In the paper we will employ set theory to study the formal aspects of quantum mechanics without explicitly making use of space-time. It is demonstrated that von Neuman and Zermelo numeral sets, previously efectively used in the explanation of Hardy’s paradox, follow a Heisenberg quantum form. Here monadic union plays the role of time derivative. The logical counterpart of monadic union plays the part of the Hamiltonian in the commutator. The use of numerals and monadic union in (...) the classical probability resolution of Hardy’s paradox [1] is supported with the present derivation of a commutator for sets. (shrink)

The notion of equality between two observables will play many important roles in foundations of quantum theory. However, the standard probabilistic interpretation based on the conventional Born formula does not give the probability of equality between two arbitrary observables, since the Born formula gives the probability distribution only for a commuting family of observables. In this paper, quantum set theory developed by Takeuti and the present author is used to systematically extend the standard probabilistic interpretation of (...) quantum theory to define the probability of equality between two arbitrary observables in an arbitrary state. We apply this new interpretation to quantum measurement theory, and establish a logical basis for the difference between simultaneous measurability and simultaneous determinateness. (shrink)

Many approaches to quantum gravity -the theory that should account for quantum and gravitational phenomena under the same theoretical umbrella- seem to point at some form of spacetime emergence, i.e., the fact that spacetime is not a fundamental entity of our physical world. This tenet has sparked many philosophical discussions: from the so-called empirical incoherence problem to different accounts of emergence and mechanisms thereof. In this contribution, I focus on the partial order relation of causal set (...) class='Hi'>theory and argue that causation can be characterized as an a-temporal constraint over the kinematic space defined by the theory. The relation constrains the growth of a new element/event with respect to the other elements/events of a given set. Therefrom, the flow of time emerges from the collection of the possible growths of the given set, where each possibility is characterized by a classical probability assigned by the dynamics of the theory. (shrink)

In previous works, an ontology of properties for quantum mechanics has been proposed, according to which quantum systems are bundles of properties with no principle of individuality. The aim of the present article is to show that, since quasi-set theory is particularly suited for dealing with aggregates of items that do not belong to the traditional category of individual, it supplies an adequate meta-language to speak of the proposed ontology of properties and its structure.

The file on this site provides the slides for a lecture given in Hangzhou in May 2018, and the lecture itself is available at the URL beginning 'sms' in the set of links provided in connection with this item. -/- It is commonly assumed that regular physics underpins biology. Here it is proposed, in a synthesis of ideas by various authors, that in reality structures and mechanisms of a biological character underpin the world studied by physicists, in principle supplying detail (...) in the domain that according to regular physics is of an indeterminate character. In regular physics mathematical equations are primary, but this constraint leads to problems with reconciling theory and reality. Biology on the other hand typically does not characterise nature in quantitative terms, instead investigating in detail important complex interrelationships between parts, leading to an understanding of the systems concerned that is in some respects beyond that which prevails in regular physics. It makes contact with quantum physics in various ways, for example in that both involve interactions between observer and observed, an insight that explains what is special about processes involving observation, justifying in the quantum physics context the replacement of the unphysical many-worlds picture by one involving collapse. The link with biology furthermore clarifies Wheeler’s suggestion that a multiplicity of observations can lead to the ‘fabrication of form’, including the insight that this process depends on very specific ‘structures with power’ related to the 'semiotic scaffolding' of the application of sign theory to biology known as biosemiotics. -/- The observer-observed 'circle' of Wheeler and Yardley is a special case of a more general phenomenon, oppositional dynamics, related to the 'intra-action' of Barad's Agential Realism, involving cooperating systems such as mind and matter, abstract and concrete, observer and observed, that preserve their identities while interacting with one another in such a way as to act as a unit. A third system may also be involved, the mediating system of Peirce linking the two together. Such a situation of changing connections and separations may plausibly lead in the future to an understanding of how complex systems are able to evolve to produce 'life, the universe and everything'. -/- (Added 1 July 2018) The general structure proposed here as an alternative to a mathematics-based physics can be usefully characterised by relating it to different disciplines and the specialised concepts utilised therein. In theoretical physics, the test for the correctness of a theory typically involves numerical predictions, corresponding to which theories are expressed in terms of equations, that is to say assertions that two quantities have identical values. Equations have a lesser significance in biology which typically talks in terms of functional mechanisms, dependent for example on details of chemistry and concepts such as genes, natural selection, signals and geometrical or topologically motivated concepts such as the interconnections between systems and the unfolding of DNA. Biosemiotics adds to this the concept of signs and their interpretation, implying novel concepts such as semiotic scaffolding and the semiosphere, code duality, and appreciation of the different types of signs, including symbols and their capacity for abstraction and use in language systems. Circular Theory adds to this picture, as do the ideas of Barad, considerations such as the idea of oppositional dynamics. The proposals in this lecture can be regarded as the idea that concepts such as those deriving from biosemiotics have more general applicability than just conventional biology and may apply, in some circumstances, to nonlinear systems generally, including the domain new to science hypothesised to underlie the phenomena of present-day physics. -/- The task then has to be to restore the mathematical aspect presumed, in this picture, not to be fundamental as it is in conventional theory. Deacon has invoked a complex sequence of evolutionary steps to account for the emergence over time of human language systems, and correspondingly mathematical behaviour can be subsumed under the general evolutionary mechanisms of biosemiotics (cf. also the proposals of Davis and Hersh regarding the nature of mathematics), so that the mathematical behaviour of physical systems is consistent with the proposed scheme. In conclusion, it is suggested that theoretical physicists should cease expecting to find some universal mathematical ‘theory of everything’, and focus instead on understanding in more detail complex systems exhibiting behaviour of a biological character, extending existing understanding. This may in time provide a more fruitful understanding of the natural world than does the regular approach. The essential concepts have an observational basis from both biology and the little-known discipline of cymatics (a discipline concerned with the remarkable patterns that specific waveforms can give rise to), while again computer simulations also offer promise in providing insight into the complex behaviours involved in the above proposals. -/- References -/- Jesper Hoffmeyer, Semiotic Scaffolding of Living Systems. Commens, a Digital Companion to C. S. Peirce (on Commens web site). Terrence Deacon, The Symbolic Species, W.W. Norton & Co. Karen Barad, Meeting the Universe Halfway: Quantum Physics and the Entanglement of Matter and Meaning, Duke University Press. Philip Davis and Reuben Hersh, The Mathematical Experience, Penguin. Ilexa Yardley, Circular Theory. (shrink)

Rudolf Haag’s Local Quantum Physics (LQP) is an alternative framework to conventional relativistic quantum field theory for combining special relativity and quantum theory based on first principles, making it of great interest for the purposes of conceptual analysis despite currently being relatively limited as a tool for making experimental predictions. In LQP, the elementary particles are defined as species of causal link between interaction events, together with which they comprise its most fundamental entities. This notion (...) of particle has yet to be independently assessed as such. Here, it is captured via a set of propositions specifying particle characteristics and then compared to previous particle notions. Haag’s particle differs decisively with respect to mechanical intuitions about particles by lacking, among other things, even an approximate independent space–time location. This notion is thus found to differ greatly even from those of relativistic quantum mechanics and quantum field theory, which have been applied to the known elementary particles. (shrink)

We look into the ontology of quantum theory as distinct from that of the classical theory in the sciences. Theories carry with them their own ontology while the metaphysics may remain the same in the background. We follow a broadly Kantian tradition, distinguishing between the noumenal and phenomenal realities where the former is independent of our perception while the latter is assembled from the former by means of fragmentary bits of interpretation. Theories do not tell us how (...) the noumenal world is constituted but are conceptual constructs applying to models generated in the phenomenal world within limited contexts. -/- The ontology of quantum theory principally rests on the view that entities in the world are pervasively correlated with one another not by means of probabilities as in the case of the classical theory, but by means of probability amplitudes involving finely tuned phases characterising the oscillatory behaviour of quantum mechanical states. -/- The amplitude-dependent correlations (quantum entanglement) exist over and above the classical ones expressed in terms of probabilities. While the classical correlations are essentially local in nature, quantum correlations are shared globally in the process of environment-induced decoherence. The decoherence is an effectively random process that removes local correlations in the course of global sharing of entanglement—the removal being especially manifest in the case of systems that appear as classical ones. It is this aspect of the decoherence process that makes the so-called measurement postulate (one relating to wave function collapse) cohere with the rest of the principles of quantum theory where the latter implies a Schrodinger type time evolution of quantum mechanical systems. -/- The mathematical basis of the quantum correlations consists of the description of pure states of a system in terms of vectors in a linear vector space (a mixed state appears as an admixture of pure states with some probability distribution associated with it) and, in addition, the description of (pure) states of composite systems as vectors in the product space arising from the component sub-systems. -/- The crucial aspect of the decoherence process, of significance in the context of the apparent incompatibility of the measurement postulate with the unitary Schrodinger evolution, relates to the fact that it is almost instantaneous in the case of a classical object (one that is nevertheless amenable to a quantum description). Indeed, the decoherence time is, in all likelihood, of the order of the Planck scale, being driven by field fluctuations in the Planck regime. This points to factors of an unknown nature determining the finest details of the decoherence process since Planck scale physics remains an obscure terrain. -/- In other words, quantum theory is, to all intents and purposes, in the need of a radical revision, in keeping with the fact that all theories are defeasible and need revision as our domains of experience expand and get realigned in a complex manner. The context within which quantum theory (and quantum field theory too) is defined is set precisely by the Planck scale, across which a novel theoretical framework is likely to emerge. However, as in the case of theory revisions in general, that emerging theory will stand in an asymmetric relation of incommensurability with the present-day quantum theory, where the concepts of the latter will be comprehensible in terms of those of the former, but the converse will not hold. (shrink)

The original purpose of the present study, 2011, started with a preprint «On the Probable Failure of the Uncountable Power Set Axiom», 1988, is to save from the transfinite deadlock of higher set theory the jewel of mathematical Continuum — this genuine, even if mostly forgotten today raison d’être of all traditional set-theoretical enterprises to Infinity and beyond, from Georg Cantor to David Hilbert to Kurt Gödel to W. Hugh Woodin to Buzz Lightyear.

