We study the general problem of axiomatizing structures in the framework of modal logic and present a uniform method for complete axiomatization of the modal logics determined by a large family of classes of structures of any signature.

We study the general problem of axiomatizing structures in the framework of modal logic and present a uniform method for complete axiomatization of the modal logics determined by a large family of classes of structures of any signature.

Multialgebras have been much studied in mathematics and in computer science. In 2016 Carnielli and Coniglio introduced a class of multialgebras called swap structures, as a semantic framework for dealing with several Logics of Formal Inconsistency that cannot be semantically characterized by a single finite matrix. In particular, these LFIs are not algebraizable by the standard tools of abstract algebraic logic. In this paper, the first steps towards a theory of non-deterministic algebraization of logics by swap structures are given. Specifically, (...) a formal study of swap structures for LFIs is developed, by adapting concepts of universal algebra to multialgebras in a suitable way. A decomposition theorem similar to Birkhoff’s representation theorem is obtained for each class of swap structures. Moreover, when applied to the 3-valued algebraizable logics J3 and Ciore, their classes of algebraic models are retrieved, and the swap structures semantics become twist structures semantics. This fact, together with the existence of a functor from the category of Boolean algebras to the category of swap structures for each LFI, suggests that swap structures can be seen as non-deterministic twist structures. This opens new avenues for dealing with non-algebraizable logics by the more general methodology of multialgebraic semantics. (shrink)

In the recent years, problem-solving become a central topic that discussed by educators or researchers in mathematics education. it’s not only as the ability or as a method of teaching. but also, it is a little in reviewing about the components of the support to succeed in problem-solving, such as student's belief and attitude towards mathematics, algebraic thinking skills, resources and teaching materials. In this paper, examines the algebraic thinking skills as a foundation for problem-solving, and learning cycle as a (...) breath of continuous learning. In this paper, learning cycle to be used is a modified type of 5E based on beliefs. (shrink)

The classification of algebraic surfaces by the Italian School of algebraic geometry is universally recognized as a breakthrough in 20th-century mathematics. The methods by which it was achieved do not, however, meet the modern standard of rigor and therefore appear dubious from a contemporary viewpoint. In this article, we offer a glimpse into the mathematical practice of the three leading exponents of the Italian School of algebraic geometry: Castelnuovo, Enriques, and Severi. We then bring into focus their distinctive conception of (...) rigor and intuition. Unlike what is often assumed today, from their perspective, rigor is neither opposed to intuition nor understood as a unitary phenomenon – Enriques distinguishes between small-scale rigor and large-scale rigor and Severi between formal rigor and substantial rigor. Finally, we turn to the notion of mathematical objectivity. We draw from our case study in order to advance a multi-dimensional analysis of objectivity. Specifically, we suggest that various types of rigor may be associated with different conceptions of objectivity: namely, objectivity as faithfulness to facts and objectivity as intersubjectivity. (shrink)

Any logic is represented as a certain collection of well-orderings admitting or not some algebraic structure such as a generalized lattice. Then universal logic should refer to the class of all subclasses of all well-orderings. One can construct a mapping between Hilbert space and the class of all logics. Thus there exists a correspondence between universal logic and the world if the latter is considered a collection of wave functions, as which the points in Hilbert space can be (...) interpreted. The correspondence can be further extended to the foundation of mathematics by set theory and arithmetic, and thus to all mathematics. (shrink)

An endomorphism on an algebra \ is said to be strong if it is compatible with every congruence on \; and \ is said to have the strong endomorphism kernel property if every congruence on \, other than the universal congruence, is the kernel of a strong endomorphism on \. Here we characterise the structure of those double MS-algebras that have this property by the way of Priestley duality.

On a very natural conception of sets, every set has an absolute complement. The ordinary cumulative hierarchy dismisses this idea outright. But we can rectify this, whilst retaining classical logic. Indeed, we can develop a boolean algebra of sets arranged in well-ordered levels. I show this by presenting Boolean Level Theory, which fuses ordinary Level Theory (from Part 1) with ideas due to Thomas Forster, Alonzo Church, and Urs Oswald. BLT neatly implement Conway’s games and surreal numbers; and a (...) natural extension of BLT is definitionally equivalent with ZF. (shrink)

This paper aims to clarify Merleau-Ponty’s contribution to an embodied-enactive account of mathematical cognition. I first identify the main points of interest in the current discussions of embodied higher cognition and explain how they relate to Merleau-Ponty and his sources, in particular Husserl’s late works. Subsequently, I explain these convergences in greater detail by more specifically discussing the domains of geometry and algebra and by clarifying the role of gestalt psychology in Merleau-Ponty’s account. Beyond that, I explain how, for (...) Merleau-Ponty, mathematical cognition requires not only the presence and actual manipulation of some concrete perceptible symbols but, more strongly, how it is fundamentally linked to the structural transformation of the concrete configurations of symbolic systems to which these symbols appertain. Furthemore, I fill a gap in the literature by explaining Merleau-Ponty’s claim that these structural transformations are operated through motor intentionality. This makes it possible, in turn, to contrast Merleau-Ponty’s approach to ontologically idealistic and realistic views on mathematical objects. On Merleau-Ponty’s account, mathematical objects are relational entities, that is, gestalts that necessarily imply situated cognizers to whom they afford a specific type of engagement in the world and on whom they depend in their eventual structural transformations. I argue that, by attributing a strongly constitutive role to phenomenal configurations and their motor transformation in mathematical thinking, Merleau-Ponty contributes to clarifying the worldly, historical, and socio-cultural aspects of mathematical truths without compromising what we perceive as their universality, certainty, and necessity. (shrink)

Various changes and advances have taken place in the ways of teaching and learning. With the encountered global pandemic crisis, flexible learning has been practiced particularly in state universities and colleges. This study sought to determine the level of procedural fluency in College Algebra as well as the difficulties and interventions applied by tertiary students of Cotabato State University (CSU) in the flexible learning set-up. The period covered by the study was during the first semester of the school year (...) 2021-2022. The respondents were a sample of 30 first-year mathematics major students of CSU. It used mixed methods; the quantitative approach was utilized to know the students’ knowledge of procedures in solving mathematical problems while the qualitative approach was utilized to explore the difficulties of the students in learning mathematics as well as the coping strategies that can serve as basis for mathematics intervention program. The findings revealed that most of the respondents have a low level of procedural fluency in College Algebra, particularly emerging in the understanding the problem, devising a strategy, and solving steps while inept in the problem-solving checking process. On the part of new normal learning, difficulties were experienced such as poor knowledge of the prerequisite skills and unstable internet connection. As part of their interventions, facing the encountered problems, practicing repeatedly, and watching online video tutorials were applied. Based on the findings of the study, it is concluded that the first-year students have low level of procedural fluency in College Algebra problem-solving particularly in the Polya’s principles of understanding the problem, devising a plan, carrying out the plan, and checking the answer. Because of the learning challenges encountered by the students, it is recommended to develop step-by-step video tutorials that can address their problem-solving difficulties and supplement their learning in the flexible mode of delivery. (shrink)

In this paper I apply the concept of _inter-Model Inconsistency in Set Theory_ (MIST), introduced by Carolin Antos (this volume), to select positions in the current universe-multiverse debate in philosophy of set theory: I reinterpret H. Woodin’s _Ultimate L_, J. D. Hamkins’ multiverse, S.-D. Friedman’s hyperuniverse and the algebraic multiverse as normative strategies to deal with the situation of de facto inconsistency toleration in set theory as described by MIST. In particular, my aim is to situate these positions on the (...) spectrum from inconsistency avoidance to inconsistency toleration. By doing so, I connect a debate in philosophy of set theory with a debate in philosophy of science about the role of inconsistencies in the natural sciences. While there are important differences, like the lack of threatening explosive inferences, I show how specific philosophical positions in the philosophy of set theory can be interpreted as reactions to a state of inconsistency similar to analogous reactions studied in the philosophy of science literature. My hope is that this transfer operation from philosophy of science to mathematics sheds a new light on the current discussion in philosophy of set theory; and that it can help to bring philosophy of mathematics and philosophy of science closer together. (shrink)

Consider the set of inferences that are acceptable to use in all our theory building endeavors. Call this set of inferences the universal theory building toolkit, or just ’the toolkit’ for short. It is clear that the toolkit is tightly connected to logic in a variety of ways. Beall, for example, has argued that logic just is the toolkit. This paper avoids making a stand on that issue and instead investigates reasons for thinking that, logic or not, the toolkit (...) is substructural. It is presented as a dialogue for the simple reason that it summarizes a range of dialogues on this subject that the author has had with various folks over the past few years. The method I use to investigate the toolkit is inspired in both philosophical and technical details by Alasdair Urquhart’s work on semantics for relevance logics from the early 1970s. (shrink)

The ability to solve problems is a prerequisite in preparing mathematics preservice teachers. This study assessed preservice teachers’ problem-solving difficulties and performance, particularly in worded problems on number sense, measurement, geometry, algebra, and probability. Also, academic profile differences in the preservice teacher’s problem-solving performance and common errors were determined. A descriptive-comparative research design was employed with 158 random respondents. Data were gathered face-to-face during the first semester of the school year 2022-2023, and data were analyzed with the aid of (...) jamovi software, ensuring ethical measures. Overall findings revealed that the preservice teachers experienced average difficulty in solving problems. The low performance of the preservice teachers on the given problems was also demonstrated. Further analysis revealed a significant difference between the preservice teachers’ problem-solving performance based on their subject preference and program. Moreover, the error analysis revealed that the preservice teachers incurred comprehension errors in misrepresentation, misinterpretation, and miscalculation. These results will serve as a measure for policymakers and curriculum developers of the teacher education institution concerned to make relevant enhancements to the math courses offered in the elementary and secondary education programs. (shrink)

Như thế, hôm nay chính là ngày đầu tiên bước sang tuổi lên 7 của ISR, mặc dù nếu đếm năm, thì đã sang năm thứ 7 kể từ 1-1-2023 rồi.

