Results for 'classical mathematics, constructive mathematics, L. Carnot's mechancis, S. Carnot's thermodynamics, geometry'

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  1. Alternative mathematics and alternative theoretical physics: The method for linking them together.Antonino Drago - 1996 - Epistemologia 19 (1):33-50.
    I characterize Bishop's constructive mathematics as an alternative to classical mathematics, which makes use of the actual infinity. From the history an accurate investigation of past physical theories I obtianed some ones - mainly Lazare Carnot's mechanics and Sadi Carnot's thermodynamics - which are alternative to the dominant theories - e.g. Newtopn's mechanics. The way to link together mathematics to theoretical physics is generalized and some general considerations, in particualr on the geoemtry in theoretical physics, are (...)
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  2.  16
    The Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines, Rigorously Constructed upon the Foundation Laid by S. Carnot and F. ReechC. Truesdell S. Bharatha. [REVIEW]Edward E. Daub - 1979 - Isis 70 (3):478-479.
  3. Two Approaches to Modelling the Universe: Synthetic Differential Geometry and Frame-Valued Sets.John L. Bell - unknown
    I describe two approaches to modelling the universe, the one having its origin in topos theory and differential geometry, the other in set theory. The first is synthetic differential geometry. Traditionally, there have been two methods of deriving the theorems of geometry: the analytic and the synthetic. While the analytical method is based on the introduction of numerical coordinates, and so on the theory of real numbers, the idea behind the synthetic approach is to furnish the subject (...)
     
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  4. The Incredible Shrinking Manifold.John L. Bell - unknown
    Traditionally, there have been two methods of deriving the theorems of geometry: the analytic and the synthetic. While the analytical method is based on the introduction of numerical coordinates, and so on the theory of real numbers, the idea behind the synthetic approach is to furnish the subject of geometry with a purely geometric foundation in which the theorems are then deduced by purely logical means from an initial body of postulates. The most familiar examples of the synthetic (...)
     
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  5.  46
    On Principles In Sadi Carnot’s Theory (1824). Epistemological reflections.Raffaele Pisano - 2010 - Almagest 2/1:128–179 2 (1):128-179.
    In 1824 Sadi Carnot published Réflexions sur la Puissance Motrice du Feu in which he founded almost the entire thermodynamics theory. Two years after his death, his friend Clapeyron introduced the famous diagram PV for analytically representing the famous Carnot’s cycle: one of the main and crucial ideas presented by Carnot in his booklet. Twenty-five years later, in order to achieve the modern version of the theory, Kelvin and Clausius had to reject the caloric hypothesis, which had influenced a few (...)
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  6.  45
    Essay on Machines in General (1786): Text, Translations and Commentaries. Lazare Carnot’s Mechanics—Volume 1.Raffaele Pisano, Jennifer Coopersmith & Murray Peake - 2021 - Springer.
    This book offers insights relevant to modern history and epistemology of physics, mathematics and, indeed, to all the sciences and engineering disciplines emerging of 19th century. This research volume is the first of a set of three Springer books on Lazare Nicolas Marguérite Carnot’s (1753–1823) remarkable work: Essay on Machines in General (Essai sur les machines en général [1783] 1786). The other two forthcoming volumes are: Principes fondamentaux de l’équilibre et du mouvement (1803) and Géométrie de position (1803). Lazare Carnot (...)
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  7.  38
    The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics.John L. Bell - 2019 - Springer Verlag.
    This book explores and articulates the concepts of the continuous and the infinitesimal from two points of view: the philosophical and the mathematical. The first section covers the history of these ideas in philosophy. Chapter one, entitled ‘The continuous and the discrete in Ancient Greece, the Orient and the European Middle Ages,’ reviews the work of Plato, Aristotle, Epicurus, and other Ancient Greeks; the elements of early Chinese, Indian and Islamic thought; and early Europeans including Henry of Harclay, Nicholas of (...)
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  8.  39
    Kant, Riemann, and Reichenbach on Space and Geometry.William L. Harper - 1995 - Proceedings of the Eighth International Kant Congress 1:423-454.
    Classic examples of ostensive geometrical constructions are used to clarify Kant’s account of how they provide knowledge of claims about rigid bodies we can observe and manipulate. It is argued that on Kant’s account claims warranted by ostensive constructions must be limited to scales and tolerances corresponding to our perceptual competencies. This limitation opens the way to view Riemann’s work as contributing valuable conceptual resources for extending geometrical knowledge beyond the bounds of observation. It is argued that neither Reichenbach’s descriptions (...)
