Results for 'non-Archimedean mathematics'

969 found
Order:
  1. Non-Archimedean Preferences Over Countable Lotteries.Jeffrey Sanford Russell - 2020 - Journal of Mathematical Economics 88 (May 2020):180-186.
    We prove a representation theorem for preference relations over countably infinite lotteries that satisfy a generalized form of the Independence axiom, without assuming Continuity. The representing space consists of lexicographically ordered transfinite sequences of bounded real numbers. This result is generalized to preference orders on abstract superconvex spaces.
    Download  
     
    Export citation  
     
    Bookmark   5 citations  
  2. Non-archimedean analysis on the extended hyperreal line *R_d and the solution of some very old transcendence conjectures over the field Q.Jaykov Foukzon - 2015 - Advances in Pure Mathematics 5 (10):587-628.
    In 1980 F. Wattenberg constructed the Dedekind completiond of the Robinson non-archimedean field  and established basic algebraic properties of d [6]. In 1985 H. Gonshor established further fundamental properties of d [7].In [4] important construction of summation of countable sequence of Wattenberg numbers was proposed and corresponding basic properties of such summation were considered. In this paper the important applications of the Dedekind completiond in transcendental number theory were considered. We dealing using set theory ZFC  (-model of (...)
    Download  
     
    Export citation  
     
    Bookmark  
  3. The Solution of the Invariant Subspace Problem. Complex Hilbert space. Part I.External non-Archimedean field {∗}^ℝ_{c}^{#} by Cauchy completion of the internal non-Archimedean field {∗}^ℝ.Jaykov Foukzon - 2022 - Journal of Advances in Mathematics and Computer Science 37 (10):51-89.
    Our main result will be that: if T is a bounded linear operator on an infinite-dimensional separable complex Hilbert space H,it follow that T has a non-trivial closed invariant subspace.
    Download  
     
    Export citation  
     
    Bookmark  
  4. Natorp's mathematical philosophy of science.Thomas Mormann - 2022 - Studia Kantiana 20 (2):65 - 82.
    This paper deals with Natorp’s version of the Marburg mathematical philosophy of science characterized by the following three features: The core of Natorp’s mathematical philosophy of science is contained in his “knowledge equation” that may be considered as a mathematical model of the “transcendental method” conceived by Natorp as the essence of the Marburg Neo-Kantianism. For Natorp, the object of knowledge was an infinite task. This can be elucidated in two different ways: Carnap, in the Aufbau, contended that this endeavor (...)
    Download  
     
    Export citation  
     
    Bookmark  
  5. Big-Oh Notations, Elections, and Hyperreal Numbers: A Socratic Dialogue.Samuel Alexander & Bryan Dawson - 2023 - Proceedings of the ACMS 23.
    We provide an intuitive motivation for the hyperreal numbers via electoral axioms. We do so in the form of a Socratic dialogue, in which Protagoras suggests replacing big-oh complexity classes by real numbers, and Socrates asks some troubling questions about what would happen if one tried to do that. The dialogue is followed by an appendix containing additional commentary and a more formal proof.
    Download  
     
    Export citation  
     
    Bookmark  
  6. The role of epistemological models in Veronese's and Bettazzi's theory of magnitudes.Paola Cantù - 2010 - In Marcello D'Agostino, Federico Laudisa, Giulio Giorello, Telmo Pievani & Corrado Sinigaglia (eds.), New Essays in Logic and Philosophy of Science. College Publications.
    The philosophy of mathematics has been accused of paying insufficient attention to mathematical practice: one way to cope with the problem, the one we will follow in this paper on extensive magnitudes, is to combine the `history of ideas' and the `philosophy of models' in a logical and epistemological perspective. The history of ideas allows the reconstruction of the theory of extensive magnitudes as a theory of ordered algebraic structures; the philosophy of models allows an investigation into the way (...)
    Download  
     
    Export citation  
     
    Bookmark  
  7. Non-Archimedean population axiologies.Calvin Baker - forthcoming - Economics and Philosophy:1-22.
    Non-Archimedean population axiologies – also known as lexical views – claim (i) that a sufficient number of lives at a very high positive welfare level would be better than any number of lives at a very low positive welfare level and/or (ii) that a sufficient number of lives at a very low negative welfare level would be worse than any number of lives at a very high negative welfare level. Such axiologies are popular because they can avoid the (Negative) (...)
    Download  
     
