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O programa newtoniano de matematização da natureza fornece os elementos necessários para a investigação de uma filosofia da matemática no pensamento de Isaac Newton, especialmente ao se considerar a relação entre as práticas matemáticas de Newton e Descartes. Tal relação suscita uma gama bastante ampla de problemas os quais interessam para esta pesquisa, particularmente, aqueles que dizem respeito ao realismo matemático. O que pretendemos é extrair dessa discussão aspectos relevantes da filosofia e da prática matemática de Newton, com vistas a esclarecer em que medida suas opções matemáticas o comprometem com a ontologia dos objetos matemáticos / The Newtonian program of mathematization of nature provides the necessary elements for investigating the philosophy of mathematics in Isaac Newtons thought, especially when considering the relationship between the mathematical practices of Newton and Descartes. This relationship evokes a wide range of problems which are of interest for this research, in particular issues concerning mathematical realism. The goal is to extract from this discussion relevant aspects of the Newtons mathematical philosophy and practice, with a view to clarify how his mathematical options commit him to the ontology of mathematical objects

The purpose of this dissertation is to provide an account of the metaphysical status of mathematical entities in Aristotle. Aristotle endorses a form of realism about mathematical entities: for him as well as for Platonists, anti-realism, the view that mathematical objects do not exist, is not a viable option. The thesis consists of two main parts: a part dedicated to the objects of geometry, and a part dedicated to numbers. Furthermore, I have included an introductory chapter about a passage in the second chapter of Book B of the Physics (193b31- 194a7) where Aristotle endorses a form of naïve realism with regard to mathematical entities. Many of the passages that give us an insight into Aristotle's philosophy of mathematics are to be found in the third chapter of Book M of the Metaphysics. Aristotle's primary concern there, however, is not so much to present his own positive account as to provide answers to a series of (not so obvious) Platonic arguments. In the second chapter of my thesis, I discuss some of those arguments and highlight their role in Aristotle's own position about the metaphysical status of geometrical entities. In a passage that is of crucial importance to understand Aristotle's views regarding the mode of existence of the objects of mathematics (Meta. M.3, 1078a25-31), Aristotle allows for the potential existence of them. I argue that Aristotle's sketchy remarks in Meta. M.3 point towards a geometry based on the commonsensical notion of the solid. This account can be further developed if we take into consideration the purpose of the preceding chapter M.2: to refute Platonic arguments that attribute greater metaphysical status to 'limit entities' (entities bounding and within a physical body), that is, to points, lines, and surfaces. According to Aristotle, such 'limit entities' have only a potential existence-what does this claim amount to? To answer this question, I will explore a more traditional reading of this claim and I will also put forward a more radical one: from a contemporary perspective, this reading makes Aristotelian geometry a distant cousin of modern Whiteheadian or Tarskian geometries. Providing an account of the metaphysical status of number in Aristotle poses quite a few challenges. On the one hand, the scarcity of the evidence forces commentators to rely on a few scattered remarks (primarily from the Physics) and to extract Aristotle's own views from heavily polemical contexts (such as the convoluted arguments that occupy much of books M and N of the Metaphysics). On the other hand, the Fregean tradition casts a great shadow upon the majority of the interpretations; indeed, a great amount of the relevant scholarship is dominated by Fregean tendencies: it is, for example, widely held that numbers for Aristotle are not supposed to be properties of objects, much like colour, say, or shape, but second-order properties (properties-of-properties) of objects. The scope of the third chapter is to critically examine some of the Fregean-inspired arguments that have led to a thoroughly Fregean depiction of Aristotle, and to lay the foundations for an alternative reading of the crucial texts.

O programa newtoniano de matematização da natureza fornece os elementos necessários para a investigação de uma filosofia da matemática no pensamento de Isaac Newton, especialmente ao se considerar a relação entre as práticas matemáticas de Newton e Descartes. Tal relação suscita uma gama bastante ampla de problemas os quais interessam para esta pesquisa, particularmente, aqueles que dizem respeito ao realismo matemático. O que pretendemos é extrair dessa discussão aspectos relevantes da filosofia e da prática matemática de Newton, com vistas a esclarecer em que medida suas opções matemáticas o comprometem com a ontologia dos objetos matemáticos / The Newtonian program of mathematization of nature provides the necessary elements for investigating the philosophy of mathematics in Isaac Newtons thought, especially when considering the relationship between the mathematical practices of Newton and Descartes. This relationship evokes a wide range of problems which are of interest for this research, in particular issues concerning mathematical realism. The goal is to extract from this discussion relevant aspects of the Newtons mathematical philosophy and practice, with a view to clarify how his mathematical options commit him to the ontology of mathematical objects

Among the various theses in the philosophy of mathematics, intuitionism is the thesis that numbers are constructs of the human mind. In this thesis, a historical account of intuitionism will be exposited- - from its beginnings in Kant's classic work, Critique of Pure Reason, to contemporary treatments by Brouwer and other intuitionists who have developed his position further. In chapter II, I examine the ontology of Kant's philosophy of arithmetic. The issue at hand is to explore how Kant, using intuition and time, argues for numbers as mental constructs. In chapter III, I examine how mathematics for Kant yields synthetic a priori truth, which is to say an informative statement about the world whose truth can be known independently of observation. In chapter IV, I examine how intuitionism developed under the care of Brouwer and others (e.g. Dummett) and how Hilbert sought to address issues in Kantian philosophy of mathematics with his finitist approach. In conclusion, I examine briefly what intuitionism resolves and what it leaves to be desired.