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  9.  26
    The swap of integral and limit in constructive mathematics.Rudolf Taschner - 2010 - Mathematical Logic Quarterly 56 (5):533-540.
    Integration within constructive, especially intuitionistic mathematics in the sense of L. E. J. Brouwer, slightly differs from formal integration theories: Some classical results, especially Lebesgue's dominated convergence theorem, have tobe substituted by appropriate alternatives. Although there exist sophisticated, but rather laborious proposals, e.g. by E. Bishop and D. S. Bridges , the reference to partitions and the Riemann-integral, also with regard to the results obtained by R. Henstock and J. Kurzweil , seems to give a better direction. Especially, (...)
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  10. Perception-Action Mutuality Obviates Mental Construction.M. F. Fultot, L. Nie & C. Carello - 2016 - Constructivist Foundations 11 (2):298-307.
    Context: The dominant approach to the study of perception is representational/computational, with an emphasis on the achievements of the brain and the nervous system, which are taken to construct internal models of the world. Alternatives include ecological, embedded, embodied, and enactivist approaches, all of which emphasize the centrality of action in understanding perception. Problem: Despite sharing many theoretical commitments that lead to a rejection of the classical approach, the alternatives are characterized by important contrasts and points of divergence. Here (...)
     
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  11.  28
    Constructive geometry and the parallel postulate.Michael Beeson - 2016 - Bulletin of Symbolic Logic 22 (1):1-104.
    Euclidean geometry, as presented by Euclid, consists of straightedge-and-compass constructions and rigorous reasoning about the results of those constructions. We show that Euclidean geometry can be developed using only intuitionistic logic. This involves finding “uniform” constructions where normally a case distinction is used. For example, in finding a perpendicular to line L through point p, one usually uses two different constructions, “erecting” a perpendicular when p is on L, and “dropping” a perpendicular when p is not on L, (...)
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  12.  24
    The Ethics of Geometry[REVIEW]Dennis L. Sepper - 1990 - Review of Metaphysics 44 (1):149-151.
    The foci of this penetrating study are Euclid's geometry and Descartes' mathematics. It is a contribution to the history of mathematics, but it is much more, for the differing approaches to mathematics in the ancient and the modern worlds is shown to have deep consequences for both doing and knowing. The investigation is centered on the nature of geometrical construction in ancient and modern mathematics, and, by extension, the crucial importance of construction to the reality and self-understanding of modernity.
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  13.  12
    Carnot’s theory of transversals and its applications by Servois and Brianchon: the awakening of synthetic geometry in France.Andrea Del Centina - 2021 - Archive for History of Exact Sciences 76 (1):45-128.
    In this paper we discuss in some depth the main theorems pertaining to Carnot’s theory of transversals, their initial reception by Servois, and the applications that Brianchon made of them to the theory of conic sections. The contributions of these authors brought the long-forgotten theorems of Desargues and Pascal fully to light, renewed the interest in synthetic geometry in France, and prepared the ground from which projective geometry later developed.
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  14.  36
    G. W. F. Hegel: Geometrical Studies Introduction.Alan L. T. Paterson - 2008 - Hegel Bulletin 29 (1-2):118-131.
    Throughout his life, Hegel showed great interest in physics and mathematics. His most sustained, surviving treatment of Euclidean geometry is his early work ‘Geometrische Studien’, which he completed while he was a private tutor [Hoffmeister] in Frankfurt, shortly before leaving for Jena to join Schelling.GSis not easy reading, but despite that, it seems to me that Hegel presents in it a remarkably erudite as well as interesting and insightful critique of geometry. He investigates some of the themes from (...)
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  15.  27
    Discussion. Applied constructive mathematics: on Hellman's 'mathematical constructivism in spacetime'.H. Billinge - 2000 - British Journal for the Philosophy of Science 51 (2):299-318.
    claims that constructive mathematics is inadequate for spacetime physics and hence that constructive mathematics cannot be considered as an alternative to classical mathematics. He also argues that the contructivist must be guilty of a form of a priorism unless she adopts a strong form of anti-realism for science. Here I want to dispute both claims. First, even if there are non-constructive results in physics this does not show that adequate constructive alternatives could not be formulated. (...)
