Results for 'foundations of mathematics'

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  1. Physical Foundations of Mathematics (In Russian).Andrey Smirnov - manuscript
    The physical foundations of mathematics in the theory of emergent space-time-matter were considered. It is shown that mathematics, including logic, is a consequence of equation which describes the fundamental field. If the most fundamental level were described not by mathematics, but something else, then instead of mathematics there would be consequences of this something else.
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  2. Categorical foundations of mathematics or how to provide foundations for abstract mathematics.Jean-Pierre Marquis - 2013 - Review of Symbolic Logic 6 (1):51-75.
    Fefermans argument is indeed convincing in a certain context, it can be dissolved entirely by modifying the context appropriately.
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  3. The foundations of mathematics from a historical viewpoint.Antonino Drago - 2015 - Epistemologia 38 (1):133-151.
    A new hypothesis on the basic features characterising the Foundations of Mathematics is suggested. By means of them the entire historical development of Mathematics before the 20th Century is summarised through a table. Also the several programs, launched around the year 1900, on the Foundations of Mathematics are characterised by a corresponding table. The major difficulty that these programs met was to recognize an alternative to the basic feature of the deductive organization of a theory (...)
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  4. Hyperintensional Foundations of Mathematical Platonism.David Elohim - manuscript
    This paper aims to provide hyperintensional foundations for mathematical platonism. I examine Hale and Wright's (2009) objections to the merits and need, in the defense of mathematical platonism and its epistemology, of the thesis of Necessitism. In response to Hale and Wright's objections to the role of epistemic and metaphysical modalities in providing justification for both the truth of abstraction principles and the success of mathematical predicate reference, I examine the Necessitist commitments of the abundant conception of properties endorsed (...)
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  5. Category theory and the foundations of mathematics: Philosophical excavations.Jean-Pierre Marquis - 1995 - Synthese 103 (3):421 - 447.
    The aim of this paper is to clarify the role of category theory in the foundations of mathematics. There is a good deal of confusion surrounding this issue. A standard philosophical strategy in the face of a situation of this kind is to draw various distinctions and in this way show that the confusion rests on divergent conceptions of what the foundations of mathematics ought to be. This is the strategy adopted in the present paper. It (...)
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  6. Wittgenstein on Gödelian 'Incompleteness', Proofs and Mathematical Practice: Reading Remarks on the Foundations of Mathematics, Part I, Appendix III, Carefully.Wolfgang Kienzler & Sebastian Sunday Grève - 2016 - In Sebastian Sunday Grève & Jakub Mácha (eds.), Wittgenstein and the Creativity of Language. Palgrave Macmillan. pp. 76-116.
    We argue that Wittgenstein’s philosophical perspective on Gödel’s most famous theorem is even more radical than has commonly been assumed. Wittgenstein shows in detail that there is no way that the Gödelian construct of a string of signs could be assigned a useful function within (ordinary) mathematics. — The focus is on Appendix III to Part I of Remarks on the Foundations of Mathematics. The present reading highlights the exceptional importance of this particular set of remarks and, (...)
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  7. Descriptive Complexity, Computational Tractability, and the Logical and Cognitive Foundations of Mathematics.Markus Pantsar - 2020 - Minds and Machines 31 (1):75-98.
    In computational complexity theory, decision problems are divided into complexity classes based on the amount of computational resources it takes for algorithms to solve them. In theoretical computer science, it is commonly accepted that only functions for solving problems in the complexity class P, solvable by a deterministic Turing machine in polynomial time, are considered to be tractable. In cognitive science and philosophy, this tractability result has been used to argue that only functions in P can feasibly work as computational (...)
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  8.  77
    Wittgenstein on the Foundations of Mathematics.Andrew McLean-Inglis - 1992 - Dissertation, Oxford University
    In Part I, an attempt is made to survey the original source material on which any detailed assessment of Wittgenstein's remarks on the foundations of mathematics from his middle and later periods ought to be based. This survey is presented within the context of a sketch of Wittgenstein's biography, which also mentions some of the major developments in his thinking. In addition, certain main themes are emphasized; these have to do primarily with the Kantian aspects of Wittgenstein's thought (...)
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  9. Axioms, Definitions, and the Pragmatic a priori: Peirce and Dewey on the “Foundations” of Mathematical Science.Bradley C. Dart - 2024 - European Journal of Pragmatism and American Philosophy 16 (1).