    Export citation  
     
    Bookmark  
  8. Basic non-Archimedean functional analysis over non-Archimedean field c #. Applications to constructive quantum field theory.Jaykov Foukzon - 2024 - HAL Id: hal-04583394.
    Functional analysis works with TVS (Topological Vector Spaces), classically over archimedean fields like  and .Canonical non-Archimedean functional analysis, where alternative but equally valid number systems such as p-adic numbers p etc. are fundamental, is a fast-growing discipline. This paper deals with TVS over non-classical non-Archimedean fields.
    Download  
     
    Export citation  
     
    Bookmark  
  9. Totality, Regularity, and Cardinality in Probability Theory.Paolo Mancosu & Guillaume Massas - 2024 - Philosophy of Science 91 (3):721-740.
    Recent developments in generalized probability theory have renewed a debate about whether regularity (i.e., the constraint that only logical contradictions get assigned probability 0) should be a necessary feature of both chances and credences. Crucial to this debate, however, are some mathematical facts regarding the interplay between the existence of regular generalized probability measures and various cardinality assumptions. We improve on several known results in the literature regarding the existence of regular generalized probability measures. In particular, we give necessary and (...)
    Download  
     
    Export citation  
     
    Bookmark  
  10. Explicit Legg-Hutter intelligence calculations which suggest non-Archimedean intelligence.Samuel Allen Alexander & Arthur Paul Pedersen - forthcoming - Lecture Notes in Computer Science.
    Are the real numbers rich enough to measure intelligence? We generalize a result of Alexander and Hutter about the so-called Legg-Hutter intelligence measures of reinforcement learning agents. Using the generalized result, we exhibit a paradox: in one particular version of the Legg-Hutter intelligence measure, certain agents all have intelligence 0, even though in a certain sense some of them outperform others. We show that this paradox disappears if we vary the Legg-Hutter intelligence measure to be hyperreal-valued rather than real-valued.
    Download  
     
    Export citation  
     
    Bookmark  
  11. (1 other version)Non-deductive justification in mathematics.A. C. Paseau - 2023 - Handbook of the History and Philosophy of Mathematical Practice.
    In mathematics, the deductive method reigns. Without proof, a claim remains unsolved, a mere conjecture, not something that can be simply assumed; when a proof is found, the problem is solved, it turns into a “result,” something that can be relied on. So mathematicians think. But is there more to mathematical justification than proof? -/- The answer is an emphatic yes, as I explain in this article. I argue that non-deductive justification is in fact pervasive in mathematics, and (...)
    Download  
     
    Export citation  
     
    Bookmark  
  12. Non-mathematical dimensions of randomness: Implications for problem gambling.Catalin Barboianu - 2024 - Journal of Gambling Issues 36.
    Randomness, a core concept of gambling, is seen in problem gambling as responsible for the formation of the math-related cognitive distortions, especially the Gambler’s Fallacy. In problem-gambling research, the concept of randomness was traditionally referred to as having a mathematical nature and categorized and approached as such. Randomness is not a mathematical concept, and I argue that its weak mathematical dimension is not decisive at all for the randomness-related issues in gambling and problem gambling, including the correction of the misconceptions (...)
    Download  
     
    Export citation  
     
    Bookmark  
  13. 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 (...)
    Download  
     