Transnational, global trades, investments, and travels, amongst other drivers of globalization, helps to reverberate the deadly coronavirus pandemic from Wuhan, China, across the world like whirl fire. In order to contain the infectious spread of the pandemic, and mitigate its negative effects on macro-economic variables, the World Health Organization, (WHO) designed Covid-19 protocols that are being enforced by governments and people of the world. Based on the above account, the study examined the Second wave effect of Covid’19 pandemic on transportation (...) business: Keke-napep and Motorcycle transport systems in Asaba metropolis, Nigeria. The main objective is to empirically determine the real impact of the pandemic on rural land transport business. The study employed structured questionnaire and interview methods to generate data from the 250 respondents selected using the census sampling technique. Out of the 250 sets of questionnaires administered, 235 (94%) were retrieved while 15 (6%) were rejected. The statistical tools used for analyzing data were correlation and multiple regression techniques. Findings were that covid’19 pandemic negatively affected rural land transport business. The study concludes that, the coronavirus pandemic caused a contraction in transport business. It is therefore recommended that, there is a need for government to relax the stringent covid’19 protocols to allow economic activities to resume again. Campaigns on safety and hygiene should be intensified so that the populace will imbibe the culture of general cleanliness and personal hygiene, amongst others. (shrink)

Scientific journal presented by Bogdan Khmelnitsky Melitopol State Pedagogical University, Ukraine, Melitopol. Main points: 1. Actual Problems of Modern Philosophy 2. Researches in Philosophy connected with natural components, sociological aspects and self - identity development.

In abstract algebraic logic, many systems, such as those paraconsistent logics taking inspiration from da Costa's hierarchy, are not algebraizable by even the broadest standard methodologies, as that of Blok and Pigozzi. However, these logics can be semantically characterized by means of non-deterministic algebraic structures such as Nmatrices, RNmatrices and swap structures. These structures are based on multialgebras, which generalize algebras by allowing the result of an operation to assume a non-empty set of values. This leads to an interest in (...) exploring the foundations of multialgebras applied to the study of logic systems. -/- It is well known from universal algebra that, for every signature Sigma, there exist algebras over Sigma which are absolutely free, meaning that they do not satisfy any identities or, alternatively, satisfy the universal mapping property for the class of Sigma-algebras. Furthermore, once we fix a cardinality of the generating set, they are, up to isomorphisms, unique, and equal to algebras of terms (or propositional formulas, in the context of logic). Equivalently, the forgetful functor, from the category of Sigma-algebras to Set, has a left adjoint. This result does not extend to multialgebras. Not only multialgebras satisfying the universal mapping property do not exist, but the forgetful functor U, from the category of Sigma-multialgebras to Set, does not have a left adjoint. -/- In this paper we generalize, in a natural way, algebras of terms to multialgebras of terms, whose family of submultialgebras enjoys many properties of the former. One example is that, to every pair consisting of a function, from a submultialgebra of a multialgebra of terms to another multialgebra, and a collection of choices (which selects how a homomorphism approaches indeterminacies), there corresponds a unique homomorphism, what resembles the universal mapping property. Another example is that the multialgebras of terms are generated by a set that may be viewed as a strong basis, which we call the ground of the multialgebra. Submultialgebras of multialgebras of terms are what we call weakly free multialgebras. Finally, with these definitions at hand, we offer a simple proof that multialgebras with the universal mapping property for the class of all multialgebras do not exist and that U does not have a left adjoint. (shrink)

What is cognition? Equivalently, what is cognition good for? Or, what is it that would not be but for human cognition? But for human cognition, there would not be science. Based on this kinship between individual cognition and collective science, here we put forward Isbell conjugacy---the adjointness between objective geometry and subjective algebra---as a scientific method for developing cognitive science. We begin with the correspondence between categorical perception and category theory. Next, we show how the Gestalt maxim is subsumed (...) by the mathematical construct of colimit, a generalization of summation. The universal mapping property definitions of mathematical constructs, by virtue of being the best with respect to the universe of discourse, can be learned using reinforcement learning algorithms, which raises the possibility of abstracting the architecture of mathematics by artificial intelligence. Subsequently, we present naturality (to be contrasted with miracles), understood as 'Becoming consistent with Being', which governs the transformations of both things and their theories, as the zeroth law of change. Furthermore, the contrast---physical [mechanism] vs. biological [organism]---is smoothed via natural transformation, wherein transformations are respectful of the cohesion of the objects transformed. In closing, upon recognizing the scientific value of learning difficult-to-master differential calculus by physicists, of learning a strange four-letter language by biologists, and of learning the grammar of our respective mother tongues, we make a case for learning the theory of naturality / category theory for developing cognitive science. (shrink)

John Corcoran. 1979 Review of Hintikka and Remes. The Method of Analysis (Reidel, 1974). Mathematical Reviews 58 3202 #21388. -/- The “method of analysis” is a technique used by ancient Greek mathematicians (and perhaps by Descartes, Newton, and others) in connection with discovery of proofs of difficult theorems and in connection with discovery of constructions of elusive geometric figures. Although this method was originally applied in geometry, its later application to number played an important role in the early development of (...) algebra [Jacob Klein, English translation, Greek mathematical thought and the origin of algebra, especially pp. 154–157, M.I.T. Press, Cambridge, Mass., 1968]. -/- It is universally agreed that the method of analysis begins by “assuming the thing sought after” (e.g., in geometry, the truth of the proposition to be proved or the existence of the geometric figure to be constructed). Aside from this, little else can be taken for granted. There is disagreement concerning the “direction of analysis”, i.e. whether one is to seek implications of the assumption or whether one is to seek implicants of it. There is also disagreement concerning what is to be “anatomized” (analyzed), i.e., whether one analyzes mathematical objects (figures), mathematical propositions (the axioms, known theorems, and analytic assumption) or an imagined proof (of the analytic assumption from axioms and known theorems). (shrink)

The paper discusses Hilbert mathematics, a kind of Pythagorean mathematics, to which the physical world is a particular case. The parameter of the “distance between finiteness and infinity” is crucial. Any nonzero finite value of it features the particular case in the frameworks of Hilbert mathematics where the physical world appears “ex nihilo” by virtue of an only mathematical necessity or quantum information conservation physically. One does not need the mythical Big Bang which serves to concentrate all the violations of (...) energy conservation in a “safe”, maximally remote point in the alleged “beginning of the universe”. On the contrary, an omnipresent and omnitemporal medium obeying quantum information conservation rather than energy conservation permanently generates action and thus the physical world. The utilization of that creation “ex nihilo” is accessible to humankind, at least theoretically, as long as one observes the physical laws, which admit it in their new and wider generalization. One can oppose Hilbert mathematics to Gödel mathematics, which can be identified as all the standard mathematics until now featureable by the Gödel dichotomy of arithmetic to set theory: and then, “dialectic”, “intuitionistic”, and “Gödelian” mathematics within the former, according to a negative, positive, or zero value of the distance between finiteness and infinity. A mapping of Hilbert mathematics into pseudo-Riemannian space corresponds, therefore allowing for gravitation to be interpreted purely mathematically and ontologically in a Pythagorean sense. Information and quantum information can be involved in the foundations of mathematics and linked to the axiom of choice or alternatively, to the field of all rational numbers, from which the pair of both dual and anti-isometric Peano arithmetics featuring Hilbert arithmetic are immediately inferable. Noether’s theorems (1918) imply quantum information conservation as the maximally possible generalization of the pair of the conservation of a physical quantity and the corresponding Lie group of its conjugate. Hilbert mathematics can be interpreted from their viewpoint also after an algebraic generalization of them. Following the ideas of Noether’s theorem (1918), locality and nonlocality can be realized both physically and mathematically. The “light phase of the universe” can be linked to the gap of mathematics and physics in the Cartesian organization of cognition in Modernity and then opposed to its “dark phase”, in which physics and mathematics are merged. All physical quantities can be deduced from only mathematical premises by the mediation of the most fundamental physical constants such as the speed of light in a vacuum, the Planck and gravitational constants once they have been interpreted by the relation of locality and nonlocality. (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)

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)

Multiverse Views in set theory advocate the claim that there are many universes of sets, no-one of which is canonical, and have risen to prominence over the last few years. One motivating factor is that such positions are often argued to account very elegantly for technical practice. While there is much discussion of the technical aspects of these views, in this paper I analyse a radical form of Multiversism on largely philosophical grounds. Of particular importance will be an account of (...) reference on the Multiversist conception, and the relativism that it implies. I argue that analysis of this central issue in the Philosophy of Mathematics indicates that Radical Multiversism must be algebraic, and cannot be viewed as an attempt to provide an account of reference without a softening of the position. (shrink)

This article is a preliminary draft for initiating and commencing a new pioneer dimension of expression. To deal with higher-dimensional data or information flowing in this modern era of information technology and artificial intelligence, some innovative super algebraic structures are essential to be formulated. In this paper, we have introduced such matrices that have multiple layers and clusters of layers to portray multi-dimensional data or massively dispersed information of the plithogenic universe made up of numerous subjects their attributes, and sub-attributes. (...) For grasping that field of parallel information, events, and realities flowing from the micro to the macro level of universes, we have constructed hypersoft and hyper-super-soft matrices in a Plithogenic Fuzzy environment. These Matrices classify the non-physical attributes by accumulating the physical subjects and further sort the physical subjects by accumulating their non-physical attributes. We presented themasPlithogenicAttributiveSubjectivelyWholeHyper-Super-Soft-Matrix(PASWHSS-Matrix)andPlithogenic Subjective Attributively Whole-Hyper-Super-Soft-Matrix (PSAWHSS-Matrix). Several types of views and level-layers of these matrices are described. In addition, some local aggregation operators for Plithogenic Fuzzy Hypersoft Set (PPFHS-Set) are developed. Finally, few applications of these matrices and operators are used as numerical examples of COVID-19 data structures. (shrink)

