Relevant bibliographies by topics / Ontology, philosophy of mathematics, nominalism

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters

Academic literature on the topic 'Ontology, philosophy of mathematics, nominalism'

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Published: 9 March 2023
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Journal articles on the topic "Ontology, philosophy of mathematics, nominalism"

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Akhlaghi-Ghaffarokh, Farbod. "Existence, Mathematical Nominalism, and Meta-Ontology: An Objection to Azzouni on Criteria for Existence†." Philosophia Mathematica 26, no. 2 (April 3, 2018): 251–65. http://dx.doi.org/10.1093/philmat/nky006.

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Rodríguez Consuegra, Francisco. "Números, objetos y estructuras." Crítica (México D. F. En línea) 23, no. 68 (December 13, 1991): 7–86. http://dx.doi.org/10.22201/iifs.18704905e.1991.801.

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Introduction. This paper is principally a critical exposition of the celebrated article by Benacerraf, indicating briefly íts antecedents, emphasizing its accomplishments, problems and basic insufficiencies, followed by an evaluation of the main criticisms to which Benacerraf's article has been subjected, as well as a study of the historical framework in which a new global criticism is meaningful. The paperends with the examination of a possible connection with the structuralist philosophy of mathematics, which is in part inspired by the work of Benacerraf. Mathematical reduction according to Benacerraf (sections 2 and 3). It is here shown that the fundamental "objective" antecedents to Benacerraf's work are Quine, along with Parsons, and the nominalism of Goddard; and sorne important differences are also pointed out. Discussed is Benacerraf's rejection of the identification of numbers and objects, and its substitution by progressions in the framework of the typically Quinean argument of set polymorphism, as well as his difficult theory of identity, all of which without a clearly relativist ontological contexto His reduction of numbers to positions in a progression is situated in the now old debate between the cardinal and the ordinal, and is a step in the direction of the nascent structuralism, although it lacks sufficient justification. Some criticisms [sections 4 and 5). An evaluative study is made of the criticisms that seem to me most accurate, or the most revealing of underlying problems. Reviewed are the most relevant among such criticisms in the literature: Steiner, Resnik, Maddy, Wright, and Hale along with others of the enormous quantity of articles discussing this topic that have appeared over the last twenty years, Common lines are traced out, and some possible defenses of Benaeerraf are indicated, although again the weaknesses of his position are pointed out, weaknesses stemming from its unresolved problems (the historieal framework, the ill-defined ontology, the nascent structuralism, etc.). Essential criticisms (section 6). Beginning with the problem of counting, the axis of Benaeerraf's work, an historical excursion is presented, in which it is shown that the problem pointed out above (cardinal versus ordinal) can be seen as the center of the indicated difficulties. The theory of Dedekind-Peano is compared with that of Cantor, and the epistemological and constructive advantages of the latter are noted. It is shown how positions very similar to Benacerraf's were already held by Cassirer and Weyl (without mentioning Berkeley!); meanwhile, the Cantorian approach of Couturat and RusseU is shown to be superior, at least from the point of view of a global coneeption. Finally, the eonneetion between eonstructions and polymorphism---a problem shared by logic, mathematics and physics---is pointed out. The structuralist tendency and platonism (section 7). The antecedents of the structuralism of Resnik and Shapiro are traced to Benacerraf himself ---and the historical trace is further extended back to the ordinalists, Bourbaki and Quine--- in the hope of shedding light on the basic problem: the supposed antithesis between terms and relations (already familiar in Bradley and Russell). Further, I suggest and examine a parallelism with the relativism of mathematical entities such as this appears after the limitations of (at least first order) axiomatization. The paper ends making a connection of the subject with the theory of categories, which, surprisingly, still has not been eonsidered by the strueturalist, despite the fact that it is quite clearly a natural extension of the structuralist point of view. [Traducción de Raúl Orayen y Mark Rollins]
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Pavlov, Alexey S. "Property Nominalism in the Contemporary Analytic Metaphysics." History of Philosophy Yearbook 27 (December 28, 2022): 181–208. http://dx.doi.org/10.21146/0134-8655-2022-37-181-208.

