En 1928 la théorie des form constants par Klüver catégorisait les hallucinations visuelles en quatre grandes catégories. Alors que la notion de form constants venait à s’appliquer virtuellement à toutes les figures entoptiques, comme l’avait prévu son auteur, dont les photismes synesthésiques, Cytowic a décrit et Carol Steen représenté ce que voient réellement des synesthètes visuels. L’une des caractéristiques d’œuvres de peintres synesthètes serait justement la présence de ces formes classées par Klüver. L’anthropologie avec la thèse de l’externalisation (...) a montré que certaines cultures ont repris les form constants perçues dans le premier stade de l’intoxication pour en faire les patterns de leur art. Les modèles physiologico-mathématiques permettent de localiser en V1 l’origine de production des photismes et d’expliquer leur morphogénèse. Mais une explication des circuits neuraux empruntés pour passer des différents types de déclencheur sensoriel à l’apparition de photismes synesthésiques reste aujourd’hui au stade de l’hypothèse. La synesthésie est l’objet de recherches alliant les sciences les plus contemporaines dont la génomique. Il est possible que les études sur la synesthésie transforment son statut de phénomène marginal en un élément essentiel dans la compréhension du système physiologique sous-jacent à l’esprit humain. In 1928, Klüver’s theory of form constants placed visual hallucinations into four broad categories. Whereas, as predicted by its author, the notion of form constants was virtually applicable to all entoptic figures, including synesthetic photisms, the very notion of entoptics was definitively extended. Cytowic has described and Carol Steen has represented what visual synesthetes actually see. The presence of these forms, as they have been classified by Klüver, is precisely one of the characteristics of the works of synesthetic painters. With the thesis of externalization, the field of Anthropology has shown that some cultures have adopted the constant forms perceived in the first stage of intoxication as patterns for their art. Physiological-mathematical models allow us to locate the origin of photism production in V1, and to explain these photisms’ morphogenesis. But, even today, an explanation of the neural pathways used to pass from different types of sensory triggers to the appearance of synesthetic photisms remains at the hypothesis stage. Synesthesia is a research topic that combines some of the most contemporary sciences, including genomics. Studies on synesthesia may just transform its status as that of a marginal phenomenon into that of an essential element in understanding the physiological system underlying the human mind. (shrink)

This is the first book-length analysis of the techniques and procedures of ancient mathematical commentaries. It focuses on examples in Chinese, Sanskrit, Akkadian and Sumerian, and Ancient Greek, presenting the general issues by constant detailed reference to these commentaries, of which substantial extracts are included in the original languages and in translation, sometimes for the first time. This makes the issues accessible to readers without specialized training in mathematics or in the languages involved. The result is a much richer (...) understanding than was hitherto possible of the crucial role of commentaries in the history of mathematics in four different linguistic areas, of the nature of mathematical commentaries in general, of the contribution that the study of mathematical commentaries can make to the history of science and to the study of commentaries in general, and of the ways in which mathematical commentaries are like and unlike other kinds of commentaries. (shrink)

The paper argues that a philosophically informative and mathematically precise characterization is possible by (i) describing a particular proposal for such a characterization, (ii) showing that certain criticisms of this proposal are incorrect, and (iii) discussing the general issue of what a characterization of logical constants aims at achieving.

We present a technique to extend a Kripke structure into an elementary extension satisfying some property which can be “axiomatized” by a family of sets of sentences, where, most often, many constant symbols occur. To that end, we prove extended theorems of completeness and compactness. Also, a section of the paper is devoted to the back-and-forth construction of isomorphisms between Kripke structures.

The constantly accelerating progress of contemporary natural science is indissolubly associated with the development and use of mathematics and with the processes of mathematical modeling of the phenomena of nature. The essence of this diverse and highly fertile interaction of mathematics and natural science and the dialectics of this interaction can only be disclosed through analysis of the nature of theoretical notions in general. Today, above all in the ranks of materialistically minded researchers, it is generally accepted that theory (...) possesses a value of its own. Contrary to positivist concepts of the nature of our knowledge, the meaning and significance of theory do not consist merely of the recording of experimental data, their classification and notation in abbreviated form. The meaning of theory is considerably more significant and is revealed, above all, in its explanatory and predictive functions. Contemporary theoretical knowledge is quite advanced and comprises a highly complex system, relatively self-contained and capable of internal development. Mathematics is the most interesting and distinctive phenomenon in the system of theoretical knowledge, in the system of science in general. It is in its dialectical development that the internal force and dynamics of the development of theory, in the very broadest sense of that word, find expression. Paul Lafargue tells us that Karl Marx held that "science only attains perfection when it succeeds in making use of mathematics" . In many fields of research the formulation of new ideas and concepts rests upon mathematics, its concepts and notions, and "is suggested" by the latter. "Mathematics," writes F. Dyson, "is the main source of the notions and principles out of which new theories are created" . The basic significance of mathematics to theory has been noted by outstanding thinkers over the course of the entire history of development of science. Today the expressive phrase, "the mathematization of knowledge," has come into general currency as a description of the principal direction of growth of theoretical notions in natural science. However, with respect to the growth of modern theoretical physics in general, the statement that it develops primarily by the method of the mathematical hypothesis has general validity. For example, the principal findings and discoveries of quantum theory and particle physics, starting from the corpuscularwave dualism and terminating with the omega minus hyperon and hypothetical quarks, were obtained or made "at the tip of a mathematician's pen.". (shrink)

