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Geoffrey Hellman [96]Geoffrey P. Hellman [3]Geoffrey Paul Hellman [2]
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Geoffrey Hellman
University of Minnesota
  1. Mathematics Without Numbers: Towards a Modal-Structural Interpretation.Geoffrey Hellman - 1989 - Oxford University Press.
    Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...
  2.  22
    Mathematics Without Numbers: Towards a Modal-Structural Interpretation.Bob Hale & Geoffrey Hellman - 1992 - Philosophical Review 101 (4):919.
  3. Does Category Theory Provide a Framework for Mathematical Structuralism?Geoffrey Hellman - 2003 - 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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  4. Varieties of Continua: From Regions to Points and Back.Geoffrey Hellman & Stewart Shapiro - 2018 - Oxford University Press.
    Hellman and Shapiro explore the development of the idea of the continuous, from the Aristotelian view that a true continuum cannot be composed of points to the now standard, entirely punctiform frameworks for analysis and geometry. They then investigate the underlying metaphysical issues concerning the nature of space or space-time.
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  5. Three Varieties of Mathematical Structuralism.Geoffrey Hellman - 2001 - 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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  6. Structuralism.Geoffrey Hellman - manuscript
    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. The Classical Continuum Without Points.Geoffrey Hellman & Stewart Shapiro - 2013 - 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 logic of plural quantification. In some respects this realizes ideas going back to Aristotle,although, unlike Aristotle, we make free use of classical "actual infinity". Also, in contrast to intuitionistic, Bishop, and smooth infinitesimal analysis, we follow classical analysis in allowing partitioning of our "gunky line" into mutually exclusive and exhaustive disjoint parts, thereby demonstrating the independence (...)
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  8. Physicalism: Ontology, Determination and Reduction.Geoffrey Paul Hellman & Frank Wilson Thompson - 1975 - Journal of Philosophy 72 (October):551-64.
  9. Aristotelian Continua.Øystein Linnebo, Stewart Shapiro & Geoffrey Hellman - 2016 - Philosophia Mathematica 24 (2):214-246.
    In previous work, Hellman and Shapiro present a regions-based account of a one-dimensional continuum. This paper produces a more Aristotelian theory, eschewing the existence of points and the use of infinite sets or pluralities. We first show how to modify the original theory. There are a number of theorems that have to be added as axioms. Building on some work by Linnebo, we then show how to take the ‘potential’ nature of the usual operations seriously, by using a modal language, (...)
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  10. Mathematics without Numbers. Towards a Modal-Structural Interpretation.Geoffrey Hellman - 1991 - Tijdschrift Voor Filosofie 53 (4):726-727.
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  11. Frege Meets Aristotle: Points as Abstracts.Stewart Shapiro & Geoffrey Hellman - 2015 - Philosophia Mathematica:nkv021.
    There are a number of regions-based accounts of space/time, due to Whitehead, Roeper, Menger, Tarski, the present authors, and others. They all follow the Aristotelian theme that continua are not composed of points: each region has a proper part. The purpose of this note is to show how to recapture ‘points’ in such frameworks via Scottish neo-logicist abstraction principles. The results recapitulate some Aristotelian themes. A second agenda is to provide a new arena to help decide what is at stake (...)
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  12.  95
    Mathematical Pluralism: The Case of Smooth Infinitesimal Analysis.Geoffrey Hellman - 2006 - 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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  13. Towards a Point-Free Account of the Continuous.Geoffrey Hellman & Stewart Shapiro - 2012 - Iyyun 61:263.
  14. Mathematical Structuralism.Geoffrey Hellman & Stewart Shapiro - 2018 - Cambridge University Press.
    The present work is a systematic study of five frameworks or perspectives articulating mathematical structuralism, whose core idea is that mathematics is concerned primarily with interrelations in abstraction from the nature of objects. The first two, set-theoretic and category-theoretic, arose within mathematics itself. After exposing a number of problems, the book considers three further perspectives formulated by logicians and philosophers of mathematics: sui generis, treating structures as abstract universals, modal, eliminating structures as objects in favor of freely entertained logical possibilities, (...)