Modern categorical logic as well as the Kripke and topological models of intuitionistic logic suggest that the interpretation of ordinary “propositional” logic should in general be the logic of subsets of a given universe set. Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms—which is reflected in the duality between quotient objects and subobjects throughout algebra. If “propositional” logic is thus seen as the logic of subsets (...) of a universe set, then the question naturally arises of a dual logic of partitions on a universe set. This paper is an introduction to that logic of partitions dual to classical subset logic. The paper goes from basic concepts up through the correctness and completeness theorems for a tableau system of partition logic. (shrink)

An inquiry on the training needs in Mathematics was conducted to Laura Vicuña Center - Palawan (LVC-P) learners. Specifically, this aimed to determine their level of performance in numbers, measurement, geometry, algebra, and statistics, identify the difficulties they encountered in solving word problems and enumerate topics where they needed coaching. -/- To identify specific training needs, the study employed a descriptive research design where 36 participants were sampled purposively. The data were gathered through a problem set test and focus (...) group discussion. Findings revealed that LVC-P learners had an unsatisfactory performance in numbers, measurement, and statistics while alarmingly poor in geometry and algebra. They also faced difficulties in remembering, understanding, applying, and analyzing mathematical concepts when solving problems. Further, the learners were able to name certain topics subjected to tutorials. -/- The above facts and observations suggest that learners of LVC-P have urgent training needs in Mathematics. It is recommended that the Western Philippines University - College of Education RDE Unit continue its research, development, and extension services to LVC-P. More importantly, the results of this inquiry regarding the training needs of the learners will serve as bases for conducting an extension project and development program for LVC-P. Series of tutorial sessions is deemed necessary to address the needs and difficulties of the learners. (shrink)

Gravitation is interpreted to be an “ontomathematical” force or interaction rather than an only physical one. That approach restores Newton’s original design of universal gravitation in the framework of “The Mathematical Principles of Natural Philosophy”, which allows for Einstein’s special and general relativity to be also reinterpreted ontomathematically. The entanglement theory of quantum gravitation is inherently involved also ontomathematically by virtue of the consideration of the qubit Hilbert space after entanglement as the Fourier counterpart of pseudo-Riemannian space. Gravitation can (...) be also interpreted as purely mathematical or logical “force” or “interaction” as a corollary from its ontomathematical (rather than physical) realization. The ontomathematical approach to gravitation is implicit in general relativity equating it to operators in pseudo-Riemannian space obeying the Einstein field equation and also well-known by the “geometrization of physics”. Quantum mechanics shares the same by the separable complex Hilbert space and defining “physical quantity” by the Hermitian operators on it. One can interpret special Minkowski space involved by special relativity and the qubit Hilbert space of quantum information as Fourier counterparts immediately noticing that general relativity means gravitation as the Fourier counterpart of non-Hermitian operators implying non-unitarity and the violation of energy conservation and thus destroying Pauli’s particle paradigm. Since the Standard model obeys it, this explains the impossibility of “quantum gravitation” in any framework conservatively generalizing the Standard model so that it would include gravitation along with electromagnetic, weak, and strong interactions. Einstein’s geometrization of gravitation can be continued into a purely mathematical theory of it following Euclid’s realization for geometry to be exhaustively built in a deductive and axiomatic way as well as Riemann’s parametrization of all the class of Euclidean and non-Euclidean geometries by “space curvature”, then being generalized to Minkowski space as the operators on pseudo-Riemannian space as the Einstein field equation means gravitation. The transition from mathematical gravitation to logical one can rely on the historical lesson of the pair of Lobachevski’s and Riemann’s approaches now “reversely”, i.e., from the latter to the former. Logical gravitation is linkable to Hegel’s dialectical logic and ontological dialectics abandoning their interpretations as a new zero logic substituting classical propionyl logic. The approach of ontomathematics generalizing that of ontology, traceable even to Aristotle’s reformation of Plato’s doctrine, needs Hegel’s doctrine to be formalized as a first-order logic naturally containing Boolean algebra, isomorphic to both classical propositional logic and set theory being the class of all first-order logics, as a sub-logic along with Peano arithmetic as another sub-logic. The first-order logic at issue is called Hilbert arithmetic and elaborated in detail in other papers. It allows for both self-foundation of mathematics to be internally proved as complete and furthermore, quantum mechanics reinterpreted as quantum information to be included by the qubit Hilbert space interpretable in turn as a dual and physical counterpart of Hilbert arithmetic in a narrow sense, that is, both counterparts constitute Hilbert arithmetic in a wide sense, being mathematical and physical simultaneously and thus overcoming the Cartesian dualism of “body” gapped from “mind” by an abyss. Then, the proper philosophical interpretation of gravitation to be the fundamental ontomathematical force or interaction overcomes the ridiculous belief of the Big Bang wrongly alleged to be a scientific theory. Ontomathematical gravitation suggests an omnipresent and omnitemporal medium of “God’s” creation “ex nihilo” following only the natural necessity of quantum-information conservation particularly and locally manifested as energy conservation. (shrink)

A graph-theoretic account of fibring of logics is developed, capitalizing on the interleaving characteristics of fibring at the linguistic, semantic and proof levels. Fibring of two signatures is seen as a multi-graph (m-graph) where the nodes and the m-edges include the sorts and the constructors of the signatures at hand. Fibring of two models is a multi-graph (m-graph) where the nodes and the m-edges are the values and the operations in the models, respectively. Fibring of two deductive systems is an (...) m-graph whose nodes are language expressions and the m-edges represent the inference rules of the two original systems. The sobriety of the approach is confirmed by proving that all the fibring notions are universal constructions. This graph-theoretic view is general enough to accommodate very different fibrings of propositional based logics encompassing logics with non-deterministic semantics, logics with an algebraic semantics, logics with partial semantics and substructural logics, among others. Soundness and weak completeness are proved to be preserved under very general conditions. Strong completeness is also shown to be preserved under tighter conditions. In this setting, the collapsing problem appearing in several combinations of logic systems can be avoided. (shrink)

A possible world is a junky world if and only if each thing in it is a proper part. The possibility of junky worlds contradicts the principle of general fusion. Bohn (2009) argues for the possibility of junky worlds, Watson (2010) suggests that Bohn‘s arguments are flawed. This paper shows that the arguments of both authors leave much to be desired. First, relying on the classical results of Cantor, Zermelo, Fraenkel, and von Neumann, this paper proves the possibility of junky (...) worlds for certain weak set theories. Second, the paradox of Burali-Forti shows that according to the Zermelo-Fraenkel set theory ZF, junky worlds are possible. Finally, it is shown that set theories are not the only sources for designing plausible models of junky worlds: Topology (and possibly other "algebraic" mathematical theories) may be used to construct models of junky worlds. In sum, junkyness is a relatively widespread feature among possible worlds. (shrink)

Non-commuting quantities and hidden parameters – Wave-corpuscular dualism and hidden parameters – Local or nonlocal hidden parameters – Phase space in quantum mechanics – Weyl, Wigner, and Moyal – Von Neumann’s theorem about the absence of hidden parameters in quantum mechanics and Hermann – Bell’s objection – Quantum-mechanical and mathematical incommeasurability – Kochen – Specker’s idea about their equivalence – The notion of partial algebra – Embeddability of a qubit into a bit – Quantum computer is not Turing machine (...) – Is continuality universal? – Diffeomorphism and velocity – Einstein’s general principle of relativity – „Mach’s principle“ – The Skolemian relativity of the discrete and the continuous – The counterexample in § 6 of their paper – About the classical tautology which is untrue being replaced by the statements about commeasurable quantum-mechanical quantities – Logical hidden parameters – The undecidability of the hypothesis about hidden parameters – Wigner’s work and и Weyl’s previous one – Lie groups, representations, and psi-function – From a qualitative to a quantitative expression of relativity − psi-function, or the discrete by the random – Bartlett’s approach − psi-function as the characteristic function of random quantity – Discrete and/ or continual description – Quantity and its “digitalized projection“ – The idea of „velocity−probability“ – The notion of probability and the light speed postulate – Generalized probability and its physical interpretation – A quantum description of macro-world – The period of the as-sociated de Broglie wave and the length of now – Causality equivalently replaced by chance – The philosophy of quantum information and religion – Einstein’s thesis about “the consubstantiality of inertia ant weight“ – Again about the interpretation of complex velocity – The speed of time – Newton’s law of inertia and Lagrange’s formulation of mechanics – Force and effect – The theory of tachyons and general relativity – Riesz’s representation theorem – The notion of covariant world line – Encoding a world line by psi-function – Spacetime and qubit − psi-function by qubits – About the physical interpretation of both the complex axes of a qubit – The interpretation of the self-adjoint operators components – The world line of an arbitrary quantity – The invariance of the physical laws towards quantum object and apparatus – Hilbert space and that of Minkowski – The relationship between the coefficients of -function and the qubits – World line = psi-function + self-adjoint operator – Reality and description – Does a „curved“ Hilbert space exist? – The axiom of choice, or when is possible a flattening of Hilbert space? – But why not to flatten also pseudo-Riemannian space? – The commutator of conjugate quantities – Relative mass – The strokes of self-movement and its philosophical interpretation – The self-perfection of the universe – The generalization of quantity in quantum physics – An analogy of the Feynman formalism – Feynman and many-world interpretation – The psi-function of various objects – Countable and uncountable basis – Generalized continuum and arithmetization – Field and entanglement – Function as coding – The idea of „curved“ Descartes product – The environment of a function – Another view to the notion of velocity-probability – Reality and description – Hilbert space as a model both of object and description – The notion of holistic logic – Physical quantity as the information about it – Cross-temporal correlations – The forecasting of future – Description in separable and inseparable Hilbert space – „Forces“ or „miracles“ – Velocity or time – The notion of non-finite set – Dasein or Dazeit – The trajectory of the whole – Ontological and onto-theological difference – An analogy of the Feynman and many-world interpretation − psi-function as physical quantity – Things in the world and instances in time – The generation of the physi-cal by mathematical – The generalized notion of observer – Subjective or objective probability – Energy as the change of probability per the unite of time – The generalized principle of least action from a new view-point – The exception of two dimensions and Fermat’s last theorem. (shrink)