We investigate two constants cT and rT, introduced by Chaitin and Raatikainen respectively, defined for each recursively axiomatizable consistent theory T and universal Turing machine used to determine Kolmogorov complexity. Raatikainen argued that cT does not represent the complexity of T and found that for two theories S and T, one can always find a universal Turing machine such that equation image. We prove the following are equivalent: equation image for some universal Turing machine, equation image for some universal (...) Turing machine, and T proves some Π1-sentence which S cannnot prove. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

In the present paper, I go beyond these examples by bringing into play an example that I nd more experimental in nature, namely that of the use of the so-called PSLQ algorithm in researching integer relations between numerical constants. It is the purpose of this paper to combine a historical presentation with a preliminary exploration of some philosophical aspects of the notion of experiment in experimental mathematics. This dual goal will be sought by analysing these aspects as they are (...) presented by some of the protagonists of the eld and discussing them using notions from contemporary philosophy of science. (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)

Extending the language of the intuitionistic propositional logic Int with additional logical constants, we construct a wide family of extensions of Int with the following properties: (a) every member of this family is a maximal conservative extension of Int; (b) additional constants are independent in each of them.

The "mathematization" of knowledge is a historically inevitable process governed by two circumstances. In the first place there is the need for the further extension of knowledge in all areas of human activity, whether it be the study of natural phenomena or the theory of taking decisions in the economic or social sphere. Marx pointed out long ago that a science reaches its highest levels only when it succeeds in making use of mathematics. The second circumstance rendering the process of (...) deepening and broadening the mathematization of knowledge not only necessary but possible lies in the constant development both of mathematics itself and of the technical means broadening the sphere of its application. (shrink)

A general equation is derived describing the concentration of all possible complexes of a central molecule with a set of ligands bound to the central molecule. This deduction allows the reaction rate constants for the binding of a given molecule to the central molecule to depend on the species of molecules already bound and the location of the molecules already bound. The model thus allows for structural alteration of the central molecule by binding. Functions describing the concentration dependence of (...) any effect whatever depending on the distribution of complexes are deduced. Possible applications and methods of application are indicated.II est dérivé une équation qui décrit la concentration des toutes les complexes d'une molécule centrale avec une collection de ligands connectés à la molécule centrale. La déduction montre comme les coefficients de la rapidité de reaction entre la molécule centrale et les molécules de la collection dépend de l'espèce et de la location des molécules déjà connectées.La modèle dérivée de l'équation montre les changements structurelles par la connection. Les functions qui décrivent la dependence de chaque effet à la concentration, sont dérivés. Ces effects dépendent aussi à la distribution des complexes. Des applications et des méthodes sont indiqués.Eine allgemeine Vergleichung hat man bekommen welche die Konzentration aller möglichen Komplexen eines zentralen Moleküles mit einer Reihe von „Ligands” verbunden mit dem zentralen Molekül beschreibt. Diese Folgerung gestattet dass die Reaktiongeschwindigkeit Konstanten für die Bindung eines gegebenen Moleküles mit dem zentralen Molekül abhÄngt von den Arten schon gegebenen Molekülen und die Ortsbestimmung der schon gebundenen Molekülen. Das Model zeigt die strukturelle Änderungen des zentralen Moleküls durch Bindung. Funktionen, die die Konzentrationband jedes Effektes beschreiben dass abhÄngt von der Distribution der Komplexen, sind abgeleitet.Mögliche Anwendungen und Methoden von Anwendung sind dargelegt. (shrink)