    Peirce and Dewey were generally more concerned with the process of scientific activity than purely mathematical work. However, their accounts of knowledge production afford some insights into the epistemology of mathematical postulates, especially definition and axioms. Their rejection of rationalist metaphysics and their emphasis on continuity in inquiry provides the pretext for the pragmatic a priori – hypothetical and operational assumptions whose justification relies on their fruitfulness in the long run. This paper focuses on the application of this idea to (...)
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  10. The Quantum Strategy of Completeness: On the Self-Foundation of Mathematics.Vasil Penchev - 2020 - Cultural Anthropology eJournal (Elsevier: SSRN) 5 (136):1-12.
    Gentzen’s approach by transfinite induction and that of intuitionist Heyting arithmetic to completeness and the self-foundation of mathematics are compared and opposed to the Gödel incompleteness results as to Peano arithmetic. Quantum mechanics involves infinity by Hilbert space, but it is finitist as any experimental science. The absence of hidden variables in it interpretable as its completeness should resurrect Hilbert’s finitism at the cost of relevant modification of the latter already hinted by intuitionism and Gentzen’s approaches for completeness. This (...)
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  11. Hilbert's different aims for the foundations of mathematics.Besim Karakadılar - manuscript
    The foundational ideas of David Hilbert have been generally misunderstood. In this dissertation prospectus, different aims of Hilbert are summarized and a new interpretation of Hilbert's work in the foundations of mathematics is roughly sketched out. Hilbert's view of the axiomatic method, his response to criticisms of set theory and intuitionist criticisms of the classical foundations of mathematics, and his view of the role of logical inference in mathematical reasoning are briefly outlined.
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  12. Principles of mathematics.Bertrand Russell - 1931 - New York,: W.W. Norton & Company.
    Published in 1903, this book was the first comprehensive treatise on the logical foundations of mathematics written in English. It sets forth, as far as possible without mathematical and logical symbolism, the grounds in favour of the view that mathematics and logic are identical. It proposes simply that what is commonly called mathematics are merely later deductions from logical premises. It provided the thesis for which _Principia Mathematica_ provided the detailed proof, and introduced the work of (...)
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  13. A Pluralist Foundation of the Mathematics of the First Half of the Twentieth Century.Antonino Drago - 2017 - Journal of the Indian Council of Philosophical Research 34 (2):343-363.
    MethodologyA new hypothesis on the basic features characterizing the Foundations of Mathematics is suggested.Application of the methodBy means of it, the several proposals, launched around the year 1900, for discovering the FoM are characterized. It is well known that the historical evolution of these proposals was marked by some notorious failures and conflicts. Particular attention is given to Cantor's programme and its improvements. Its merits and insufficiencies are characterized in the light of the new conception of the FoM. (...)
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  14. Logic. of Descriptions. A New Approach to the Foundations of Mathematics and Science.Joanna Golińska-Pilarek & Taneli Huuskonen - 2012 - Studies in Logic, Grammar and Rhetoric 27 (40):63-94.
    We study a new formal logic LD introduced by Prof. Grzegorczyk. The logic is based on so-called descriptive equivalence, corresponding to the idea of shared meaning rather than shared truth value. We construct a semantics for LD based on a new type of algebras and prove its soundness and completeness. We further show several examples of classical laws that hold for LD as well as laws that fail. Finally, we list a number of open problems. -/- .
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  15. WHAT IS THE "x" WHICH OCCURS IN "sin x"? Being an Essay Towards a Conceptual Foundations of Mathematics.Mohamed Amer - manuscript
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  16. The "Unreasonable" Effectiveness of Mathematics: The Foundational Approach of the Theoretic Alternatives.Catalin Barboianu - 2015 - Revista de Filosofie 62 (1):58-71.
    The attempts of theoretically solving the famous puzzle-dictum of physicist Eugene Wigner regarding the “unreasonable” effectiveness of mathematics as a problem of analytical philosophy, started at the end of the 19th century, are yet far from coming out with an acceptable theoretical solution. The theories developed for explaining the empirical “miracle” of applied mathematics vary in nature, foundation and solution, from denying the existence of a genuine problem to structural theories with an advanced level of mathematical formalism. Despite (...)
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  17. Marriages of Mathematics and Physics: A Challenge for Biology.Arezoo Islami & Giuseppe Longo - 2017 - Progress in Biophysics and Molecular Biology 131:179-192.
