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Geoffrey Hellman [73]G. Hellman [8]Geoffrey P. Hellman [3]
  1. Geoffrey Hellman, In…Nite Possibilities and Possibilities of In…Nity.
    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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  2. Geoffrey Hellman, Carnap* Replies.
    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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  3. Geoffrey Hellman, Foundational Frameworks.
    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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  4. Geoffrey Hellman, Maximality Vs. Extendability: Reflections on Structuralism and Set Theory.
    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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  5. Geoffrey Hellman, Russell's Absolutism Vs.(?) Structuralism.
    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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  6. Geoffrey Hellman, Structuralism.
    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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  7. Geoffrey Hellman, Structuralism, Mathematical.
    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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  8. G. Hellman & S. Shapiro (2013). The Classical Continuum Without Points – CORRIGENDUM. Review of Symbolic Logic 6 (3):571-571.
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  9. Geoffrey Hellman (2013). Neither Categorical nor Set-Theoretic Foundations. Review of Symbolic Logic 6 (1):16-23.
    First we review highlights of the ongoing debate about foundations of category theory, beginning with Fefermantop-down” approach, where particular categories and functors need not be explicitly defined. Possible reasons for resisting the proposal are offered and countered. The upshot is to sustain a pluralism of foundations along lines actually foreseen by Feferman (1977), something that should be welcomed as a way of resolving this long-standing debate.
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  10. Geoffrey Hellman & Stewart Shapiro (2013). The Classical Continuum Without Points. Review of Symbolic Logic 6 (3):488-512.
    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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  11. G. Hellman (2011). On the Significance of the Burali-Forti Paradox. Analysis 71 (4):631-637.
    After briefly reviewing the standard set-theoretic resolutions of the Burali-Forti paradox, we examine how the paradox arises in set theory formalized with plural quantifiers. A significant choice emerges between the desirable unrestricted availability of ordinals to represent well-orderings and the sensibility of attempting to refer to ‘absolutely all ordinals’ or ‘absolutely all well-orderings’. This choice is obscured by standard set theories, which rely on type distinctions which are obliterated in the setting with plurals. Zermelo's attempt ( 1930 ) to secure (...)
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  12. Geoffrey Hellman (2009). Interpretations of Probability in Quantum Mechanics: A Case of “Experimental Metaphysics”. In Wayne C. Myrvold & Joy Christian (eds.), Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle. Springer. 211--227.
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  13. Geoffrey Hellman (2006). Against 'Absolutely Everything'! In Agustín Rayo & Gabriel Uzquiano (eds.), Absolute Generality. Clarendon Press.
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  14. Geoffrey Hellman (2006). ¸ Itekellersetal:Sp.
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  15. Geoffrey Hellman (2006). Mathematical Pluralism: The Case of Smooth Infinitesimal Analysis. [REVIEW] Journal of Philosophical Logic 35 (6):621 - 651.
    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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  16. Geoffrey Hellman (2006). Pluralism and the Foundations of Mathematics. In ¸ Itekellersetal:Sp. 65--79.
    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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  17. Geoffrey Hellman (2006). What is Categorical Structuralism? In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics. Springer. 151--161.
  18. Mirna Dzamonja, David M. Evans, Erich Grädel, Geoffrey P. Hellman, Denis Hirschfeldt, Julia Knight, Michael C. Laskowski, Roger Maddux, Volker Peckhaus & Wolfram Pohlers (2005). Books to Asl, Box 742, Vassar College, 124 Raymond Avenue, Poughkeepsie, Ny 12604, Usa. In a Review, a Reference “Jsl Xliii 148,” for Example, Refers Either to the Publication Reviewed on Page 148 of Volume 43 of the Journal, or to the Review Itself (Which Contains Full Bibliographical Information for the Reviewed Publication). Analogously, a Reference. [REVIEW] Bulletin of Symbolic Logic 11 (2).
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  19. Mirna Dzamonja, David M. Evans, Erich Gradel, Geoffrey P. Hellman, Denis Hirschfeldt, Julia Knight, Michael C. Laskowski, Roger Maddux, Volker Peckhaus & Wolfram Pohlers (2005). The Bulletin of Symbolic Logic Volume 11, Number 2, June 2005. Bulletin of Symbolic Logic 11 (2).
     
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  20. David M. Evans, Erich Grädel, Geoffrey P. Hellman, Denis Hirschfeldt, Thomas J. Jech, Julia Knight, Michael C. Laskowski, Volker Peckhaus, Wolfram Pohlers & Sławomir Solecki (2005). Vassar College, 124 Raymond Avenue, Poughkeepsie, Ny 12604, Usa. In a Review, a Reference “Jsl Xliii 148,” for Example, Refers Either to the Publication Reviewed on Page 148 of Volume 43 of the Journal, or to the Review Itself (Which Contains Full Bibliographical Information for the Reviewed Publication). Analogously, a Reference “Bsl VII 376” Refers to the Review Beginning on Page 376 in Volume 7 of This Bulletin, Or. [REVIEW] Bulletin of Symbolic Logic 11 (1):37.