This paper is a contribution to graded model theory, in the context of mathematical fuzzy logic. We study characterizations of classes of graded structures in terms of the syntactic form of their first-order axiomatization. We focus on classes given by universal and universal-existential sentences. In particular, we prove two amalgamation results using the technique of diagrams in the setting of structures valued on a finite MTL-algebra, from which analogues of the Łoś–Tarski and the Chang–Łoś–Suszko preservation theorems follow.

This is the first book in a two-volume series. The present volume introduces the basics of the conceptual foundations of quantum physics. It appeared first as a series of video lectures on the online learning platform Udemy.]There is probably no science that is as confusing as quantum theory. There's so much misleading information on the subject that for most people it is very difficult to separate science facts from pseudoscience. The goal of this book is to make you able to (...) separate facts from fiction with a comprehensive introduction to the scientific principles of a complex topic in which meaning and interpretation never cease to puzzle and surprise. An A-Z guide which is neither too advanced nor oversimplified to the weirdness and paradoxes of quantum physics explained from the first principles to modern state-of-the-art experiments and which is complete with figures and graphs that illustrate the deeper meaning of the concepts you are unlikely to find elsewhere. A guide for the autodidact or philosopher of science who is looking for general knowledge about quantum physics at intermediate level furnishing the most rigorous account that an exposition can provide and which only occasionally, in few special chapters, resorts to a mathematical level that goes no further than that of high school. It will save you a ton of time that you would have spent searching elsewhere, trying to piece together a variety of information. The author tried to span an 'arch of knowledge' without giving in to the temptation of taking an excessively one-sided account of the subject. What is this strange thing called quantum physics? What is its impact on our understanding of the world? What is ‘reality’ according to quantum physics? This book addresses these and many other questions through a step-by-step journey. The central mystery of the double-slit experiment and the wave-particle duality, the fuzzy world of Heisenberg's uncertainty principle, the weird Schrödinger's cat paradox, the 'spooky action at a distance' of quantum entanglement, the EPR paradox and much more are explained, without neglecting such main contributors as Planck, Einstein, Bohr, Feynman and others who struggled themselves to come up with the mysterious quantum realm. We also take a look at the experiments conducted in recent decades, such as the surprising "which-way" and "quantum-erasure" experiments. Some considerations on why and how quantum physics suggests a worldview based on philosophical idealism conclude this first volume. This treatise goes, at times, into technical details that demand some effort and therefore requires some basics of high school math (calculus, algebra, trigonometry, elementary statistics). However, the final payoff will be invaluable: Your knowledge of, and grasp on, the subject of the conceptual foundations of quantum physics will be deep, wide, and outstanding. Additionally, because schools, colleges, and universities teach quantum physics using a dry, mostly technical approach which furnishes only superficial insight into its foundations, this manual is recommended for all those students, physicists or philosophers of science who would like to look beyond the mere formal aspect and delve deeper into the meaning and essence of quantum mechanics. The manual is a primer that the public deserves. (shrink)

The paper considers a generalization of Peano arithmetic, Hilbert arithmetic as the basis of the world in a Pythagorean manner. Hilbert arithmetic unifies the foundations of mathematics (Peano arithmetic and set theory), foundations of physics (quantum mechanics and information), and philosophical transcendentalism (Husserl’s phenomenology) into a formal theory and mathematical structure literally following Husserl’s tracе of “philosophy as a rigorous science”. In the pathway to that objective, Hilbert arithmetic identifies by itself information related to finite sets and series and quantum (...) information referring to infinite one as both appearing in three “hypostases”: correspondingly, mathematical, physical and ontological, each of which is able to generate a relevant science and area of cognition. Scientific transcendentalism is a falsifiable counterpart of philosophical transcendentalism. The underlying concept of the totality can be interpreted accordingly also mathematically, as consistent completeness, and physically, as the universe defined not empirically or experimentally, but as that ultimate wholeness containing its externality into itself. (shrink)