Auf Grund der Vorstellung, dass die Erregungsleitung auf einer Wiedererregung der benachbarten Gewebebezirke durch lokale bioelektrische Ströme beruht, wurde vorher eine mathematische Theorie der Fortpflanzung der Erregung im Nerv entwickelt, welche einige Tatsachen befriedigend darstellt. In der vorliegenden Arbeit wird die Theorie auf den Fall angewandt, dass die Erregung von einem Gewebe auf ein anderes übertragen wird, wobei die beiden Gewebe verschiedene elektrische Eigenschaften haben. Es zeigt sich, dass dabei gewisse Bedingungen für die Möglichkeit der Übertragung der Erregung erfüllt sein (...) müssen, welche an den vonL. Lapique geforderten Isochronismus erinnern.Dans deux mémoires précédents nous avons développé une théorie mathématique de la propagation de l'excitation nerveuse, basée sur l'hypothèse, que cette propagation est due à une réexcitation par les courants bioélectriques. Dans le mémoire présent nous étudions le cas de deux tissus adjacents, différents dans leurs constantes électriques, du point de vue du passage de l'excitation d'un tissu sur l'autre. Il se trouve, que pour que ce passage put avoir lieu, certaines relations doivent être satisfaites, des relations qui rappellent l'isochronisme postulé parL. Lapique. (shrink)

Is the mathematical description of the Universe quantum, classical, both or neither? The mandated assumption of rationalism is that if an argument is inconsistent, it is flawed for a conclusion. However, suppose the structural basis of the Universe is fundamentally inconsistent. In that case, paradoxes in the frameworks of logic and mathematics would not be anomalies. A geometric model with a counter-rational framework of inconsistent relationships is applied to analyze Hardy’s paradox, the fine structure constant, and the general relationship (...) between the correlated quantum and classical EPR-type structures. The model conjectures that the well-studied paradoxes found in theoretical arguments and empirically in EPR phenomena are not anomalies and instead point to a new framework for modelling universal structures that incorporates inconsistency. (shrink)

In this paper I continue the investigation in Viaggiu, Viaggiu concerning my proposal on the nature of the cosmological constant. In particular, I study both mathematically and physically the quantum Planckian context and I provide, in order to depict quantum fluctuations and in absence of a complete quantum gravity theory, a semiclassical solution where an effective inhomogeneous metric at Planckian scales or above is averaged. In such a framework, a generalization of the well known Buchert formalism is obtained with the (...) foliation in terms of the mean value \\) of the time operator \ in a maximally localizing state \ of a quantum spacetime and in a cosmological context. As a result, after introducing a decoherence length scale \ where quantum fluctuations are averaged on, a classical de Sitter universe emerges with a small cosmological constant depending on \ and frozen in a true vacuum state, provided that the kinematical backreaction is negligible at that scale \. Finally, I analyse the case with a non-vanishing initial spatial curvature \ showing that, for a reasonable large class of models, spatial curvature and kinematical backreation \ are suppressed by the dynamical evolution of the spacetime. (shrink)

It has been observed many times before that, as yet, there are no encompassing, integrated theories of mathematical practice available.To witness, as we currently do, a variety of schools in this field elaborating their philosophical frameworks, and trying to sort out their differences in the course of doing so, is also to be constantly reminded of the fact that a lot of epistemic aspects, extremely relevant to this task, remain dramatically underexamined. This volume wants to contribute to the stock (...) of studies filling this perceived lacuna. It contains papers by established, upcoming, as well as beginning scholars, covering general, metaphilosophical themes such as naturalism, semiotics, pragmaticism, or empiricism, next to more specific topics including the unity of mathematical theories, thruth-flow in mathematics, diagrammatic reasoning, erroneous argumentation, or numerical analysis. (shrink)

The paper considers the intellectual computer mathematics system InparSolver, which is designed to automatically explore and solve basic classes of computational mathematics problems on multi-core computers with graphics accelerators. The problems of results reliability of solving problems with approximate input data are outlined. The features of the use of existing computer mathematics systems are analyzed, their weaknesses are found. The functionality of InparSolver, some innovative approaches to the implementation of effective solutions to problems in a hybrid architecture are described. Examples (...) of applied usage of InparSolver for processes mathematical modeling in various subject areas are given. Nowadays, new more complex objects and phenomena in many subject areas are constantly emerging, which are subject to mathematical research on a computer. This encourages the development of new numerical methods and technologies of mathematical modeling, as well as the creation of more powerful computers for their implementation. With the advent and constant development of supercomputers of various architectures, the problems of their effective use, expansion of tasks range should be solved, ensuring the reliability of computer results and increasing the level of intellectual information support for users ‒ specialists in various fields. Today, the issues of solving these problems are given special attention by many specialists in the fields of information technology and parallel programming. The world's leadingscientists in the field of computer technology see the solution to the problems of efficient usage of modern supercomputers in algorithmic software creation that easily adapts to different computer architectures with different types of memory and coprocessors, supports efficient parallelism on millions of cores etc. In addition, improving the efficiency of high-performance computing on modern supercomputers is provided by their intellectualization, transferring to the computer to perform a significant part of the functions. The industry of development and usage of intelligent computer technologies is one of the main directions of science and technology development in modern society. (shrink)