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  16. Surmounting the Cartesian Cut Through Philosophy, Physics, Logic, Cybernetics, and Geometry: Self-reference, Torsion, the Klein Bottle, the Time Operator, Multivalued Logics and Quantum Mechanics. [REVIEW]Diego L. Rapoport - 2011 - Foundations of Physics 41 (1):33-76.
    In this transdisciplinary article which stems from philosophical considerations (that depart from phenomenology—after Merleau-Ponty, Heidegger and Rosen—and Hegelian dialectics), we develop a conception based on topological (the Moebius surface and the Klein bottle) and geometrical considerations (based on torsion and non-orientability of manifolds), and multivalued logics which we develop into a unified world conception that surmounts the Cartesian cut and Aristotelian logic. The role of torsion appears in a self-referential construction of space and time, which will be further related to (...)
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  17. Points as Higher-order Constructs: Whitehead’s Method of Extensive Abstraction.Achille C. Varzi - 2021 - In Stewart Shapiro & Geoffrey Hellman (eds.), The Continuous. Oxford University Press. pp. 347–378.
    Euclid’s definition of a point as “that which has no part” has been a major source of controversy in relation to the epistemological and ontological presuppositions of classical geometry, from the medieval and modern disputes on indivisibilism to the full development of point-free geometries in the 20th century. Such theories stem from the general idea that all talk of points as putative lower-dimensional entities must and can be recovered in terms of suitable higher-order constructs involving only extended regions (...)
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  18. Can constructive mathematics be applied in physics?Douglas S. Bridges - 1999 - Journal of Philosophical Logic 28 (5):439-453.
    The nature of modern constructive mathematics, and its applications, actual and potential, to classical and quantum physics, are discussed.
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  19.  29
    Peirce on Abstraction.William L. Reese - 1961 - Review of Metaphysics 14 (4):704 - 713.
    Recall, if you will, the standard objections to the traditional doctrines. While the most subtle of the competing doctrines is, in my opinion, the Aristotelian and scholastic account of abstraction, the objection to this doctrine is that it requires a realism which is too immediate, so that the forms of one's present state of knowledge are allowed to pass as the forms of nature. And although, as I understand it, Aristotelian mathematics is gained by abstraction from an already fairly abstract (...)
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  20.  32
    A Mechanistic Investigation of the Algae Growth “Droop” Model.V. Lemesle & L. Mailleret - 2008 - Acta Biotheoretica 56 (1):87-102.
    In this work a mechanistic explanation of the classical algae growth model built by M. R. Droop in the late sixties is proposed. We first recall the history of the construction of the “predictive” variable yield Droop model as well as the meaning of the introduced cell quota. We then introduce some theoretical hypotheses on the biological phenomena involved in nutrient storage by the algae that lead us to a “conceptual” model. Though more complex than Droop’s one, our model (...)
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  21.  26
    Construction and the Role of Schematism in Kant's Philosophy of Mathematics.A. T. Winterbourne - 1981 - Studies in History and Philosophy of Science Part A 12 (1):33.
    This paper argues that kant's general epistemology incorporates a theory of algebra which entails a less constricted view of kant's philosophy of mathematics than is sometimes given. To extract a plausible theory of algebra from the "critique of pure reason", It is necessary to link kant's doctrine of mathematical construction to the idea of the "schematism". Mathematical construction can be understood to accommodate algebraic symbolism as well as the more familiar spatial configurations of geometry.
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  22.  10
    L’Académie et les géomètres.Thomas El Murr Bénatouïl - 2010 - Philosophie Antique 10:41-80.
    L’article met en lumière la continuité intellectuelle de l’Académie à propos d’une question précise, les rapports entre philosophie et géométrie. On soutient d’abord que, dans les livres VI-VII de la République, Platon ne cherche pas à réformer les pratiques des géomètres mais identifie les contraintes incontournables de leurs raisonnements (constructions, hypothèses), qui constituent et limitent leur objectivité. On montre ensuite que cette analyse constitue le cadre des réflexions académiciennes ultérieures sur la géométrie. Speusippe reprend et développe l’analyse platonicienne des constructions (...)
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  23.  27
    Operationalism: An Interpretation of the Philosophy of Ancient Greek Geometry.Viktor Blåsjö - 2022 - Foundations of Science 27 (2):587-708.