RENÉ DESCARTES E O MÉTODO CARTESIANO -/- RENÉ DESCARTES AND THE CARTESIAN METHOD -/- Emanuel Isaque Cordeiro da Silva - CAP-UFPE, IFPE-BJ e UFRPE. E-mails: [email protected] e [email protected]. WhatsApp: (82)98143-8399. -/- INTRODUÇÃO -/- Antes de abordar a metafísica tal qual Descartes a propõe como uma sólida “fundamentação” das ciências e, também, antes de falar das ciências construídas para a busca desse fundamento, é necessário analisar o método cartesiano, salve que é a alma desse presente artigo. Não se trata apenas de (...) relatar os preceitos do método ao seu modelo matemático, e sim, também, para entender porque o método em si necessita de uma fundamentação alicerçada na metafísica. Mas isso é a maneira pelo qual o projeto revolucionário de Descartes substituiu a autoridade da tradição pela autoridade da razão. -/- I – UM PROJETO REVOLUCIONÁRIO -/- No se pode deixar atingir-se, lendo o Discurso do Método, pelo caráter revolucionário que Descartes queria conferir ao seu projeto. É notável, e vale salientar o cuidado que Descartes toma na primeira parte, sublinhando o que seu projeto tem de singular: “Assim, meu propósito não é ensinar aqui o método que cada um deve seguir para bem conduzir sua razão, mas somente mostrar de que modo procurei conduzir a minha” (DESCARTES, 1996. p. 5). Ainda mais notável é o fato escolhido por Descartes e destacado na segunda parte, a da casa que deve ser derrubada para reconstruí-la: -/- É verdade que não vemos demolirem-se todas as casas de uma cidade só com o propósito de refazê-las de outra forma e de tornar as ruas mais belas, mas não é incomum vermos muitos mandarem derrubar as suas para reconstruí-las, e até, por vezes, a isso são obrigados quando elas correm o risco de cair por si mesmas e os alicerces não estão muito firmes (DESCARTES, 1996. pp. 17-18). -/- Descartes enfatiza explicitamente a oposição entre tal abordagem e a estratégia reformista de “redigitar” a sua casa, organizando-a, reparando-a aqui e ali. Certamente ele está combatendo a ideia revolucionária na política, o design ambicioso daqueles que querem perturbar o Estado, porque nesta área a destruição tornaria a reconstrução futura impossível. Mas o que é insano sobre “uma cidade inteira” torna-se possível quando se trata de cada um de sua própria casa. -/- O que é possível torna-se necessário, quando a casa ameaça entrar em colapso, uma vez que seus alicerces são vulneráveis. Casa vacilante, que seria inútil só querer reparar, é assim que Descartes considera o todo “opiniões” que lhe foram transmitidas em sua educação. Essas opiniões são, acima de tudo, incertas: pode haver alguma verdade nelas, mas provavelmente também há o lado falso, e essa mistura indissociável faz com que todas sejam falsas, a verdade só tem valor para nós se pudermos distinguir com certeza do erro. -/- O julgamento negativo de Descartes de tudo o que ele tem, como escreve, “recebido em dívida”, pressupõe, é claro, uma crítica do conteúdo dessas opiniões e, acima de tudo, uma crítica aos princípios sobre os quais eles descansam. Mas isso não é o suficiente para fazer as pessoas entenderem a escolha inicial de uma abordagem revolucionária, de preferência ao empreendimento reformista, que seria criticar, uma após a outra, as opiniões recebidas, e consolidar, pouco a pouco, a sua “Casa”. Mais do que o conteúdo peculiar das opiniões recebidas, é o próprio fato que elas são recebidas, em uma aquisição desordenada e perigosa, o que importa. E se temos que revolucionar o conhecimento, se temos que destruir essas opiniões antes de substituí-las por outras é, antes de mais nada, uma questão de substituir conquista controlada da desordem e chance de sua aquisição, em vez de se limitar a correções que só aumentariam essa confusão. -/- Nisso, a decisão primeira de Descartes ultrapassa as circunstâncias históricas do seu aparecimento, e conserva em todos os tempos o valor de um modelo para o pensamento: a filosofia nasce de uma ruptura, não com esta ou aquela tradição, mas com a tradição como tal. -/- II – DESCARTES E A TRADIÇÃO -/- A tradição é um pensamento sem sujeito: pensamento que ninguém pensa, não pensa. Descartes se opõe ao requisito essencial, para toda ciência, de ser a matéria de um sujeito: não há ciência até que toda verdade seja reconhecida em seu poder de sugerir notícias, de acordo com uma sequência que deve ser reconhecida, para que eu (Descartes) possa reivindicar, de forma legítima, a ciência universal. Quando Descartes expressa essa afirmação, em seu próprio texto, em seus textos de juventude, o que nos parece ser a marca de uma excessiva arrogância, expressa, antes, uma condição inscrita na própria natureza do conhecimento. -/- O projeto revolucionário está assim ligado à uma ideia que Descartes enuncia no início da segunda parte do Discurso do Método: “[...] não há tanta perfeição nas obras compostas de várias peças, e feitas pelas mãos de vários mestres, como naquelas em que apenas um trabalhou” (DESCARTES, 1996. p. 15). Certamente, por “perfeição, Descartes entende menos a magnitude e a riqueza essa ordem e consistência. Ele reconhece que algumas áreas são, irremediavelmente, entregues à imperfeição. Este é o caso do Estado e das instituições políticas em geral: resultados não intencionais de uma multidão de ações históricas entrelaçadas, nossas instituições nunca correspondem a um projeto consciente, e não seria razoável empreender sua reforma de acordo com tal projeto. Descartes não ignora as virtudes da cooperação para a execução de atividades materiais complexas. Porque as aptidões corporais dos homens são distintas e capazes de se especializar, os indivíduos podem interferir uns com os outros e se harmonizar visando a colaboração do todo. -/- Não obstante, as obras de pensamento, as ciências em particular, não conseguem encontrar a perfeição de que são susceptíveis por serem construídas por sucessivas sedimentações históricas, de acordo com a descoberta uma das outras, fazendo com que todas essas descobertas, a ciência, não seria obra de ninguém. Se o exercício de uma atividade manual nos impede de praticar outra (porque as mesmas mãos dificilmente podem se exercitar ao mesmo tempo para cultivar os campos e tocar a cítara), o conhecimento de uma verdade nos ajuda a encontrar outras verdades. Finja que esse conhecimento é apenas uma contribuição fragmentada para uma síntese coletiva da qual o princípio nos escapa, é desconsiderar sua fecundidade, isto é, o que faz dele um verdadeiro conhecimento. -/- O que Descartes está lutando, essa necessidade de que todos participem de uma instituição já iniciada e que ninguém pode dominar, essa herança de aceitar, enriquecer e transmitir é o que chamamos de tradição: então ele escreve o Discurso do Método não em latim, linguagem de transmissão histórica do conhecimento, mas em francês, linguagem “vulgar”. Certamente, Descartes luta contra a tradição científica e filosófica em vigor em seu tempo, a herança já enfraquecida do pensamento de Aristóteles, transmitida durante a Idade Média pelas Universidades (o que denominamos de “escolasticismo”), mas ele luta acima de tudo em sua pretensão de manter seu valor de ser, precisamente, uma tradição. Esta não é uma luta em nome da “novidade” ou da “modernidade”, que são sempre definidas em relação a uma tradição, mas uma luta em nome da verdade, o que é mais antiga do que qualquer tradição. -/- III – A LIGAÇÃO ENTRE AUTORIDADE E A VERDADE -/- Descartes assim fórmula seu projeto, na segunda parte do Discurso do Método: [...] mas, quanto às opiniões que até então eu aceitara, o melhor que podia fazer era suprimi-las de uma vez por todas, a fim de substituí-las depois, ou por outras melhores, ou então pelas mesmas, quando eu as tivesse ajustado ao nível da razão (DESCARTES, 1996. p. 18). -/- Que uma opinião poderia ter sido recebido por ele em sua vida, sem ter sido ajustada ao nível da razão, significa que foi considerada a sua verdade, sem que ele tenha que se pronunciar sobre o julgamento do que declarava verdadeiro. Então foi um preconceito, que ele achava que era verdade apenas para tê-lo recebido, isto é, por causa de sua fonte. Que a afirmação de uma opinião como verdadeira é validada por sua fonte, e isso corresponde ao que denominamos como “argumento de autoridade”: é verdade porque Platão, ou Aristóteles, disseram isso. Agora é possível, é claro, descobrir verdades lendo Platão ou Aristóteles, desde que essas descobertas sejam não apenas para conferir Platão, Aristóteles, ou qualquer outro autor (“autoritário”), mas para encontrar a si mesmo com as verdades em questão. Parece, portanto, que a revolução cartesiana visa romper o elo entre as concepções de verdade e autoridade. -/- -/- Todavia, olhando mais de perto, veremos que esse não é o caso. Quebrar o elo entre autoridade e verdade é ser crítico, enquanto Descartes encara sim o “espírito da dúvida”. Ser crítico é estar interessado apenas no conteúdo das ideias e ser perfeitamente indiferente à sua fonte. Descartes, como supracitado, não se compromete a criticar suas antigas opiniões uma após a outra, o que seria o passo mais oposto ao princípio de autoridade, quanto mais indiferente à questão da fonte. O que Descartes propõe é bem diferente: ele propõe substituir, com todas as autoridades que estão fora dele (em seu exterior), pela autoridade que está nele (em seu interior), e que muitas vezes é até confundida, veremos isso. Agora é sobre “ajustar as opiniões ao nível da razão”: a razão, isto é, a razão de Descartes, tal qual o fala e presente em cada um, deve ser autoritária em matéria de verdade, deve ser o tribunal diante do qual tudo aparece. É uma interiorização do argumento de autoridade, não é uma rejeição deste argumento. -/- A interiorização da autoridade explica o lugar central da certeza na filosofia de Descartes. O que importa na verdade é a certeza de possuí-la, e essa certeza é uma ato de confiança em relação às fonte autorizada da verdade: tudo que é “ajustado ao nível da razão” será certo, se a razão for o que é autoritário. -/- Duas questões orientadoras sucessivas inspirarão a investigação da certeza: -/- 1 – A primeira consistirá em estabelecer o que ele realmente tem certeza, delimitando o domínio da certeza autêntica daquilo que é garantido pela autoridade da razão. 2 – A segunda consistirá em se perguntar se ele está certo em ter certeza sobre algo, porque a autoridade internalizada deve se questionar. -/- A primeira questão, considerada isoladamente, leva à ideia cartesiana de método. O conjunto das duas questões, a passagem da primeira para a segunda, depois da segunda para a primeira, constitui a metafísica de Descartes. -/- IV – O QUE É A RAZÃO ? -/- O que significa ter razão em um tribunal? As primeiras linhas do Discurso do Método definem a razão como sendo um senso comum, isto é, a faculdade de distinguir a verdade da mentira, o que nos diferencia dos animais. Em outras palavras, a razão é o poder, não para racionar, mas degenerar, pensando na medida em que se pronuncia sobre qualquer objeto que se apresente. Separado em um sentido de outras faculdades do pensamento, este poder está envolvido em qualquer operação de pensamento: a memória supõe um julgamento sobre o passado, o desejo um julgamento sobre o que procurar, o amor um julgamento sobre o que é digno de ser amado... Nesse sentido, a razão se confunde com o próprio pensamento. É até confuso, para todos, consigo mesmo, pois ninguém pode julgar sua razão sem voltar a agir: mesmo aqueles que estão acostumados a reclamar do que foi dado a eles nunca reivindicam, nota Descartes, uma razão de melhor qualidade: pois isso equivaleria a desqualificar essa afirmação conforme a expressam. -/- Coextensiva com o pensamento, a razão está em nós como uma luz natural. Como a luz do Sol que ilumina a variedade das coisas no mundo sem perder sua identidade, permanecendo a mesma luz, o pensamento julgador mantém sua unidade sem nunca se quebrar ou se perder, seja qual for os objetos aos quais se aplica: seja o que for que eu julgue, é sempre eu quem julgo. A ciência deve ser difundida mediante essa ideia, isto é, como um certo modo de julgar, ou pensar, e não dá diversidade de seus objetos. -/- -/- Rejeitar as opiniões recebidas, e julgadas boas porque foram recebidas, é recuperar o poder soberano de julgar aquilo que até então passou despercebido. Colocar a razão no tribunal não é uma novidade, mas uma revolução no sentido estrito, isto é, um retorno. A razão é o senso comum, o poder, não apenas para julgar, mas julgar bem, para distinguir o verdadeiro do falso. Julgar mal, tomar o falso pela verdade, não deve ser dotado de um “poder de julgar mal”, é errado usar nosso poder para julgar bem. A diferença entre o bem e o mal não está no poder, mas no uso desse poder, na maneira de usá-lo, de proceder quando se julga, em uma palavra, o método. É necessário romper com uma aquisição sem método de conhecimento, de acordo com as influências sofridas, e conferir a si mesmo um método firme, que será como o procedimento do tribunal da razão. E como nossa capacidade de comer é saudável por si só, esse método não deve ser tomado como uma técnica destinada para melhorá-lo, para torná-lo mais eficaz, e sim como uma higiene projetada para preservá-lo. -/- Mas onde encontrar as regras desse método? Em um domínio, precisamente, onde nosso poder de regulação ou gerenciamento tem sido relativamente preservado, fazendo com que se possa explorar esse domínio, descobri-lo e extrair dele um procedimento universal. Porque o pensamento sendo esta luz que ilumina de igual forma as variedades de tudo que se apresenta a ela, o que torna-se verdadeiro nesse domínio deve ser de todos os outros. Essa área de referência é o domínio da matemática. -/- V – O MODELO MATEMÁTICO -/- Descartes acredita que é no domínio da matemática e, precisamente da geometria, que os antigos fizeram descobertas difíceis e duradouras. A geometria dos antigos e a dos modernos e, especialmente a de François Viète (1540-1603), acrescentou a análise na matemática, o que denominamos de “álgebra”. A descoberta em 1631 da geometria analítica, que relaciona curvas geométricas com equações algébricas, é também o principal título de fama de Descartes como matemático. No entanto, não devemos superestimar a importância que esta descoberta teve para o próprio Descartes. Seu principal interesse era, em sua opinião, metodológico: permitia libertar a mente para simplificar as técnicas de cálculo. -/- Porque na matemática Descartes é compartilhado entre admiração pelo que eles permitem, e desprezo pelo que fazem com ele. A ideia de facilidade é o motivo comum dessa admiração e desprezo. Matemática é fácil, e sua virtude nos mostrar como o conhecimento é fácil, é fácil para o homem saber tudo o que ele pode saber, desde que o mesmo tenha tempo. Todavia, essa facilidade faz com que a atividade habitual dos matemáticos seja um pouco ridícula, a resolução de “problemas matemáticos”, que Descartes considerava como um mero divertimento, é dificilmente digna de consideração. É novamente em nome desse caso que Descartes mais uma vez repreende os antigos matemáticos por terem dado seus resultados mas obstruindo ou ocultando seu método, a fim de impressionar os crédulos, dificultando a aparência do que não é. O método está em contraste com o essencial do que pode ser aprendido com a matemática. -/- -/- Para identificar corretamente este método, é necessário abstrair os próprios objetos matemáticos (números, figuras etc.), que nos faz acreditar que a matemática é uma ciência especial ao lado das outras ciências, o que nos leva a uma divisão em ramos separados da mesma (aritmética, geometria, álgebra etc.). Então aparece uma matemática universal, consistindo no estudo de “vários relatórios ou proporções” que podem ser encontrados entre quaisquer objetos, para que possamos constituí-los em séries contínuas e deduzi-los uns dos outros, e chegar, daqueles que são conhecidos, àqueles que ainda não estão distantes, tão distantes como estão, tão ocultos como parecem ser. -/- Não há nada difícil em tal conhecimento, há apenas o simples e o complexo, e o tempo necessário para alcançar o complexo mediante o simples. Não há nada difícil porque não há nada oculto: se “conhecer” era a perigosa empreitada de se aventurar na dimensão secreta das coisas, seria necessário um gênio e o uso de poderes ocultos. Mas “conhecer” é um exercício do senso comum que direciona sua luz para tudo que pode esclarecer: é, portanto, suficiente para um método. -/- Nós todos sabemos que as primeiras verdades matemáticas são simples. Mas por que elas são? Porque elas nos contam tudo sem reservas, sem nada opaco, sobre tudo que estão falando e tratando: seu objeto é completamente iluminado, não há a necessidade de voltar à ele para aprofundá-lo indefinidamente, devemos passar para outro. Estas são representações estritamente adequadas ao que elas alegam, para representar o que Descartes denominou de ideais claras e distintas: claro, porque percebemos todos os elementos distintos, porque não podemos confundi-los com os outros. -/- Se a primeira verdade é considerada simples, a segunda, por sua vez, não parece ser assim. Seria, sem dúvida, considerada difícil, reservando uma parte do desconhecido se o enfrentássemos de frente. Mas é apenas complexo, isto é, só podemos entendê-lo depois de ter compreendido o primeiro, e depois entender que não há mais nada para entender no primeiro, e sim passar adiante: basta respeitar essa ordem, que é a ordem do nosso entendimento, para que seja também considerado uma ideia claro e distinta. -/- Até onde podemos prosseguir? Até o fim, de acordo com Descartes, até a conclusão do que podemos saber, uma vez que nossa mente, que agora é ordenada por nosso conhecimento, é limitada: limite claro, limite esse que não é misteriosamente abrigado no mundo ou na espessura da realidade. Abaixo desse limite, nosso conhecimento pode ser perfeito se for metódico. Abaixo do limite, nada oculto, nada profundo, tudo toma o seu lugar nas “longas cadeias de razões”; além do limite, nada escondido, apenas o que é não é nossa responsabilidade. -/- Se não há nada escondido, a realidade que conhecemos não vai além do que sabemos sobre ela. Logo, o método não é apenas uma maneira humana de pedir objetos a nossa conveniência: atribuindo-lhes o seu lugar em nossa ordem, tornando-os claros para nós, envolve-os em seu verdadeiro ser. A revolução cartesiana consiste, então, no fato de que as condições de nosso conhecimento definem a verdade do ser conhecido. -/- Não obstante, sempre perpetua-se uma “dificuldade”, que nada mais é do que o tempo. Se não há a necessidade de o gênio saber perfeitamente, leva tempo para ir do simples ao complexo. O homem só pode saber se tem tempo, muito tempo antes dele: essa preocupação para com a necessidade de tempo, retorna com uma obsessão sob a caneta e a perspectiva de Descartes. Essa necessidade, então, de ter tempo diferencia radicalmente, como veremos, o problema do conhecimento e da ação, da conduta a ser mantida na vida, da moralidade, uma vez que “as ações da vida muitas vezes não sofrem demora...”. -/- VI – REGRAS DO MÉTODO: OS QUATRO PRECEITOS -/- Podemos agora compreender os quatro preceitos aos quais, segundo Descartes, o método é reduzido (Discurso, segunda parte). -/- VI.I - Regra de evidência -/- O primeiro era de nunca aceitar coisa alguma como verdadeira sem que a conhecesse evidentemente como tal; ou seja, evitar cuidadosamente a precipitação e a prevenção, e não incluir em meus juízos nada além daquilo que se apresentasse tão clara e distintamente a meu espírito, que eu não tivesse nenhuma ocasião de pô-lo em dúvida (DESCARTES, 1996. p. 23). -/- Mesmo que sua declaração o duplique, é de fato um mesmo preceito acerca do que deve ser concebido como verdadeiro. Receber com método é esperar até que a coisa se apresente para ele: essa paciência, esse “cuidado”, remove os preconceitos característicos das ideias recebidas antes da hora, a precipitação e a prevenção que fazem parecer obscuro o que é o tempo. Se, nessa apresentação para ele, ele tomou um cuidado de aderir (nada mais) ao que é claro e distinto, é sem reserva que uma “coisa” pode ser recebida, sem poder surgir o risco, a “oportunidade”, de ter que voltar mais tarde: ele teria visto a sua verdade, e está evidência foi o suficiente para que ele pudesse duvidar disso. -/- VI.II - Regra de análise -/- “O segundo, dividir cada uma das dificuldades que examinasse em tantas parcelas quantas fosse possível e necessário para melhor resolvê-las” (DESCARTES, 1996. p. 23). Ora, o que é apresentado como difícil de examinar, na verdade não o é, é apenas complexo, composto de múltiplas parcelas. Quantas parcelas? É a resolução da dificuldade que deve decidir essa questão: quando um matemático consegue estabelecer um teorema “difícil”, ligando-o às proposições mais simples por uma cadeia contínua de deduções, ele sabe que não há mais nada a saber em seu objeto, que contém exatamente tantas divisões quantas forem necessárias para formar está série. Logo, não precisamos nos preocupar com mais nada. -/- VI.III - Regra de ordem -/- O terceiro, conduzir por ordem meus pensamentos, começando pelos objetos mais simples e mais fáceis de conhecer , para subir pouco a pouco, como por degraus, até o conhecimento dos mais compostos; e supondo certa ordem mesmo entre aqueles que não se precedem naturalmente uns aos outros (DESCARTES, 1996. p. 23). -/- Essa é a ordem, de acordo com os pensamentos de Descartes, o que pode ser visto no fato de que, entre os objetos mais simples e os mais complexos, a distinção está na ordem de facilidade, isto é, os primeiros são os mais fáceis de saber, enquanto o segundo, por sua vez, só pode ser conhecido quando dominamos a totalidade do primeiro. O importante é passar de um para o outro como por graus, numa série contínua, sem estar diante de um hiato como o conhecimento prévio não seria suficiente para deixar claro o novo conhecimento. E desde que está ordem de razões, fixando cada objeto em seu devido lugar, nos entrega em sua totalidade, é também a ordem objetiva, a ordem dos próprios objetos, tanto que não devemos hesitar em supor isso entre eles, mesmo quando o modo como se apresentam não revela ordem. -/- VI.IV - Regra de enumeração -/- “E, o último, fazer em tudo enumerações tão completas, e revisões tão gerais, que eu tivesse certeza de nada omitir” (DESCARTES, 1996. p. 23). Omitir alguma coisa séria destruir o próprio princípio da ordem das razões, uma vez que qualquer objeto só pode ser visto clara e distintamente se todos que o precederam também tiverem sido vistos em seu lugar. Se a dedução atingir este objeto, significa que nada foi omitido até então. Porém, quando a dedução torna-se muito longa, torna-se, também, mais difícil de mantê-la inteira, como deveria, sob o olhar da mente; somos então tentados a confiar apenas em sua memória. O preceito de enumeração não deve ser entendido como uma preocupação retrospectiva (certifique-se de que nada o tenha esquecido), mas no sentido atual, como os meios de manter em mente todas as evidências do passado necessárias para a evidência presente. Dos quatro preceitos do método, é o único a ter um caráter de técnica e não de simples higiene: é aqui que se confronta a única dificuldade real do conhecimento, o fato de que leva tempo, e sua consequência, recorrer à memória, à faculdade pela qual as velhas ideias descartenses foram “concebidas em sua dívida”. -/- -/- VII – O MÉTODO EXIGE UM FUNDAMENTO METAFÍSICO -/- O método permite responder à uma questão: do que tenho certeza? É certo que homens que seguem ideias aceitas também conhecem esse estado psicológico chamado “certeza”. Todavia, entre esse estado psicológico e a verdadeira certeza do conhecimento metódico, há uma diferença de que o primeiro está imune a “oportunidades” de duvidar, já que ele não tem autoconfiança. Toda oportunidade de dúvida que se apresenta transformaria essa “certeza” em incerteza: por isso que é sensato duvidar em princípio das opiniões recebidas, sem esperar, por assim dizer, oportunidades de duvidar, e decretar antecipadamente seu conteúdo incerto, não necessariamente substituí-lo por outro conteúdo, mas aceitar apenas o que lhe fora assegurado. -/- Poderia ser o mesmo com a certeza do conhecimento metódico? Pode ser apresentado a ele como uma oportunidade para duvidar, e seria sábio também duvidá-lo antecipadamente, submetendo-o a uma dúvida de princípio? Seria então uma dúvida muito diferente da anterior. Descartes exclamou que: “Se alguma vez duvidei um dia do que me aperfeiçoei seguramente, tal dúvida deveria permanecer fora da certeza: não a destruiria, nem mesmo a enfraqueceria, a colocaria em suspenso no vazio; ele não sugeriria para mim que eu talvez esteja menos certo do que pensava, mas sim que posso estar errado em ter tanta certeza. Eu vou manter minha certeza enquanto duvidoso, porque não haveria nenhuma esperança para eu encontrar melhor do que ela: minha única esperança seria descobrir que ela é verdadeiramente fundamentada, que eu confio com razão, e não porque eu não posso contrário.” -/- Mas por que uma oportunidade para duvidar dessa maneira? A própria razão, na medida em que é incontestavelmente autoritária, deve levantar essa dúvida por sua própria autoridade. É porque é autoritária que tem o direito de se perguntar sobre o que o autoriza, sobre o que o investe com tal autoridade. Afinal, a certeza de sua autoridade é uma certeza, que não pode ser classificada entre as certezas que ela nos ensina, uma vez que reconhecemos essa autoridade. -/- Por isso, é importante encontrar, para essa certeza, um fundamento para abrigá-lo, de uma vez por todas, desta suspeita sobre si mesmo. A busca por esse fundamento deve ser metódica, mas deve ao mesmo tempo abalar nossa confiança no método, a fim de encontrar o que o justifica. Essa busca paradoxal é o que Descartes denomina de "metafísica". Desdobra-se principalmente nas seis Meditações da Primeira Filosofia de 1641, geralmente chamadas de Meditações Metafísicas. Seu exame pode ser dividido em duas partes: -/- • o primeiro diz respeito à dúvida e ao seu trabalho; • o segundo diz respeito à fundação de certas ciências, ou seja, a existência de um Deus verdadeiro. -/- REFERENCIAL TEÓRICO -/- DESCARTES, R. Discurso do método. Trad. Maria Ermantina Galvão. 2ª ed. São Paulo: Martins Fontes, 1996. ______________. Discurso do método. Coleção Os pensadores, vol. XV. Trad. J. Guinsburg e Bento Prado Jr. 1ª ed. São Paulo: Abril Cultural, 1973. -/- Estimule a criatividade, respeite o direito autoral. -/- Emanuel Isaque Cordeiro da Silva © 2019. (shrink)