How fast? How strong? How many? So what? Why do numbers matter in biology? Chromatin binding proteins are forever in motion, exchanging rapidly between bound and free pools. How do regulatory systems whose components are in constant flux ensure stability and flexibility? This review explores the application of quantitative and mathematical approaches to mechanisms of epigenetic regulation. We discuss methods for measuring kinetic parameters and protein quantities in living cells, and explore the insights that have been gained by quantifying (...) and modelling dynamics of chromatin binding proteins. (shrink)

We prove that a countable, complete, first-order theory with infinite dcl( $ \theta $ ) and precisely three non-isomorphic countable models interprets a variant of Ehrenfeucht’s or Peretyatkin’s example.

The fine structure constant α ≡ e2/ c ≈ 1/137 is one of the fundamental parameters of the standard model of particle physics. There is a long history of attempts to derive the measured value of α from an underlying theory, or exhibit it in the form of a compact mathematical expression [2–4, 6, 8, 14–16]. The most significant advance in this endeavour was made by Dirac, who showed that if magnetic monopoles exist, with magnetic charge μ, then..

The fine structure constant appears in several fields of physics and its value is experimentally obtained with a high accuracy. Its physical origin however is unsolved long-standing problem. Richard Feynman expressed the idea that it could be similar to the natural irrational numbers, pi, and e. Amongst the proposed theoretical expressions with values closer to the experimental one is the formula of I. Gorelik which is based on rotating dipole with two empirically suggested coefficients, while the physical origin is unknown. (...) BSM-Supergravitation Unified Theory suggests a physical mechanism defining the fine structure constant. The mathematical expression is based on a spatial precession mode of vibrations in a tetrahedron of spheres with specific properties. The value from the derived expression differs from the CODATA 98 value by 0.000008% only. The conclusion is that the fine structure constant is a natural irrational number. While the irrational number pi, is defined in a 2D space, the fine structure constant is related to a vibration property of a specific formation in 3D space. (shrink)

Let T be a complete, countable, first-order theory with a finite number of countable models. Assuming that dcl(∅) is infinite we show that T has the strict order property.

The word ‘logic’ has many senses. Here we will understand it as meaning an account of what follows from what and why. With contemporary methodology, logic in this sense – though it may not always have been thought of in this way – is a branch of applied mathematics. This has various implications for how one understands a number of issues concerning validity. In this paper I will explain this perspective of logic, and explore some of its consequences with respect (...) to the notion of logical form. (shrink)

The received view on the development of the correspondence rules in Carnap’s philosophy of science is that at first, Carnap assumed the explicit definability of all theoretical terms in observational terms and later weakened this assumption. In the end, he conjectured that all observational terms can be explicitly defined in in theoretical terms, but not vice versa. I argue that from the very beginning, Carnap implicitly held this last view, albeit at times in contradiction to his professed position. To establish (...) this point I argue that, first, Carnap’s ‘Über die Aufgabe der Physik’ is a contribution to the philosophy of science of logical empiricism, contrary to Thomas Mormann and in agreement with Herbert Feigl. Second, Michael Friedman misunderstands the ‘Aufgabe’ with his claim that it describes a method for arriving at explicit definitions for theoretical terms. Another received view on Carnap’s philosophy of science is that it assumed a formalization of physical theories that was too detached from actual physics and thus justly disavowed in favor of the semantic view as, for example, developed by van Fraassen. But the ‘Aufgabe’ and related works including the Aufbau show that from the very beginning to his last works, Carnap suggested formalizing physical theories as restrictions in mathematical spaces, as in the state-space conception of scientific theories favored by van Fraassen. (shrink)

In this paper a number of oppositions which have haunted mathematics and philosophy are described and analyzed. These include the Continuous and the Discrete, the One and the Many, the Finite and the Infinite, the Whole and the Part, and the Constant and the Variable.