    The human attempts to access, measure and organize physical phenomena have led to a manifold construction of mathematical and physical spaces. We will survey the evolution of geometries from Euclid to the Algebraic Geometry of the 20th century. The role of Persian/Arabic Algebra in this transition and its Western symbolic development is emphasized. In this relation, we will also discuss changes in the ontological attitudes toward mathematics and its applications. Historically, the encounter of geometric and algebraic perspectives enriched the (...)
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  18. Computability. Computable functions, logic, and the foundations of mathematics[REVIEW]R. Zach - 2002 - History and Philosophy of Logic 23 (1):67-69.
    Epstein and Carnielli's fine textbook on logic and computability is now in its second edition. The readers of this journal might be particularly interested in the timeline `Computability and Undecidability' added in this edition, and the included wall-poster of the same title. The text itself, however, has some aspects which are worth commenting on.
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  19.  88
    Foundations of Relational Realism: A Topological Approach to Quantum Mechanics and the Philosophy of Nature.Michael Epperson & Elias Zafiris - 2013 - Lanham: Lexington Books. Edited by Elias Zafiris.
    Foundations of Relational Realism presents an intuitive interpretation of quantum mechanics, based on a revised decoherent histories interpretation, structured within a category theoretic topological formalism. -/- If there is a central conceptual framework that has reliably borne the weight of modern physics as it ascends into the twenty-first century, it is the framework of quantum mechanics. Because of its enduring stability in experimental application, physics has today reached heights that not only inspire wonder, but arguably exceed the limits of (...)
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  20. A NEW PHILOSOPHICAL FOUNDATION OF CONSTRUCTIVE MATHEMATICS.Antonino Drago - manuscript
    The current definition of Constructive mathematics as “mathematics within intuitionist logic” ignores two fundamental issues. First, the kind of organization of the theory at issue. I show that intuitionist logic governs a problem-based organization, whose model is alternative to that of the deductive-axiomatic organization, governed by classical logic. Moreover, this dichotomy is independent of that of the kind of infinity, either potential or actual, to which respectively correspond constructive mathematical and classical mathematical tools. According to this view a (...)
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  21. Reconstructing the Unity of Mathematics circa 1900.David J. Stump - 1997 - Perspectives on Science 5 (3):383-417.
    Standard histories of mathematics and of analytic philosophy contend that work on the foundations of mathematics was motivated by a crisis such as the discovery of paradoxes in set theory or the discovery of non-Euclidean geometries. Recent scholarship, however, casts doubt on the standard histories, opening the way for consideration of an alternative motive for the study of the foundations of mathematics—unification. Work on foundations has shown that diverse mathematical practices could be integrated into (...)
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  22. Poincaré’s Philosophy of Mathematics.A. P. Bird - 2021 - Cantor's Paradise (00):00.
    It is undeniable Poincaré was a very famous and influential scientist. So, possibly because of it, it was relatively easy for him to participate in the heated discussions of the foundations of mathematics in the early 20th century. We can say it was “easy” because he didn't get involved in this subject by writing great treatises, or entire books about his own philosophy of mathematics (as other authors from the same period did). Poincaré contributed to the philosophy (...)
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  23. The Axiom of choice in Quine's New Foundations for Mathematical Logic.Ernst P. Specker - 1954 - Journal of Symbolic Logic 19 (2):127-128.
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  24. Make It So: Imperatival Foundations for Mathematics.Neil Barton, Ethan Russo & Chris Scambler - manuscript
    This article articulates and assesses an imperatival approach to the foundations of mathematics. The core idea for the program is that mathematical domains of interest can fruitfully be viewed as the outputs of construction procedures. We apply this idea to provide a novel formalisation of arithmetic and set theory in terms of such procedures, and discuss the significance of this perspective for the philosophy of mathematics.
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  25. Sofia A. Yanovskaya: The Marxist Pioneer of Mathematical Logic in the Soviet Union.Dimitris Kilakos - 2019 - Transversal: International Journal for the Historiography of Science 6:49-64.