    I present a systematic interpretation of the foundational purpose of constructions in ancient Greek geometry. I argue that Greek geometers were committed to an operationalist foundational program, according to which all of mathematics—including its entire ontology and epistemology—is based entirely on concrete physical constructions. On this reading, key foundational aspects of Greek geometry are analogous to core tenets of 20th-century operationalist/positivist/constructivist/intuitionist philosophy of science and mathematics. Operationalism provides coherent answers to a range of traditional philosophical problems regarding (...) mathematics, such as the epistemic warrant and generality of diagrammatic reasoning, superposition, and the relation between constructivism and proof by contradiction. Alleged logical flaws in Euclid can be interpreted as sound operationalist reasoning. Operationalism also provides a compelling philosophical motivation for the otherwise inexplicable Greek obsession with cube duplication, angle trisection, and circle quadrature. Operationalism makes coherent sense of numerous specific choices made in this tradition, and suggests new interpretations of several solutions to these problems. In particular, I argue that: Archytas’s cube duplication was originally a single-motion machine; Diocles’s cissoid was originally traced by a linkage device; Greek conic section theory was thoroughly constructive, based on the conic compass; in a few cases, string-based constructions of conic sections were used instead; pointwise constructions of curves were rejected in foundational contexts by Greek mathematicians, with good reason. Operationalism enables us to view the classical geometrical tradition as a more unified and philosophically aware enterprise than has hitherto been recognised. (shrink)
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  24.  11
    Book Review: The Poetics of Perspective. [REVIEW]Harvey L. Hix - 1995 - Philosophy and Literature 19 (2):368-370.
    In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Poetics of PerspectiveHarvey L. HixThe Poetics of Perspective, by James Elkins; xv & 324 pp. Ithaca: Cornell University Press, 1994, $39.95.The Poetics of Perspective does not mention that Leonardo was born more than 100 years before Galileo and nearly 200 before Newton, but doing so would underscore its thesis. According to James Elkins, our anachronistic view of perspective, invented in the Enlightenment, systematically distorts our understanding of (...)
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  25.  27
    S. Feferman and W. Sieg Inductive definitions and subsystems of analysis. Iterated inductive definitions and subsystems of analysis: recent proof-theoretical studies, by Wilfried Buchholz, Solomon Feferman, Wolfram Pohlers, and Wilfried Sieg. Lecture notes in mathematics, vol. 897, Springer-Verlag, Berlin, Heidelberg, and New York, 1981, pp. 16–77. - Solomon Feferman and Wilfried Sieg. Proof theoretic equivalences between classical and constructive theories for analysis. Iterated inductive definitions and subsystems of analysis: recent proof-theoretical studies, by Wilfried Buchholz, Solomon Feferman, Wolfram Pohlers, and Wilfried Sieg. Lecture notes in mathematics, vol. 897, Springer-Verlag, Berlin, Heidelberg, and New York, 1981, pp. 78–142. - Solomon Feferman. Iterated inductive fixed-point theories: application to Hancock's conjecture. Patras logic symposion, Proceedings of the logic symposion held at Patras, Greece, August 18–22, 1980, edited by George Metakides, Studies in logic. [REVIEW]Helmut Pfeiffer - 1994 - Journal of Symbolic Logic 59 (2):668-670.
  26.  19
    Complements of Intersections in Constructive Mathematics.Douglas S. Bridges & Hajime Ishihara - 1994 - Mathematical Logic Quarterly 40 (1):35-43.
    We examine, from a constructive perspective, the relation between the complements of S, T, and S ∩ T in X, where X is either a metric space or a normed linear space. The fundamental question addressed is: If x is distinct from each element of S ∩ T, if s ϵ S, and if t ϵ T, is x distinct from s or from t? Although the classical answer to this question is trivially affirmative, constructive answers involve (...)
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  27.  19
    Constructive notions of set: Part I. Sets in Martin–Löf type theory.Laura Crosilla - 2005 - Annali Del Dipartimento di Filosofia 11:347-387.
    This is the first of two articles dedicated to the notion of constructive set. In them we attempt a comparison between two different notions of set which occur in the context of the foundations for constructive mathematics. We also put them under perspective by stressing analogies and differences with the notion of set as codified in the classical theory Zermelo–Fraenkel. In the current article we illustrate in some detail the notion of set as expressed in Martin–L¨of type (...)
