Philosophy of the Faculty of Humanities (National Research University Higher School of Economics)
30 April 2016
The “improper interpretation” of an infinite set-theory structure founds the “proper interpretation” and thus that structure self-founds itself as the one interpretation of it can found the other
https://arxiv.org/abs/2005.01412?fbclid=IwAR2tLJPQ-lmfhX3mkPQlV7_SLmHVTSmtA4dTFxDD9OMwNRbkAKtS61fd8KU
използван „Принстънският дух” и приятелството между Айнщайн и Гьодел. Набедената непълнота на квантовата механика и доказаната непълнота на аритметиката са преплетени и „сдвоени”, така че взаимни отблясъци осветяват аритметиката и математиката с
онтологичен пламък, но и квантовата механика и информация – с философска фундаменталност и способност да обосновава.
Неразрешими твърдения ли са самите т. нар. теореми на Гьодел за непълнотата, ако те се отнесат към самите себе си? Може ли парадоксът на Скулем да се използва за обобщаване на Айнщайновия „принцип на относителността” (1918) от дифеоморфизми и за
дискретни морфизми? Как следва да се тълкуват явленията на сдвояване (entanglement), квантовият компютър и квантовата информация аритметически и логически?
Книгата е предназначена за научни работници в областта на физиката, математиката и философията, за докторанти и студенти, за всеки, който се интересува от този съвсем нов отрасъл на знанието.
The French mathematician Henri Poincaré offered a statement known as “Poincaré’s conjecture” without a proof. It states that any 4-dimensional ball is equivalent to 3-dimensional Euclidean space topologically: a continuous mapping exists so that it maps the former ball into the latter space one-to-one.
At first glance, it seems to be too paradoxical for the following mismatches: the former is 4-dimensional and as if “closed” unlike the latter, 3-dimensional and as if “open” according to common sense. So, any mapping seemed to be necessarily discrete to be able to overcome those mismatches, and being discrete impies for the conjecture to be false.
Anyway, nobody managed neither to prove nor to reject rigorously the conjecture about one century. It was included even in the Millennium Prize Problems by the Clay Mathematics Institute.
It was proved by Grigory Perelman in 2003 using the concept of information.
Physical interpretation in terms of special relativity:
One may notice that the 4-ball is almost equivalent topologically to the “imaginary domain” of Minkowski space in the following sense of “almost”: that “half” of Minkowski space is equivalent topologically to the unfolding of a 4-ball. Then, the conjecture means the topological equivalence of the physical 3-space and its model in special relativity. In turn, that topological equivalence means their equivalence as to causality physically. So, Perelman has proved the adequacy of Minkowski space as a model of the physical 3-dimensional space rigorously. Of course, all experiments confirm the same empirically, but not mathematically as he did.
An idea of another proof of the conjecture based on that physical interpretation:
Topologically seen, the problem turns out to be reformulated so: one needs a proof of the topological equivalence of a 4-ball and its unfolding by 3-balls (what the “half” of Minkowski space is, topologically).
If one adds a complementary, second unfolding to link both ends of the first unfolding, the problem would be resolved: 4-ball would be equivalent to two 3-spaces topologically. Two 3-spaces are equivalent to a single one as follows: one divides a 3-space into two parts by a certain plane (that plane does not belong to any of them). Any part is equivalent topologically to a 3-space for any open neighborhood is transformed into an open one by the mapping of each part (excluding the boundary of the plane) into the complete 3-space.
That idea is linked to the original proof of Perelman by the concept of information. It means that any bit of information interpreted physically conserves causality. In other words, the choice of any of both states of a bit (e.g. designated as “0” and “1” recorded in a cell) does not violate causality (the cell, either “0” or “1”, or both “0” and “1” are equivalent to each other topologically and to a 3-space).
6th WORLD CONGRESS ON THE SQUARE OF OPPOSITION
http://www.square-of-opposition.org/square2018.html
into the state of being
❖ This represents the “creation”. The transition of nothing into being is mathematically necessary
❖ The choice (which can be interpreted philosophically as “free will”) appears necessary in mathematical reasons
❖ The choice generates asymmetry, which is the beginning of time and thus, of the physical word
❖ Information is the quantity of choices and linked to time intimately
2. Quantum information is what is conserved, action is what is changed.
3. The gap between mathematical models and physical reality, needing truth as adequacy to be overcome, is substituted by the openness of choice.
4. That openness in turn can be interpreted as the openness of the present as a different concept of truth recollecting Heidegger’s one as “unhiddeness”.
5. Quantum information as what is conserved can be thought philoso[hically as the conservation of that openness.
(The Venue: Sala Lubrańskiego (Lubrański’s Hall at the Collegium Minus), Adam Mickiewicz University, Address: ul. Wieniawskiego 1)
The presentation: 24 May, 15:30
« Arbtrariness of the sign », Suitzerland, Geneva, University of Geneva, 10-12 January 2017: 11 January, 14:40-15:10
http://www.square-of-opposition.org/Rapanui2016.html
Karlsruhe, Germany, 2-6 November 2016
Zentrum für Kunst und Medientechnologie
(ZKM, Kube, 3 Nov, 11:20-12:50)
September 22-23, 2016 (22 Sep, 12:20 – 14:00)
Yekaterinburg, Russia (Lenin av. 51,Ural Federal University, Room 314)
2nd Annual Conference of The European Network of Japanese Philosophy (ENOJP)
Université libre de Bruxelles 2016, December 7-10
8 December 15:30 (room 3)