    K. Marx’s 200th jubilee coincides with the celebration of the 85 years from the first publication of his “Mathematical Manuscripts” in 1933. Its editor, Sofia Alexandrovna Yanovskaya (1896–1966), was a renowned Soviet mathematician, whose significant studies on the foundations of mathematics and mathematical logic, as well as on the history and philosophy of mathematics are unduly neglected nowadays. Yanovskaya, as a militant Marxist, was actively engaged in the ideological confrontation with idealism and its influence on modern (...) and their interpretation. Concomitantly, she was one of the pioneers of mathematical logic in the Soviet Union, in an era of fierce disputes on its compatibility with Marxist philosophy. Yanovskaya managed to embrace in an originally Marxist spirit the contemporary level of logico-philosophical research of her time. Due to her highly esteemed status within Soviet academia, she became one of the most significant pillars for the culmination of modern mathematics in the Soviet Union. In this paper, I attempt to trace the influence of the complex socio-cultural context of the first decades of the Soviet Union on Yanovskaya’s work. Among the several issues I discuss, her encounter with L. Wittgenstein is striking. (shrink)
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  26. Frege on the Foundation of Geometry in Intuition.Jeremy Shipley - 2015 - Journal for the History of Analytical Philosophy 3 (6).
    I investigate the role of geometric intuition in Frege’s early mathematical works and the significance of his view of the role of intuition in geometry to properly understanding the aims of his logicist project. I critically evaluate the interpretations of Mark Wilson, Jamie Tappenden, and Michael Dummett. The final analysis that I provide clarifies the relationship of Frege’s restricted logicist project to dominant trends in German mathematical research, in particular to Weierstrassian arithmetization and to the Riemannian conceptual/geometrical tradition at Göttingen. (...)
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  27. Univalent Foundations as a Foundation for Mathematical Practice.Harry Crane - 2018
    I prove that invoking the univalence axiom is equivalent to arguing 'without loss of generality' (WLOG) within Propositional Univalent Foundations (PropUF), the fragment of Univalent Foundations (UF) in which all homotopy types are mere propositions. As a consequence, I argue that practicing mathematicians, in accepting WLOG as a valid form of argument, implicitly accept the univalence axiom and that UF rightly serves as a Foundation for Mathematical Practice. By contrast, ZFC is inconsistent with WLOG as it is applied, (...)
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  28. Artificial Evil and the Foundation of Computer Ethics.Luciano Floridi & J. W. Sanders - 2001 - Springer Netherlands. Edited by Luciano Floridi & J. W. Sanders.
    Moral reasoning traditionally distinguishes two types of evil:moral (ME) and natural (NE). The standard view is that ME is the product of human agency and so includes phenomena such as war,torture and psychological cruelty; that NE is the product of nonhuman agency, and so includes natural disasters such as earthquakes, floods, disease and famine; and finally, that more complex cases are appropriately analysed as a combination of ME and NE. Recently, as a result of developments in autonomous agents in cyberspace, (...)
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  29. Artificial evil and the foundation of computer ethics.L. Floridi & J. Sanders - 2000 - Etica E Politica 2 (2).
    Moral reasoning traditionally distinguishes two types of evil: moral and natural. The standard view is that ME is the product of human agency and so includes phenomena such as war, torture and psychological cruelty; that NE is the product of nonhuman agency, and so includes natural disasters such as earthquakes, floods, disease and famine; and finally, that more complex cases are appropriately analysed as a combination of ME and NE. Recently, as a result of developments in autonomous agents in cyberspace, (...)
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  30. Expanding the notion of inconsistency in mathematics: the theoretical foundations of mutual inconsistency.Carolin Antos - forthcoming - From Contradiction to Defectiveness to Pluralism in Science: Philosophical and Formal Analyses.
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  31. Anti-Realism and Anti-Revisionism in Wittgenstein’s Philosophy of Mathematics.Anderson Nakano - 2020 - Grazer Philosophische Studien 97 (3):451-474.
    Since the publication of the Remarks on the Foundations of Mathematics, Wittgenstein’s interpreters have endeavored to reconcile his general constructivist/anti-realist attitude towards mathematics with his confessed anti-revisionary philosophy. In this article, the author revisits the issue and presents a solution. The basic idea consists in exploring the fact that the so-called “non-constructive results” could be interpreted so that they do not appear non-constructive at all. The author substantiates this solution by showing how the translation of mathematical results, (...)
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  32. Foundations of Intensional Logic.David Kaplan - 1964 - Dissertation, Ucla
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  33. A Structuralist Proposal for the Foundations of the Natural Numbers.Desmond Alan Ford - manuscript
    This paper introduces a novel object that has less structure than, and is ontologically prior to the natural numbers. As such it is a candidate model of the foundation that lies beneath the natural numbers. The implications for the construction of mathematical objects built upon that foundation are discussed.