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  28.  15
    The L.E.J. Brouwer Centenary Symposium: proceedings of the conference held in Noordwijkerhout, 8-13 June 1981.L. E. J. Brouwer, A. S. Troelstra & D. van Dalen (eds.) - 1982 - New York, N.Y.: Sole distributors for the U.S.A. and Canada, Elsevier Science Pub. Co..
  29.  44
    Two Mathematically Equivalent Versions of Maxwell’s Equations.Tepper L. Gill & Woodford W. Zachary - 2011 - Foundations of Physics 41 (1):99-128.
    This paper is a review of the canonical proper-time approach to relativistic mechanics and classical electrodynamics. The purpose is to provide a physically complete classical background for a new approach to relativistic quantum theory. Here, we first show that there are two versions of Maxwell’s equations. The new version fixes the clock of the field source for all inertial observers. However now, the (natural definition of the effective) speed of light is no longer an invariant for all observers, (...)
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  30.  33
    Plato’s mathematical construction.Reviel Netz - 2003 - Classical Quarterly 53 (2):500-509.
  31.  21
    Intuitionistic Proof Versus Classical Truth: The Role of Brouwer’s Creative Subject in Intuitionistic Mathematics.Enrico Martino - 2018 - Cham, Switzerland: Springer Verlag.
    This book examines the role of acts of choice in classical and intuitionistic mathematics. Featuring fifteen papers - both new and previously published - it offers a fresh analysis of concepts developed by the mathematician and philosopher L.E.J. Brouwer, the founder of intuitionism. The author explores Brouwer's idealization of the creative subject as the basis for intuitionistic truth, and in the process he also discusses an important, related question: to what extent does the intuitionistic perspective succeed in avoiding the (...)
  32.  9
    “Hume’s Guillotine” in the Interdisciplinary Context.G. G. Malinetsky & A. A. Skurlyagin - 2019 - Russian Journal of Philosophical Sciences 12:7-25.
    The authors deal with the classic paradox of ethical theories, “Hume’s guillotine,” based on the contradiction in morality between what is and what should be. At the same time, the theological justification of moral principles is beyond this criticism, because the sacred commandments “by definition” combine what is and what should be, overcoming the “secular” gap between being and duty, and thus “the problem of transition from description to evaluation is removed.” Modern ways of removing this contradiction, revealed by Hume, (...)
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  33.  9
    Physics, mathematics, and all that quantum jazz.Shu Tanaka, Masamitsu Bando & Utkan Güngördü (eds.) - 2014 - New Jersey: World Scientific.
    My life as a quantum physicist / M. Nakahara -- A review on operator quantum error correction - Dedicated to Professor Mikio Nakahara on the occasion of his 60th birthday / C.-K. Li, Y.-T. Poon and N.-S. Sze -- Implementing measurement operators in linear optical and solid-state qubits / Y. Ota, S. Ashhab and F. Nori -- Fast and accurate simulation of quantum computing by multi-precision MPS: Recent development / A. Saitoh -- Entanglement properties of a quantum lattice-gas model on (...)
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  34. Kant's Philosophy of Geometry--On the Road to a Final Assessment.L. Kvasz - 2011 - Philosophia Mathematica 19 (2):139-166.
    The paper attempts to summarize the debate on Kant’s philosophy of geometry and to offer a restricted area of mathematical practice for which Kant’s philosophy would be a reasonable account. Geometrical theories can be characterized using Wittgenstein’s notion of pictorial form . Kant’s philosophy of geometry can be interpreted as a reconstruction of geometry based on one of these forms — the projective form . If this is correct, Kant’s philosophy is a reasonable reconstruction of such theories (...)
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  35. On the Role of Constructivism in Mathematical Epistemology.A. Quale - 2012 - Constructivist Foundations 7 (2):104-111.
    Context: the position of pure and applied mathematics in the epistemic conflict between realism and relativism. Problem: To investigate the change in the status of mathematical knowledge over historical time: specifically, the shift from a realist epistemology to a relativist epistemology. Method: Two examples are discussed: geometry and number theory. It is demonstrated how the initially realist epistemic framework – with mathematics situated in a platonic ideal reality from where it governs our physical world – became untenable, with the (...)
     
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  36.  23
    Handbook of Constructive Mathematics.Douglas Bridges, Hajime Ishihara, Michael Rathjen & Helmut Schwichtenberg (eds.) - 2023 - Cambridge: Cambridge University Press.