Quantum information is discussed as the universal substance of the world. It is interpreted as that generalization of classical information, which includes both finite and transfinite ordinal numbers. On the other hand, any wave function and thus any state of any quantum system is just one value of quantum information. Information and its generalization as quantum information are considered as quantities of elementary choices. Their units are correspondingly a bit and a qubit. The course of time (...) is what generates choices by itself, thus quantum information and any item in the world in final analysis. The course of time generates necessarily choices so: The future is absolutely unorderable in principle while the past is always well-ordered and thus unchangeable. The present as the mediation between them needs the well-ordered theorem equivalent to the axiom of choice. The latter guarantees the choice even among the elements of an infinite set, which is the case of quantum information. The concrete and abstract objects share information as their common base, which is quantum as to the formers and classical as to the latter. The general quantities of matter in physics, mass and energy can be considered as particular cases of quantum information. The link between choice and abstraction in set theory allows of “Hume’s principle” to be interpreted in terms of quantum mechanics as equivalence of “many” and “much” underlying quantum information. Quantum information as the universal substance of the world calls for the unity of physics and mathematics rather than that of the concrete and abstract objects and thus for a form of quantum neo-Pythagoreanism in final analysis. (shrink)

We introduce a new theorem in social choice theory built on a path integral approach which will show that, under some reasonable conditions, there is a unique way to aggregate individual preferences based on fuzzy sets into a social preference based on probabilities, and that this way is invariant under any permutation of alternatives. We then apply this theorem to the case of democratic decision making with data of the behaviour and voting preferences of voting agents and show that (...) there is a tradeoff between fairness and efficiency and that no voting system can achieve both simultaneously. (shrink)

This paper introduces the theory QST of quasets as a formal basis for the Nmatrices. The main aim is to construct a system of Nmatrices by substituting standard sets by quasets. Since QST is a conservative extension of ZFA (the Zermelo-Fraenkel set theory with Atoms), it is possible to obtain generalized Nmatrices (Q-Nmatrices). Since the original formulation of QST is not completely adequate for the developments we advance here, some possible amendments to the theory are also considered. (...) One of the most interesting traits of such an extension is the existence of complementary quasets which admit elements with undetermined membership. Such elements can be interpreted as quantum systems in superposed states. We also present a relationship of QST with the theory of Rough Sets RST, which grants the existence of models for SQT formed by rough sets. Some consequences of the given formalism for the relation of logical consequence are also analysed. (shrink)

In this contribution, I explore the possibility of characterizing the emergence of time in causal set theory (CST) in terms of the growing block universe (GBU) metaphysics. I show that although GBU seems to be the most intuitive time metaphysics for CST, it leaves us with a number of interpretation problems, independently of which dynamics we choose to favor for the theory —here I shall consider the Classical Sequential Growth and the Covariant model. Discrete general covariance of the (...) CSG dynamics does not allow us to individuate a single history of the universe (defined by a causal history of different causal sets), thereby making the claim that ‘the past exists’ at best problematic. In addition, because the evolution of the universe in CSG dynamics leads to an outward branching causal tree, it becomes impossible to determine a proper ‘line of becoming’, thereby blurring the presentists’ claim that only the present exists. Similarly, the covariant approach runs into the same, if not even more severe problems, since each configuration of the universe would amount to a set of possible causal sets, thereby making the individuation of a single configuration of the universe —and thus the physical interpretation of the theory— implausible. (shrink)

The concepts of choice, negation, and infinity are considered jointly. The link is the quantity of information interpreted as the quantity of choices measured in units of elementary choice: a bit is an elementary choice between two equally probable alternatives. “Negation” supposes a choice between it and confirmation. Thus quantity of information can be also interpreted as quantity of negations. The disjunctive choice between confirmation and negation as to infinity can be chosen or not in turn: This corresponds to set- (...) class='Hi'>theory or intuitionist approach to the foundation of mathematics and to Peano or Heyting arithmetic. Quantum mechanics can be reformulated in terms of information introducing the concept and quantity of quantum information. A qubit can be equivalently interpreted as that generalization of “bit” where the choice is among an infinite set or series of alternatives. The complex Hilbert space can be represented as both series of qubits and value of quantum information. The complex Hilbert space is that generalization of Peano arithmetic where any natural number is substituted by a qubit. “Negation”, “choice”, and “infinity” can be inherently linked to each other both in the foundation of mathematics and quantum mechanics by the meditation of “information” and “quantum information”. (shrink)

1. The no form action theory is a novel philosophical theory that transcends traditional dualisms such as matter and spirit, form and substance. It establishes a two-dimensional theoretical framework composed of form and no form, and uses this two-dimensional theory to understand various aspects of the world. 2. The three key actions of no form are: isolation action (e.g., distinguishing between different things), motive force action (e.g., driving changes in things), and manifestation action (e.g., allowing us to (...) perceive things). These three actions have strong "self-reflexivity". For example, the relationships between these three actions can be divided into: isolation relationship, motive force relationship, and manifestation relationship. These three relationships, in turn, belong to isolation action, motive force action, and manifestation action respectively. 3. The no form action theory have strong universality, capable of providing unified theoretical explanations across various disciplines. For example, it can be used to explain quantum entanglement phenomena in physics, as well as the mechanism of consciousness generation in psychology; it can explain ontology, as well as time and space; it can explain emotions, as well as logic. It can explain language, as well as the macroscopic world. 4. The completion of the core ideas of the no form action theory marks the beginning of a new historical stage in philosophical research. It provides a completely new set of methods and perspectives to understand the world, and has achieved a series of important results: (1) It offers new interpretations of the most fundamental philosophical concepts such as "being", "existence", "is", "freedom", "consciousness", "beauty", "space", "time", "basis", "cause", "self", "identity", etc., and introduces many new concepts. It binds these concepts technically within the framework of the no form action theory. This establishes a highly logical and explanatory philosophical system. As a result, philosophical research now has clearer, more rigorous, and more reliable technical operational methods. (2) It derives and explains the three basic laws of formal logic, and demonstrates that they are interdependent and form an inseparable whole. (3) It provides a new solution to Descartes' "mind-body problem" and also explains difficult issues in physics such as quantum entanglement. (4) It discovers a third dimension in the world besides time and space: "now". This discovery provides a completely new perspective for understanding reality, consciousness, and the nature of the universe. (shrink)

The paper shows how the Bohmian approach to quantum physics can be applied to develop a clear and coherent ontology of non-perturbative quantum gravity. We suggest retaining discrete objects as the primitive ontology also when it comes to a quantum theory of space-time and therefore focus on loop quantum gravity. We conceive atoms of space, represented in terms of nodes linked by edges in a graph, as the primitive ontology of the theory and (...) show how a non-local law in which a universal and stationary wave-function figures can provide an order of configurations of such atoms of space such that the classical space-time of general relativity is approximated. Although there is as yet no fully worked out physical theory of quantum gravity, we regard the Bohmian approach as setting up a standard that proposals for a serious ontology in this field should meet and as opening up a route for fruitful physical and mathematical investigations. (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. David Elohim examines the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, (...) undecidable propositions, and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal and hyperintensional cognitivism and modal and hyperintensional expressivism. Elohim develops a novel, topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel, dynamic two-dimensional semantics grounded in two-dimensional hyperintensional Turing machines. Chapter \textbf{3} provides an abstraction principle for epistemic (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between Elohim's hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. Elohim countenances a hyperintensional semantics for novel epistemic abstractionist modalities. Elohim suggests, too, that higher observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for two-dimensional hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between interpretational and objective modalities and the truthmakers thereof. This yields the first hyperintensional category theory in the literature. Elohim invents a new mathematical trick in which first-order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals in the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Elohim provides a counter-example to epistemic closure for logical deduction. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

Quantum information is discussed as the universal substance of the world. It is interpreted as that generalization of classical information, which includes both finite and transfinite ordinal numbers. On the other hand, any wave function and thus any state of any quantum system is just one value of quantum information. Information and its generalization as quantum information are considered as quantities of elementary choices. Their units are correspondingly a bit and a qubit. The course of time (...) is what generates choices by itself, thus quantum information and any item in the world in final analysis. The course of time generates necessarily choices so: The future is absolutely unorderable in principle while the past is always well-ordered and thus unchangeable. The present as the mediation between them needs the well-ordered theorem equivalent to the axiom of choice. The latter guarantees the choice even among the elements of an infinite set, which is the case of quantum information. The concrete and abstract objects share information as their common base, which is quantum as to the formers and classical as to the latters. The general quantities of matter in physics, mass and energy can be considered as particular cases of quantum information. The link between choice and abstraction in set theory allows of “Hume’s principle” to be interpreted in terms of quantum mechanics as equivalence of “many” and “much” underlying quantum information. Quantum information as the universal substance of the world calls for the unity of physics and mathematics rather than that of the concrete and abstract objects and thus for a form of quantum neo-Pythagoreanism in final analysis. (shrink)

A practical viewpoint links reality, representation, and language to calculation by the concept of Turing (1936) machine being the mathematical model of our computers. After the Gödel incompleteness theorems (1931) or the insolvability of the so-called halting problem (Turing 1936; Church 1936) as to a classical machine of Turing, one of the simplest hypotheses is completeness to be suggested for two ones. That is consistent with the provability of completeness by means of two independent Peano arithmetics discussed in Section I. (...) Many modifications of Turing machines cum quantum ones are researched in Section II for the Halting problem and completeness, and the model of two independent Turing machines seems to generalize them. Then, that pair can be postulated as the formal definition of reality therefore being complete unlike any of them standalone, remaining incomplete without its complementary counterpart. Representation is formal defined as a one-to-one mapping between the two Turing machines, and the set of all those mappings can be considered as “language” therefore including metaphors as mappings different than representation. Section III investigates that formal relation of “reality”, “representation”, and “language” modeled by (at least two) Turing machines. The independence of (two) Turing machines is interpreted by means of game theory and especially of the Nash equilibrium in Section IV. Choice and information as the quantity of choices are involved. That approach seems to be equivalent to that based on set theory and the concept of actual infinity in mathematics and allowing of practical implementations. (shrink)

Quantum physicists have made many attempts to solve the quantum measurement problem, but no solution seems to have received widespread acceptance. The time has come for a new approach. In Sense Perception and Reality: A Theory of Perceptual Relativity, Quantum Mechanics and the Observer Dependent Universe I suggest the quantum measurement problem is caused by a failure to understand that each species has its own sensory world and that when we say the wave function collapses (...) and brings a particle into existence we mean the particle is brought into existence in the human sensory world by the combined operation of the human sensory apparatus, particle detectors and the experimental set up. This is similar to the Copenhagen Interpretation suggested by Niels Bohr and others, but the understanding that the collapse of the wave function brings a particle into existence in the human sensory world removes the need for a dividing line between the quantum world and the macro world. The same rules can apply to both worlds and the ideas stated in this paper considerably strengthen the Copenhagen Interpretation of quantum mechanics. (shrink)