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  34. Foundation of paralogical nonstandard analysis and its application to some famous problems of trigonometrical and orthogonal series.Jaykov Foukzon - manuscript
    FOURTH EUROPEAN CONGRESS OF MATHEMATICS STOCKHOLM,SWEDEN JUNE27 ­ - JULY 2, 2004 Contributed papers L. Carleson’s celebrated theorem of 1965 [1] asserts the pointwise convergence of the partial Fourier sums of square integrable functions. The Fourier transform has a formulation on each of the Euclidean groups R , Z and Τ .Carleson’s original proof worked on Τ . Fefferman’s proof translates very easily to R . M´at´e [2] extended Carleson’s proof to Z . Each of the statements of the (...)
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  35. The Mereological Foundation of Megethology.Massimiliano Carrara & Enrico Martino - 2016 - Journal of Philosophical Logic 45 (2):227-235.
    In Mathematics is megethology. Philosophia Mathematica, 1, 3–23) David K. Lewis proposes a structuralist reconstruction of classical set theory based on mereology. In order to formulate suitable hypotheses about the size of the universe of individuals without the help of set-theoretical notions, he uses the device of Boolos’ plural quantification for treating second order logic without commitment to set-theoretical entities. In this paper we show how, assuming the existence of a pairing function on atoms, as the unique assumption non (...)
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  36. Hilbert mathematics versus (or rather “without”) Gödel mathematics: V. Ontomathematics!Vasil Penchev - 2024 - Metaphysics eJournal (Elsevier: SSRN) 17 (10):1-57.
    The paper is the final, fifth part of a series of studies introducing the new conceptions of “Hilbert mathematics” and “ontomathematics”. The specific subject of the present investigation is the proper philosophical sense of both, including philosophy of mathematics and philosophy of physics not less than the traditional “first philosophy” (as far as ontomathematics is a conservative generalization of ontology as well as of Heidegger’s “fundamental ontology” though in a sense) and history of philosophy (deepening Heidegger’s destruction of (...)
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  37. Skolem’s “paradox” as logic of ground: The mutual foundation of both proper and improper interpretations.Vasil Penchev - 2020 - Epistemology eJournal (Elsevier: SSRN) 13 (19):1-16.
    A principle, according to which any scientific theory can be mathematized, is investigated. That theory is presupposed to be a consistent text, which can be exhaustedly represented by a certain mathematical structure constructively. In thus used, the term “theory” includes all hypotheses as yet unconfirmed as already rejected. The investigation of the sketch of a possible proof of the principle demonstrates that it should be accepted rather a metamathematical axiom about the relation of mathematics and reality. Its investigation needs (...)
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  38. Mathematics as a science of non-abstract reality: Aristotelian realist philosophies of mathematics.James Franklin - 2022 - Foundations of Science 27 (2):327-344.
    There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays (...)
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  39. The case of quantum mechanics mathematizing reality: the “superposition” of mathematically modelled and mathematical reality: Is there any room for gravity?Vasil Penchev - 2020 - Cosmology and Large-Scale Structure eJournal (Elsevier: SSRN) 2 (24):1-15.
    A case study of quantum mechanics is investigated in the framework of the philosophical opposition “mathematical model – reality”. All classical science obeys the postulate about the fundamental difference of model and reality, and thus distinguishing epistemology from ontology fundamentally. The theorems about the absence of hidden variables in quantum mechanics imply for it to be “complete” (versus Einstein’s opinion). That consistent completeness (unlike arithmetic to set theory in the foundations of mathematics in Gödel’s opinion) can be interpreted (...)
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  40. Sociocultural Foundations of Modern Science.Rinat M. Nugayev - 2012 - Journal of Culture Studies 2 (8):1-16.
    It is argued that the origins of modern science can be revealed due to joint account of external and internal factors. The author tries to keep it in mind applying his scientific revolution model according to which the growth of knowledge consists in interaction, interpenetration and even unification of different scientific research programmes. Hence the Copernican Revolution as a matter of fact consisted in realization and elimination of the gap between the mathematical astronomy and Aristotelian qualitative physics in Ptolemaic cosmology. (...)
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  41.  40
    ONE AND THE MULTIPLE ON THE PHILOSOPHY OF MATHEMATICS - ALEXIS KARPOUZOS.Alexis Karpouzos - 2025 - Comsic Spirit 1:6.
    The relationship between the One and the Multiple in mystic philosophy is a profound and central theme that explores the nature of existence, the cosmos, and the divine. This theme is present in various mystical traditions, including those of the East and West, and it addresses the paradoxical coexistence of the unity and multiplicity of all things. -/- In mystic philosophy, the **One** often represents the ultimate reality, the source from which all things emanate and to which all things return. (...)