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This article aims to give an introductory overview of Property Nominalism in contemporary analytic metaphysics for a Russian-speaking reader. The question of the level and order relations between different properties has been much debated in analytic philosophy over the last 100 years. However, analytic philosophers have often given much less attention to more fundamental questions of the ontological status of properties and how properties relate to the objects that instantiate them. The most plausible answer to these questions would be Property Nominalism which is well compatible with Physicalism – the Weltanschauung currently dominant among analytic philosophers. As in the case of medieval nominalism, the pathos of Property Nominalism is in eliminating redundant entities in our ontology. However, it should be distinguished from the nominalism of Willard V.O. Quine, Alfred Tarski and Nelson Goodman, which extends not only to universals but to the entire class of abstract objects. Property Nominalism is subdivided into Non-reductive (Ostrich) and Reductive Nominalism. The latter includes Predicative (or Conceptual) Nominalism, Class Nominalism, Tropes Nominalism and Resemblance Nominalism, presented in the works of Gonzalo Rodriguez-Pereyra. In the opinion of the author of this article, we will arrive at an optimal solution to the problem of universals if we adhere to the explanatory models of Ostrich Nominalism and Resemblance Nominalism. The advantages of these theories are that they do not postulate any additional entities and do not require us to develop a specialized theory of substance.
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LO, TIEN-CHUN. "Theist concept-nominalism and the regress problem." Religious Studies 55, no. 02 (July 2, 2018): 199–213. http://dx.doi.org/10.1017/s003441251800046x.

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AbstractLeftow's theist concept-nominalism is proposed as a theory of properties which is compatible with God's aseity and sovereignty. In this article, I focus on the question of whether theist concept-nominalism is successful in answering a notorious problem in the literature on properties, i.e. the regress problem. In the second section, I summarize TCN by illustrating what its ontology is and how its theory works. In the third section, the regress problem is recast within the framework of TCN. In the fourth section, I present my solution to this problem. In the final section, several objections to my solution are addressed and replied.
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Rayo, A. "Nominalism, Trivialism, Logicism." Philosophia Mathematica 23, no. 1 (June 16, 2014): 65–86. http://dx.doi.org/10.1093/philmat/nku013.

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Hochberg, Herbert. "Nominalism and Idealism." Axiomathes 23, no. 2 (March 25, 2011): 213–34. http://dx.doi.org/10.1007/s10516-011-9150-3.

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Smith, R. Scott. "Tropes and Some Ontological Prerequisites for Knowledge." Metaphysica 20, no. 2 (October 25, 2019): 223–37. http://dx.doi.org/10.1515/mp-2019-2013.

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Abstract Many have written about trope ontology, but relatively few have considered its implications for some of the ontological conditions needed for us to have knowledge. I explore the resources of trope ontology to meet those conditions. With J. P. Moreland, I argue that, being simple, we can eliminate tropes’ qualitative contents without ontological loss, resulting in bare individuators. Then I extend Moreland’s argument, arguing that tropes undermine some of the needed ontological conditions for knowledge. Yet, we do know many things, and trope nominalists presuppose that too. Therefore, I consider three counter-arguments, starting with David Lewis’s rebuttal based on appeal to brute facts. Second, I explore Jeffrey Brower’s recent proposal as a possible solution. Last, I consider Robert Garcia’s recent distinction between module and modifier tropes, to see if it can be of assistance. I conclude, however, that trope nominalism cannot preserve some of the needed ontology to have knowledge.
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Rizza, D. "Mathematical Nominalism and Measurement." Philosophia Mathematica 18, no. 1 (June 29, 2009): 53–73. http://dx.doi.org/10.1093/philmat/nkp010.

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Kovacs, David Mark. "Deflationary Nominalism and Puzzle Avoidance†." Philosophia Mathematica 27, no. 1 (October 1, 2018): 88–104. http://dx.doi.org/10.1093/philmat/nky019.

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Sepkoski, David. "Nominalism and constructivism in seventeenth-century mathematical philosophy." Historia Mathematica 32, no. 1 (February 2005): 33–59. http://dx.doi.org/10.1016/j.hm.2003.09.002.