As quantum technologies (QT) advance, their potential impact on and relation with society has been developing into an important issue for exploration. In this paper, we investigate the topic of democratization in the context of QT, particularly quantum computing. The paper contains three main sections. First, we briefly introduce different theories of democracy (participatory, representative, and deliberative) and how the concept of democratization can be formulated with respect to whether democracy is taken as an intrinsic or instrumental value. (...) Second, we give an overview of how the concept of democratization is used in the QT field. Democratization is mainly adopted by companies working on quantum computing and used in a very narrow understanding of the concept. Third, we explore various narratives and counter-narratives concerning democratization in QT. Finally, we explore the general efforts of democratization in QT such as different forms of access, formation of grassroot communities and special interest groups, the emerging culture of manifesto writing, and how these can be located within the different theories of democracy. In conclusion, we argue that although the ongoing efforts in the democratization of QT are necessary steps towards the democratization of this set of emerging technologies, they should not be accepted as sufficient to argue that QT is a democratized field. We argue that more reflexivity and responsiveness regarding the narratives and actions adopted by the actors in the QT field and making the underlying assumptions of ongoing efforts on democratization of QT explicit, can result in a better technology for society. (shrink)

Applied Evolutionary Epistemology is a scientific-philosophical theory that defines evolution as the set of phenomena whereby units evolve at levels of ontological hierarchies by mechanisms and processes. This theory also provides a methodology to study evolution, namely, studying evolution involves identifying the units that evolve, the levels at which they evolve, and the mechanisms and processes whereby they evolve. Identifying units and levels of evolution in turn requires the development of ontological hierarchy theories, and examining mechanisms and (...) processes necessitates theorizing about causality. Together, hierarchy and causality theories explain how biorealities form and diversify with time. This paper analyzes how Applied EE redefines both hierarchy and causality theories in the light of the recent explosion of network approaches to causal reasoning associated with studies on reticulate and macroevolution. Causality theories have often been framed from within a rigid, ladder-like hierarchy theory where the rungs of the ladder represent the different levels, and the elements on the rungs represent the evolving units. Causality then is either defined reductionistically as an upward movement along the strands of a singular hierarchy, or holistically as a downward movement along that same hierarchy. Upward causation theories thereby analyze causal processes in time, i.e. over the course of natural history or phylogenetically, as Darwin and the founders of the Modern Synthesis intended. Downward causation theories analyze causal processes in space, ontogenetically or ecologically, as the current eco-evo-devo schools are evidencing. This work demonstrates how macroevolution and reticulate evolution theories add to the complexity by examining reticulate causal processes in space–time, and the interactional hierarchies that such studies bring forth introduce a new form of causation that is here called reticulate causation. Reticulate causation occurs between units and levels belonging to different as well as to the same ontological hierarchies. This article concludes that beyond recognizing the existence of multiple units, levels, and mechanisms or processes of evolution, also the existence of multiple kinds of evolutionary causation as well as the existence of multiple evolutionary hierarchies needs to be acknowledged. This furthermore implies that evolution is a pluralistic process divisible into different kinds. (shrink)

Although Fuzzy logic and Fuzzy Mathematics is a widespread subject and there is a vast literature about it, yet the use of Fuzzy issues like Fuzzy sets and Fuzzy numbers was relatively rare in time concept. This could be seen in the Fuzzy time series. In addition, some attempts are done in fuzzing Turing Machines but seemingly there is no need to fuzzy time. Throughout this article, we try to change this picture and show why it is helpful to consider (...) the instants of time as Fuzzy numbers. In physics, though there are revolutionary ideas on the time concept like B theories in contrast to A theory also about central concepts like space, momentum… it is a long time that these concepts are changed, but time is considered classically in all well-known and established physics theories. Seemingly, we stick to the classical time concept in all fields of science and we have a vast inertia to change it. Our goal in this article is to provide some bases why it is rational and reasonable to change and modify this picture. Here, the central point is the modified version of “Unexpected Hanging” paradox as it is described in "Is classical Mathematics appropriate for theory of Computation".This modified version leads us to a contradiction and based on that it is presented there why some problems in Theory of Computation are not solved yet. To resolve the difficulties arising there, we have two choices. Either “choosing” a new type of Logic like “Para-consistent Logic” to tolerate contradiction or changing and improving the time concept and consequently to modify the “Turing Computational Model”. Throughout this paper, we select the second way for benefiting from saving some aspects of Classical Logic. In chapter 2, by applying quantum Mechanics and Schrodinger equation we compute the associated fuzzy number to time. These, provides a new interpretation of Quantum Mechanics.More exactly what we see here is "Particle-Fuzzy time" interpretation of quantum Mechanics, in contrast to some other interpretations of Quantum Mechanics like " Wave-Particle" interpretation. At the end, we propound a question about the possible solution of a paradox in Physics, the contradiction between General Relativity and Quantum Mechanics. (shrink)

We introduce the notion of complexity, first at an intuitive level and then in relatively more concrete terms, explaining the various characteristic features of complex systems with examples. There exists a vast literature on complexity, and our exposition is intended to be an elementary introduction, meant for a broad audience. -/- Briefly, a complex system is one whose description involves a hierarchy of levels, where each level is made of a large number of components interacting among themselves. The time evolution (...) of such a system is of a complex nature, depending on the interactions among subsystems in the next level below the one under consideration and, at the same time, conditioned by the level above, where the latter sets the context for the evolution. Generally speaking, the interactions among the constituents of the various levels lead to a dynamics characterized by numerous characteristic scales, each level having its own set of scales. What is more, a level commonly exhibits ‘emergent properties’ that cannot be derived from considerations relating to its component systems taken in isolation or to those in a different contextual setting. In the dynamic evolution of some particular level, there occurs a self-organized emergence of a higher level and the process is repeated at still higher levels. -/- The interaction and self-organization of the components of a complex system follow the principle commonly expressed by saying that the ‘whole is different from the sum of the parts’. In the case of systems whose behavior can be expressed mathematically in terms of differential equations this means that the interactions are nonlinear in nature. -/- While all of the above features are not universally exhibited by complex systems, these are nevertheless indicative of a broad commonness relative to which individual systems can be described and analyzed. There exist measures of complexity which, once again, are not of universal applicability, being more heuristic than exact. The present state of knowledge and understanding of complex systems is itself an emerging one. Still, a large number of results on various systems can be related to their complex character, making complexity an immensely fertile concept in the study of natural, biological, and social phenomena. -/- All this puts a very definite limitation on the complete description of a complex system as a whole since such a system can be precisely described only contextually, relative to some particular level, where emergent properties rule out an exact description of more than one levels within a common framework. -/- We discuss the implications of these observations in the context of our conception of the so-called noumenal reality that has a mind-independent existence and is perceived by us in the form of the phenomenal reality. The latter is derived from the former by means of our perceptions and interpretations, and our efforts at sorting out and making sense of the bewildering complexity of reality takes the form of incessant processes of inference that lead to theories. Strictly speaking, theories apply to models that are constructed as idealized versions of parts of reality, within which inferences and abstractions can be carried out meaningfully, enabling us to construct the theories. -/- There exists a correspondence between the phenomenal and the noumenal realities in terms of events and their correlations, where these are experienced as the complex behavior of systems or entities of various descriptions. The infinite diversity of behavior of systems in the phenomenal world are explained within specified contexts by theories. The latter are constructs generated in our ceaseless attempts at interpreting the world, and the question arises as to whether these are reflections of `laws of nature' residing in the noumenal world. This is a fundamental concern of scientific realism, within the fold of which there exists a trend towards the assumption that theories express truths about the noumenal reality. We examine this assumption (referred to as a ‘point of view’ in the present essay) closely and indicate that an alternative point of view is also consistent within the broad framework of scientific realism. This is the view that theories are domain-specific and contextual, and that these are arrived at by independent processes of inference and abstractions in the various domains of experience. Theories in contiguous domains of experience dovetail and interpenetrate with one another, and bear the responsibility of correctly explaining our observations within these domains. -/- With accumulating experience, theories get revised and the network of our theories of the world acquires a complex structure, exhibiting a complex evolution. There exists a tendency within the fold of scientific realism of interpreting this complex evolution in rather simple terms, where one assumes (this, again, is a point of view) that theories tend more and more closely to truths about Nature and, what is more, progress towards an all-embracing ‘ultimate theory’ -- a foundational one in respect of all our inquiries into nature. We examine this point of view closely and outline the alternative view -- one broadly consistent with scientific realism -- that there is no ‘ultimate’ law of nature, that theories do not correspond to truths inherent in reality, and that successive revisions in theory do not lead monotonically to some ultimate truth. Instead, the theories generated in succession are incommensurate with each other, testifying to the fact that a theory gives us a perspective view of some part of reality, arrived at contextually. Instead of resembling a monotonically converging series successive theories are analogous to asymptotic series. -/- Before we summarize all the above considerations, we briefly address the issue of the complexity of the {\it human mind} -- one as pervasive as the complexity of Nature at large. The complexity of the mind is related to the complexity of the underlying neuronal organization in the brain, which operates within a larger biological context, its activities being modulated by other physiological systems, notably the one involving a host of chemical messengers. The mind, with no materiality of its own, is nevertheless emergent from the activity of interacting neuronal assemblies in the brain. As in the case of reality at large, there can be no ultimate theory of the mind, from which one can explain and predict the entire spectrum of human behavior, which is an infinitely rich and diverse one. (shrink)

This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of ℂ replaced by ℤ₂. Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability (...) calculus. The previous attempts all required the brackets to take values in ℤ₂. But the usual QM brackets <ψ|ϕ> give the "overlap" between states ψ and ϕ, so for subsets S,T⊆U, the natural definition is <S|T>=|S∩T| (taking values in the natural numbers). This allows QM/sets to be developed with a full probability calculus that turns out to be a non-commutative extension of classical Laplace-Boole finite probability theory. The pedagogical model is illustrated by giving simple treatments of the indeterminacy principle, the double-slit experiment, Bell's Theorem, and identical particles in QM/Sets. A more technical appendix explains the mathematics behind carrying some vector space structures between QM over ℂ and QM/Sets over ℤ₂. (shrink)