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  15.  14
    Physicalism.Geoffrey Hellman - 1980 - Philosophical Review 89 (4):625.
  16. Pluralism and the Foundations of Mathematics.Geoffrey Hellman - 2006 - In ¸ Itekellersetal:Sp. pp. 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.  45
    Bayes and Beyond.Geoffrey Hellman - 1997 - 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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  18.  97
    Determination and Logical Truth.Geoffrey Hellman - 1985 - Journal of Philosophy 82 (November):607-16.
    Some remarks on determination, physicalism, model theory, and logical truth.//An attempt to defend physicalism against objections that its bases are indeterminate.
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  19. Predicative Foundations of Arithmetic.Solomon Feferman & Geoffrey Hellman - 1995 - Journal of Philosophical Logic 24 (1):1 - 17.
  20.  63
    Regions-Based Two Dimensional Continua: The Euclidean Case.Geoffrey Hellman & Stewart Shapiro - 2015 - Logic and Logical Philosophy 24 (4).
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  21.  40
    Robert L. Martin and Peter W. Woodruff. On Representing ‘True-in-L' in L. Philosophia , Vol. 5 No. 3 , Pp. 213–217. - Saul Kripke. Outline of a Theory of Truth. The Journal of Philosophy, Vol. 72 , Pp. 690–716. - Anil Gupta. Truth and Paradox. Journal of Philosophical Logic, Vol. 11 , Pp. 1–60. - Hans G. Herzberger. Notes on Naive Semantics. Journal of Philosophical Logic, Vol. 11 , Pp. 61–102. [REVIEW]Geoffrey Hellman - 1985 - Journal of Symbolic Logic 50 (4):1068-1071.
  22.  53
    Physicalist Materialism.Geoffrey Hellman & Frank Wilson Thompson - 1977 - Noûs 11 (4):309-45.
  23. Against 'Absolutely Everything'!Geoffrey Hellman - 2006 - In Agustín Rayo & Gabriel Uzquiano (eds.), Absolute Generality. Clarendon Press.
     
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  24.  41
    Constructive Mathematics and Quantum Mechanics: Unbounded Operators and the Spectral Theorem. [REVIEW]Geoffrey Hellman - 1993 - Journal of Philosophical Logic 22 (3):221 - 248.
  25.  64
    Carnap* Replies.Geoffrey Hellman - 2018 - The Monist 101 (4):388-393.
    In an imagined dialogue between two figures called “Carnap*” and “Quine*” that appeared in the Library of Living Philosophers volume in 1986, certain proposals and clarifications of the linguistic doctrine were offered by Carnap* answering Quinean objections, but these were brushed aside rather breezily in a reply to this dialogue in the same volume by Quine himself. After a brief summary of the questions at issue in that earlier dialogue, Carnap* is here allowed a final reply, introducing yet another variant (...)
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  26.  27
    Gleason's Theorem is Not Constructively Provable.Geoffrey Hellman - 1993 - Journal of Philosophical Logic 22 (2):193 - 203.
  27. Maximality Vs. Extendability: Reflections on Structuralism and Set Theory.Geoffrey Hellman - unknown
    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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  28.  53
    Stochastic Einstein-Locality and the Bell Theorems.Geoffrey Hellman - 1982 - Synthese 53 (3):461 - 504.
    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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  29.  46
    Hartry Field. Science Without Numbers: A Defense of Nominalism 2nd Ed. [REVIEW]Geoffrey Hellman & Mary Leng - 2019 - Philosophia Mathematica 27 (1):139-148.
    FieldHartry. Science Without Numbers: A Defense of Nominalism 2nd ed.Oxford University Press, 2016. ISBN 978-0-19-877792-2. Pp. vi + 56 + vi + 111.
