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

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 1999.
Includes bibliographical references.
"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.

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.

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.

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.

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.

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

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.

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

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

El objetivo de este trabajo es examinar el pensamiento filosófico del jesuita valenciano Benet Perera (1535-1610), dividiendo dicho examen en tres secciones temáticas, dedicadas a la reflexión de este autor sobre las matemáticas, la filosofía natural y la metafísica. Célebre entre sus contemporáneos por sus monumentales obras de exégesis bíblica, en el latín humanista de Perera se unen una inagotable erudición con la agudeza necesaria para dar cuenta de los grandes problemas filosóficos de su época en consonancia con el espíritu reformista y militante que emana del Concilio de Trento. Su De communibus omnium rerum naturalium, tratado de filosofía natural publicado en 1576, ha despertado últimamente un creciente interés entre los historiadores de la filosofía del renacimiento, no solamente por representar un exhaustivo compendio de las cuestiones filosóficas que constituyen el fermento en el que se fraguó la naciente filosofía moderna, sino también por constituir un texto de referencia para figuras como Giordano Bruno, Galileo o Leibniz. Compartiendo las coordenadas metodológicas y las claves hermenéuticas de las más recientes investigaciones sobre la obra de Perera, en nuestro trabajo hemos tratado de ofrecer una visión integral del pensamiento del valenciano, entendiéndolo como el intento de construir una cosmovisión adecuada a las exigencias religiosas de la Compañía de Jesús. De esta manera, mostraremos cómo la intervención de Perera en la renacentista quaestio de certitudine mathematicarum no sólo responde a un intento de conservar el paradigma aristotélico tradicional, sino que también se debe vincular con las tendencias anti-astrológicas presentes en el proyecto reformista católico. Examinaremos, asimismo, cómo la filosofía natural desplegada en la obra de nuestro autor se construye en franca oposición a lo que podríamos considerar una «física protestante», desarrollada en torno al círculo de Melanchthon. En nuestra tercera sección, en fin, analizaremos de qué modo la disolución de la estructura onto-teológica de la metafísica, que Perera opera al distinguir entre una filosofía primera dedicada al ens y una metafísica consagrada al estudio de la realidad suprasensible, constituye también el reflejo de la estructura jerárquica que la joven Compañía de Jesús trató de imprimir a su visión del mundo. En todo este recorrido, trataremos de rastrear la influencia que las tesis filosóficas de Perera ejercieron en autores contemporáneos y posteriores, con el fin de probar que el peso de nuestro autor en el devenir de la historia de la filosofía justifica con creces la necesidad de estudiar a fondo sus textos publicados e inéditos.
The aim of this work is to examine the philosophical works of the Valencian Jesuit Benet Perera (1535-1610) on philosophy of mathematics, natural philosophy and metaphysics. Celebrated among his contemporaries for his monumental works of biblical exegesis, in the humanist latin of Perera an inexhaustible erudition is joined with the acuity necessary to account for the great philosophical problems of his time in line with the reformist and militant spirit emanating from the Council of Trent. His De communibus omnium rerum naturalium (1576) represents an exhaustive compendium of the early modern philosophical issues later used by Giordano Bruno, Galileo or Leibniz. Sharing the methodological coordinates and hermeneutic keys of the most recent research on Perera's work, we offer here a comprehensive vision of his thought as an effort to build an appropriate weltanschauung to the religious demands of the Society of Jesus. We will show how Perera's intervention in the Renaissance Quaestio de certitudine mathematicarum not only aims to preserve the traditional Aristotelian paradigm but must also be linked to the anti-astrological tendencies of the Catholic reformist project. We will also examine how Perera’s natural philosophy is deployed in opposition to what we might consider a "Protestant physics", developed around Melanchthon's circle. Finally, we study how the dissolution of the onto-theological structure of metaphysics, which Perera operates by distinguishing between a first philosophy dedicated to ens and a metaphysics of suprasensitive reality, is also a reflection of the hierarchical structure that the young Society of Jesus sought to print into her world-view. Throughout our work, we will trace the influence that Perera exerted on contemporary and later authors, in order to prove that the weight of our author in the history of modern philosophy more than justifies a careful read of his published and unpublished texts.