Classical logic is usually interpreted as the logic of propositions. But from Boole's original development up to modern categorical logic, there has always been the alternative interpretation of classical logic as the logic of subsets of any given (nonempty) universe set. Partitions on a universe set are dual to subsets of a universe set in the sense of the reverse-the-arrows category-theoretic duality--which is reflected in the duality between quotient objects and subobjects throughout algebra. Hence the idea arises of a dual (...) logic of partitions. That dual logic is described here. Partition logic is at the same mathematical level as subset logic since models for both are constructed from (partitions on or subsets of) arbitrary unstructured sets with no ordering relations, compatibility or accessibility relations, or topologies on the sets. Just as Boole developed logical finite probability theory as a quantitative treatment of subset logic, applying the analogous mathematical steps to partition logic yields a logical notion of entropy so that information theory can be refounded on partition logic. But the biggest application is that when partition logic and the accompanying logical information theory are "lifted" to complex vector spaces, then the mathematical framework of quantum mechanics is obtained. Partition logic models indefiniteness (i.e., numerical attributes on a set become more definite as the inverse-image partition becomes more refined) while subset logic models the definiteness of classical physics (an entity either definitely has a property or definitely does not). Hence partition logic provides the backstory so the old idea of "objective indefiniteness" in QM can be fleshed out to a full interpretation of quantum mechanics. (shrink)

In this article, we apply the notion of interval neutrosophic sets to ideal theory in BCK/BCI-algebras.

With the present paper I maintain that the group field theory (GFT) approach to quantum gravity can help us clarify and distinguish the problems of spacetime emergence from the questions about the nature of the quanta of space. I will show that the mechanism of phase transition suggests a form of indifference between scales (or phases) and that such an indifference allows us to black-box questions about the nature of the ontology of the fundamental levels of the (...) class='Hi'>theory. I consider the GFT approach to quantum gravity which derives spacetime as an emergent property from abstract non-geometrical tetrahedra through a processes of phase transition. Because the physical interpretation of the tetrahedra is problematic in view of their (non)-spatiotemporal nature, I will apply the considerations about phase transitions to GFT and conclude that the fundamental ontology of the theory and the mechanism of spacetime emergence can and should be treated separately. (shrink)

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. (...) In defense of this claim, I offer evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences. (shrink)

A principle, according to which any scientific theory can be mathematized, is investigated. Social science, liberal arts, history, and philosophy are meant first of all. That kind of 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. The main statement is formulated as follows: Any scientific theory admits isomorphism to some mathematical structure in a way constructive. 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. The sketch of the proof is organized in five steps: a generalization of epoché; involving transfinite induction in the transition between Peano arithmetic and set theory; discussing the finiteness of Peano arithmetic; applying transfinite induction to Peano arithmetic; discussing an arithmetical model of reality. 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. The present paper follows a pathway grounded on Husserl’s phenomenology and “bracketing reality” to achieve the generalized arithmetic necessary for the principle to be founded in alternative ontology, in which there is no reality external to mathematics: reality is included within mathematics. That latter mathematics is able to self-found itself and can be called Hilbert mathematics in honour of Hilbert’s program for self-founding mathematics on the base of arithmetic. The principle of universal mathematizability is consistent to Hilbert mathematics, but not to Gödel mathematics. Consequently, its validity or rejection would resolve the problem which mathematics refers to our being; and vice versa: the choice between them for different reasons would confirm or refuse the principle as to the being. An information interpretation of Hilbert mathematics is involved. It is a kind of ontology of information. The Schrödinger equation in quantum mechanics is involved to illustrate that ontology. Thus the problem which of the two mathematics is more relevant to our being is discussed again in a new way A few directions for future work can be: a rigorous formal proof of the principle as an independent axiom; the further development of information ontology consistent to both kinds of mathematics, but much more natural for Hilbert mathematics; the development of the information interpretation of quantum mechanics as a mathematical one for information ontology and thus Hilbert mathematics; the description of consciousness in terms of information ontology. (shrink)

To this day, a hundred and fifty years after Mendeleev's discovery, the overal structure of the periodic system remains unaccounted for in quantum-mechanical terms. Given this dire situation, a handful of scientists in the 1970s embarked on a quest for the symmetries that lie hidden in the periodic table. Their goal was to explain the table's structure in group-theoretical terms. We argue that this symmetry program required an important paradigm shift in the understanding of the nature of chemical elements. (...) The idea, in essence, consisted of treating the chemical elements, not as particles, but as states of a superparticle. We show that the inspiration for this came from elementary particle physics, and in particular from Heisenberg's suggestion to treat the proton and neutron as different states of the nucleon. We provide a careful study of Heisenberg's last paper on the nature of elementary particles, and explain why the Democritean picture of matter no longer applied in modern physics and a Platonic symmetry-based picture was called for instead. We show how Heisenberg's Platonic philosophy came to dominate the field of elementary particle physics, and how it found its culmination point in Gell-Mann's classification of the hadrons in the eightfold way. We argue that it was the success of Heisenberg's approach in elementary particle physics that sparked the group-theoretical approach to the periodic table. We explain how it was applied to the set of chemical elements via a critical examination of the work of the Russian mathematician Abram Ilyich Fet the Turkish-American physicist Asim Orhan Barut, before giving some final reflections. (shrink)

Triadic (systemical) logic can provide an interpretive paradigm for understanding how quantum indeterminacy is a consequence of the formal nature of light in relativity theory. This interpretive paradigm is coherent and constitutionally open to ethical and theological interests. -/- In this statement: -/- (1) Triadic logic refers to a formal pattern that describes systemic (collaborative) processes involving signs that mediate between interiority (individuation) and exteriority (generalized worldview or Umwelt). It is also called systemical logic or the logic of (...) relatives. The term "triadic logic" emphasizes that this logic involves mediation of dualities through an irreducibly triadic formalism. The term "systemical logic" emphasizes that this logic applies to systems in contrast to traditional binary logic which applies to classes. The term "logic of relatives" emphasizes that this logic is background independent (in the sense discussed by Smolin ). -/- (2) An interpretive paradigm refers to a way of thinking that generates an understanding through concepts, their inter-relationships and their connections with experience. -/- (3) Coherence refers to holistic integrity or continuity in the meaning of concepts that form an interpretation or understanding. -/- (4) Constitutionally open refers to an inherent dependence in principle of an interpretation or understanding on something outside of a specific discipline's discourse or domain of inquiry (epistemic system). Interpretations that are constitutionally open are incomplete in themselves and open to responsive, interdisciplinary discourse and collaborative learning. (shrink)

Relationships between current theories, and relationships between current theories and the sought theory of quantum gravity (QG), play an essential role in motivating the need for QG, aiding the search for QG, and defining what would count as QG. Correspondence is the broad class of inter-theory relationships intended to demonstrate the necessary compatibility of two theories whose domains of validity overlap, in the overlap regions. The variety of roles that correspondence plays in the search for QG are (...) illustrated, using examples from specific QG approaches. Reduction is argued to be a special case of correspondence, and to form part of the definition of QG. Finally, the appropriate account of emergence in the context of QG is presented, and compared to conceptions of emergence in the broader philosophy literature. It is argued that, while emergence is likely to hold between QG and general relativity, emergence is not part of the definition of QG, and nor can it serve usefully in the development and justification of the new theory. (shrink)