    Export citation  
     
    Bookmark   2 citations  
  14. The Archimedean trap: Why traditional reinforcement learning will probably not yield AGI.Samuel Allen Alexander - 2020 - Journal of Artificial General Intelligence 11 (1):70-85.
    After generalizing the Archimedean property of real numbers in such a way as to make it adaptable to non-numeric structures, we demonstrate that the real numbers cannot be used to accurately measure non-Archimedean structures. We argue that, since an agent with Artificial General Intelligence (AGI) should have no problem engaging in tasks that inherently involve non-Archimedean rewards, and since traditional reinforcement learning rewards are real numbers, therefore traditional reinforcement learning probably will not lead to AGI. We indicate (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  15. Extended mathematical cognition: external representations with non-derived content.Karina Vold & Dirk Schlimm - 2020 - Synthese 197 (9):3757-3777.
    Vehicle externalism maintains that the vehicles of our mental representations can be located outside of the head, that is, they need not be instantiated by neurons located inside the brain of the cogniser. But some disagree, insisting that ‘non-derived’, or ‘original’, content is the mark of the cognitive and that only biologically instantiated representational vehicles can have non-derived content, while the contents of all extra-neural representational vehicles are derived and thus lie outside the scope of the cognitive. In this paper (...)
    Download  
     
    Export citation  
     
    Bookmark   7 citations  
  16. Mathematical and Non-causal Explanations: an Introduction.Daniel Kostić - 2019 - Perspectives on Science 1 (27):1-6.
    In the last couple of years, a few seemingly independent debates on scientific explanation have emerged, with several key questions that take different forms in different areas. For example, the questions what makes an explanation distinctly mathematical and are there any non-causal explanations in sciences (i.e., explanations that don’t cite causes in the explanans) sometimes take a form of the question of what makes mathematical models explanatory, especially whether highly idealized models in science can be explanatory and in virtue of (...)
    Download  
     
    Export citation  
     
    Bookmark  
  17. Non-deductive Logic in Mathematics: The Probability of Conjectures.James Franklin - 2013 - In Andrew Aberdein & Ian J. Dove (eds.), The Argument of Mathematics. Dordrecht, Netherland: Springer. pp. 11--29.
    Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective (...)
    Download  
     
    Export citation  
     
    Bookmark  
  18. After Non-Euclidean Geometry: Intuition, Truth and the Autonomy of Mathematics.Janet Folina - 2018 - Journal for the History of Analytical Philosophy 6 (3).
    The mathematical developments of the 19th century seemed to undermine Kant’s philosophy. Non-Euclidean geometries challenged Kant’s view that there is a spatial intuition rich enough to yield the truth of Euclidean geometry. Similarly, advancements in algebra challenged the view that temporal intuition provides a foundation for both it and arithmetic. Mathematics seemed increasingly detached from experience as well as its form; moreover, with advances in symbolic logic, mathematical inference also seemed independent of intuition. This paper considers various philosophical responses (...)
    Download  
     
    Export citation  
     
    Bookmark  
  19. Non-mathematical Content by Mathematical Means.Sam Adam-Day - manuscript
    In this paper, I consider the use of mathematical results in philosophical arguments arriving at conclusions with non-mathematical content, with the view that in general such usage requires additional justification. As a cautionary example, I examine Kreisel’s arguments that the Continuum Hypothesis is determined by the axioms of Zermelo-Fraenkel set theory, and interpret Weston’s 1976 reply as showing that Kreisel fails to provide sufficient justification for the use of his main technical result. If we take the perspective that mathematical results (...)
    Download  
     
    Export citation  
     
    Bookmark  
  20. Are mathematical explanations causal explanations in disguise?A. Jha, Douglas Campbell, Clemency Montelle & Phillip L. Wilson - 2024 - Philosophy of Science 91 (4):887-905.
    There is a major debate as to whether there are non-causal mathematical explanations of physical facts that show how the facts under question arise from a degree of mathematical necessity considered stronger than that of contingent causal laws. We focus on Marc Lange’s account of distinctively mathematical explanations to argue that purported mathematical explanations are essentially causal explanations in disguise and are no different from ordinary applications of mathematics. This is because these explanations work not by appealing to what (...)
    Download  
     
    Export citation  
     
    Bookmark  
  21. Archimedean Ethics (10th edition).Pedro Brea - 2020 - Texasphilosophical.
    What effect has finding the Archimedean point in ourselves had on how we look at ethics? The modern era of philosophy began with Descartes finding within himself an unshakable point from which to pursue knowledge of the world and himself. This intellectual alienation from the world into the universal mathematical structures of the human mind has led to a reversal where, henceforth, production, rather than contemplation, of knowledge became epistemologically superior. Guided by Hannah Arendt’s discussion of the Archimedean (...)
    Download  
     