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Dissertations / Theses on the topic "Ontology, philosophy of mathematics, nominalism"

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Uzquiano, Gabriel 1968. "Ontology and the foundations of mathematics." Thesis, Massachusetts Institute of Technology, 1999. http://hdl.handle.net/1721.1/9370.

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Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 1999.
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"Ontology and the Foundations of Mathematics" consists of three papers concerned with ontological issues in the foundations of mathematics. Chapter 1, "Numbers and Persons," confronts the problem of the inscrutability of numerical reference and argues that, even if inscrutable, the reference of the numerals, as we ordinarily use them, is determined much more, precisely than up to isomorphism. We argue that the truth conditions of a variety of numerical modal and counterfactual sentences (whose acceptance plays a crucial role in applications) place serious constraints on the sorts of items to which numerals, as we ordinarily use them, can be taken to refer: Numerals cannot be taken to refer to objects that exist contingently such as people, mountains, or rivers, but rather must be taken to refer to objects that exist necessarily such as abstracta. Chapter 2, "Modern Set Theory and Replacement," takes up a challenge to explain the reasons one should accept the axiom of replacement of Zermelo-Fraenkel set theory, when its applications within ordinary mathematics and the rest of science are often described as rare and recondite. We argue that this is not a question one should be interested in; replacement is required to ensure that the element-set relation is well-founded as well as to ensure that the cumulation of sets described by set theory reaches and proceeds beyond the level w of the cumulative hierarchy. A more interesting question is whether we should accept instances of replacement on uncountable sets, for these are indeed rarely used outside higher set theory. We argue that the best case for (uncountable) replacement comes not from direct, intuitive considerations, but from the role replacement plays in the formulation of transfinite recursion and the theory of ordinals, and from the fact that it permits us to express and assert the (first-order) content of the modern cumulative view of the set theoretic universe as arrayed in a cumulative hierarchy of levels. Chapter 3, "A No-Class Theory of Classes," makes use of the apparatus of plural quantification to construe talk of classes as plural talk about sets, and thus provide an interpretation of both one- and two-sorted versions of first-order Morse-Kelley set theory, an impredicative theory of classes. We argue that the plural interpretation of impredicative theories of classes has a number of advantages over more traditional interpretations of the language of classes as involving singular reference to gigantic set-like entities, only too encompassing to be sets, the most important of these being perhaps that it makes the machinery of classes available for the formalization of much recent and very interesting work in set theory without threatening the universality of the theory as the most comprehensive theory of collections, when these are understood as objects.
by Gabriel Uzquiano.
Ph.D.
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Knowles, Robert Frazer. "Towards a fictionalist philosophy of mathematics." Thesis, University of Manchester, 2015. https://www.research.manchester.ac.uk/portal/en/theses/towards-a-fictionalist-philosophy-of-mathematics(e078d675-7f4c-45e7-a1a0-baf8d899940d).html.

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In this thesis, I aim to motivate a particular philosophy of mathematics characterised by the following three claims. First, mathematical sentences are generally speaking false because mathematical objects do not exist. Second, people typically use mathematical sentences to communicate content the truth of which does not require mathematical objects to exist. Finally, in using mathematical language in this way, speakers are not doing anything out of the ordinary: they are performing straightforward assertions. In Part I, I argue that the role played by mathematics in our scientific explanations is a purely expressive one, merely allowing us to say more than we otherwise would be able to about, or yielding a greater understanding of, the physical world. Mathematical objects to not need to exist for mathematical language to play this role. This proposal puts a normative constraint on our use of mathematical language: we ought to use mathematically presented theories to express belief only in the consequences they have for non-mathematical things. In Part II, I will argue that what the normative proposal recommends is in fact what people generally do in both pure and applied mathematical contexts. I motivate this claim by showing that it is predicted by our best general means of analysing natural language. I provide a semantic theory of applied arithmetical sentences and show that they do not purport to refer to numbers, as well as a pragmatic theory for pure mathematical language use which shows that pure mathematical utterances do not typically communicate content that implies the existence of mathematical objects. In conclusion, I show the hermeneutic fictionalist position that emerges is preferable to any alternative which interprets mathematical discourse as aimed at describing a domain of independently existing abstract mathematical objects.
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Gan, Nathaniel. "A Functional Approach to Ontology." Thesis, The University of Sydney, 2021. https://hdl.handle.net/2123/24947.