Arthur Kaufmann is one of the most prominent figures among the contemporary philosophers of law in German speaking countries. For many years he was a director of the Institute of Philosophy of Law and Computer Sciences for Law at the University in Munich. Presently, he is a retired professor of this university. Rare in the contemporary legal thought, Arthur Kaufmann's philosophy of law is one with the highest ambitions — it aspires to pinpoint the ultimate foundations of law by explicitly (...) proposing an ontology, a general theory of knowledge and concept of a person. Kaufmann's work derives, first of all, from the thinking of Gustav Radburch, his teacher, and then from ideas of Karl Engish and Hans-Georg Gadamer. The philosophy undertakes to pursue the ultimate foundation of law, law which is understood by Kaufmann, first of all, as a "concrete judgement" that is, what is right in a concrete situation. Justice belongs to the essence of law and "unjust law" is contradictio in adiectio. Kaufmann opposes all those theories, which as the only foundation for establishing just law (Recht) adopt legal norms (Gesetz). In Kaufmann's opinion , such theories are powerless in the face of all types of distortions of law rendered by political forces. He suggests that the basic phenomenon which needs to be explained and which cannot be disregarded by a philosopher of law is so-called "legal lawlessness" ("Gestzliches Unrecht"). "Legal lawlessness" which forms a part of life experience for the people of twentieth century totalitarian states. It proved "with the accuracy of scientific experiment" that the reality of law consists of something more than bare conformity with legal norms. The existence of lex corrupta indicates that law contains something "non-dispositive" which requires acknowledgment of both law-maker and judge. Kaufmann, accepting the convergent concept of truth and cognition, assumes that "non-dispositive" content, emerging as the conformity of a number of cognitive acts of different subjects (inter-subjective communicativeness and verifiability), indicates the presence of being in this cognition. The questions "What is law?" and "What are the principles of a just solution?" lead straight to the ontology of law, to the question about the ontological foundations of law. Kaufmann discerns the ontological foundations of law in the specifically understood "nature of things" and, ultimately, in a "person". He proposes a procedural theory of justice, founded on a "person". In my work, I undertake to reconstruct the train of thought which led Kaufmann to the recognition of a "person" as the ontological foundation of law. In the first part, the conception of philosophy adopted by Kaufmann, initial characteristics of law — of reality which is the subject of analysis, as well as, the requirements for proper philosophical explanation of law posed by Kaufmann are introduced. In the second, Kaufmann's reconstruction of the process of the realisation of law is presented. Next, the conception of analogy which Kaufmann uses when explaining law is analyzed. In the fourth part, Kaufmann's conception of ontological foundations of law is discussed. A critical analysis is carried out in which I demonstrate that the theory of the ontological foundation of law proposed by Kaufmann and the concept of a person included in it do not allow a satisfac¬tory explanation of the phenomenon of "legal lawlessness" and lead to a number of difficulties in the philosophical explanation of law. Finally, the perspectives of a proper formulation of the issue of the ontological foundations of law are drafted in the context of the analyzed theory. My interest is centered on the conception of philosophy adopted by Kaufmann, according to which the existence of the reality is inferred on the basis of a certain configuration of the content of consciousness, whereas at the point of departure of philosophy of law, the data to be explained is a certain process, which is, basically, a process of cognition, while the reality appears only as a condition for the possibility of the occurrence of the process. I wish to argue that the difficulties which appear in the explanation of law are a consequence of the assumed fundamental philosophical solu¬tions, which seem to be characteristic, though usually not assumed explicitly, in philoso¬phy and theory of law dominant at present in continental Europe. Thereby, I wish to show the significance of ontological and epistemological solutions to the possibility of a proper formulation of the problems posed by philosophy and theory of law. Kaufmann proclaims himself in favour of a philosophy which poses questions about the ultimate foundations of understanding of the reality. In epistemology, he assumes that answers to the questions "What is reality like?" and ultimately "What is real?" are inferred on the basis of uniformity of a cognitive acts of different subjects. Cognition of the reality is accomplished exclusively through the content of conceptual material. The two fundamental questions posed by philosophy of law are "What is just law?" and "How is the just law enacted?" The latter is a question about the process of achieving a solution to a concrete case. Since, in Kaufmann's opinion, law does not exist apart from the process of its realisation, an answer to the question about the manner of realisation of law is of fundamental significance to answering the question: "What is law?" and to the explanation of the question about the ontological grounding of law, which is, as well, the foundation of justice. The proper solution has to take into account the moment of "non-dispositive" content of law; its positiveness understood as the reality and, at the same time, it has to point to the principles of the historical transformation of the content. Law, in the primary meaning of the word, always pertains, in Kaufmann's opinion, to a concrete case. A legal norm is solely the "possibility" of law and the entirely real law is ipsa res iusta, that which is just in a given situation. Determination of what is just takes place in a certain type of process performed by a judge (or by man confronted with a choice). Kaufmann aims to reconstruct this process. A question about the ontological foundation of law is a question about the ontological foundations of this process. In the analyzed theory it is formulated as a question about the transcendental conditions, necessary for the possibility of the occurrence of the process: how the reality should be thought to make possible the reconstructed process of the realisation of law. Kaufmann rejects the model for finding a concrete solution based on simple subsump¬tion and proposes a model in which concrete law ensues, based on inference by analogy, through the process of "bringing to conformity" that which is normative with that which is factual. Kaufmann distinguishes three levels in the process of the realisation of law. On the highest level, there are the fundamental legal principles, on the second legal norms, on the third — concrete solutions. The fundamental principles of law are general inasmuch as they cannot be "applied" directly to concrete conditions of life, however, they play an important part in establishing norms. A judge encounters a concrete situation and a system of legal norms. A life situation and norms are situated on inherently different levels of factuality and normativeness. In order to acquire a definite law both a norm (system of norms) and a life situation (Lebenssachverhalt) should undergo a kind of "treatment" which would allow a mutual conformity to be brought to them. Legal norms and definite conditions of life come together in the process of analogical inference in which the "factual state" ("Tatbestand") — which represents a norm, and in the "state of things" ("Sachverhalt") — which represents a specific situation are constructed. A "factual state" is a sense interpreted from a norm with respect to specific conditions of life. The "state of things" is a sense constructed on the basis of concrete conditions of life with respect to norms (system of norms). Legal norms and concrete conditions of life meet in one common sense established during the process of realisation of law. Mutatis mutandis the same refers to the process of composition of legal norms: as the acquisition of concrete law consists in a mutual "synchronization" of norms and concrete conditions of life, so acquisition of legal norms consists of bringing to conformity fundamental principles and possible conditions of life. According to Kaufmann, both of these processes are based on inference through analogy. As this inference is the heart of these processes it is simultaneously a foundation finding just law and justice. How does Kaufmann understand such an inference? As the basis for all justice he assumes a specifically interpreted distributive justice grounded on proportionality. Equality of relations is required between life conditions and their normative qualification. Concrete conditions of life are ascribed normative qualification not through simple application of a general norm. More likely, when we look for a solution we go from one concrete normative qualified case to another, through already known "applications" of norms to a new "application". The relation between life conditions and their normative qualifica¬tion has to be proportional to other, earlier or possible (thought of) assignments of that which is factual to that which is normative. Law as a whole does not consist of a set of norms, but only of a unity of relations. Since law is a, based on proportion, relative unity of a norm and conditions of life, in order to explain law in philosophical manner, the question about ontological base of this unity has to be asked. What is it that makes the relation between a norm and conditions of life "non-dispositive"? What is the basis for such an interpretation of a norm and case which makes it possible to bring a norm and conditions of life into mutual "conformity"? This is a question about a third thing (next to norms and conditions of life), with respect to which the relative identity between a norm and conditions of life occurs, about the intermediary between that which is normative and that which is factual and which provides for the process of establishing of norms, as well as, finding solutions. It is the "sense" in which the idea of law or legal norm and conditions of life have to be identical to be brought to mutual "conformity". In Kaufmann's opinion such a sense is nothing else but the "nature of things" which determines the normative qualification of the reality. Since establishment of this "sense" appears to be "non-dispositive" and controlled inter-subjectively (namely, other subjects will reach a similar result) so, in conformity with the convergent concept of truth, the "nature of things" must be assigned a certain ontological status. According to Kaufmann this is a real relation which occurs between being and obligation, between the conditions of life and normative quality. However, it should be underlined that from the point of view of the analyzed system the "nature of things" is a correlate of constructed sense, a result of a construction which is based on the principle of consistent understanding of senses ("non-normative" and "normative") and is not a reality which is transcendent against the arrangement of senses. In Kaufmann's theory, inference from analogy appears to be a process of reshaping the concepts (senses) governed by tendency to understand the contents appearing in relations between that which is factual and that which is normative in a consistent way. The analogical structure of language (concepts) and recognition of being as composed of an essence and existence is an indispensable requirement for the possibility of the realisation of law, based on specifically understood inference from analogy. It is necessary to assume a moment of existence without content which ensures unity of cognition. Existence emerges thus as a condition of the possibility of cognition. According to Kaufmann, the "nature of things" is the heart of inference through analogy and the basis for establishment of finding of law. Inference from the "state of things" to a norm or from a norm to the "state of things" always means inference through the "nature of things". The "nature of things" is the proper medium of objective legal sense sought in every cognition of law. In Kaufmann's view, the question whether the "nature of things" is ultima ratio of interpretation of law or is only a means of supplement gaps in law or whether it is one of the sources of law, is posed wrongly. The "nature of things" serves neither to supplement the gaps nor is it a source of law as, for example, a legal norm may be. It is a certain kind of "catalyst" necessary in every act of making law and solving a concrete case. Owing to "nature of things" it is possible to bring to a mutual conformity the idea of law and possible conditions of life or legal norms and concrete conditions of life. In Kaufmann's conception the "nature of things" is not yet the ultimate basis for understanding the "non-dispositiveness" of law. The relation between obligation and being is determined in the process of the realisation of law. Both the process itself and that which is transformed in this process are given. A question about the ontological bases of "material" contents undergoing "treatment" in the process of the realisation of law and about being which is the basis of regularity of the occurrence of the process arises. Only this will allow an explanation that the result of the process is not optional. Thus, a question about reality to which law refers and about the subject realising the law has to be formed. To this, Kaufmann gives the following answer: that which is missing is man but not "empirical man" but man as a "person". A "person" understood as a set of relations between man and other people and things. A "person" is the intermediary between those things which are different — norm and case are brought to conformity. A "person" is that which is given and permanent in the process of the realisation of law. It determines the content of law, is "subject" of law; this aspect is described by Kaufmann as the "what" of the process of realisation of law. A "person" consists of precisely just these relations which undergo "treatment" in the process. On the other hand, a "person" is "a place" in which the processes of realisation of law occur, it is the "how" of normative discourse, a "person" is that which determines the procedure of the process, being "outside" of it. This aspect of a "person" is connected with the formal moment of law. A "person" being, at the same time, the "how" and the "what" of the process of the realisation of law, is also, to put it differently, a structural unity of relation and that which constitutes this relation (unity of relatio and relata). According to this approach a "person" is neither an object nor a subject. It exists only "in between". It is not substance. Law is the relation between being and obligation. That which is obligatory is connected with that which is general. That which is general does not exist on its own, it is not completely real. Accordingly, a "person" as such is also not real. It is relational, dynamic and historical. A "person" is not a state but an event. In Kaufmann's opinion, such a concept of a "person" helps to avoid the difficulties connected with the fungibility of law in classical legal positivism. A "person" is that which is given, which is not at free disposal and secures the moment of "non-dispositiveness" of law. Kaufmann concludes: "The idea (»nature«) of law is either the idea of a personal man or is nothing". Theory points at the structure of realising law and explains the process of adoption of general legal norms for a concrete situation. The analysis has shown however, that in this theory a satisfactory answer to the question about the ultimate foundations of law is not given. It seems that in the analyzed theory the understanding of human being takes place through understanding of law. What is good for man as a "person", what is just, what a "person" deserves may be determined only against the existing system of law. A "per¬son" adopted as a basis of law is the reality postulated in the analysis of the process of the realisation of law. It is a condition of possibility of this process ( explaining, on one hand, its unity and, on the other hand, the non-dispositive moments stated in this process). A "person" in the discussed theory is entirely defined by the structure of law, it can be nothing more than that which is given in law, what law refers to, what law is about. Being, which is a "person", is constituted by relations between people and objects, the relations which are based on fundamental links between norms and conditions of life established in a process of bringing them to conformity. It has to be assumed that man as a "person" is a subject of law only as far as realising law "treats" given senses according to their current configuration. The system of law is a starting point and it describes in content what man is as a "person". Moreover, being a "person" is the condition for entering legal relations. Consistently, Kaufmann writes that "empirical man" is not the subject of law, man is not "out of nature" a "person". People become "persons" due to the fact that they acknowledge each other as "persons" — acknowledging, at the same time, law. This acknowledgement is a con¬dition of existence, of the possibility of the occurrence of process of realisation of law and of constituting legal relations which ultimately constitute a "person". Kaufmann assumes, that law tends towards a moral aim: it may and must create an external freedom, without which the internal freedom to fulfil moral obligations cannot develop. However, this postulate is not based on the necessary structure of human being. From the point of view of his system, it is nothing more than only a condition for the possibility of the occurrence of the process of the realisation of law — lack of freedom would destroy the "how" of this process. Thus, the postulate to protect the freedom of personal acts has to be interpreted, in accordance with the analyzed theory, as a postulate, the fulfilment of which aims ultimately at the accomplishment of the very same process of realisation of law itself and not the realisation of a given man. Kaufmann considers a "person" to be an element which unites the system of law as a whole. Law is a structure of relations, which are interdependent and inter-contingent. Consequently, a "person" which is to form the ontological basis of law has to be entity consisting of all relations. Being also the "how" of the process of realisation of law, if a "person" is to warrant its unity, it has to be a common source for all procedures. Hence, a single "person" would constitute a subject of law. Man appears to be only a moment of a certain entirety, realisation of which should be an aim of his actions. Law, creating a "person" as an object and subject of law becomes a primary entity. In the analyzed theory, the basis for determination of aims which law sets to man is not the allocation of man-subject to something which improves him but rather, such relation is only just constituted by law. A question appears, why should aims set in law also be the aims of "empirical man"? Why is this "empirical man" to be punished in the name of a "person" understood in such a way? If, however, it is assumed that what is man is determined by a system which is superior to him, then man has to be understood only as a part of a whole and there are no grounds to prohibit istrumental treatment of man and so the road to all aspects of totalitarianism might be opened. A problem of the application of created theory to the reality arises, the reality which the theory pretends to explain. Ultimately in his theory Kaufmann does not give any systemic grounds for a radical questioning of the validity of any legal norms. Every new norm becomes an equal part of system of norms. It is only its interpretation and application to given conditions of life that may be disputable, however, this refers to all norms without exception. Cohesive inter-pretation of norms and applications is necessary and sufficient for the acquisition of just law. New norms have to be interpreted in the light of others, correspondingly, the other norms require reinterpretation in the light of the new ones. Contradiction in interpretation of a norm does not form a basis for questioning norms but may serve only to question the manner of their interpretation (understanding). Therefore, no grounds exist to assume any legal norm as criminal or unjust, and in consequence, to question any consistently realised system based on formally, properly established norms, as "legal lawlessness". As law and a "person" do not exist without the process of realisation of law, the role of legal safety becomes crucial as the condition for the possibility of the occurrence of the process of realisation of law. Denying legal safety would be tantamount to negation of law in general (also of moral law) as negation of safety takes away, at the same time, the basis for occurrence of the process of realisation of law. Moreover, any lack of legal safety would also mean lack of a basis for the existence of man as a "person". Kaufmann's thesis, that civil disobedience is legalized only when it has a chance to lead to success, consistent with his concept of the foundations of law, seems to point directly to conclusions which deny the facts taken under consideration and doubtlessly Kaufmann's own intentions, since it would have to be assumed that accordingly there are no grounds to question a legal system in force based on violence which secures its operation. Force finally seems to determine which one of the mutually irreconcilable normative systems constitute law and which does not. A legitimate position is one which leads to success, it is the weaker system which is negated. If so, then basically violent imposition of law is not an act directed against the law in force but, to the contrary, realisation of law. In the context of the new system the former system of law may be talked about as unjust solely in the sense of being incapable of being consistently united with the new. However, at the base, ultimately, lies force which reaffirms differences and excludes from the process of realisation of law certain norms and their interpretations. Kaufmann was aiming at grounding of that which is "non-dispositive" in a certain given framework of interpretation. Nevertheless, he does not provide foundations for the understanding of phenomena, which he undertakes to explain at a point of departure. Instead of explaining them the theory negates the possibility of their existence. The reality postulated in regard to "non-dispositive" moments of the reconstructed process of acquiring law consist of a specifically understood "person", which appears in Kaufmann's conceptions as a condition of the possibility of the realisation of law. According to this approach understanding of a "person" may be only a function of law. To understand "legal lawlessness" and foundations of justice it is necessary to look for such theory of law in which understanding of man as a "person" and being is not a function of understanding of law (in which a "person" is not only a condition for possibility of reconstructed process of realisation of law; for possibility of cognition processes). It seems necessary to start from theory of being and a "person" based on broader experience than the one assumed by Kaufmann and reconstruct the ontological foundations of the process of realisation of law only in such perspective. Kaufmann points out that that to which law refers is ipsa res iusta a concrete relation of man to other people and things. This relation, in his theory, appears to be basically only just constituted by law (normative senses "applied" to conditions of life). Therefore, understanding the relation between a given man and other people and things which constitute the aim of his actions, that is understanding of good, is enacted against the background of constitution of senses; constitution which is a result of a process aiming towards consistent understanding of particular contents (of nor¬mative and non-normative senses). "Being" is secondary towards constructed senses it is only their correlate. The primary relation consists of relation of a man to law (system of norms), while the secondary relation is one of man to something which is the aim of his action (relation between man and good). Considering such approach it is difficult to envision a satisfying answer to the fundamental question: why does law put concrete man under any obligation to obey it? The source of this problem can be seen in reduction of the base for understanding good to content of obligation formulated in auto-reflection. Such reduction seems to be a consequence of Kaufmann's adoption of "convergent concept of truth" and in con¬sequence his recognition of indirect, essentialistic grasp of reality formulated in concepts as the basic and only foundation of theory of being and of law. In view of such an approach, analogy of law, concepts and being is the condition for the possibility of the process of transformation of senses which aims at consistent interpretation of all law. Existence is postulated with respect to the possibility of unity of experience and cognition. However, also a different approach to understanding of the problem of being and good is possible. In spontaneous cognition being is affirmed, first of all, not as a certain, non-contradictory, determined content, but as something existing. Together with a cer¬tain content (passed indirectly through notions) existence of being is co-given. The basis for unity of being is not formed by the consistence of content, as it is in the case of the theories departing from the analysis of cognition processes, but by an act of existence realising content (essence). Such an approach makes it also possible to go beyond the convergent concept of truth. It is worth mentioning that allocation of an agent to good is realised not only by the content of duty. A statement that something is good is primary with respect to determination of this good in content. The recognised good always bears some content, however, there are no reasons to base the concept of good exclusively on indirect, formulated in concepts cognition. As primary, can be adopted the relation of man to good and not of man to law. Determination in content appears to be only an articulation of aspectual cognition of being, as an object of action. In such a case the basis for relative unity of norm and conditions of life is not the "nature of things" understood as correlate of sense but it is relation to good based on internal constitution of man as potential, not self-sufficient being. It does not mean, that the moments of the process of realisation of law singled out by Kaufmann are not important to determination of what is just. He, quite rightly, points to significant role played by norms in the evaluation of concrete situations, in man's search for closer specification in content of good innate to him. The structure of process of determining law for a concrete situation, to a great degree corresponds to the processes of determining law which take place not only in the legal sciences. Kaufmann's analyses of the process of realisation of law show the complexity of the structure of these processes and point towards important moments allowing a better understanding of law and man. Nevertheless, these analyses cannot be a basis for construction of philosophical theory of law, theory which hopes to point out the ultimate, ontological foundations for understanding law. Kaufmann's results may become fully valid only in a more general perspective including broader experience at the point of departure. (shrink)