    Export citation  
     
    Bookmark  
  22. Because without Cause: Non-Causal Explanations in Science and Mathematics[REVIEW]Mark Povich & Carl F. Craver - 2018 - Philosophical Review 127 (3):422-426.
    Lange’s collection of expanded, mostly previously published essays, packed with numerous, beautiful examples of putatively non-causal explanations from biology, physics, and mathematics, challenges the increasingly ossified causal consensus about scientific explanation, and, in so doing, launches a new field of philosophic investigation. However, those who embraced causal monism about explanation have done so because appeal to causal factors sorts good from bad scientific explanations and because the explanatory force of good explanations seems to derive from revealing the relevant causal (...)
    Download  
     
    Export citation  
     
    Bookmark   8 citations  
  23. Expressivism, Anti-Archimedeanism and Supervenience.Christine Tiefensee - 2014 - Res Publica 20 (2):163-181.
    Metaethics is traditionally understood as a non-moral discipline that examines moral judgements from a standpoint outside of ethics. This orthodox understanding has recently come under pressure from anti-Archimedeans, such as Ronald Dworkin and Matthew Kramer, who proclaim that rather than assessing morality from an external perspective, metaethical theses are themselves substantive moral claims. In this paper, I scrutinise this anti-Archimedean challenge as applied to the metaethical position of expressivism. More precisely, I examine the claim that expressivists do not avoid (...)
    Download  
     
    Export citation  
     
    Bookmark   3 citations  
  24. Gödel mathematics versus Hilbert mathematics. I. The Gödel incompleteness (1931) statement: axiom or theorem?Vasil Penchev - 2022 - Logic and Philosophy of Mathematics eJournal (Elsevier: SSRN) 14 (9):1-56.
    The present first part about the eventual completeness of mathematics (called “Hilbert mathematics”) is concentrated on the Gödel incompleteness (1931) statement: if it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic, this would pioneer the pathway to Hilbert mathematics. One of the main arguments that it is an axiom consists in the direct contradiction of the axiom of induction in arithmetic and the axiom of infinity (...)
    Download  
     
    Export citation  
     
    Bookmark  
  25. Mathematical Explanation by Law.Sam Baron - 2019 - British Journal for the Philosophy of Science 70 (3):683-717.
    Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that the basic theory faces. The final view (...)
    Download  
     
    Export citation  
     
    Bookmark   19 citations  
  26. Applied Mathematics without Numbers.Jack Himelright - 2023 - Philosophia Mathematica 31 (2):147-175.
    In this paper, I develop a "safety result" for applied mathematics. I show that whenever a theory in natural science entails some non-mathematical conclusion via an application of mathematics, there is a counterpart theory that carries no commitment to mathematical objects, entails the same conclusion, and the claims of which are true if the claims of the original theory are "correct": roughly, true given the assumption that mathematical objects exist. The framework used for proving the safety result has (...)
    Download  
     
    Export citation  
     
    Bookmark   2 citations  
  27. Non-deterministic algebraization of logics by swap structures1.Marcelo E. Coniglio, Aldo Figallo-Orellano & Ana Claudia Golzio - 2020 - Logic Journal of the IGPL 28 (5):1021-1059.
    Multialgebras have been much studied in mathematics and in computer science. In 2016 Carnielli and Coniglio introduced a class of multialgebras called swap structures, as a semantic framework for dealing with several Logics of Formal Inconsistency that cannot be semantically characterized by a single finite matrix. In particular, these LFIs are not algebraizable by the standard tools of abstract algebraic logic. In this paper, the first steps towards a theory of non-deterministic algebraization of logics by swap structures are given. (...)
    Download  
     
    Export citation  
     
    Bookmark   12 citations  
  28. Exposing some points of interest about non-exposed points of desirability.Arthur Van Camp & Teddy Seidenfeld - 2022 - International Journal of Approximate Reasoning 144:129-159.
    We study the representation of sets of desirable gambles by sets of probability mass functions. Sets of desirable gambles are a very general uncertainty model, that may be non-Archimedean, and therefore not representable by a set of probability mass functions. Recently, Cozman (2018) has shown that imposing the additional requirement of even convexity on sets of desirable gambles guarantees that they are representable by a set of probability mass functions. Already more that 20 years earlier, Seidenfeld et al. (1995) (...)
    Download  
     