The method of Ramsey sentences has been proposed for handling theoretical constructs within a scientific system. Essentially it consists of constructing a certain "monolithic" sentence for an entire theory. In this present paper several improvements are suggested which help to overcome some of the awkward features of the method. In particular we have here many Ramsey sentences rather than just one, each erstwhile primitive theoretical term giving rise to a Ramsey sentence. Such a sentence in effect defines what we call (...) a Ramsey constant. Using Ramsey constants, we attempt to improve the method in important logical and semantical respects. It is suggested also that such constants are of interest for the philosophy of mathematics. (shrink)

The context for this paper is Feferman's theory of explicit mathematics, a formal framework serving many purposes. It is suitable for representing Bishop-style constructive mathematics as well as generalized recursion, including direct expression of structural concepts which admit self-application. The object of investigation here is the theory of explicit mathematics augmented by the monotone fixed point principle, which asserts that any monotone operation on classifications (Feferman's notion of set) possesses a least fixed point. To be more precise, the new axiom (...) not merely postulates the existence of a least solution, but, by adjoining a new functional constant to the language, it is ensured that a fixed point is uniformly presentable as a function of the monotone operation. The upshot of the paper is that the latter extension of explicit mathematics (when based on classical logic) embodies considerable proof-theoretic strength. It is shown that it has at least the strength of the subsystem of second order arithmetic based on Π 1 2 comprehension. (shrink)

My discussion will concern itself with mathematics, medicine and the possible relations between the two. It will be an exercise in logical analysis, a review of some sad, sad facts, and in some sense a promise of glad tidings. In short, it will be an effort to bring the immortal inhabitants of the mathematical heaven into harmonious relations with the mortal ills of man's vale of tears.As to the curious role of mathematics with respect to the natural sciences, several (...) preliminary observations insist on prompt consideration. It is a matter of common knowledge, for example, that mathematics is being constantly applied to physics. With great success, Galileo wedded the two in his study of motion down an inclined plane. The mathematics he used to describe the physical phenomenon involved the equation of a parabola. His daring genius thus got far beyond the conventional circles that were permitted to motion by theology. His courageous use of the then “higher mathematics” was followed by centuries of application of still higher and higher mathematics, till now a small army of mathematicians is busy almost exclusively with the problem of inventing or discovering new varieties of mathematics which could serve physics better and better. The mathematical physicist of to-day is a specialist, nine-tenths mathematician, one-tenth physicist, with the nine-tenths serving the one-tenth. (shrink)

Foundational questions in logic, mathematics, computer science and physics are constant sources of epistemological debate in contemporary philosophy. To what extent is the transfinite part of mathematics completely trustworthy? Why is there a general 'malaise' concerning the logical approach to the foundations of mathematics? What is the role of symmetry in physics? Is it possible to build a coherent worldview compatible with a macroobjectivistic position and based on the quantum picture of the world? What account can be given of opinion (...) change in the light of new evidence? These are some of the questions discussed in this volume, which collects 14 lectures on the foundations of science given at the School of Philosophy of Science, Trieste, October 1989. The volume will be of particular interest to any student or scholar engaged in interdisciplinary research into the foundations of science in the context of contemporary debates. (shrink)

This paper continues investigations of the monotone fixed point principle in the context of Feferman's explicit mathematics begun in [14]. Explicit mathematics is a versatile formal framework for representing Bishop-style constructive mathematics and generalized recursion theory. The object of investigation here is the theory of explicit mathematics augmented by the monotone fixed point principle, which asserts that any monotone operation on classifications (Feferman's notion of set) possesses a least fixed point. To be more precise, the new axiom not merely postulates (...) the existence of a least solution, but, by adjoining a new constant to the language, it is ensured that a fixed point is uniformly presentable as a function of the monotone operation. Let T 0 + UMID denote this extension of explicit mathematics. [14] gave lower bounds for the strength of two subtheories of T 0 + UMID in relating them to fragments of second order arithmetic based on Π 1 2 comprehension. [14] showed that T $_0 \upharpoonright$ + UMID and T $_0 \upharpoonright$ + IND N + UMID have at least the strength of (Π 1 2 - CA) $\upharpoonright$ and (Π 1 2 - CA), respectively. Here we are concerned with the exact reversals. Let UMID N be the monotone fixed-point principle for subclassifications of the natural numbers. Among other results, it is shown that T $_0 \upharpoonright$ + UMID N and T $_0 \upharpoonright$ + IND N + UMID N have the same strength as (Π 1 2 - CA) $\upharpoonright$ and (Π 1 2 - CA), respectively. The results are achieved by constructing set-theoretic models for the aforementioned systems of explicit mathematics in certain extensions of Kripke-Platek set theory and subsequently relating these set theories to subsystems of second arithmetic. (shrink)