In the quantum-Bayesian approach to quantum foundations, a quantum state is viewed as an expression of an agent’s personalist Bayesian degrees of belief, or probabilities, concerning the results of measurements. These probabilities obey the usual probability rules as required by Dutch-book coherence, but quantum mechanics imposes additional constraints upon them. In this paper, we explore the question of deriving the structure of quantum-state space from a set of assumptions in the spirit of quantum Bayesianism. (...) The starting point is the representation of quantum states induced by a symmetric informationally complete measurement or SIC. In this representation, the Born rule takes the form of a particularly simple modification of the law of total probability. We show how to derive key features of quantum-state space from (i) the requirement that the Born rule arises as a simple modification of the law of total probability and (ii) a limited number of additional assumptions of a strong Bayesian flavor. (shrink)

In the contemporary philosophy of set theory, discussion of new axioms that purport to resolve independence necessitates an explanation of how they come to be justified. Ordinarily, justification is divided into two broad kinds: intrinsic justification relates to how `intuitively plausible' an axiom is, whereas extrinsic justification supports an axiom by identifying certain `desirable' consequences. This paper puts pressure on how this distinction is formulated and construed. In particular, we argue that the distinction as often presented is neither well-demarcated (...) nor sufficiently precise. Instead, we suggest that the process of justification in set theory should not be thought of as neatly divisible in this way, but should rather be understood as a conceptually indivisible notion linked to the goal of explanation. (shrink)

(...) However, whether we chose a weak or strong approximation, the set would not make any sense at all, if (once more) this choice would not be justified in either temporal or spatial sense or given the context of possible applicability of the set in different circumstances. This would obviously represent a dualism in itself as we would (for instance) posit and apply a full identity-equality-equivalence of x and y when applying Newtonian physics to certain observations we make (it would (...) be the case of neural correlates), and we would posit and apply a non - identity-equality-equivalence of x and y when applying Quantum mechanics to other observations. Following this dualism in and of theories, the same sate would need to be slightly modified: U2:(x~y)∪(x≈y); U3:(x~y)∩(x≈y); U4a:(x~y)⊆(x≈y) vs. U4b:(x~y)⊇(x≈y). (shrink)