    Export citation  
     
    Bookmark  
  29. Mathematical Explanations and the Piecemeal Approach to Thinking About Explanation.Gabriel Târziu - 2018 - Logique Et Analyse 61 (244):457-487.
    A new trend in the philosophical literature on scientific explanation is that of starting from a case that has been somehow identified as an explanation and then proceed to bringing to light its characteristic features and to constructing an account for the type of explanation it exemplifies. A type of this approach to thinking about explanation – the piecemeal approach, as I will call it – is used, among others, by Lange (2013) and Pincock (2015) in the context of their (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  30. Who's Afraid of Mathematical Diagrams?Silvia De Toffoli - 2023 - Philosophers' Imprint 23 (1).
    Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for non-diagrammatic expressions. Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs. In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic proofs, one from topology and one from algebra, I show (...)
    Download  
     
    Export citation  
     
    Bookmark   3 citations  
  31. Mathematical representation: playing a role.Kate Hodesdon - 2014 - Philosophical Studies 168 (3):769-782.
    The primary justification for mathematical structuralism is its capacity to explain two observations about mathematical objects, typically natural numbers. Non-eliminative structuralism attributes these features to the particular ontology of mathematics. I argue that attributing the features to an ontology of structural objects conflicts with claims often made by structuralists to the effect that their structuralist theses are versions of Quine’s ontological relativity or Putnam’s internal realism. I describe and argue for an alternative explanation for these features which instead explains (...)
    Download  
     
    Export citation  
     
    Bookmark   3 citations  
  32. Mathematical Explanations in Evolutionary Biology or Naturalism? A Challenge for the Statisticalist.Fabio Sterpetti - 2021 - Foundations of Science 27 (3):1073-1105.
    This article presents a challenge that those philosophers who deny the causal interpretation of explanations provided by population genetics might have to address. Indeed, some philosophers, known as statisticalists, claim that the concept of natural selection is statistical in character and cannot be construed in causal terms. On the contrary, other philosophers, known as causalists, argue against the statistical view and support the causal interpretation of natural selection. The problem I am concerned with here arises for the statisticalists because the (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  33.  82
    Foundations of Mathematics.Kliment Babushkovski - manuscript
    Analytical philosophy defines mathematics as an extension of logic. This research will restructure the progress in mathematical philosophy made by analytical thinkers like Wittgenstein, Russell, and Frege. We are setting up a new theory of mathematics and arithmetic’s familiar to Wittgenstein’s philosophy of language. The analytical theory proposed here proves that mathematics can be defined with non-logical terms, like numbers, theorems, and operators. We’ll explain the role of the arithmetical operators and geometrical theorems to be foundational in (...)
    Download  
     
    Export citation  
     
    Bookmark  
  34. Mathematical instrumentalism, Gödel’s theorem, and inductive evidence.Alexander Paseau - 2011 - Studies in History and Philosophy of Science Part A 42 (1):140-149.
    Mathematical instrumentalism construes some parts of mathematics, typically the abstract ones, as an instrument for establishing statements in other parts of mathematics, typically the elementary ones. Gödel’s second incompleteness theorem seems to show that one cannot prove the consistency of all of mathematics from within elementary mathematics. It is therefore generally thought to defeat instrumentalisms that insist on a proof of the consistency of abstract mathematics from within the elementary portion. This article argues that though (...)
    Download  
     
    Export citation  
     
    Bookmark   2 citations  
  35. Platonism and Intra-mathematical Explanation.Sam Baron - forthcoming - Philosophical Quarterly.
    I introduce an argument for Platonism based on intra-mathematical explanation: the explanation of one mathematical fact by another. The argument is important for two reasons. First, if the argument succeeds then it provides a basis for Platonism that does not proceed via standard indispensability considerations. Second, if the argument fails it can only do so for one of three reasons: either because there are no intra-mathematical explanations, or because not all explanations are backed by dependence relations, or because some form (...)
    Download  
     
