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Ontology of Sets

Edited by Rafal Urbaniak (University of Ghent, University of Gdansk)
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  1. F. G. Asenjo (1965). Theory of Multiplicities. Logique Et Analyse 8:105-110.
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  2. Zdzisław Augustynek (1995). Natura czasoprzestrzeni a istnienie zbiorów. Filozofia Nauki 1.
    This paper tries to prove two statements. Firstly, that set-theoretic positions in the controversy on the ontic nature of space-time logically imply set-theoretic realism. Secondly, thatmereological positions in this controversy give set-theoretic nominalism an appearance of verisimilitude.
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  3. George Boolos (1998). Must We Believe in Set Theory? In Richard Jeffrey (ed.), Logic, Logic, and Logic. Harvard University Press. 120-132.
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  4. George Boolos (1998). Reply to Charles Parsons' ``Sets and Classes''. In Richard Jeffrey (ed.), Logic, Logic, and Logic. Harvard University Press. 30-36.
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  5. Manuel Bremer (2010). Universality in Set Theories. Ontos.
    The book discusses the fate of universality and a universal set in several set theories.
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  6. Salvatore Florio & Stewart Shapiro (2014). Set Theory, Type Theory, and Absolute Generality. Mind 123 (489):157-174.
    In light of the close connection between the ontological hierarchy of set theory and the ideological hierarchy of type theory, Øystein Linnebo and Agustín Rayo have recently offered an argument in favour of the view that the set-theoretic universe is open-ended. In this paper, we argue that, since the connection between the two hierarchies is indeed tight, any philosophical conclusions cut both ways. One should either hold that both the ontological hierarchy and the ideological hierarchy are open-ended, or that neither (...)
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  7. James Hawthorne (1996). Mathematical Instrumentalism Meets the Conjunction Objection. Journal of Philosophical Logic 25 (4):363-397.
    Scientific realists often appeal to some version of the conjunction objection to argue that scientific instrumentalism fails to do justice to the full empirical import of scientific theories. Whereas the conjunction objection provides a powerful critique of scientific instrumentalism, I will show that mathematical instnrunentalism escapes the conjunction objection unscathed.
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  8. Harold T. Hodes (1991). Where Do Sets Come From? Journal of Symbolic Logic 56 (1):150-175.
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  9. Richard Jeffrey (ed.) (1998). Logic, Logic, and Logic. Harvard University Press.
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  10. Czesław Lejewski (1985). Accommodating the Informal Notion of Class Within the Framework of Lesniewski's Ontology. Dialectica 39 (3):217-241.
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  11. Alex Levine (2005). Conjoining Mathematical Empiricism with Mathematical Realism: Maddy's Account of Set Perception Revisited. Synthese 145 (3):425 - 448.
    Penelope Maddy’s original solution to the dilemma posed by Benacerraf in his (1973) ‘Mathematical Truth’ was to reconcile mathematical empiricism with mathematical realism by arguing that we can perceive realistically construed sets. Though her hypothesis has attracted considerable critical attention, much of it, in my view, misses the point. In this paper I vigorously defend Maddy’s (1990) account against published criticisms, not because I think it is true, but because these criticisms have functioned to obscure a more fundamental issue that (...)
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  12. David Lewis (1991). Parts of Classes. Blackwell.
  13. Øystein Linnebo & Richard Pettigrew (2011). Category Theory as an Autonomous Foundation. Philosophia Mathematica 19 (3):227-254.
    Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can be made for its logical and conceptual autonomy. Its justificatory autonomy turns on whether the objects of a foundation for mathematics should be specified only up to isomorphism, as is customary in other (...)
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  14. Toby Meadows (2013). WHAT CAN A CATEGORICITY THEOREM TELL US? Review of Symbolic Logic (3):524-544.
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  15. Christopher Menzel (2014). Wide Sets, ZFCU, and the Iterative Conception. Journal of Philosophy 111 (2):57-83.
    The iterative conception of set is typically considered to provide the intuitive underpinnings for ZFCU (ZFC+Urelements). It is an easy theorem of ZFCU that all sets have a definite cardinality. But the iterative conception seems to be entirely consistent with the existence of “wide” sets, sets (of, in particular, urelements) that are larger than any cardinal. This paper diagnoses the source of the apparent disconnect here and proposes modifications of the Replacement and Powerset axioms so as to allow for the (...)
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  16. Anne Newstead (2009). Cantor on Infinity in Nature, Number, and the Divine Mind. American Catholic Philosophical Quarterly 83 (4):533-553.
