Search results for 'Hellman Geoffrey' (try it on Scholar)

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  1. Geoffrey Hellman (1997). Quantum Mechanical Unbounded Operators and Constructive Mathematics – a Rejoinder to Bridges. Journal of Philosophical Logic 26 (2):121-127.score: 360.0
    As argued in Hellman (1993), the theorem of Pour-El and Richards (1983) can be seen by the classicist as limiting constructivist efforts to recover the mathematics for quantum mechanics. Although Bridges (1995) may be right that the constructivist would work with a different definition of 'closed operator', this does not affect my point that neither the classical unbounded operators standardly recognized in quantum mechanics nor their restrictions to constructive arguments are recognizable as objects by the constructivist. Constructive substitutes that (...)
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  2. Geoffrey Hellman (2001). Three Varieties of Mathematical Structuralism. Philosophia Mathematica 9 (2):184-211.score: 300.0
    Three principal varieties of mathematical structuralism are compared: set-theoretic structuralism (‘STS’) using model theory, Shapiro's ante rem structuralism invoking sui generis universals (‘SGS’), and the author's modal-structuralism (‘MS’) invoking logical possibility. Several problems affecting STS are discussed concerning, e.g., multiplicity of universes. SGS overcomes these; but it faces further problems of its own, concerning, e.g., the very intelligibility of purely structural objects and relations. MS, in contrast, overcomes or avoids both sets of problems. Finally, it is argued that the modality (...)
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  3. Geoffrey Hellman (2003). Does Category Theory Provide a Framework for Mathematical Structuralism? Philosophia Mathematica 11 (2):129-157.score: 300.0
    Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis-a-vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell's many-topoi view and modal-structuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and set theory can be carried out (...)
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  4. Geoffrey Hellman (2004). Predicativism as a Philosophical Position. Revue Internationale de Philosophie 3:295-312.score: 300.0
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  5. Geoffrey Hellman (1980). Quantum Logic and Meaning. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1980:493 - 511.score: 300.0
    Quantum logic as genuine non-classical logic provides no solution to the "paradoxes" of quantum mechanics. From the minimal condition that synonyms be substitutable salva veritate, it follows that synonymous sentential connectives be alike in point of truth-functionality. It is a fact of pure mathematics that any assignment Φ of (0, 1) to the subspaces of Hilbert space (dim. ≥ 3) which guarantees truth-preservation of the ordering and truth-functionality of QL negation, violates truth-functionality of QL ∨ and $\wedge $ . Thus, (...)
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  6. Geoffrey Hellman (1992). Bell-Type Inequalities in the Nonideal Case: Proof of a Conjecture of Bell. [REVIEW] Foundations of Physics 22 (6):807-817.score: 300.0
    Recently Bell has conjectured that, with “epsilonics,” one should be able to argue, à la EPR, from “almost ideal correlations” (in parallel Bohm-Bell pair experiments) to “almost determinism,” and that this should suffice to derive an approximate Bell-type inequality. Here we prove that this is indeed the case. Such an inequality—in principle testable—is derived employing only weak locality conditions, imperfect correlation, and a propensity interpretation of certain conditional probabilities. Outcome-independence (Jarrett's “completeness” condition), hence “factorability” of joint probabilities, is not assumed, (...)
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  7. Geoffrey Hellman (2001). On Nominalism. Philosophy and Phenomenological Research 62 (3):691-705.score: 300.0
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  8. Geoffrey Hellman & Stewart Shapiro (2013). The Classical Continuum Without Points. Review of Symbolic Logic 6 (3):488-512.score: 300.0
    We develop a point-free construction of the classical one-dimensional continuum, with an interval structure based on mereology and either a weak set theory or a logic of plural quantification. In some respects, this realizes ideas going back to Aristotle, although, unlike Aristotle, we make free use of contemporary . Also, in contrast to intuitionistic analysis, smooth infinitesimal analysis, and Eret Bishopgunky lineindecomposabilityCantor structure of ℝ as a complete, separable, ordered field. We also present some simple topological models of our system, (...)
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  9. Geoffrey Hellman (1981). How to Godel a Frege-Russell: Godel's Incompleteness Theorems and Logicism. Noûs 15 (4):451-468.score: 300.0
  10. Geoffrey Hellman (2006). What is Categorical Structuralism?. In. In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics. Springer. 151--161.score: 300.0
  11. Geoffrey Hellman, Structuralism.score: 300.0
    With the rise of multiple geometries in the nineteenth century, and in the last century the rise of abstract algebra, of the axiomatic method, the set-theoretic foundations of mathematics, and the influential work of the Bourbaki, certain views called “structuralist” have become commonplace. Mathematics is seen as the investigation, by more or less rigorous deductive means, of “abstract structures”, systems of objects fulfilling certain structural relations among themselves and in relation to other systems, without regard to the particular nature of (...)