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This thesis develops the functional approach to ontology, a conceptual-analytical approach that seeks to adjudicate ontological debates by analysing the functions of our concepts corresponding to the disputed entities. The functional approach suggests a method of determining when we should posit ontological facts regarding disputed entities and, if we should, when we should also affirm the existence of those entities. Indispensability arguments, which are commonly employed in ontological debates, can be interpreted in line with the functional approach; thus, this approach can accommodate what presently goes on in ontological debates better than deflationary approaches. We have an evaluative method by which to assess indispensability arguments; thus, the functional approach allows a clearer explication of ontological facts (when such facts are posited) than under mainstream approaches to ontology. The functional approach is applied to mathematics. It is argued that two functions of our mathematical concepts (pertaining to our mathematical discourse and scientific explanations), potentially bear on the debate over mathematical ontology and connect with some popular arguments for Platonism. Non-realist views on mathematical ontology are also examined, and the ontological position of mathematical fictionalism (and similar views) is clarified. Then, some popular anti-Platonist arguments are considered. It is argued that although some popular proposed solutions to the Benacerrafian problems do not succeed, the problems are not insurmountable. This puts Platonism in a favourable dialectical position compared to other views on mathematical ontology. The debate over mathematical ontology is analogous to other ontological debates at several points, so we have reason to favour realism in other domains. Thus, the functional approach affords progress in long-standing ontological debates and assuages worries that ontological debates might be misguided.
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Collin, James Henry. "Nominalist's credo." Thesis, University of Edinburgh, 2013. http://hdl.handle.net/1842/7997.

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Introduction: I lay out the broad contours of my thesis: a defence of mathematical nominalism, and nominalism more generally. I discuss the possibility of metaphysics, and the relationship of nominalism to naturalism and pragmatism. Chapter 2: I delineate an account of abstractness. I then provide counter-arguments to claims that mathematical objects make a di erence to the concrete world, and claim that mathematical objects are abstract in the sense delineated. Chapter 3: I argue that the epistemological problem with abstract objects is not best understood as an incompatibility with a causal theory of knowledge, or as an inability to explain the reliability of our mathematical beliefs, but resides in the epistemic luck that would infect any belief about abstract objects. To this end, I develop an account of epistemic luck that can account for cases of belief in necessary truths and apply it to the mathematical case. Chapter 4: I consider objections, based on (meta)metaphysical considerations and linguistic data, to the view that the existential quantifier expresses existence. I argue that these considerations can be accommodated by an existentially committing quantifier when the pragmatics of quantified sentences are properly understood. I develop a semi-formal framework within which we can define a notion of nominalistic adequacy. I show how our notion of nominalistic adequacy can show why it is legitimate for the nominalist to make use of platonistic “assumptions” in inference-making. Chapter 5: I turn to the application of mathematics in science, including explanatory applications, and its relation to a number of indispensability arguments. I consider also issues of realism and anti-realism, and their relation to these arguments. I argue that abstraction away from pragmatic considerations has acted to skew the debate, and has obscured possibilities for a nominalistic understanding of mathematical practices. I end by explaining the notion of a pragmatic meta-vocabulary, and argue that this notion can be used to carve out a new way of locating our ontological commitments. Chapter 6: I show how the apparatus developed in earlier chapters can be utilised to roll out the nominalist project to other domains of discourse. In particular, I consider propositions and types. I claim that a unified account of nominalism across these domains is available. Conclusion: I recapitulate the claims of my thesis. I suggest that the goal of mathematical enquiry is not descriptive knowledge, but understanding.
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Mount, Beau Madison. "The kinds of mathematical objects." Thesis, University of Oxford, 2017. http://ora.ox.ac.uk/objects/uuid:47784b87-7a7b-43c0-8ce2-8983a867d560.