Thomas Hobbes est sans doute mieux connu comme philosophe politique que comme homme de science et ses longues querelles avec John Wallis en mathématiques et Robert Boyle en physique n’ont guère encouragé les historiens des sciences à prêter attention à son œuvre scientifique. Pourtant, Hobbes conçut la philosophie comme une science et se considérait comme le fondateur non seulement d’une science nouvelle : la philosophie civile, mais aussi de la science de l’optique - récemment renouvelée à la faveur de la découverte du télescope - et même des mathématiques. Mais à quoi Hobbes pense-t-il quand il parle de science ? Aux mathématiques qu’il admire tant ? A la philosophie naturelle de Galilée ? Ou à la médecine de Harvey ? En quel sens la philosophie civile est-elle une science et quel est le statut des mathématiques ? Telles sont les questions que nous abordons à partir d’une analyse du De Corpore et des dix premiers chapitres du De Homine traduits du latin. L’interprétation proposée ici consiste à réaffirmer l’unité du système des Éléments de la philosophie et à souligner la dimension matérialiste et réaliste de la science hobbesienne. Bien que Noel Malcolm ait définitivement établi que Hobbes n’est pas l’auteur du Short Tract on first principles, nous montrons que le tournant scientifique de Hobbes est profondément marqué par son intérêt pour l’optique qu’il renouvela sur la base d’une ontologie matérialiste et des principes du mécanisme hérités de Galilée
Thomas Hobbes is perhaps best known as a political philosopher than as a scientist and his too long quarrels with John Wallis in mathematics and Robert Boyle in physics did little to encourage historians of science to pay attention to his scientific work. Yet Hobbes conceived of philosophy as a science and considered himself the founder not only of a new science: civil philosophy, but also the science of optics - recently renewed thanks to the discovery of the telescope - even mathematics. But what Hobbes has in mind when he talks about science? Mathematics he so admires? Galileo’s natural philosophy? Or Harvey’s medicine? In what sense civil philosophy is a science and what is the status of mathematics? These are the issues we discuss from an analysis of De Corpore and the first ten chapters of De Homine translated from Latin. The interpretation proposed here is to underline the unity of the system of the Elements of philosophy and emphasize the materialistic and realistic nature of Hobbesian science. Although Noel Malcolm has definitively established that Hobbes is not the author of Short Tract on First Principles, we show that Hobbes’s shift to science was deeply marked by his interest in the science of optics he renewed on the basis of a materialist ontology and principles inherited from Galilee mechanism

1 Abstract Meaning of numbers between Plato and Aristotle Antonín Šíma The dissertation titled "The Transformation of the Concept of Number between Plato and the Early Academy" deals with the problem of numbers in early Platonism between Aristotle and Plato. In Plato's dialogues, within professional mathematical disciplines of knowledge, numbers fulfil a function of propaedeutic procedure to the method of thinking − dialectic. Dialectic engages in the most general structures of thinking whose centre is the problem of being and good, which is only mentioned marginally in our thesis. The philosophy of dialogues is based on the ontological and epistemological dignity of unchanging and eternally existing ideas. In Metaphysics A Aristotle describes Plato's and the Platonic doctrines of the early Academy in whose centre there are principles expressed by numbers: one and indefinite two, which are assessed according to Aristotelian principles doctrine as form and matter. Aristotle mentions Platonic dialectical method which focuses on researching the general in speech. This method distinguishes Platonic thought from Pythagorean philosophy in Aristotle's precursors' philosophy overview. In the criticized doctrine, numbers have the same meaning as ideas or ideal numbers standing on the scale of ontological dignity...

(in English): The aim of the work is to reconstruct and interpret the method of historicized analysis and its employment to examine the phenomenon of "making up people". The concept is Hacking's description for the impact scientific classifications can have on classified people. The point of departure for the examination in the work is the thesis that historicized analysis employs the elements of philosophical conceptual analysis together with historical tools philosophy of science corroborates and whose strategies are often in opposition to the analytical tradition. As a follow-up of the main thesis the work also examines whether the historicized analysis can be understood as a history of the present. Moreover, it asks questions that come up in connection with the project of "making up people" such as: "What are the conditions for a scientific category to emerge? When categories emerge do new kinds of people emerge as well? What is the specific structure that enables the mutual interaction and effect scientific categories and classified people make? One of the aims will therefore be to elucidate to what extend the historicized analysis is able to answer those questions. Last but not least the work looks into the critical implications and usefulness of the method of historicized analysis.

Tomáš Pivoda, The Multiplicity of Being: The Ontology of Alain Badiou PhD thesis Abstract The thesis introduces for the first time in the Czech philosophical context the ontology of the French philosopher Alain Badiou, as he set it out in his fundamental work Being and Event (L'être et l'événement, 1988). It first presents the starting point of Badiou's philosophy as well as the reasons of his identification of ontology with the set theory, and it points out Badiou's importance for contemporary philosophy, especially for the so called speculative realism around Quentin Meillassoux. The main axis of the exposition is then built around Badiou's four fundamental "Ideas": the multiplicity, the event, the truths and the subject, in connection with which it is shown how Badiou constructs his conceptual apparatus out of individual axioms of the set theory, whereby he follows the basic formal definition of multiplicity based on the operator . In connection with∈ the first Idea of multiplicity, the thesis exposes - with references to Martin Heidegger and Plato - Badiou's conceptual transposition of the couple one/multiple on the couple existence/being and defines the fundamental concepts of his ontology - the situation, the presentation, the representation and the void, with the help of which Badiou interprets...