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  12. Geoffrey Hellman (2006). Mathematical Pluralism: The Case of Smooth Infinitesimal Analysis. [REVIEW] Journal of Philosophical Logic 35 (6):621 - 651.score: 300.0
    A remarkable development in twentieth-century mathematics is smooth infinitesimal analysis ('SIA'), introducing nilsquare and nilpotent infinitesimals, recovering the bulk of scientifically applicable classical analysis ('CA') without resort to the method of limits. Formally, however, unlike Robinsonian 'nonstandard analysis', SIA conflicts with CA, deriving, e.g., 'not every quantity is either = 0 or not = 0.' Internally, consistency is maintained by using intuitionistic logic (without the law of excluded middle). This paper examines problems of interpretation resulting from this 'change of logic', (...)
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  13. Geoffrey Hellman (1998). Mathematical Constructivism in Spacetime. British Journal for the Philosophy of Science 49 (3):425-450.score: 300.0
    To what extent can constructive mathematics based on intuitionistc logic recover the mathematics needed for spacetime physics? Certain aspects of this important question are examined, both technical and philosophical. On the technical side, order, connectivity, and extremization properties of the continuum are reviewed, and attention is called to certain striking results concerning causal structure in General Relativity Theory, in particular the singularity theorems of Hawking and Penrose. As they stand, these results appear to elude constructivization. On the philosophical side, it (...)
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  14. Geoffrey Hellman, Maximality Vs. Extendability: Reflections on Structuralism and Set Theory.score: 300.0
    In a recent paper, while discussing the role of the notion of analyticity in Carnap’s thought, Howard Stein wrote: “The primitive view–surely that of Kant–was that whatever is trivial is obvious. We know that this is wrong; and I would put it that the nature of mathematical knowledge appears more deeply mysterious today than it ever did in earlier centuries – that one of the advances we have made in philosophy has been to come to an understanding of just ∗I (...)
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  15. Geoffrey Hellman (1983). Realist Principles. Philosophy of Science 50 (2):227-249.score: 300.0
    We list, with discussions, various principles of scientific realism, in order to exhibit their diversity and to emphasize certain serious problems of formulation. Ontological and epistemological principles are distinguished. Within the former category, some framed in semantic terms (truth, reference) serve their purpose vis-a-vis instrumentalism (Part 1). They fail, however, to distinguish the realist from a wide variety of (constructional) empiricists. Part 2 seeks purely ontological formulations, so devised that the empiricist cannot reconstruct them from within. The main task here (...)
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  16. Geoffrey Hellman, Russell's Absolutism Vs.(?) Structuralism.score: 300.0
    Along with Frege, Russell maintained an absolutist stance regarding the subject matter of mathematics, revealed rather than imposed, or proposed, by logical analysis. The Fregean definition of cardinal number, for example, is viewed as (essentially) correct, not merely adequate for mathematics. And Dedekind’s “structuralist” views come in for criticism in the Principles. But, on reflection, Russell also flirted with views very close to a (different) version of structuralism. Main varieties of modern structuralism and their challenges are reviewed, taking account of (...)
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  17. Geoffrey Hellman (1989). Mathematics Without Numbers: Towards a Modal-Structural Interpretation. Oxford University Press.score: 300.0
    Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...
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  18. Solomon Feferman & Geoffrey Hellman (1995). Predicative Foundations of Arithmetic. Journal of Philosophical Logic 24 (1):1 - 17.score: 300.0
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  19. Geoffrey Hellman, Foundational Frameworks.score: 300.0
    After some metatheoretic preliminaries on questions of justification and rational reconstruction, we lay out some key desiderata for foundational frameworks for mathematics, some of which reflect recent discussions of pluralism and structuralism. Next we draw out some implications (pro and con) bearing on set theory and category and topos therory. Finally, we sketch a variant of a modal-structural core system, incorporating elements of predicativism and the systems of reverse mathematics, and consider how it fares with respect to the desiderata highlighted (...)
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  20. Geoffrey Hellman (2006). Pluralism and the Foundations of Mathematics. In ¸ Itekellersetal:Sp. 65--79.score: 300.0
    A plurality of approaches to foundational aspects of mathematics is a fact of life. Two loci of this are discussed here, the classicism/constructivism controversy over standards of proof, and the plurality of universes of discourse for mathematics arising in set theory and in category theory, whose problematic relationship is discussed. The first case illustrates the hypothesis that a sufficiently rich subject matter may require a multiplicity of approaches. The second case, while in some respects special to mathematics, raises issues of (...)