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The Kinds of Mathematical Objects is an exploration of the taxonomy of the mathematical realm and the metaphysics of mathematical objects. I defend antireductionism about cardinals and ordinals: the view that no cardinal number and no ordinal number is a set. Instead, I suggest, cardinals and ordinals are sui generis abstract objects, essentially linked to specific abstraction functors (higher-order functions corresponding to operators in abstraction principles). Sets, in contrast, are not essentially values of abstraction functors: the best explanation of the nature of sethood is given by a variation on the standard iterative account. I further defend the theses that no cardinal number is an ordinal number and that the natural numbers are, as Frege maintained, all and only the finite cardinal numbers. My case for these conclusions relies not on the well-known antireductionist argument developed by Paul Benacerraf, but on considerations about ontological dependence. I argue that, given generally accepted principles about the dependence of a set on its elements, ordinal and cardinal numbers have dependence profiles that are not compatible with any version of set-theoretic ontological reductionism. In addition, a formal framework for set theory with sui generis abstract objects is developed on a type-theoretical basis. I give a philosophical defence of the choice of type theory and discuss various questions relating to the nature of its models.
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Cole, Julian C. "Practice-dependent realism and mathematics." Connect to resource, 2005. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1124122328.

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Thesis (Ph. D.)--Ohio State University, 2005.
Title from first page of PDF file. Document formatted into pages; contains xi, 248 p. Includes bibliographical references (p. 244-248). Available online via OhioLINK's ETD Center
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Lawrence, Nicholas. "A Brief Introduction to Transcendental Phenomenology and Conceptual Mathematics." Thesis, Södertörns högskola, Filosofi, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:sh:diva-32873.

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By extending Husserl’s own historico-critical study to include the conceptual mathematics of more contemporary times – specifically category theory and its emphatic development since the second half of the 20th century – this paper claims that the delineation between mathematics and philosophy must be completely revisited. It will be contended that Husserl’s phenomenological work was very much influenced by the discoveries and limitations of the formal mathematics being developed at Göttingen during his tenure there and that, subsequently, the rôle he envisaged for his material a priori science is heavily dependent upon his conception of the definite manifold. Motivating these contentions is the idea of a mathematics which would go beyond the constraints of formal ontology and subsequently achieve coherence with the full sense of transcendental phenomenology. While this final point will be by no means proven within the confines of this paper it is hoped that the very fact of opening up for the possibility of such an idea will act as a supporting argument to the overriding thesis that the relationship between mathematics and phenomenology must be problematised.
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Declos, Alexandre. "La métaphysique de Nelson Goodman." Thesis, Université de Lorraine, 2017. http://www.theses.fr/2017LORR0238/document.

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Cette thèse de doctorat est consacrée à la pensée du philosophe américain Nelson Goodman (1906-1998). Nous y défendons, à l’encontre de la plus grande partie de la critique, une lecture métaphysicienne de son œuvre. Il est démontré que Goodman, dans tous ses travaux, développe une métaphysique technique et méconnue, dont les piliers sont le nominalisme, le pluralisme, le perdurantisme, l’actualisme, et l’universalisme méréologique. Cette lecture permet de réévaluer l’ensemble de la pensée goodmanienne. Elle établit aussi ses liens insoupçonnés avec la métaphysique analytique contemporaine
This PhD dissertation is dedicated to the philosophy of Nelson Goodman (1906-1998). We defend, against most critics, a metaphysical interpretation of Goodman’s works. It will be shown that the latter developed a technical and often overlooked metaphysics, whose pillars are nominalism, pluralism, perdurantism, actualism, and mereological universalism. This reading allows for a critical reevaluation of Goodman’s views. It also brings to light his unexpected links with contemporary analytic metaphysics
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Melanson, William Jason. "Justified existential belief an investigation of the justifiability of believing in the existence of abstract mathematical objects /." Columbus, Ohio : Ohio State University, 2006. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1140465070.

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Tricard, Julien. "Les quantités dans la nature : les conditions ontologiques de l’applicabilité des mathématiques." Thesis, Sorbonne université, 2019. http://www.theses.fr/2019SORUL132.