    Export citation  
     
    Bookmark  
  36. Logic, mathematics, physics: from a loose thread to the close link: Or what gravity is for both logic and mathematics rather than only for physics.Vasil Penchev - 2023 - Astrophysics, Cosmology and Gravitation Ejournal 2 (52):1-82.
    Gravitation is interpreted to be an “ontomathematical” force or interaction rather than an only physical one. That approach restores Newton’s original design of universal gravitation in the framework of “The Mathematical Principles of Natural Philosophy”, which allows for Einstein’s special and general relativity to be also reinterpreted ontomathematically. The entanglement theory of quantum gravitation is inherently involved also ontomathematically by virtue of the consideration of the qubit Hilbert space after entanglement as the Fourier counterpart of pseudo-Riemannian space. Gravitation can be (...)
    Download  
     
    Export citation  
     
    Bookmark  
  37. John Barwise & Lawrence Moss, Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena[REVIEW]Varol Akman - 1997 - Journal of Logic, Language and Information 6 (4):460-464.
    This is a review of Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena, written by Jon Barwise and Lawrence Moss and published by CSLI Publications in 1996.
    Download  
     
    Export citation  
     
    Bookmark  
  38. (1 other version)Not so distinctively mathematical explanations: topology and dynamical systems.Aditya Jha, Douglas Campbell, Clemency Montelle & Phillip L. Wilson - 2022 - Synthese 200 (3):1-40.
    So-called ‘distinctively mathematical explanations’ (DMEs) are said to explain physical phenomena, not in terms of contingent causal laws, but rather in terms of mathematical necessities that constrain the physical system in question. Lange argues that the existence of four or more equilibrium positions of any double pendulum has a DME. Here we refute both Lange’s claim itself and a strengthened and extended version of the claim that would pertain to any n-tuple pendulum system on the ground that such explanations are (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  39. Non-Commutative Scalar Fields.Parker Emmerson - 2024 - Journal of Liberated Mathematics 1:9.
    In this paper, we explore numerical methods for simulating scalar field con-figurations in non-commutative two-dimensional spaces. We focus on the finite difference techniques employed to compute mixed partial derivatives and the action functional in the presence of non-commutative corrections. The methods presented address the challenges posed by non-commutative geometry, specifically in computing the mixed derivative terms that arise due to the deformation of spatial coordinates. We introduce semi-implicit time-stepping schemes to en-sure numerical stability when dealing with stiff nonlinear terms. The (...)
    Download  
     
    Export citation  
     
    Bookmark  
  40. Bolzano’s Mathematical Infinite.Anna Bellomo & Guillaume Massas - 2021 - Review of Symbolic Logic:1-55.
    Bernard Bolzano (1781–1848) is commonly thought to have attempted to develop a theory of size for infinite collections that follows the so-called part–whole principle, according to which the whole is always greater than any of its proper parts. In this paper, we develop a novel interpretation of Bolzano’s mature theory of the infinite and show that, contrary to mainstream interpretations, it is best understood as a theory of infinite sums. Our formal results show that Bolzano’s infinite sums can be equipped (...)
    Download  
     
    Export citation  
     
    Bookmark   2 citations  
  41. Consequences of Assigning Non-Measurable Sets Imprecise Probabilities.Joshua Thong - 2024 - Mind (531):793-804.
    This paper is a discussion note on Isaacs et al. (2022), who have claimed to offer a new motivation for imprecise probabilities, based on the mathematical phenomenon of non-measurability. In this note, I clarify some consequences of their proposal. In particular, I show that if their proposal is applied to a bounded 3-dimensional space, then they have to reject at least one of the following: (i) If A is at most as probable as B and B is at most as (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  42. Computational reverse mathematics and foundational analysis.Benedict Eastaugh - manuscript
    Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational analysis, which explores the limits of different foundations for mathematics in a formally precise manner. This paper gives a detailed account of the motivations and methodology of foundational analysis, which have heretofore been largely left implicit in the practice. It then shows how this account can be (...)
    Download  
     