The logic CD is an intermediate logic which exactly corresponds to the Kripke models with constant domains. It is known that the logic CD has a Gentzen-type formulation called LD and rules are replaced by the corresponding intuitionistic rules) and that the cut-elimination theorem does not hold for LD. In this paper we present a modification of LD and prove the cut-elimination theorem for it. Moreover we prove a “weak” version of cut-elimination theorem for LD, saying that all “cuts” except (...) some special forms can be eliminated from a proof in LD. From these cut-elimination theorems we obtain some corollaries on syntactical properties of CD: fragments collapsing into intuitionistic logic. Harrop disjunction and existence properties, and a fact on the number of logical symbols in the axiom of CD. (shrink)

Unitary field theories and “SUPER-GUT” theories work with an universal continuum, the structured spacetime of R. Descartes, B. Spinoza, B. Riemann, and A. Einstein, or a (Machian (1–3) ) structured vacuum according the quantum theory of unitary fields (Dirac, (4,5) and Heisenberg (6–8) ). The atomistic aspect of the substantial world is represented by the fundamental constants which are invariant against “all transformations” and which “depend on nothings” (Planck (9–11) ). A satisfactory unitary theory has to involve these (...) class='Hi'>constants like the mathematical numbers. Today, Planck's conception of the three elementary constants ħ, c, and G may be the key to general relativistic quantum field theory like unitary theory. However, the elementary constants are a question of measurement-theory, also.According to Popper's theory (12–16) of induction, such unitary theories are “universal explaining theories.” The fundamental constants involve the complementarity between the universal statements in unitary theory and the “basic statements” in the language of classical observables. (shrink)

Exact measurements are a central practice of modern physics. In certain cases, they are essential for determining values of coefficients, for confirming theories, and for detecting the existence of effects. The history of piezoelectricity at the end of the nineteenth century reveals two different methods of exact measurement: a mathematical versus an “artisanal” approach. In the former, a scientist first carried out the experiment and later employed mathematical methods to reduce error. In the latter, a scientist physically manipulated (...) the experimental apparatus to bypass possible sources of error before its performance. These two approaches were related to German theoreticians and to French experimentalists, respectively. However, affiliation with a particular school rather than nationality was the decisive factor in the differences between the two approaches. Despite differences, adherents of both approaches sought to attain high precision and to eliminate even small experimental errors. This history exhibits the complexity and flexibility of experiments and their analysis. It supports the claim of the “New Experimentalism” that theory does not supply complete directions for practical experimental decisions. An example for this is found in the way an experimental error was discovered only by a comparison to an earlier experiment rather than to a theory. (shrink)

This paper discusses a deterministic model of the spread of an infectious disease in a closed population that was proposed byKermack &McKendrick . The mathematical assumptions on which the model is based are listed and criticized. The ‘threshold theorem’ according to which an epidemic develops if, and only if, the initial population density exceeds a certain value determined by the parameters of the model, is discussed. It is shown that the theorem is not true. A weaker result is stated (...) and proved.A new simplified version of the model is discussed. In this model it is assumed that an infected individual becomes infectious after a constant time and that it will removed after a constant time. Numerical simulations of this model show that the form of the fluctuations in the number of diseased individuals depends on the values of the parameters. The number of infectious individuals may oscillate up and down under certain circumstances. Necessary conditions for the number of infectious individuals to rise above previous levels are stated and proved. (shrink)

We consider the Heisenberg uncertainty principle of position and momentum in 3-dimensional spaces of constant curvature K. The uncertainty of position is defined coordinate independent by the geodesic radius of spherical domains in which the particle is localized after a von Neumann–Lüders projection. By applying mathematical standard results from spectral analysis on manifolds, we obtain the largest lower bound of the momentum deviation in terms of the geodesic radius and K. For hyperbolic spaces, we also obtain a global lower (...) bound \, which is non-zero and independent of the uncertainty in position. Finally, the lower bound for the Schwarzschild radius of a static black hole is derived and given by \, where \ is the Planck length. (shrink)

The in vitro spontaneous contractions of human myometrium samples can be described using a phenomenological model involving different cell states and adjustable parameters. In patients not receiving hormone treatment, the dynamic behavior could be described using a three-state model similar to the one we have already used to explain the oscillations of intra-uterine pressure during parturition. However, the shape of the spontaneous contractions of myometrium from patients on progestin treatment was different, due to a two-step relaxation regime including a latched (...) phase which cannot be simulated using the previous model without introducing an ad hoc mechanism to account for the extra energy involved in this sustained contraction. One way to do this is to introduce an anomalous rate of ATP consumption, the biochemical reasons for which have not yet been elucidated and which cannot be mathematically simulated using our experimental data. An alternative explanation is the reduced cycling rate of actin-myosin cross-bridges known to occur during the latch-phase. Our experimental findings suggest a third possibility, namely a sol-gel transition with a specific relaxation time constant, which would maintain a significant part of the cell population in the contracted-state until the intracellular-medium returns to its normal fluid behavior. (shrink)

The main aim of this paper is to introduce first-order versions of logics of evidence and truth, together with corresponding sound and complete Kripke semantics with variable and constant domains. According to the intuitive interpretation proposed here, these logics intend to represent possibly inconsistent and incomplete information bases over time. The paper also discusses the connections between Belnap-Dunn’s and da Costa’s approaches to paraconsistency, and argues that the logics of evidence and truth combine them in a very natural way.