    Constructive mathematics – mathematics in which ‘there exists’ always means ‘we can construct’ – is enjoying a renaissance. Fifty years on from Bishop’s groundbreaking account of constructive analysis, constructive mathematics has spread out to touch almost all areas of mathematics and to have profound influence in theoretical computer science. This handbook gives the most complete overview of modern constructive mathematics, with contributions from leading specialists surveying the subject’s myriad aspects. Major themes include: constructive algebra and (...)
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  37.  54
    On probability theory and probabilistic physics—Axiomatics and methodology.L. S. Mayants - 1973 - Foundations of Physics 3 (4):413-433.
    A new formulation involving fulfillment of all the Kolmogorov axioms is suggested for acomplete probability theory. This proves to be not a purely mathematical discipline. Probability theory deals with abstract objects—images of various classes of concrete objects—whereas experimental statistics deals with concrete objects alone. Both have to be taken into account. Quantum physics and classical statistical physics prove to be different aspects ofone probabilistic physics. The connection of quantum mechanics with classical statistical mechanics is examined and the origin (...)
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  38.  9
    The Philosophy of Physics (review). [REVIEW]Martin Curd - 2000 - Journal of the History of Philosophy 38 (4):602-603.
    In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Philosophy of PhysicsMartin CurdRoberto Torretti. The Philosophy of Physics. Cambridge: Cambridge University Press, 1999. Pp. xvi + 512. Cloth, $64.95. Paper, $23.95.This is the first volume in a new Cambridge series, "The Evolution of Modern Philosophy." It is a historical work, tracing the interaction between physics and philosophy from the scientific revolution of the seventeenth century through general relativity and quantum mechanics in the twentieth century. The (...)
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  39.  25
    Philosophy of Geometry from Riemann to Poincaré. [REVIEW]S. L. - 1982 - Review of Metaphysics 35 (3):633-634.
    This deeply researched, carefully constructed and very thoughtful book is fascinating in its own right as well as being indispensable background material for anyone interested in current philosophical thought about space, time, and geometry.
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  40.  24
    The Ideal and the Real: An Outline of Kant's Theory of Space, Time and Mathematical Construction. By Anthony Winterbourne. [REVIEW]John L. Treloar - 1991 - Modern Schoolman 68 (3):265-267.
  41. De-Psychologizing Intuitionism: The Anti-Realist Rejection of Classical Logic.Sanford Shieh - 1993 - Dissertation, Harvard University
    The most puzzling and intriguing aspect of intuitionism as a philosophy of mathematics is its claim that classical deductive reasoning in mathematics is illegitimate. The two most well-known proponents of this position are L. E. J. Brouwer and Michael Dummett. Both of their criticisms of the use of classical logic in mathematics have, by and large, been taken to depend on the thesis that the principle of bivalence does not apply to mathematical statements; and the difference between these (...)
     
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  42.  31
    Blick Von der intuitionistischen warte.von A. Heyting - 1958 - Dialectica 12 (3‐4):332-345.
    ZusammenfassungDie Arbeit enthält Bemerkungen über den Intuitionismus and über seine Beziehungen zu anderen Gebieten der Grundlagenforschung. Innerhalb der intuitionistischen Mathematik werden, im Anschluss an die Kritik von Griss gegen den Gebrauch der Negation, Evidenzstufen unterschieden, abhängend von der Art, in der bedingte Konstruktionen zugelassen werden. Auch werden gewisse Schwierigkeiten in der Theorie der endlichen Spezies diskutiert. Was die Grundlagenforschung im Aligemeinen betrifft, wird bemerkt, dass sie die klassische Mathematik weitgehend in ihre intuitiven, formalen and platonischen Bestandteile zerlegt hat. Es wird (...)
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  43.  45
    Probabilistics: A lost science.L. S. Mayants - 1982 - Foundations of Physics 12 (8):797-811.
    For certain methodological and historical reasons, the science of probability (probabilistics) had never been constructed before as a single whole, and it has basically split into probability theory and into statistics. One of the reasons was the neglect of an extremely important methodological principle which reads: It is necessary to distinguish strictly between concrete objects and abstract objects. This principle is displayed and exemplified. Its use has made it possible to discover the basic phenomenon of probalilistics and to construct the (...)
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  44.  68
    The Extended Relativity Theory in Born-Clifford Phase Spaces with a Lower and Upper Length Scales and Clifford Group Geometric Unification.Carlos Castro - 2005 - Foundations of Physics 35 (6):971-1041.