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Si nos théories physiques peuvent décrire les traits les plus généraux de la réalité, on sait aussi que pour le faire, elles utilisent le langage des mathématiques. On peut alors légitimement se demander si notre capacité à décrire, sinon la nature intime des objets et phénomènes physiques, du moins les relations et structures qu’ils instancient, ne vient pas de cette application des mathématiques. Dans cette thèse, nous soutenons que les mathématiques sont si efficacement applicables en physique tout simplement parce que la réalité décrite par les physiciens est de nature quantitative. Pour cela, nous proposons d’abord une ontologie des quantités, puis des lois de la nature, qui s’inscrit dans les débats contemporains sur la nature des propriétés (théorie des universaux, théorie des tropes, ou nominalisme), et des lois (régularités, ou relations entre universaux). Ensuite, nous examinons deux sortes d’application des mathématiques : la mathématisation des phénomènes par la mesure, puis la formulation mathématique des équations reliant des grandeurs physiques. Nous montrons alors que les propriétés et les lois doivent être comme notre ontologie les décrit, pour que les mathématiques soient légitimement, et si efficacement, applicables. L’intérêt de ce travail est d’articuler des discussions purement ontologiques (et très anciennes, comme la querelle des universaux) avec des exigences épistémologiques rigoureuses qui émanent de la physique actuelle. Cette articulation est conçue de manière transcendantale, car la nature quantitative de la réalité (des propriétés et des lois) y est défendue comme condition d’applicabilité des mathématiques en physique
Assuming that our best physical theories succeed in describing the most general features of reality, one can only be struck by the effectiveness of mathematics in physics, and wonder whether our ability to describe, if not the very nature of physical entities, at least their relations and the fundamental structures they enter, does not result from applying mathematics. In this dissertation, we claim that mathematical theories are so effectively applicable in physics merely because physical reality is of quantitative nature. We begin by displaying and supporting an ontology of quantities and laws of nature, in the context of current philosophical debates on the nature of properties (universals, classes of tropes, or even nominalistic resemblance classes) and of laws (as mere regularities or as relations among universals). Then we consider two main ways mathematics are applied: first, the way measurement mathematizes physical phenomena, second, the way mathematical concepts are used to formulate equations linking physical quantities. Our reasoning has eventually a transcendental flavor: properties and laws of nature must be as described by the ontology we first support with purely a priori arguments, if mathematical theories are to be legitimately and so effectively applied in measurements and equations. What could make this work valuable is its attempt to link purely ontological (and often very ancient) discussions with rigorous epistemological requirements of modern and contemporary physics. The quantitative nature of being (properties and laws) is thus supported on a transcendental basis: as a necessary condition for mathematics to be legitimately applicable in physics
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Books on the topic "Ontology, philosophy of mathematics, nominalism"

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Deflating existential consequence: A case for nominalism. New York: Oxford University Press, 2004.

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Sepkoski, David. Nominalism and constructivism in seventeenth-century mathematical philosophy. London : New York: Routledge, 2007.

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Universals, qualities, and quality-instances: A defense of realism. Lanham, MD: University Press of America, 1985.

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Philosophy of mathematics: Structure and ontology. New York: Oxford University Press, 1997.

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Azzouni, Jody. Metaphysical myths, mathematical practice: The ontology and epistemology of the exact sciences. Cambridge [England]: Cambridge University Press, 1994.

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Dembiński, Bogdan. Późna nauka Platona: Związki ontologii i matematyki. Katowice: Wydawn. Uniwersytetu Śląskiego, 2003.

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Charles, Sayward, and Garavaso Pieranna, eds. Arithmetic and ontology: A non-realist philosophy of arithmetic. Amsterdam: Rodopi, 2006.

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Gillespie, Sam. The mathematics of novelty: Badiou's minimalist metaphysics. Melbourne: Re.Press, 2008.

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The mathematics of novelty: Badiou's minimalist metaphysics. Melbourne: Re.Press, 2008.

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Jedrzejewski, Franck. Ontologie des catégories. Paris: L'Harmattan, 2011.

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Book chapters on the topic "Ontology, philosophy of mathematics, nominalism"

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Çevik, Ahmet. "Mathematical Nominalism." In Philosophy of Mathematics, 287–304. Boca Raton: Chapman and Hall/CRC, 2021. http://dx.doi.org/10.1201/9781003223191-18.