Word problems in mathematics seem to constantly pose learning difficulties for all kinds of students. Recent work in math education (for example, [Lakoff, G. & Nuñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books]) suggests that the difficulties stem from an inability on the part of students to decipher the metaphorical properties of the language in which such problems are cast. A 2003 pilot study [Danesi, M. (2003a). Semiotica, 145, (...) 71–83] confirmed this hypothesis in an anecdotal way. This paper reviews the implications of that study and of a follow-up one that is described here as well, in the light of how the metaphorical analysis of word problems allows learners to overcome typical difficulties in word problem-solving by teaching them how to flesh out the underlying concepts and convert them into appropriate representations. (shrink)

In his Essay concerning Human Understanding, John Locke explicitly refers to Newton’s Philosophiae naturalis principia mathematica in laudatory but restrained terms: “Mr. Newton, in his never enough to be admired Book, has demonstrated several Propositions, which are so many new Truths, before unknown to the World, and are farther Advances in Mathematical Knowledge” (Essay, 4.7.3). The mathematica of the Principia are thus acknowledged. But what of philosophia naturalis? Locke maintains that natural philosophy, conceived as natural science (as opposed to (...) natural history), would give us demonstrations of the necessary connection between the (ultimately, simple) ideas constitutive of our complex ideas of various natural kinds of substances (e.g., gold). Indeed Locke goes so far as to suggest that a completely adequate natural science would also realize (perhaps, per impossibile) the goal of transforming the corpuscularian hypothesis into knowledge by demonstrating the necessary connection between the ‘microstructure’ (primary qualities of insensible corpuscles) of a particular natural kind of substance (e.g., gold) and the ideas of secondary qualities constitutive of the complex idea of that kind of substance. Locke’s conclusion concerning the possibility of the development of a natural science thus conceived is pessimistic: In vain therefore shall we endeavor to discover by our Ideas, (the only true way of certain and universal knowledge,) what other Ideas are to be found constantly joined with that of our complex Idea of any Substance: since we neither know the real Constitution of the minute Parts, on which their Qualities do depend; not, did we know them could we discover any necessary connexion between them, and any of their Secondary Qualities: which is necessary to be done, before we can certainly know their necessary co-existence (Essay, 4.3.14). It is understandable that, with such a conception of the science of nature, Locke found little of it in Newton’s Principia. In this paper, I further explore what might, perhaps with some hyperbole, be termed Locke’s ‘disappointment’ with the Prinicipia as a contribution to natural science. In particular, I argue that Locke’s adherence to the idealist epistemology of the Way of Ideas entails that mathematics cannot lend its certainty as a scientia to natural philosophy. Consequently, he finds more mathematics than natural philosophy in the Principia. (shrink)

Space and time are geometrical notions that Sophie Germain, a French mathematician, discusses on several occasions in her Pensées diverses, however not only in a geometrical way but also in terms of a philosophical and moral understanding: she speaks of a human’s lifespan, the space they occupy, their place in creation and the knowledge toward which they always aim. This mixture of mathematical and philosophical thinking brings out Germain’s dream: she wants to apply the language of numbers to moral (...) and political issues. As such this chapter aims to examine Germain’s mathematical understanding of space and time in order to discover her moral theory. Furthermore, in a purely hypothetical context (because one does not know which authors she has read), I compare Germain’s moral thinking to Pascal’s moral and mathematical thoughts in Pensées. We underline the possible inspiration by Pascal by highlighting the similarities between these authors concerning space and time, which both treat mathematically and philosophically. These authors agree that time and space can be measured, and thus provide constant and mathematically uniform elements. At the same time, time and space provide the framework for moral thinking. The latter is not “capable” of enjoying the present moment; he is on a quest for the future. For Germain, this results into knowledge, but for Pascal, this is a sign of an unhappy life in which people do not find rest and are constantly looking for a diversion. (shrink)