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  21. Geoffrey Hellman (1982). Stochastic Einstein-Locality and the Bell Theorems. Synthese 53 (3):461 - 504.score: 300.0
    Standard proofs of generalized Bell theorems, aiming to restrict stochastic, local hidden-variable theories for quantum correlation phenomena, employ as a locality condition the requirement of conditional stochastic independence. The connection between this and the no-superluminary-action requirement of the special theory of relativity has been a topic of controversy. In this paper, we introduce an alternative locality condition for stochastic theories, framed in terms of the models of such a theory (§2). It is a natural generalization of a light-cone determination condition (...)
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  22. Geoffrey Hellman (1998). Maoist Mathematics? Philosophia Mathematica 6 (3):334-345.score: 300.0
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  23. Geoffrey Hellman (1999). Reduction(?) To What? Philosophical Studies 95 (1-2):203-214.score: 300.0
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  24. Geoffrey Hellman, Carnap* Replies.score: 300.0
    Despite my concerted efforts to formulate the linguistic doctrine of (first-order) logical truth, explicitly not as a claim that stipulations governing logical particles suffice to generate the logical truths (LD(I)), but as a determination thesis (LD(III))--that stipulations that certain particles behave as the classical logical particles suffice to determine uniquely the class of logically valid sentences, whose emptiness is clear and relatively unproblematic--, Quine2 nevertheless managed to read me as having claimed “that the logical truths can be generated (sic!) by (...)
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  25. Geoffrey Hellman (1981). Quantum Logic and the Projection Postulate. Philosophy of Science 48 (3):469-486.score: 300.0
    This paper explores the status of the von Neumann-Luders state transition rule (the "projection postulate") within "real-logic" quantum logic. The entire discussion proceeds from a reading of the Luders rule according to which, although idealized in applying only to "minimally disturbing" measurements, it nevertheless makes empirical claims and is not a purely mathematical theorem. An argument (due to Friedman and Putnam) is examined to the effect that QL has an explanatory advantage over Copenhagen and other interpretations which relativize truth-value assignments (...)
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  26. Geoffrey Hellman (1978). Randomness and Reality. In Peter D. Asquith & Ian Hacking (eds.), PSA 1978. University of Chicago Press. 79--97.score: 300.0
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  27. Geoffrey Hellman (1994). Real Analysis Without Classes. Philosophia Mathematica 2 (3):228-250.score: 300.0
    This paper explores strengths and limitations of both predicativism and nominalism, especially in connection with the problem of characterizing the continuum. Although the natural number structure can be recovered predicatively (despite appearances), no predicative system can characterize even the full predicative continuum which the classicist can recognize. It is shown, however, that the classical second-order theory of continua (third-order number theory) can be recovered nominalistically, by synthesizing mereology, plural quantification, and a modal-structured approach with essentially just the assumption that an (...)
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  28. Geoffrey Hellman (1997). Bayes and Beyond. Philosophy of Science 64 (2):191-221.score: 300.0
    Several leading topics outstanding after John Earman's Bayes or Bust? are investigated further, with emphasis on the relevance of Bayesian explication in epistemology of science, despite certain limitations. (1) Dutch Book arguments are reformulated so that their independence from utility and preference in epistemic contexts is evident. (2) The Bayesian analysis of the Quine-Duhem problem is pursued; the phenomenon of a "protective belt" of auxiliary statements around reasonably successful theories is explicated. (3) The Bayesian approach to understanding the superiority of (...)
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  29. Geoffrey Hellman (1993). Constructive Mathematics and Quantum Mechanics: Unbounded Operators and the Spectral Theorem. [REVIEW] Journal of Philosophical Logic 22 (3):221 - 248.score: 300.0
  30. Geoffrey Hellman (1990). Toward a Modal-Structural Interpretation of Set Theory. Synthese 84 (3):409 - 443.score: 300.0
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  31. Geoffrey Hellman (1987). EPR, Bell, and Collapse: A Route Around "Stochastic" Hidden Variables. Philosophy of Science 54 (4):558-576.score: 300.0
    Two EPR arguments are reviewed, for their own sake, and for the purpose of clarifying the status of "stochastic" hidden variables. The first is a streamlined version of the EPR argument for the incompleteness of quantum mechanics. The role of an anti-instrumentalist ("realist") interpretation of certain probability statements is emphasized. The second traces out one horn of a central foundational dilemma, the collapse dilemma; complex modal reasoning, similar to the original EPR, is used to derive determinateness (of all spin components (...)