Among the various theses in the philosophy of mathematics, intuitionism is the thesis that numbers are constructs of the human mind. In this thesis, a historical account of intuitionism will be exposited- - from its beginnings in Kant's classic work, Critique of Pure Reason, to contemporary treatments by Brouwer and other intuitionists who have developed his position further. In chapter II, I examine the ontology of Kant's philosophy of arithmetic. The issue at hand is to explore how Kant, using intuition and time, argues for numbers as mental constructs. In chapter III, I examine how mathematics for Kant yields synthetic a priori truth, which is to say an informative statement about the world whose truth can be known independently of observation. In chapter IV, I examine how intuitionism developed under the care of Brouwer and others (e.g. Dummett) and how Hilbert sought to address issues in Kantian philosophy of mathematics with his finitist approach. In conclusion, I examine briefly what intuitionism resolves and what it leaves to be desired.

Through philosophical means, this thesis investigates the question: What mathematics can mean to students philosophically and psychologically? It is reasonable to assume that students may touch upon philosophical issues in trying to make sense of mathematics, since, in a sense, all individuals philosophise while searching for meaning in their own activities. Moreover, the existing literature indicates a substantial gap in our understanding of the meaning of mathematics and its philosophy in education. The thesis is based on a hermeneutical perspective. In this context, in-depth interviews were conducted with 17-year-old students in a Greek school. This method allowed me to obtain data which illuminated the objective meaning of students’ philosophical beliefs by way of the subjective, psychological meaning that they attributed to mathematics. The sample consisted of 28 students comprising both sexes and all levels of engagement with mathematics. The main issues that were examined were: whether mathematics exists; whether mathematical knowledge is certain, objective, true and immutable; whether mathematics consists of rules; and whether mathematical knowledge is based on logic or on experience. A thematic analysis helped me to move within the hermeneutical circle of understanding. As well as organising the objective meaning of students’ philosophical beliefs into themes and subthemes, analysis showed how for each student, there was an emergent a story which illustrating how they could combine such beliefs in order to find subjective meaning in mathematics. The most important finding of the study suggests that the students’ beliefs were influenced by common sense, and that students were able to find positive subjective meaning in mathematics when they were able to relate aspects of mathematical reasoning (e.g. certainty, subjectivity, rules, experience) to the operation of their everyday common sense. The study therefore shows that discussing philosophical issues, and in particular mathematical reasoning, could be of considerable benefit for students learning mathematics.

Ano da defesa é 2010 / Orientador: Irineu Bicudo / Banca: Inocêncio Fernandes Balieiro Filho / Banca: Paulo Isamo Hiratsuka / Resumo: O objetivo dessa pesquisa é participar da discussão acerca das diferentes concepções de Platão e Aristóteles a respeito da natureza e do estatuto ontológico dos entes matemáticos. Enquanto Platão situa o âmbito ontológico dos entes matemáticos entre dois mundos, o sensível e o inteligível, Aristóteles nega o caráter supra-sensível dos objetos matemáticos e oferece como resposta a sua filosofia empirista da matemática. Aristóteles teria dirigido duras críticas contra Platão e os acadêmicos nos dois últimos livros da Metafísica, M e N, respectivamente. Desde a antiguidade, vários autores sustentam que tais críticas referem-se às "doutrinas não-escritas" de Platão, que seriam cursos por ele ministrados na Academia, cujo teor ele não quis escrever por considerar que somente à dialética oral caberia o ensinamento dos primeiros princípios. Utilizando uma metodologia de pesquisa filosófica e também a história da filosofia e da matemática, foram abordados diversos textos, que vão desde livros e artigos atuais, até as próprias obras de Platão e Aristóteles relacionadas ao tema. Como parte das reflexões finais, o presente trabalho destaca a importância da exegese para uma correta interpretação das filosofias da matemática de Platão e Aristóteles e ainda das relações entre elas. / Abstract: The research aim is the discussion about Plato and Aristotle's different conceiving about the nature and the ontological status of mathematical entities. While Plato located the ontological scope of mathematical entities between two worlds, the sensible and the intelligible, Aristotle denies the character "super-sensible" of the mathematical entities and offers in response his own empiricist philosophy of mathematics. Aristotle would have direct harsh criticism to Plato and the academics in two last books of his Metaphysics, M and N, respectively. Since ancient times several authors argue that these criticism refer to "unwritten doctrines" of Plato, that they would be courses that he taught at the Academy, whose contents he did not want to write because he had believe that only oral dialectic should teach the first principles. Using a philosophical methodology of research and also the history of philosophy and mathematics several texts were discussed, like current books and articles as well as works of Plato and Aristotle about the theme. As part of final reflection, the present work highlights the exegesis importance for a correct interpretation of the mathematics philosophy from Plato and Aristotle and even the relationships between them. / Mestre

Thesis (Ph. D.)--University of Texas at Austin, 1998. / Vita. Includes bibliographical references (leaves 266-274). Available also in a digital version from Dissertation Abstracts.