    The mathematician Georg Cantor strongly believed in the existence of actually infinite numbers and sets. Cantor’s “actualism” went against the Aristotelian tradition in metaphysics and mathematics. Under the pressures to defend his theory, his metaphysics changed from Spinozistic monism to Leibnizian voluntarist dualism. The factor motivating this change was two-fold: the desire to avoid antinomies associated with the notion of a universal collection and the desire to avoid the heresy of necessitarian pantheism. We document the changes in Cantor’s thought with (...)
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  17. Anne Newstead (2008). Intertwining Metaphysics and Mathematics: The Development of Georg Cantor's Set Theory 1871-1887. Review of Contemporary Philosophy 7:35-55.
  18. Anne Newstead (2001). Aristotle and Modern Mathematical Theories of the Continuum. In Demetra Sfendoni-Mentzou & James Brown (eds.), Aristotle and Contemporary Philosophy of Science. Peter Lang.
    This paper is on Aristotle's conception of the continuum. It is argued that although Aristotle did not have the modern conception of real numbers, his account of the continuum does mirror the topology of the real number continuum in modern mathematics especially as seen in the work of Georg Cantor. Some differences are noted, particularly as regards Aristotle's conception of number and the modern conception of real numbers. The issue of whether Aristotle had the notion of open versus closed intervals (...)
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  19. Anne Newstead (1997). Actual Versus Potential Infinity (BPhil Manuscript.). Dissertation, University of Oxford
    Does mathematical practice require the existence of actual infinities, or are potential infinities enough? Contrasting points of view are examined in depth, concentrating on Aristotle’s arguments against actual infinities, Cantor’s attempts to refute Aristotle, and concluding with Zermelo’s assertion of the primacy of potential infinity in mathematics.
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  20. Judith M. Prakel (1983). A Leśniewskian Re-Examination of Goodman's Nominalistic Rejection of Classes. Topoi 2 (1):87-98.
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  21. Wilfrid Sellars (1963). Classes as Abstract Entities and the Russell Paradox. Review of Metaphysics 17 (1):67 - 90.
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  22. Michael J. Shaffer (2006). Some Recent Appeals to Mathematical Experience. Principia 10 (2):143-170.
    ome recent work by philosophers of mathematics has been aimed at showing that our knowledge of the existence of at least some mathematical objects and/or sets can be epistemically grounded by appealing to perceptual experience. The sensory capacity that they refer to in doing so is the ability to perceive numbers, mathematical properties and/or sets. The chief defense of this view as it applies to the perception of sets is found in Penelope Maddy’s Realism in Mathematics, but a number of (...)
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  23. Barry Smith (2009). Biometaphysics. In Robin Le Poidevin (ed.), The Routledge Companion to Metaphysics. Routledge.
    While Darwin is commonly supposed to have demonstrated the inapplicability of the Aristotelian ontology of species to biological science, recent developments, especially in the wake of the Human Genome Project, have given rise to a new golden age of classification in which ontological ideas -- as for example in the Gene Ontology, the Cell Ontology, the Protein Ontology, and so forth -- are once again playing an important role. In regard to species, on the other hand, matters are more complex. (...)
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  24. Eric Steinhart (2002). Why Numbers Are Sets. Synthese 133 (3):343 - 361.
    I follow standard mathematical practice and theory to argue that the natural numbers are the finite von Neumann ordinals. I present the reasons standardly given for identifying the natural numbers with the finite von Neumann's (e.g., recursiveness; well-ordering principles; continuity at transfinite limits; minimality; and identification of n with the set of all numbers less than n). I give a detailed mathematical demonstration that 0 is { } and for every natural number n, n is the set of all natural (...)
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  25. Rafal Urbaniak (2010). Neologicist nominalism. Studia Logica 96 (2):149-173.
    The goal is to sketch a nominalist approach to mathematics which just like neologicism employs abstraction principles, but unlike neologicism is not committed to the idea that mathematical objects exist and does not insist that abstraction principles establish the reference of abstract terms. It is well-known that neologicism runs into certain philosophical problems and faces the technical difficulty of finding appropriate acceptability criteria for abstraction principles. I will argue that a modal and iterative nominalist approach to abstraction principles circumvents those (...)
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  26. Rafal Urbaniak (2008). Lesniewski's Systems of Logic and Mereology; History and Re-Evaluation. Dissertation, University of Calgary
  27. Krzysztof Wójtowicz (2008). Redukcje ontologiczne w matematyce. Część I. Filozofia Nauki 3.
    The article is the first part of a series of papers devoted to the problem of ontological reductions in mathematics – in particular, of choosing the basic category of mathematical entities. The received view is that such a category is provided by set theory, which serves as the ontological framework for the whole of mathematics (as all mathematical entities can be represented as sets). However, from the point of view of "naive mathematical realism" we should rather think of the mathematical (...)
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