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Quesada, Francisco Miró. "Logic, Mathematics, Ontology." In Philosophy of Mathematics Today, 3–37. Dordrecht: Springer Netherlands, 1997. http://dx.doi.org/10.1007/978-94-011-5690-5_1.

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de Freitas, Elizabeth. "Deleuze, Ontology, and Mathematics." In Encyclopedia of Educational Philosophy and Theory, 1–7. Singapore: Springer Singapore, 2016. http://dx.doi.org/10.1007/978-981-287-532-7_374-1.

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de Freitas, Elizabeth. "Deleuze, Ontology, and Mathematics." In Encyclopedia of Educational Philosophy and Theory, 412–19. Singapore: Springer Singapore, 2017. http://dx.doi.org/10.1007/978-981-287-588-4_374.

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Ishimoto, Arata. "Logicism Revisited in the Propositional Fragment of Leśniewski’s Ontology." In Philosophy of Mathematics Today, 219–32. Dordrecht: Springer Netherlands, 1997. http://dx.doi.org/10.1007/978-94-011-5690-5_12.

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Cotnoir, A. J., and Achille C. Varzi. "What is Mereology?" In Mereology, xvi—20. Oxford University Press, 2021. http://dx.doi.org/10.1093/oso/9780198749004.003.0001.

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This chapter provides a brief illustration of the centrality of part-whole inquiry throughout the history of philosophy, West and East. It explains two original motivations for the contemporary formal explorations of mereological systems. Husserl’s approach, stemming from Brentano, sought to treat part-whole relations as formal ontology – comprising a set of general structural principles applying to any objects whatsoever. Leśniewski’s approach was motivated by nominalism and the search for an alternative foundation for mathematics not beset by the paradoxes of naïve set theory. Some attention is paid to the different uses of ‘part’ in natural language and to whether mereology should be thought of as providing a single, overarching account. The final section details the logical machinery used throughout the book.
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"Chapter Seven. Nominalism." In Philosophy of Mathematics, 101–15. Princeton University Press, 2017. http://dx.doi.org/10.1515/9781400885244-009.

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8

Chihara, Charles. "Nominalism." In The Oxford Handbook of Philosophy of Mathematics and Logic, 483–514. Oxford University Press, 2005. http://dx.doi.org/10.1093/0195148770.003.0015.

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Guigon, Ghislain. "Nominalism." In Routledge Encyclopedia of Philosophy. London: Routledge, 2019. http://dx.doi.org/10.4324/9780415249126-n038-2.

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Abstract:
‘Nominalism’ refers to a family of views about what there is. The objects we are familiar with (e.g. hands, laptops, cookies, and trees) can be characterized as concrete and particular. Nominalists agree that there are such things. But one group of nominalists denies that anything is nonparticular and another group denies that anything is nonconcrete. These two sorts of nominalism, referred to as ‘nominalism about universals’ and ‘nominalism about abstract objects’, have common motivations in contemporary philosophy. According to nominalists, universals and abstract objects are mysterious entities whose claim to existence is suspicious or ad hoc. This gives them reasons to want to reject universals and abstract objects but leads to an explanatory challenge: nominalists must explain away the appearance of the universal or the abstract. Varieties of nominalism differ with respect to how they address this challenge. Universals are sui generis entities that are typically thought to be what properties are. Nominalist theories about universals can thus be divided between those that admit that there are sui generis properties but maintain that they are particular instead of universal; those that use constructions out of concrete particular objects to play the role of properties, and those that reject properties altogether. Each of these nominalist strategies has its own merits and difficulties. Theories that seem to posit the existence of abstract objects, like mathematics and set theory, are successfully used in natural sciences. The utility of these theories challenges nominalism about abstract objects and has shaped the current debate about whether these objects exist. According to some nominalists, theories that posit abstract objects are useful but false. Other nominalists maintain that these theories are true but that their truth does not entail that abstract objects exist.
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Rosen, Gideon, and John P. Burgess. "Nominalism Reconsidered." In The Oxford Handbook of Philosophy of Mathematics and Logic, 515–35. Oxford University Press, 2005. http://dx.doi.org/10.1093/0195148770.003.0016.

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