FROM THE HISTORY OF PHYSICS TO THE DISCOVERY OF THE FOUNDATIONS OF PHYSICS By Antonino Drago, formerly at Naples University “Federico II”, Italy – drago@unina,.it (Size : 391.800 bytes 75,400 words) The book summarizes a half a century author’s work on the foundations of physics. For the forst time is established a level of discourse on theoretical physics which at the same time is philosophical in nature (kinds of infinity, kinds of organization) and formal (kinds of mathematics, kinds of logic). (...) This double side of the two dichotomies assures a deep comprehension of theoretical physics by avoiding both a purely philosophical analysis and a purely formal analysis, each manifestly insufficient for representing all the aspects of theoretical physics. Among the many formulations of a physical theory, e.g. Mechanics, also Mach(1) recognized technical differences only, Instead my suggestion is to consider their differences as revealing the characteristic features of their foundations. No more radical differences may exist than those concerning both their kinds of Mathematics and their kinds of Logic. As a fact, after the 20th Century within each of these sciences there exist two antagonist foundations, within Mathematics either the classical one or the constructive one(2); within Mathematical Logic either the classical logic or the intuitionist one(3). Let us take Mechanics as an instance. Being the choices of the Newton’s formulation respectively the actual infinity of the differential equations relying on the infinitesimals and the classical logic, a formulation based on the alternative choices is Lazare Carnot’s,(4) whose mathematics is no more than algebraic-trigonometric and the logic is the intuitionist one, because this formulation makes use of doubly negated propositions, whose corresponding affirmative propositions are idealistic or false; i.e. in these cases the law of double negation fails, as in intuitionist logic occurs. By considering these two dichotomies as the foundations of the physical theories, each couple of choices upon them fashions one out four models of scientific theory, which are baptized through the authors of their most representative theories: Newtonian, Carnotian, Lagrangian, Descartesian. In correspondence four versions of the principle of inertia are recognized, as suggested by respectively: Descartes. Lazare Carnot, Enriques and Cavalieri.(5) All that suggests that the four models of scientific theory may be graphically represented as a compass orientating our minds amid the so numerous physical theories. By taking these dichotomies as interpretative categories, a new interpretative history of Physics including both classical and modern physics results according to a quadrilinear development. The birth of modern physics is characterized by the two theories - Einstein’s paper on special relativity and Einstein’s first paper on quanta - whose choices are the alternative ones to those of the previously dominant physical theory, Newtonian mechanics: in the latter paper Einstein i) declares the dichotomy on the kinds of mathematics (“Introduction”) and then he chooses the discrete mathematics; ii) by using doubly negated propositions he organizes his theory in an alternative way to the deductive-axiomatic one.(6) Moreover, a new history of Philosophy of knowledge results. Kant’s first two a priori transcendental categories represent the physical interpretation of the two choices out of the four pairs of choices on the two dichotomies, i.e. the choices of the dominant theory of mechanics, Newton’s. Hegel’s dialectic was a misleading attempt of anticipating the subsequent intuitionist logic. The book starts by a comparison of different formulations of thermodynamics in order to recognize the first dichotomy on the kind of organization. In chapter 2 the dichotomy on the kind of infinity is recognized through a comparison of Newrton’s mechanics and Lazare Carnot’s. A quickly history of the other classical physical theories is illustrated in chapter 3; it results a foundational conflict between the last two theories, and hence a conflict between their opposite couples of choices. The two main theories of modern physics, special realtivity and quantum mechanics confirm the foundational role played by the two dichotomies which gives reason for the radical change in the history of theoretical physics. Indeed, the opposition of the two main pairs of choices gives reason of the crisis in theoretical physics in the early 1900s. As a global result, the book represents the first interpretation of the entire history of Physics, both classical and modern; This fact proves that the four models of scientific theories can works as a compass (that of the front cover) suggesting a direction among the many physical theories. This new history of physics leads to re-visit both Koyré’s and Kuhn’s historiographies. Their interpretative categories are interpreted and explained through the two dichotomies so that it is possible to suggest à la Koyré categories based on the alternative choices theories to Newton’s as the suitable categories for interpreting the alternative theories to Newton’s mechanics. All that suggests a new interpretation of the crucial step in the history of Western philosophy of knowledge, i.e. the passage from Leibniz to Kant: Leibniz’s suggestion of the two labyrinths of human minds may be seen as a close anticipation of the two above dichotomies. Kant applied them in order to construct the well-known four cosmological proofs, which however he misinterpreted as leading to contradictions, instead to tow different methods of investigation corresponding to the two different kinds of logic.(7) Bibliography 1. Mach E. (1883), Die Mechanik, Leipzig: Brockhaus, Chap. IV, Sect. III. 2. Bishop E. (1967), Constructive Analysis, New York: Mc Graw-Hill. 3. Heyting A. (1962), Intuitionism. An Introduction, Amsterdam: North-Holland. 4. L. Carnot (1783), Essai on the Machines en général, Defay, Dijion (Enlish translation: Springer, Berlin, 2020). Drago A. (2004), “A new appraisal of old formulations of mechanics”, Am. J. Phys., 72(3), pp. 407-9. 5. Vella M.R. (2007), “Le quattro versioni del principio di inerzia”, in M. Leone et al. (eds.), L’eredità di Fermi, Majorana e altri Temi. Napoli: ESI, pp. 147-151. 6. Drago A. (2014), “The emergence of two options from Einstein°s first paper on Quanta”, in Pisano R., Capecchi D., Lukesova A. (eds.), Physics, Astronomy and Engineering. Critical Problems in the History of Science and Society, Scientia Socialis P., Siauliai, 2013, pp. 227-234. 7. This and all in the above points are illustrated by the book Drago A. (2017), Dalla Storia della Fisica ai Fondamenti della Scienza, Roma: Aracne. (shrink)