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  32. Geoffrey Hellman, In…Nite Possibilities and Possibilities of In…Nity.score: 300.0
    In the …rst part of this paper, the origins of modal-structuralism are traced from Hilary Putnam’s seminal article, "Mathematics without Foundations" (1967) to its transformation and development into the author’s modal-structural approach. The addition of a logic of plurals is highlighted for its recovery (in combination with the resources of mereology) of full, second-order logic, essential for articulating a good theory of mathematical structures. The second part concentrates on the motivation of large trans…nite cardinal numbers, arising naturally from the second-order (...)
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  33. Geoffrey Hellman (1977). Symbol Systems and Artistic Styles. Journal of Aesthetics and Art Criticism 35 (3):279-292.score: 300.0
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  34. Geoffrey Hellman (1974). The New Riddle of Radical Translation. Philosophy of Science 41 (3):227-246.score: 300.0
    This paper presents parts of a theory of radical translation with applications to the problem of construing reference. First, in sections 1 to 4 the general standpoint, inspired by Goodman's approach to induction, is set forth. Codification of sound translational practice replaces the aim of behavioral reduction of semantic notions. The need for a theory of translational projection (manual construction on the basis of a finite empirical correlation of sentences) is established by showing the anomalies otherwise resulting (e.g. from Quine's (...)
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  35. Geoffrey Hellman (1993). Book Reviews. [REVIEW] Philosophia Mathematica 1 (1):75-88.score: 300.0
  36. Geoffrey Hellman & Keith Hossack (1992). Constructivist Mathematics, Quantum Physics and Quantifiers. Aristotelian Society Supplementary Volume 66:61 - 97.score: 300.0
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  37. Geoffrey Hellman (1982). Einstein and Bell: Strengthening the Case for Microphysical Randomness. Synthese 53 (3):445 - 460.score: 300.0
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  38. Geoffrey Hellman (1984). Introduction. Noûs 18 (4):557-567.score: 300.0
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  39. Geoffrey Hellman (1997). Responses to Maher, and to Kelly, Schulte, and Juhl. Philosophy of Science 64 (2):317-322.score: 300.0
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  40. Geoffrey Hellman, Structuralism, Mathematical.score: 300.0
    Structuralism is a view about the subject matter of mathematics according to which what matters are structural relationships in abstraction from the intrinsic nature of the related objects. Mathematics is seen as the free exploration of structural possibilities, primarily through creative concept formation, postulation, and deduction. The items making up any particular system exemplifying the structure in question are of no importance; all that matters is that they satisfy certain general conditions—typically spelled out in axioms defining the structure or structures (...)
     
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  41. Geoffrey Hellman (1979). Against Bad Method. Metaphilosophy 10 (2):190–202.score: 300.0
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  42. Geoffrey Hellman (1980). A Probabilistic Version of the Kochen-Specker No-Hidden-Variable Proof. Synthese 44 (3):495 - 500.score: 300.0
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  43. Geoffrey Hellman (2001). Critical Studies/Book Review. Philosophia Mathematica 9 (2):231-237.score: 300.0
  44. Geoffrey Hellman (1969). Finitude, Infinitude, and Isomorphism of Interpretations in Some Nominalistic Calculi. Noûs 3 (4):413-425.score: 300.0
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  45. Geoffrey Hellman (1999). Reduction(?) to What? Comments on L. Sklar's "The Reduction (?) of Thermodynamics to Statistical Mechanics". Philosophical Studies 95 (1/2):203 - 214.score: 300.0
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  46. J. Michael Dunn & Geoffrey Hellman (1986). Dualling: A Critique of an Argument of Popper and Miller. British Journal for the Philosophy of Science 37 (2):220-223.score: 300.0
  47. Geoffrey Hellman (1993). Gleason's Theorem is Not Constructively Provable. Journal of Philosophical Logic 22 (2):193 - 203.score: 300.0
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  48. Geoffrey Hellman (1989). Never Say “Never”! Philosophical Topics 17 (2):47-67.score: 300.0
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  49. Geoffrey Hellman (1988). The Many Worlds Interpretation of Set Theory. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1988:445 - 455.score: 300.0
    Standard presentations of axioms for set theory as truths simpliciter about actual-objects the sets-confront a number of puzzles associated with platonism and foundationalism. In his classic (1930), Zermelo suggested an alternative "many worlds" view. Independently, Putnam (1967) proposed something similar, explicitly incorporating modality. A modal-structural synthesis of these ideas is sketched in which obstacles to their formalization are overcome. Extendability principles are formulated and used to motivate many small large cardinals. The use of second-order logic as a coherent (...)
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  50. Geoffrey Hellman (1979). Reply to Feyerabend: From Bad to Worse. Metaphilosophy 10 (2):206–207.score: 300.0
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