When this article was first planned, writing was going to be exclusively about two things - the origin of life and human evolution. But it turned out to be out of the question for the author to restrict himself to these biological and anthropological topics. A proper understanding of them required answering questions like “What is the nature of the universe – the home of life – and how did it originate?”, “How can time travel be removed from fantasy and (...) science fiction, to be made scientific and practical?”, and “How can the proposed young age of genus Homo be made to actually be reasonable – when simply stating it would be solid ground for instant rejection and dismissal?” The result is that the article also talks about subjects like Artificial Intelligence, General Relativity, and cosmology. -/- From where did life originate? God? Evolution? Panspermia? If the tendency of humans and scientists to regard undiscovered science as pseudoscience can be overcome, Einstein gave another alternative to consider when he introduced General Relativity. Time isn’t linear – progressing in a straight line from past to present to future. That assumption ignores Relativity which states that space AND TIME are curved. Where did life and the genetic code come from? Can the answer build AI? -/- The first question can be answered by the section of this article titled SETI, Evolution, and Time which says life (possibly multicellular and intelligent) and the genetic code came from humans acquiring knowledge of these things over the centuries, then applying that knowledge – via terraforming, accumulation of raw materials like amino acids and nucleic acids, genetic engineering - to a time in the past when life didn’t exist. From that origin, life evolved through innumerable mutations and adaptations, with humans once again acquiring knowledge of it in cyclic (nonlinear) time. -/- The second question is answered by saying artificial intelligence (AI) as the product of life is only half of the equation. The other half refers to Relativity’s curved space-time and violation of the notion that time always travels from past to future. We have always lived in an artificially intelligent, non-probabilistic universe where everything in time and space is connected into one thing by quantum entanglement – making the brain and genes products of binary-digit activity or artificial intelligence (life is not merely dependent on biology’s “lock and key” mechanisms but also possesses AI). -/- The earliest documented representative of the genus Homo is Homo habilis, which evolved around 2.8 million years ago. Scientists used to believe there was a straight line from H. habilis to us, Homo sapiens. This article will use the “advanced” waves loved by Physics Nobel laureate Richard Feynman, view the history of science through the lens of Conic Sections applied to Relativity’s curved space-time, and incorporate the necessity of so-called imaginary time * – popularized by Prof. Stephen Hawking. While the evolutionary proposals are more in agreement with this early straight line than with modern theories, Albert Einstein’s General Relativity is used to transform the straight line into a curved line, ultimately concluding that Homo habilis (H. habilis) originated only (and unbelievably, as far as today’s science and technology is concerned) ~250,000 years ago. Other branches and dead ends of Homo – e.g. Neanderthals – are the result of mutations and adaptations, with the resultant modifications to anatomy and physiology. The surprisingly young age of H. habilis allows nearly 200,000 years for habilis, or one of its descendants, to reach Australia … if this country’s indigenous Aboriginal population did, as claimed, reach this “island continent” 60,000 years ago. -/- * The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, is a failure of classical physics to predict observed phenomena: it can be shown that a blackbody - a hypothetical perfect absorber and radiator of energy - would release an infinite amount of energy, contradicting the principles of conservation of energy and indicating that a new model for the behaviour of blackbodies was needed. At the start of the 20th century, physicist Max Planck derived the correct solution by making some strange (for the time) assumptions. In particular, Planck assumed that electromagnetic radiation can only be emitted or absorbed in discrete packets, called quanta. Albert Einstein postulated that Planck's quanta were real physical particles (what we now call photons), not just a mathematical fiction. From there, Einstein developed his explanation of the photoelectric effect (when quanta or photons of light shine on certain metals, electrons are released and can form an electric current). So it appears entirely possible that another supposed mathematical trickery (imaginary time and the y-axis of Wick rotation) will find practical application in the future. -/- The article includes mathematical references to cosmology (spoiler alert – you’ll read about things like Vector-Tensor-Scalar Geometry, topology, the “eternal present”, Einstein’s Unified Field, the inverse-square law, and there being no Big Bang and no multiverse - but there will also be no equations). -/- The other subheadings in this essay are – -/- NONLINEAR TIME AND ELECTRICAL ENGINEERING (about a 2009 electrical engineering experiment at America’s Yale University, and cosmic wormholes) -/- BITS AND TOPOLOGY (base-2 maths aka Binary digiTS, Mobius strips, and figure-8 Klein bottles) -/- WICK ROTATION, CAUSALITY, AND UNITING TIME (do past, present and future co-exist in an “eternal present”?) -/- DIGITAL BRAIN, DIGITAL UNIVERSE (if the brain and the universe are ultimately composed of binary digits, we'll someday be able to do the same things with the brain and universe that we now do with computers) -/- PROPOSAL: HUMAN AND ANIMAL INSTINCTS ARE THE RESULT OF THE UNIVERSE BEING UNIFIED BY BINARY DIGITS (AND TOPOLOGY) (If everything in the universe is ultimately composed of electronic BITS, then the universe must possess Artificial Intelligence - some prefer the term Cosmic Consciousness) -/- INFORMATION THEORY CONQUERS A RED GIANT (preserving Earth by keeping the Sun near today’s level of activity forever). 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“Applying the Heisenberg Uncertainty Principle to 21st Century Art” was delivered to the 2009 Congress of the International Association of Art Critics (AICA) in Dublin as a guide to critical thinking through a paradigm shift. This new paper uncovers a new critical theory in the form of a formula that has been successfully applied to a universal appraisal of arts across all boundaries, whether they be gender, discipline or culture. The configuration predicted by Pauli as arising from under (...) the collapsed quantum wave is sourced in an ancient icon: the hieros gamos. As her functional role as newspaper critic shifted to a more esoteric passage as performance artist/curator/blogger, the author spent 14 years theorizing; and another 14 years applying a new critical theory to contemporary art. Her (R)evolution series for the Huffington Post places the containment of the hieros gamos at the forefront of contemporary art practice. The goal is the embodiment of Jean Gebser’s 360-degree aperspectival awareness, whereas the shadow is contacted and absorbed into the very body of art. This paper will reveal the strategy for determining holistic art forms containing the 21st century icon that Pauli understood would arise from under the collapsed quantum wave. (shrink)

Although Fuzzy logic and Fuzzy Mathematics is a widespread subject and there is a vast literature about it, yet the use of Fuzzy issues like Fuzzy sets and Fuzzy numbers was relatively rare in time concept. This could be seen in the Fuzzy time series. In addition, some attempts are done in fuzzing Turing Machines but seemingly there is no need to fuzzy time. Throughout this article, we try to change this picture and show why it is helpful to consider (...) the instants of time as Fuzzy numbers. In physics, though there are revolutionary ideas on the time concept like B theories in contrast to A theory also about central concepts like space, momentum… it is a long time that these concepts are changed, but time is considered classically in all well-known and established physics theories. Seemingly, we stick to the classical time concept in all fields of science and we have a vast inertia to change it. Our goal in this article is to provide some bases why it is rational and reasonable to change and modify this picture. Here, the central point is the modified version of “Unexpected Hanging” paradox as it is described in "Is classical Mathematics appropriate for theory of Computation".This modified version leads us to a contradiction and based on that it is presented there why some problems in Theory of Computation are not solved yet. To resolve the difficulties arising there, we have two choices. Either “choosing” a new type of Logic like “Para-consistent Logic” to tolerate contradiction or changing and improving the time concept and consequently to modify the “Turing Computational Model”. Throughout this paper, we select the second way for benefiting from saving some aspects of Classical Logic. In chapter 2, by applying quantum Mechanics and Schrodinger equation we compute the associated fuzzy number to time. (shrink)

The paper addresses Leon Hen.kin's proposition as a " lighthouse", which can elucidate a vast territory of knowledge uniformly: logic, set theory, information theory, and quantum mechanics: Two strategies 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 to be unified and thus reduced as a joint property of infinity and of (...) all infinite sets. However, only Henkin's proposition equivalent to an internal position to 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 introduce infinity implicitly by Hilbert space, on which is founded its formalism. One can demonstrate that some essential properties of quantum information, entanglement, and quantum computer originate directly from infinity once it is involved in quantum mechanics. Thus, these phenomena can be elucidated as both complete and incomplete, after which choice is the border between them. A special kind of invariance to the 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 entirely in the same terms only of set theory. Quantum computer can demonstrate especially clearly the privilege of the 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 contemporary knowledge may be synthesized from a single viewpoint. (shrink)

Students using Ritalin in preparation for their exams is a hotly debated issue, while meditating or drinking coffee before those same exams is deemed uncontroversial. However, taking Ritalin, meditating and drinking coffee or even education in general, can all be considered forms of cognitive enhancement. Although social acceptance might change in the future, it is interesting to examine the current reasons that are used to distinguish cases deemed problematic or unproblematic. Why are some forms of cognitive enhancement considered (...) problematic, while others are not? In this paper, we consider cognitive enhancement as the amplification or extension of core capacities of the mind, using augmentation or improvements of our information-processing systems. We will analyse cognitive enhancement in an educational setting in order to clarify the fuzzy distinction between problematic and unproblematic forms of cognitive enhancement. We will show that the apparent distinction made by many people between problematic and unproblematic enhancement is not based on any fundamental difference between these two categories. (shrink)

A resolution to the Russell Paradox is presented that is similar to Russell's “theory of types” method but is instead based on the definition of why a thing exists as described in previous work by this author. In that work, it was proposed that a thing exists if it is a grouping tying "stuff" together into a new unit whole. In tying stuff together, this grouping defines what is contained within the new existent entity. A corollary is that a (...) thing, such as a set, does not exist until after the stuff is tied together, or said another way, until what is contained within is completely defined. A second corollary is that after a grouping defining what is contained within is present and the thing exists, if one then alters what is tied together (e.g., alters what is contained within), the first existent entity is destroyed and a different existent entity is created. A third corollary is that a thing exists only where and when its grouping exists. Based on this, the Russell Paradox's set R of all sets that aren't members of themselves does not even exist until after the list of the elements it contains (e.g. the list of all sets that aren't members of themselves) is defined. Once this list of elements is completely defined, R then springs into existence. Therefore, because it doesn't exist until after its list of elements is defined, R obviously can't be in this list of elements and, thus, cannot be a member of itself; so, the paradox is resolved. This same type of reasoning is then applied to Godel's first Incompleteness Theorem. Briefly, while writing a Godel Sentence, one makes reference to a future, not yet completed and not yet existent sentence, G, that claims its unprovability. However, only once the sentence is finished does it become a new unit whole and existent entity called sentence G. If one then goes back in and replaces the reference to the future sentence with the future sentence itself, a totally different sentence, G1, is created. This new sentence G1 does not assert its unprovability. An objection might be that all the possibly infinite number of possible G-type sentences or their corresponding Godel numbers already exist somehow, so one doesn't have to worry about references to future sentences and springing into existence. But, if so, where do they exist? If they exist in a Platonic realm, where is this realm? If they exist pre-formed in the mind, this would seem to require a possibly infinite-sized brain to hold all these sentences. This is not the case. What does exist in the mind is the system for creating G-type sentences and their corresponding numbers. This mental system for making a G-type sentence is not the same as the G-type sentence itself just as an assembly line is not the same as a finished car. In conclusion, a new resolution of the Russell Paradox and some issues with proofs of Godel's First Incompleteness Theorem are described. (shrink)