We develop a simple framework called ‘natural topology’, which can serve as a theoretical and applicable basis for dealing with real-world phenomena.Natural topology is tailored to make pointwise and pointfree notions go together naturally. As a constructive theory in BISH, it gives a classical mathematician a faithful idea of important concepts and results in intuitionism. -/- Natural topology is well-suited for practical and computational purposes. We give several examples relevant for applied mathematics, such as the decision-support system Hawk-Eye, and various (...) real-number representations. -/- We compare classical mathematics (CLASS), intuitionistic mathematics (INT), recursive mathematics (RUSS), Bishop-style mathematics (BISH) and formal topology, aiming to reduce the mutual differences to their essence. To do so, our mathematical foundation must be precise and simple. There are links with physics, regarding the topological character of our physical universe. -/- Any natural space is isomorphic to a quotient space of Baire space, which therefore is universal. We develop an elegant and concise ‘genetic induction’ scheme, and prove its equivalence on natural spaces to a formal-topological induction style. The inductive Heine-Borel property holds for ‘compact’ or ‘fanlike’ natural subspaces, including the real interval [g, h]. Inductive morphisms respect this Heine-Borel property, inversely. This partly solves the continuous-function problem for BISH, yet pointwise problems persist in the absence of Brouwer’s Thesis. -/- By inductivizing the definitions, a direct correspondence with INT is obtained which allows for a translation of many intuitionistic results into BISH. We thus prove a constructive star-finitary metrization theorem which parallels the classical metrization theorem for strongly paracompact spaces. We also obtain non-metrizable Silva spaces, in infinite-dimensional topology. Natural topology gives a solid basis, we think, for further constructive study of topological lattice theory, algebraic topology and infinite-dimensional topology. The final section reconsiders the question of which mathematics to choose for physics. Compactness issues also play a role here, since the question ‘can Nature produce a non-recursive sequence?’ finds a negative answer in CTphys . CTphys , if true, would seem at first glance to point to RUSS as the mathematics of choice for physics. To discuss this issue, we wax more philosophical. We also present a simple model of INT in RUSS, in the two-players game LIfE. (shrink)

The complete arithmetization of the bivalued propositional logic is here presented and extended with original functions not hitherto included in other interpretations of propositional logic, as the algebraic logic or sets theory. An historical review of the former attempts of arithmetical representation of the propositional logic is presented.