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  21. Warren Goldfarb, Jeremy Avigad, Andrew Arana, Geoffrey Hellman, Dana Scott & Michael Kremer (2004). Of the Association for Symbolic Logic. Bulletin of Symbolic Logic 10 (3):438.
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  22. Warren Goldfarb, Erich Reck, Jeremy Avigad, Andrew Arana, Geoffrey Hellman, Colin McLarty, Dana Scott & Michael Kremer (2004). Palmer House Hilton Hotel, Chicago, Illinois April 23–24, 2004. Bulletin of Symbolic Logic 10 (3).
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  23. Geoffrey Hellman (2004). Predicativism as a Philosophical Position. Revue Internationale de Philosophie 3:295-312.
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  24. Geoffrey Hellman (2004). Reply to Comments of Solomon Ferferman. Revue Internationale de Philosophie 3:325-328.
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  25. Geoffrey Hellman (2003). Does Category Theory Provide a Framework for Mathematical Structuralism? Philosophia Mathematica 11 (2):129-157.
    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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  26. Geoffrey Hellman, Jeremy Avigad & Paolo Mancosu (2002). The Westin Seattle, Seattle, Washington March 28–29, 2002. Bulletin of Symbolic Logic 8 (3).
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  27. G. Hellman (2001). Solomon Feferman, in the Light of Logic. Philosophia Mathematica 9 (2):231-237.
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  28. Geoffrey Hellman (2001). Critical Studies/Book Review. Philosophia Mathematica 9 (2):231-237.
  29. Geoffrey Hellman (2001). On Nominalism. Philosophy and Phenomenological Research 62 (3):691-705.
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  30. Geoffrey Hellman (2001). Three Varieties of Mathematical Structuralism. Philosophia Mathematica 9 (2):184-211.
    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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  31. Geoffrey Hellman (1999). ARTMANN, BENNO. Euclid—The Creation of Mathematics. Reviewed by EMILY R. GROSHOLZ 246. Notre Dame Journal of Formal Logic 40:6-30.
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  32. Geoffrey Hellman (1999). How to Godel a Frege-Russell. In A. D. Irvine (ed.), Bertrand Russell: Critical Assessments. Routledge. 154.
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  33. Geoffrey Hellman (1999). Review: Stewart Shapiro, Philosophy of Mathematics. Structure and Ontology. [REVIEW] Journal of Symbolic Logic 64 (2):923-926.
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  34. Geoffrey Hellman (1999). Reduction(?) To What? Philosophical Studies 95 (1-2):203-214.
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  35. 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.
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  36. Richard Healey & Geoffrey Hellman (1998). Quantum Measurement, Decoherence and Modal Interpretations. Minnesota Studies in the Philosophy of Science 17.
     
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  37. Geoffrey Hellman (1998). Beyond Definitionism—but Not Too Far Beyond. In Matthias Schirn (ed.), The Philosophy of Mathematics Today. Clarendon Press.
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  38. Geoffrey Hellman (1998). Mathematical Constructivism in Spacetime. British Journal for the Philosophy of Science 49 (3):425-450.
    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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  39. Geoffrey Hellman (1998). Maoist Mathematics? Philosophia Mathematica 6 (3):334-345.
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  40. Geoffrey Hellman (1997). Bayes and Beyond. Philosophy of Science 64 (2):191-221.
    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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  41. Geoffrey Hellman (1997). From Constructive to Predicative Mathematics. In John Earman & John Norton (eds.), The Cosmos of Science. University of Pittsburgh Press. 6--153.
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  42. Geoffrey Hellman (1997). Quantum Mechanical Unbounded Operators and Constructive Mathematics – a Rejoinder to Bridges. Journal of Philosophical Logic 26 (2):121-127.
    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 may (...)
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  43. Geoffrey Hellman (1997). Responses to Maher, and to Kelly, Schulte, and Juhl. Philosophy of Science 64 (2):317-322.
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  44. Tomek Bartoszynski, Harvey Friedman, Geoffrey Hellman, Bakhadyr Khoussainov, Phokion G. Kolaitis, Richard Shore, Charles Steinhorn, Mirna Dzamonja, Itay Neeman & Slawomir Solecki (1996). 1995–1996 Annual Meeting of the Association for Symbolic Logic. Bulletin of Symbolic Logic 2 (4).
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  45. Solomon Feferman & Geoffrey Hellman (1995). Predicative Foundations of Arithmetic. Journal of Philosophical Logic 24 (1):1 - 17.
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  46. Geoffrey Hellman (1994). Real Analysis Without Classes. Philosophia Mathematica 2 (3):228-250.
    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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  47. Geoffrey Hellman (1993). Book Reviews. [REVIEW] Philosophia Mathematica 1 (1):75-88.
  48. Geoffrey Hellman (1993). Constructive Mathematics and Quantum Mechanics: Unbounded Operators and the Spectral Theorem. [REVIEW] Journal of Philosophical Logic 22 (3):221 - 248.
  49. Geoffrey Hellman (1993). Gleason's Theorem is Not Constructively Provable. Journal of Philosophical Logic 22 (2):193 - 203.
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  50. G. Hellman (1992). Supervenience/Determination a Two-Way Street? Yes, but One of the Ways is the Wrong Way! Journal of Philosophy 89 (1):42-47.
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