Due to the wide scope of (in particular linear) homotopy type theory (using quantum natural language processing), a metatheory can be applied not just to theorizing the metatheory of scientific progress, but ordinary language or any public language defined by sociality/social agents as the precondition for the realizability of (general) intelligence via an inferential network from which judgement can be made. How this metatheory of science generalizes to public language is through the recent advances of quantum (...) natural language processing, but the traditional metalogical encodings (of Tarski) is relatively comparable through universes such as the type of types or type of types of types found in the inherent inferentialism of UF/HoTT via infinity-groupoids. The "computational trinity" of proofs=programs=algebra in HoTT also means reason is defined functionally as "what it does" by what it computes (proofs are programs). Following Reza Negarestani's recent book, through the self's self-realizability, achieving self-consciousness and consciousness beyond selfhood, "geistig manifestation" is achieved in the form of general intelligence, but only through an inferential network of social agents intrinsically encoded through computation. (shrink)

The previous Part I of the paper 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. (shrink)

Let f(1)=2, f(2)=4, and let f(n+1)=f(n)! for every integer n≥2. Edmund Landau's conjecture states that the set P(n^2+1) of primes of the form n^2+1 is infinite. Landau's conjecture implies the following unproven statement Φ: card(P(n^2+1))<ω ⇒ P(n^2+1)⊆[2,f(7)]. Let B denote the system of equations: {x_j!=x_k: i,k∈{1,...,9}}∪{x_i⋅x_j=x_k: i,j,k∈{1,...,9}}. The system of equations {x_1!=x_1, x_1 \cdot x_1=x_2, x_2!=x_3, x_3!=x_4, x_4!=x_5, x_5!=x_6, x_6!=x_7, x_7!=x_8, x_8!=x_9} has exactly two solutions in positive integers x_1,...,x_9, namely (1,...,1) and (f(1),...,f(9)). No known system S⊆B with a finite (...) number of solutions in positive integers x_1,...,x_9 has a solution (x_1,...,x_9)∈(N\{0})^9 satisfying max(x_1,...,x_9)>f(9). For every known system S⊆B, if the finiteness/infiniteness of the set {(x_1,...,x_9)∈(N\{0})^9: (x_1,...,x_9) solves S} is unknown, then the statement ∃ x_1,...,x_9∈N\{0} ((x_1,...,x_9) solves S)∧(max(x_1,...,x_9)>f(9)) remains unproven. Let Λ denote the statement: if the system of equations {x_2!=x_3, x_3!=x_4, x_5!=x_6, x_8!=x_9, x_1 \cdot x_1=x_2, x_3 \cdot x_5=x_6, x_4 \cdot x_8=x_9, x_5 \cdot x_7=x_8} has at most finitely many solutions in positive integers x_1,...,x_9, then each such solution (x_1,...,x_9) satisfies x_1,...,x_9≤f(9). The statement Λ is equivalent to the statement Φ. It heuristically justifies the statement Φ . This justification does not yield the finiteness/infiniteness of P(n^2+1). We present a new heuristic argument for the infiniteness of P(n^2+1), which is not based on the statement Φ. Algorithms always terminate. We explain the distinction between existing algorithms (i.e. algorithms whose existence is provable in ZFC) and known algorithms (i.e. algorithms whose definition is constructive and currently known). Assuming that the infiniteness of a set X⊆N is false or unproven, we define which elements of X are classified as known. No known set X⊆N satisfies Conditions (1)-(4) and is widely known in number theory or naturally defined, where this term has only informal meaning. *** (1) A known algorithm with no input returns an integer n satisfying card(X)<ω ⇒ X⊆(-∞,n]. (2) A known algorithm for every k∈N decides whether or not k∈X. (3) No known algorithm with no input returns the logical value of the statement card(X)=ω. (4) There are many elements of X and it is conjectured, though so far unproven, that X is infinite. (5) X is naturally defined. The infiniteness of X is false or unproven. X has the simplest definition among known sets Y⊆N with the same set of known elements. *** Conditions (2)-(5) hold for X=P(n^2+1). The statement Φ implies Condition (1) for X=P(n^2+1). The set X={n∈N: the interval [-1,n] contains more than 29.5+\frac{11!}{3n+1}⋅sin(n) primes of the form k!+1} satisfies Conditions (1)-(5) except the requirement that X is naturally defined. 501893∈X. Condition (1) holds with n=501893. card(X∩[0,501893])=159827. X∩[501894,∞)= {n∈N: the interval [-1,n] contains at least 30 primes of the form k!+1}. We present a table that shows satisfiable conjunctions of the form #(Condition 1) ∧ (Condition 2) ∧ #(Condition 3) ∧ (Condition 4) ∧ #(Condition 5), where # denotes the negation ¬ or the absence of any symbol. No set X⊆N will satisfy Conditions (1)-(4) forever, if for every algorithm with no input, at some future day, a computer will be able to execute this algorithm in 1 second or less. The physical limits of computation disprove this assumption. (shrink)

A definition of quantum computer is supposed: as a countable set of Turing machines on the ground of: quantum parallelism, reversibility, entanglement. Qubit is the set of all the i–th binary location cells transforming in parallel by unitary matrices. The Church thesis is suggested in the form relevat to quantum computer. The notion of the non–finite (but not infinite) potency of a set is introduced .

Instead of the half-century old foundational feud between set theory and category theory, this paper argues that they are theories about two different complementary types of universals. The set-theoretic antinomies forced naïve set theory to be reformulated using some iterative notion of a set so that a set would always have higher type or rank than its members. Then the universal u_{F}={x|F(x)} for a property F() could never be self-predicative in the sense of u_{F}∈u_{F}. But the mathematical (...) theory of categories, dating from the mid-twentieth century, includes a theory of always-self-predicative universals--which can be seen as forming the "other bookend" to the never-self-predicative universals of set theory. The self-predicative universals of category theory show that the problem in the antinomies was not self-predication per se, but negated self-predication. They also provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in Hegel and similar ideas of paradigmatic exemplars in ordinary thought. (shrink)

Suppose one has a system, the infinite set of positive integers, P, and one wants to study the characteristics of a subset (or subsystem) of that system, the infinite subset of odd positives, O, relative to the overall system. In mathematics, this is done by pairing off each odd with a positive, using a function such as O=2P+1. This puts the odds in a one-to-one correspondence with the positives, thereby, showing that the subset of odds and the original set of (...) positives are the same size, or have the same cardinality. This counter-intuitive result ignores the “natural” relationship of one odd for every two positives in the sequence of positive integers, which would suggest that O is one-half the size of P. However, in the set of axioms that constitute mathematics, it is considered valid. Fair enough. In the physical universe (i.e., the starting system), though, relationships between entities matter. In biochemistry, if you start with an organism, say a rat, and you want to study its heart, you can do this by removing some heart cells and studying them in isolation in a cell culture system. But, the results often differ compared to what occurs in the intact animal because studying the isolated cultured cells ignores the relationships in the intact body between the cells, the rest of the heart tissue and the rest of the rat. In chemistry, if a copper atom were studied in isolation, it would never be known that copper atoms in bulk can conduct electricity because the atoms share their electrons. In physics, the relationships between inertial reference frames in relativity, and observer and observed in quantum physics can't be ignored. Relationships matter in the physical world, but the mathematics of infinite sets is still used to describe it. Does this matter? It seems to, at least in physics. Infinities cause numerous problems in theoretical physics such as non-renormalizability and problems in unifying quantum mehanics and general relativity . This suggests that the pairing off method and the mathematics of infinite sets based on it are analogous to a cell culture system or studying a copper atom in isolation if they are used in studying the physical universe because they ignore the inherent relationships between entities. In the real, physical world, the natural, inherent, relationships between entities can't be ignored. Said another way, the set of axioms that constitute abstract mathematics may be similar but not identical to the set of physical axioms by which the real, physical universe runs. This suggests that the results from abstract mathematics about infinities may not apply to or should be modified for use in physics. (shrink)

In his late work (De venatione sapientiae), Cusanus unfolded basic ideas of his brilliant theology. After a long period, this ingenious teaching became clearly recognizable especially in our time. Forward with his face to the back, modern scientific theory adopts nowadays a course to which Cusanus had already pointed centuries ago. Modern thought revolves with unexpected precision and unexpected mysteriousness around two issues of his doctrine of wisdom: (i) The possibility-of-being-made is not a figment of the human brain by (...) which it organizes one's thoughts, but a fundamental and indispensable manifestation of reality. (ii) The possibility-of-being-made refers to something antecedent by which both the feasibility and the being-made get their common shape. This ultimate ground embodies the omnipotent oneness in the form of an infinite fund in which the cause of all reality and of all possibility is timelessly stored. Comparisons with the quantum ontology and the theory of quantum gravity impose themselves. (shrink)

The cognition of quantum processes raises a series of questions about ordering and information connecting the states of one and the same system before and after measurement: Quantum measurement, quantum in-variance and the non-locality of quantum information are considered in the paper from an epistemological viewpoint. The adequate generalization of ‘measurement’ is discussed to involve the discrepancy, due to the fundamental Planck constant, between any quantum coherent state and its statistical representation as a statistical ensemble (...) after measurement. Quantum in-variance designates the relation of any quantum coherent state to the corresponding statistical ensemble of measured results. A set-theory corollary is the curious in-variance to the axiom of choice: Any coherent state excludes any well-ordering and thus excludes also the axiom of choice. However the above equivalence requires it to be equated to a well-ordered set after measurement and thus requires the axiom of choice for it to be able to be obtained. Quantum in-variance underlies quantum information and reveals it as the relation of an unordered quantum “much” (i.e. a coherent state) and a well-ordered “many” of the measured results (i.e. a statistical ensemble). It opens up to a new horizon, in which all physical processes and phenomena can be interpreted as quantum computations realizing relevant operations and algorithms on quantum information. All phenomena of entanglement can be described in terms of the so defined quantum information. Quantum in-variance elucidates the link between general relativity and quantum mechanics and thus, the problem of quantum gravity. The non-locality of quantum information unifies the exact position of any space-time point of a smooth trajectory and the common possibility of all space-time points due to a quantum leap. This is deduced from quantum in-variance. Epistemology involves the relation of ordering and thus a generalized kind of information, quantum one, to explain the special features of the cognition in quantum mechanics. (shrink)