The purpose of this study is to compile a selection of the various formalisms found in conversation theory to introduce readers to Pask's discursive algebra. In this way, the text demonstrates how concept sharing and concept formation by means of the interaction of two participants may be formalized. The approach taken in this study is to examine the formal notation system used by Pask and demonstrate how such formalisms may be used to represent concept sharing and concept formation through (...) conversation. The compilation of the discursive algebra using the framework provided by conversation theory could potentially be used as an auxiliary framework to study conversational interactions in multi-agent systems theory. (shrink)

The topics approached in the 52 papers included in this book are: neutrosophic sets; neutrosophic logic; generalized neutrosophic set; neutrosophic rough set; multigranulation neutrosophic rough set (MNRS); neutrosophic cubic sets; triangular fuzzy neutrosophic sets (TFNSs); probabilistic single-valued (interval) neutrosophic hesitant fuzzy set; neutro-homomorphism; neutrosophic computation; quantum computation; neutrosophic association rule; data mining; big data; oracle Turing machines; recursive enumerability; oracle computation; interval number; dependent degree; possibility degree; power aggregation operators; multi-criteria group decision-making (MCGDM); expert set; soft sets; LA-semihypergroups; single valued (...) trapezoidal neutrosophic number; inclusion relation; Q-linguistic neutrosophic variable set; vector similarity measure; fundamental neutro-homomorphism theorem; neutro-isomorphism theorem; quasi neutrosophic triplet loop; quasi neutrosophic triplet group; BE-algebra; cloud model; Maclaurin symmetric mean; pseudo-BCI algebra; hesitant fuzzy set; photovoltaic plan; decision-making trial and evaluation laboratory (DEMATEL); Choquet integral; fuzzy measure; clustering algorithm; and many more. (shrink)

Modifying the descriptive and theoretical generalizations of Relativized Minimality, we argue that a significant subset of weak island violations arise when an extracted phrase should scope over some intervener but is unable to. Harmless interveners seem harmless because they can support an alternative reading. This paper focuses on why certain wh-phrases are poor wide scope takers, and offers an algebraic perspective on scope interaction. Each scopal element SE is associated with certain operations (e.g., not with complements). When a wh-phrase scopes (...) over some SE, the operations associated with that SE are performed in its denotation domain. The requisite operations may or may not be available in a domain, however. We present an empirical analysis of a variety of wh-phrases. It is argued that the wh-phrases that escape all weak islands (i.e., can scope over any intervener) are those that range over individuals, the reason being that all Boolean operations are defined for their domain. Collectives, manners, amounts, numbers, etc. all denote in domains with fewer operations and are thus selectively sensitive to scopal interveners—a “semantic relativized minimality effect”. (shrink)

This paper investigates a generalization of Boolean algebras which I call agglomerative algebras. It also outlines two conceptions of propositions according to which they form an agglomerative algebra but not a Boolean algebra with respect to conjunction and negation.

Hyperboolean algebras are Boolean algebras with operators, constructed as algebras of complexes (or, power structures) of Boolean algebras. They provide an algebraic semantics for a modal logic (called here a {\em hyperboolean modal logic}) with a Kripke semantics accordingly based on frames in which the worlds are elements of Boolean algebras and the relations correspond to the Boolean operations. We introduce the hyperboolean modal logic, give a complete axiomatization of it, and show that it lacks the finite model property. The (...) method of axiomatization hinges upon the fact that a "difference" operator is definable in hyperboolean algebras, and makes use of additional non-Hilbert-style rules. Finally, we discuss a number of open questions and directions for further research. (shrink)

Among Dunn’s many important contributions to relevance logic was his work on the system RM (R-mingle). Although RM is an interesting system in its own right, it is widely considered to be too strong. In this chapter, I revisit a closely related system, RM0 (sometimes known as ‘constructive mingle’), which includes the mingle axiom while not degenerating in the way that RM itself does. My main interest will be in examining this logic from two related semantical perspectives. First, I give (...) a purely operational bisemilattice semantics for it by adapting previous work of Humberstone. Second, I examine a more conventional algebraic semantics for it and discuss how this relates to the operational semantics. A novel operational semantics for J (intuitionistic logic) as well as its conventional Heyting algebraic semantics emerge as special cases of the corresponding semantics for RM0. The results of this chapter suggest that RM0 is a more interesting logic than has been appreciated and that Humberstone’s operational semantic framework similarly deserves more attention than it has received. (shrink)

The Belnap–Dunn logic is a well-known and well-studied four-valued logic, but until recently little has been known about its extensions, i.e. stronger logics in the same language, called super-Belnap logics here. We give an overview of several results on these logics which have been proved in recent works by Přenosil and Rivieccio. We present Hilbert-style axiomatizations, describe reduced matrix models, and give a description of the lattice of super-Belnap logics and its connections with graph theory. We adopt the point of (...) view ofAlgebraic Logic, exploring applications of the general theory of algebraization of logics to the super-Belnap family. In this respect we establish a number of new results, including a description of the algebraic counterparts, Leibniz filters, and strong versions of super-Belnap logics, as well as the classification of these logics within the Leibniz and Frege hierarchies. (shrink)

We present an algebraic account of the Tongan kinship terminology (TKT) that provides an insightful journey into the fabric of Tongan culture. We begin with the ethnographic account of a social event. The account provides us with the activities of that day and the centrality of kin relations in the event, but it does not inform us of the conceptual system that the participants bring with them. Rather, it is a slice in time of an ongoing dynamic process that links (...) behavior with a conceptual system of kin relations and vice versa. To understand this interplay, we need an account of the underlying conceptual system that is being activated during the event. Thus, we introduce a formal, algebraically based account of TKT. This account brings to the fore the underlying logic of TKT and allows us to distinguish between features of the kinship system that arise from the logic of TKT as a generative structure and features that must have arisen through cultural intervention. (shrink)

The reason for physics’ failure to find a final theory of the universe is examined. Problems identified are: the lack of unequivocal definitions for its fundamental elements (time, length, mass, electric charge, energy, work, matter-waves); the danger of relying too much on mathematics for solutions; especially as philosophical arguments conclude the universe cannot have a mathematical basis. It does not even need the concept of number to exist. Numbers and mathematics are human inventions arising from the human predilection for measurement. (...) Following Aristotle, a single fundamental cause is proposed to explore the efficacy of using pure non-mathematical philosophy to explain the universe, contrary to current quantum physical views. The cause is taken as Time, which is then defined, surprisingly leading automatically to a definition of a three-dimensional space answering the question into what can a universe be placed (space before space seems non-sensical). This enables the philosophical arguments to derive a universal rule giving clear-cut descriptions (definitions) of force (both gravitational and electromagnetic), motion, energy, and particles. The formation of atomic nuclei and atomic properties together with an unexpected role for neutrinos follow. In particular, a Popper-test can be prepared by introducing the concept of measurement to allow the philosophy arguments to be tested in another discipline - mathematics. The solution agrees with human physical observations to a remarkable degree of accuracy, automatically explaining and predicting values for Planck’s and the fine-structure constants. Although mathematical, it is included as confirming the efficacy of philosophy in attaining a final theory. (shrink)

This article defends the principle of sufficient reason (PSR) from a simple and direct valid argument according to which PSR implies that there is a truth that explains every truth, namely an omni-explainer. Many proponents of PSR may be willing to bite the bullet and maintain that, if PSR is true, then there is an omni-explainer. I object to this strategy by defending the principle that explanation is irreflexive. Then I argue that proponents of PSR can resist the conclusion that (...) there is an omni-explainer by denying that explanation distributes over conjunction. In the final section, I consider a plausible revision of this assumption. I argue there that, given the revised assumption, PSR seems to yield a striking picture of the explanatory structure of the universe. I explain why the resulting model does not appear utterly implausible to me. (shrink)

We propose an axiomatic approach to constructing the dynamics of systems, in which one the main elements 9e8 is the consciousness of a subject. The main axiom is the statements that the state of consciousness is completely determined by the results of measurements performed on it. In case of economic systems we propose to consider an offer of transaction as a fundamental measurement. Transactions with delayed choice, discussed in this paper, represent a logical generalization of incomplete transactions and allow for (...) a more rigorous approach to studying the properties of the algebra of economic measurements. Considering the behavior of an object as a sequence of actions and choices allows extending the obtained results to random systems, which include the subject's consciousness as one of its elements. The specifics, on which the description of fundamental measurements is based, allows easily transferring the formalism of Schwinger's theory of selective measurements (and the consequent quantum-mechanical formalism) to the economic and social systems. (shrink)