The aim of this paper is to construct a tableau decision algorithm for the modal description logic K ALC with constant domains. More precisely, we present a tableau procedure that is capable of deciding, given an ALC-formula with extra modal operators (which are applied only to concepts and TBox axioms, but not to roles), whether is satisfiable in a model with constant domains and arbitrary accessibility relations. Tableau-based algorithms have been shown to be practical even for logics of rather high (...) complexity. This gives us grounds to believe that, although the satisfiability problem for K ALC is known to be NEXPTIME-complete, by providing a tableau decision algorithm we demonstrate that highly expressive description logics with modal operators have a chance to be implementable. The paper gives a solution to an open problem of Baader and Laux [5]. (shrink)

Logicism is, roughly speaking, the doctrine that mathematics is fancy logic. So getting clear about the nature of logic is a necessary step in an assessment of logicism. Logic is the study of logical concepts, how they are expressed in languages, their semantic values, and the relationships between these things and the rest of our concepts, linguistic expressions, and their semantic values. A logical concept is what can be expressed by a logical constant in a language. So the question “What (...) is logic?” drives us to the question “What is a logical constant?” Though what follows contains some argument, limitations of space constrain me in large part to express my Credo on this topic with the broad brush of bold assertion and some promissory gestures. (shrink)

In order to resolve the cosmological constant problem, the notion of reference frame is re-examined at the quantum level. By using a quantum non-linear sigma model (Q-NLSM), a theory of quantum spacetime reference frame is proposed. The underlying mathematical structure is a new geometry endowed with intrinsic second central moment (variance) or even higher moments of its coordinates, which generalizes the classical Riemannian geometry based on only first moment (mean) of its coordinates. The second central moment of the coordinates (...) directly modifies the quadratic form distance which is the foundation of the Riemannian geometry. At semi-classical level, the second central moment introduces a flow which continuously deforms the Riemannian geometry driven by its classical Ricci curvature, which is known as the Ricci flow. A generalized equivalence principle of quantum version is also proposed to interpret the new geometry endowed with at least second moment. As a consequence, the spacetime is stabilized against quantum fluctuation, and the cosmological constant problem is resolved within the framework. With an isotropic positive curvature initial condition, the long flow time solution of the Ricci flow exists, the accelerating expansion universe at cosmic scale is an observable effect of the spacetime deformation of the normalized Ricci flow. A deceleration parameter − 0.67 consistent with measurement is obtained by using the reduced volume method introduced by Perelman. Effective theory of gravity within the framework is also discussed. (shrink)

This paper discusses Michael Dummett’s attempt to base the use of intuitionistic logic in mathematics on a proof-conditional semantics. This project is shown to face significant obstacles resulting from the existence of variants of standard intuitionistic logic. In order to overcome these obstacles, Dummett and his followers must give an intuitionistically acceptable completeness proof for intuitionistic logic relative to the BHK interpretation of the logical constants, but there are reasons to doubt that such a proof is possible. The paper (...) concludes by proposing an alternative way of thinking about why one should use intuitionistic logic when doing mathematics. (shrink)

In earlier work we have described how computer algebra may be used to derive composite rate laws for complete systems of equations, using the mathematical technique of Gröbner Bases (Bennett, Davenport and Sauro, 1988). Such composite rate laws may then be fitted to experimental data to yield estimates of kinetic parameters.Recently we have been investigating the practical application of this methodology to the estimation of kinetic parameters for the closed two enzyme system of aspartate aminotransferase (AAT) and malate dehydrogenase (...) (MDH) (Fisher 1990a; Fisher 1990b; Bennett and Fisher, 1990). (shrink)

The problem of how mathematics and physics are related at a foundational level is of interest. The approach taken here is to work towards a coherent theory of physics and mathematics together by examining the theory experiment connection. The role of an implied theory hierarchy and use of computers in comparing theory and experiment is described. The main idea of the paper is to tighten the theory experiment connection by bringing physical theories, as mathematical structures over C, the complex (...) numbers, closer to what is actually done in experimental measurements and computations. The method replaces C by C n which is the set of pairs, R n,I n, of n figure rational numbers in some basis. The properties of these numbers are based on those of numerical measurement outcomes for continuous variables. A model of space and time based on R n is discussed. The model is scale invariant with regions of constant step size interrupted by exponential jumps. A method of taking the limit n→∞ to obtain locally flat continuum-based space and time is outlined. Also R n based space is invariant under scale transformations. These correspond to expansion and contraction of space relative to a flat background. The location of the origin, which is a space and time singularity, does not change under these transformations. Some properties of quantum mechanics, based on C n and on R n space are briefly investigated. (shrink)