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  42. Time and Information in the Foundations of Physics.Vasil Penchev - 2020 - Information Theory and Research eJournal (Elsevier: SSRN) 1 (25):1-12.
    The paper justifies the following theses: The totality can found time if the latter is axiomatically represented by its “arrow” as a well-ordering. Time can found choice and thus information in turn. Quantum information and its units, the quantum bits, can be interpreted as their generalization as to infinity and underlying the physical world as well as the ultimate substance of the world both subjective and objective. Thus a pathway of interpretation between the totality via time, order, choice, and information (...)
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  43. On the epistemological foundations of the law of the lever.Maarten Van Dyck - 2009 - Studies in History and Philosophy of Science Part A 40 (3):315-318.
    In this paper I challenge Paolo Palmieri’s reading of the Mach-Vailati debate on Archimedes’s proof of the law of the lever. I argue that the actual import of the debate concerns the possible epistemic (as opposed to merely pragmatic) role of mathematical arguments in empirical physics, and that construed in this light Vailati carries the upper hand. This claim is defended by showing that Archimedes’s proof of the law of the lever is not a way of appealing to a non-empirical (...)
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  44. Poincaré on the Foundation of Geometry in the Understanding.Jeremy Shipley - 2017 - In Maria Zack & Dirk Schlimm (eds.), Research in History and Philosophy of Mathematics: The CSHPM 2016 Annual Meeting in Calgary, Alberta. New York: Birkhäuser. pp. 19-37.
    This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view that geometry is the (...)
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  45. The Importance of Developing a Foundation for Naive Category Theory.Marcoen J. T. F. Cabbolet - 2015 - Thought: A Journal of Philosophy 4 (4):237-242.
    Recently Feferman has outlined a program for the development of a foundation for naive category theory. While Ernst has shown that the resulting axiomatic system is still inconsistent, the purpose of this note is to show that nevertheless some foundation has to be developed before naive category theory can replace axiomatic set theory as a foundational theory for mathematics. It is argued that in naive category theory currently a ‘cookbook recipe’ is used for constructing categories, and it is explicitly (...)
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  46. Conceptual Origami: Unfolding the Social Construction of Mathematics.Andrew Notier - 2019 - Philosophy Now 1 (134):28-29.
    This essay presents the framework for the foundational axiom and conceptual underpinnings of mathematics and how they are applied.
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  47. Logical Foundations of Local Gauge Symmetry and Symmetry Breaking.Yingrui Yang - 2022 - Journal of Human Cognition 6 (1):18-23.
    The present paper intends to report two results. It is shown that the formula P(x)=∀y∀z[¬G(x, y)→¬M(z)] provides the logic underlying gauge symmetry, where M denotes the predicate of being massive. For the logic of spontaneous symmetry breaking, by Higgs mechanism, we have P(x)=∀y∀z[G(x, y)→M(z)]. Notice that the above two formulas are not logically equivalent. The results are obtained by integrating four components, namely, gauge symmetry and Higgs mechanism in quantum field theory, and Gödel's incompleteness theorem and Tarski's indefinability theorem in (...)
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  48. Quantum Physics: an overview of a weird world: A primer on the conceptual foundations of quantum physics.Marco Masi - 2019 - Indy Edition.
    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 (...)
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  49. Normativity and Mathematics: A Wittgensteinian Approach to the Study of Number.J. Robert Loftis - 1999 - Dissertation, Northwestern University
    I argue for the Wittgensteinian thesis that mathematical statements are expressions of norms, rather than descriptions of the world. An expression of a norm is a statement like a promise or a New Year's resolution, which says that someone is committed or entitled to a certain line of action. A expression of a norm is not a mere description of a regularity of human behavior, nor is it merely a descriptive statement which happens to entail a norms. The view can (...)
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  50. Fourteen Arguments in Favour of a Formalist Philosophy of Real Mathematics.Karlis Podnieks - 2015 - Baltic Journal of Modern Computing 3 (1):1-15.
    The formalist philosophy of mathematics (in its purest, most extreme version) is widely regarded as a “discredited position”. This pure and extreme version of formalism is called by some authors “game formalism”, because it is alleged to represent mathematics as a meaningless game with strings of symbols. Nevertheless, I would like to draw attention to some arguments in favour of game formalism as an appropriate philosophy of real mathematics. For the most part, these arguments have not yet (...)
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