    We construct the Extended Relativity Theory in Born-Clifford-Phase spaces with an upper R and lower length λ scales (infrared/ultraviolet cutoff). The invariance symmetry leads naturally to the real Clifford algebra Cl (2, 6, R) and complexified Clifford Cl C (4) algebra related to Twistors. A unified theory of all Noncommutative branes in Clifford-spaces is developed based on the Moyal-Yang star product deformation quantization whose deformation parameter involves the lower/upper scale $$(\hbar \lambda / R)$$. Previous work led us to show from (...)
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  45.  53
    Beltrami's Kantian View of Non-Euclidean Geometry.Ricardo J. Gómez - 1986 - Kant Studien 77 (1-4):102-107.
    Beltrami's first allegedly true interpretation of lobachevsky's geometry can be conceived as (i) pursuing a kantian program insofar as it shows that all the geometrical lobachevskian concepts are constructible in the euclidean space of our human representation, And (ii) proving, Even to kant, That a non-Euclidean geometry is not only logically possible (something that kant never denied) but also mathematically acceptable from a kantian point of view (something that kant would have accepted only after beltrami's interpretation).
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  46.  12
    Constructive aspects of Riemann’s permutation theorem for series.J. Berger, Douglas Bridges, Hannes Diener & Helmet Schwichtenberg - forthcoming - Logic Journal of the IGPL.
    The notions of permutable and weak-permutable convergence of a series|$\sum _{n=1}^{\infty }a_{n}$|of real numbers are introduced. Classically, these two notions are equivalent, and, by Riemann’s two main theorems on the convergence of series, a convergent series is permutably convergent if and only if it is absolutely convergent. Working within Bishop-style constructive mathematics, we prove that Ishihara’s principle BD-|$\mathbb {N}$|implies that every permutably convergent series is absolutely convergent. Since there are models of constructive mathematics in which the Riemann permutation (...)
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  47.  33
    Mathematics and God’s Point of View1.Zbigniew Król - 2016 - Studies in Logic, Grammar and Rhetoric 44 (1):81-96.
    In this paper the final stages of the historical process of the emergence of actual infinity in mathematics are considered. The application of God’s point of view – i.e. the possibility to create mathematics from a divine perspective, i.e. from the point of view of an eternal, timeless, omniscience and unlimited subject of cognition – is one of the main factors in this process. Nicole Oresme is the first man who systematically used actual infinity in mathematical reasoning, constructions and proofs (...)
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    Force, Mathematics, and Physics in Newton's Principia: A New Approach to Enduring Issues.Koffi Maglo - 2007 - Science in Context 20 (4):571-600.
    ArgumentThis paper investigates the conceptual treatment and mathematical modeling of force in Newton's Principia. It argues that, contrary to currently dominant views, Newton's concept of force is best understood as a physico-mathematical construct with theoretical underpinnings rather than a “mathematical construct” or an ontologically “neutral” concept. It uses various philosophical and historical frameworks to clarify interdisciplinary issues in the history of science and draws upon the distinction between axiomatic systems in mathematics and physics, as well as discovery patterns in science. (...)
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    The Mathematics of the Area Law: Kepler's Successful Proof in Epitome Astronomiae Copernicanae (1621).A. E. L. Davis - 2003 - Archive for History of Exact Sciences 57 (5):355-393.
    Epitome V (1621), and consisted of matching an element of area to an element of time, where each was mathematically determined. His treatment of the area depended solely on the geometry of Euclid's Elements, involving only straight-line and circle propositions – so we have to account for his deliberate avoidance of the sophisticated conic-geometry associated with Apollonius. We show also how his proof could have been made watertight according to modern standards, using methods that lay entirely within his (...)
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  50. Agent-based modeling: the right mathematics for the social sciences?Paul L. Borrill & Leigh Tesfatsion - 2011 - In J. B. Davis & D. W. Hands (eds.), Elgar Companion to Recent Economic Methodology. Edward Elgar Publishers. pp. 228.
    This study provides a basic introduction to agent-based modeling (ABM) as a powerful blend of classical and constructive mathematics, with a primary focus on its applicability for social science research. The typical goals of ABM social science researchers are discussed along with the culture-dish nature of their computer experiments. The applicability of ABM for science more generally is also considered, with special attention to physics. Finally, two distinct types of ABM applications are summarized in order to illustrate concretely (...)
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