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  30.  27
    Randomness and Reality.Geoffrey Hellman - 1978 - PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1978:79-97.
  31. Supervenience/Determination a Two-Way Street? Yes, but One of the Ways is the Wrong Way!Geoffrey Hellman - 1992 - Journal of Philosophy 89 (1):42-47.
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  32.  10
    Determination and Logical Truth.Geoffrey Hellman - 1985 - Journal of Philosophy 82 (11):607-616.
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  33. Predicativism as a Philosophical Position.Geoffrey Hellman - 2004 - Revue Internationale de Philosophie 3:295-312.
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  34.  27
    Stewart Shapiro. Philosophy of Mathematics. Structure and Ontology. Oxford University Press, New York and Oxford 1997, X + 279 Pp. [REVIEW]Geoffrey Hellman - 1999 - Journal of Symbolic Logic 64 (2):923-926.
  35.  41
    Never Say “Never”!: On the Communication Problem Between Intuitionism and Classicism.Geoffrey Hellman - 1989 - Philosophical Topics 17 (2):47-67.
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  36.  57
    Mathematical Constructivism in Spacetime.Geoffrey Hellman - 1998 - 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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  37. What is Categorical Structuralism?Geoffrey Hellman - 2006 - In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics. Springer. pp. 151--161.
  38.  80
    Quantum Mechanical Unbounded Operators and Constructive Mathematics – a Rejoinder to Bridges.Geoffrey Hellman - 1997 - 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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  39. Dualling: A Critique of an Argument of Popper and Miller.J. Michael Dunn & Geoffrey Hellman - 1986 - British Journal for the Philosophy of Science 37 (2):220-223.
  40. The New Riddle of Radical Translation.Geoffrey Hellman - 1974 - Philosophy of Science 41 (3):227-246.
    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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  41. Reduction(?) to What? Comments on L. Sklar's "The Reduction (?) of Thermodynamics to Statistical Mechanics".Geoffrey Hellman - 1999 - Philosophical Studies 95 (1/2):203 - 214.
  42.  43
    EPR, Bell, and Collapse: A Route Around "Stochastic" Hidden Variables.Geoffrey Hellman - 1987 - Philosophy of Science 54 (4):558-576.
    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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  43.  40
    Neither Categorical nor Set-Theoretic Foundations.Geoffrey Hellman - 2013 - 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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  44. The Structure of Appearance.N. Goodman & Geoffrey Hellman - 1966 - Tijdschrift Voor Filosofie 42 (4):828-829.
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  45.  98
    On Nominalism.Geoffrey Hellman - 2001 - Philosophy and Phenomenological Research 62 (3):691-705.
    Probably there is no position in Goodman’s corpus that has generated greater perplexity and criticism than Goodman’s “nominalism”. As is abundantly clear from Goodman’s writings, it is not “abstract entities” generally that he questions—indeed, he takes sensory qualia as “basic” in his Carnap-inspired constructional system in Structure—but rather just those abstracta that are so crystal clear in their identity conditions, so fundamental to our thought, so prevalent and seemingly unavoidable in our discourse and theorizing that they have come to form (...)
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  46.  65
    Toward a Modal-Structural Interpretation of Set Theory.Geoffrey Hellman - 1990 - Synthese 84 (3):409 - 443.
  47.  41
    Realist Principles.Geoffrey Hellman - 1983 - Philosophy of Science 50 (2):227-249.
    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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  48.  41
    Real Analysis Without Classes.Geoffrey Hellman - 1994 - 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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  49.  16
    Stewart Shapiro. Second-Order Languages and Mathematical Practice. The Journal of Symbolic Logic, Vol. 50 , Pp. 714–742. [REVIEW]Geoffrey Hellman - 1989 - Journal of Symbolic Logic 54 (1):291-293.
  50. Structuralism, Mathematical.Geoffrey Hellman - unknown
    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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