    Export citation  
     
    Bookmark  
  43. (1 other version)Complements, not competitors: causal and mathematical explanations.Holly Andersen - 2017 - British Journal for the Philosophy of Science 69 (2):485-508.
    A finer-grained delineation of a given explanandum reveals a nexus of closely related causal and non- causal explanations, complementing one another in ways that yield further explanatory traction on the phenomenon in question. By taking a narrower construal of what counts as a causal explanation, a new class of distinctively mathematical explanations pops into focus; Lange’s characterization of distinctively mathematical explanations can be extended to cover these. This new class of distinctively mathematical explanations is illustrated with the Lotka-Volterra equations. There (...)
    Download  
     
    Export citation  
     
    Bookmark   23 citations  
  44. Mathematical Modeling of Biological and Social Evolutionary Macrotrends.Leonid Grinin, Alexander V. Markov & Andrey V. Korotayev - 2014 - In Leonid Grinin & Andrey Korotayev (eds.), History & Mathematics: Trends and Cycles. Volgograd: "Uchitel" Publishing House. pp. 9-48.
    In the first part of this article we survey general similarities and differences between biological and social macroevolution. In the second (and main) part, we consider a concrete mathematical model capable of describing important features of both biological and social macroevolution. In mathematical models of historical macrodynamics, a hyperbolic pattern of world population growth arises from non-linear, second-order positive feedback between demographic growth and technological development. Based on diverse paleontological data and an analogy with macrosociological models, we suggest that the (...)
    Download  
     
    Export citation  
     
    Bookmark  
  45. 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 a single framework (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  46. 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. (...)
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  47. Wittgenstein on Mathematics and Certainties.Martin Kusch - 2016 - International Journal for the Study of Skepticism 6 (2-3):120-142.
    _ Source: _Volume 6, Issue 2-3, pp 120 - 142 This paper aims to contribute to the debate over epistemic versus non-epistemic readings of the ‘hinges’ in Wittgenstein’s _On Certainty_. I follow Marie McGinn’s and Daniele Moyal-Sharrock’s lead in developing an analogy between mathematical sentences and certainties, and using the former as a model for the latter. However, I disagree with McGinn’s and Moyal-Sharrock’s interpretations concerning Wittgenstein’s views of both relata. I argue that mathematical sentences as well as certainties are (...)
    Download  
     
    Export citation  
     
    Bookmark   13 citations  
  48. Safety and Pluralism in Mathematics.James Andrew Smith - forthcoming - Erkenntnis:1-19.
    A belief one has is safe if either (i) it could not easily be false or (ii) in any nearby world in which it is false, it is not formed using the method one uses to form one’s actual belief. It seems our mathematical beliefs are safe if mathematical pluralism is true: if, loosely put, almost any consistent mathematical theory is true. It seems, after all, that in any nearby world where one’s mathematical beliefs differ from one’s actual beliefs, one (...)
    Download  
     
    Export citation  
     
    Bookmark  
  49. What is Mathematics: School Guide to Conceptual Understanding of Mathematics.Catalin Barboianu - 2021 - Targu Jiu: PhilScience Press.
    This is not a mathematics book, but a book about mathematics, which addresses both student and teacher, with a goal as practical as possible, namely to initiate and smooth the way toward the student’s full understanding of the mathematics taught in school. The customary procedural-formal approach to teaching mathematics has resulted in students’ distorted vision of mathematics as a merely formal, instrumental, and computational discipline. Without the conceptual base of mathematics, students develop over time (...)
    Download  
     
    Export citation  
     
    Bookmark  
  50. Many-valued logics. A mathematical and computational introduction.Luis M. Augusto - 2020 - London: College Publications.
    2nd edition. Many-valued logics are those logics that have more than the two classical truth values, to wit, true and false; in fact, they can have from three to infinitely many truth values. This property, together with truth-functionality, provides a powerful formalism to reason in settings where classical logic—as well as other non-classical logics—is of no avail. Indeed, originally motivated by philosophical concerns, these logics soon proved relevant for a plethora of applications ranging from switching theory to cognitive modeling, and (...)
    Download  
     
    Export citation  
     
    Bookmark   2 citations  
1 — 50 / 969