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Lakatos outlined a theory of mathematical discovery and justification, which suggests ways in which concepts, conjectures and proofs gradually evolve via interaction between mathematicians. Different mathematicians may have different interpretations of a conjecture, examples or counterexamples of it, and beliefs regarding its value or theoremhood. Through discussion, concepts are refined and conjectures and proofs modified. We hypothesise that: (i) it is possible to computationally represent Lakatos's theory, and (ii) it is useful to do so. In order to test our hypotheses we have developed a computational model of his theory. Our model is a multiagent dialogue system. Each agent has a copy of a pre-existing theory formation system, which can form concepts and make conjectures which empirically hold for the objects of interest supplied. Distributing the objects of interest between agents means that they form different theories, which they communicate to each other. Agents then find counterexamples and use methods identified by Lakatos to suggest modifications to conjectures, concept definitions and proofs. Our main aim is to provide a computational reading of Lakatos's theory, by interpreting it as a series of algorithms and implementing these algorithms as a computer program. This is the first systematic automated realisation of Lakatos's theory. We contribute to the computational philosophy of science by interpreting, clarifying and extending his theory. We also contribute by evaluating his theory, using our model to test hypotheses about it, and evaluating our extended computational theory on the basis of criteria proposed by several theorists. A further contribution is to automated theory formation and automated theorem proving. The process of refining conjectures, proofs and concept definitions requires a flexibility which is inherently useful in fields which handle ill-specified problems, such as theory formation. Similarly, the ability to automatically modify an open conjecture into one which can be proved, is a valuable contribution to automated theorem proving.

[pt] O objetivo dessa tese é expor a base filosófica por trás das propostas de Wittgenstein para a filosofia da matemática. Procuramos mostrar que há um núcleo semântico das quais essas propostas são derivadas: a idéia de que o significado de uma sentença deveria ser tomado como sendo suas condições de verdade. Procuramos acompanhar esse insight desde seu aparecimento em Frege, depois no Tractatus até a última fase do pensamento de Wittgenstein. Em nossos dois últimos capítulos discutimos as mudanças que essa abordagem sofreu nessa última fase do pensamento do filósofo em conexão com a idéia de triangularização da semântica subjacente. Tratamos em com algum nível de detalhamento as interpretações propostas por Wittgenstein de alguns tipos de proposições matemáticas elementares, especialmente proposições aritméticas. / [en] This thesis deals with the philosophical basis behind Wittgenstein s proposals in the philosophy of mathematics. We try to show that there is a semantic core from which these proposals are derived: the idea that the meaning of a sentence should be taken to be its truth conditions. We sort out this insight from its inception in Frege through the Tractatus all the way to the last phase of Wittgenstein s thought. In our last two chapters we discuss the changes this approach went through in this last phase of the philosopher s thought in relation to the idea of triangularization of the underlying semantics. We then deal in some detail with Wittgenstein s proposed interpretation of some types of elementary mathematical propositions, especially arithmetical propositions.

[pt] A partir das reflexões de Frege em seus Grundlagen der Arithmetik, destaca-se como o fio condutor da presente dissertação o problema que se refere à determinação da natureza numérica. A análise que Frege dedica a este problema almeja caracterizar a noção de número com o auxílio da noção de objeto lógico, e tal aproximação receberá um intenso ataque teórico por parte de Paul Benacerraf. Este ataque teórico, por sua vez, será auxiliado pela sugestão de que, na medida em que o interesse dos matemáticos (enquanto matemáticos) permanece em um âmbito estruturalista, uma pesquisa filosófica sobre a matemática deveria, em princípio, preservar semelhante aspecto. Esta argumentação de Benacerraf, aliada aos trabalhos anteriores do grupo Bourbaki, impulsionou as mais diversas pesquisas no âmbito estruturalista, constituindo-se os trabalhos de Stewart Shapiro como um de seus mais interessantes desenvolvimentos. São assim apresentadas as bases ontológicas e epistemológicas que sustentam tal teoria – intitulada como estruturalismo ante rem – para, por fim, se delinear um histórico de tais concepções. Sublinham-se, então, alguns debates ocorridos ao final do século XIX e início do século XX, com a intenção de se iluminar as heranças conceituais de Shapiro e alguns possíveis pressupostos de Benacerraf, oferecendo-se alguma notoriedade às figuras do matemático alemão Richard Dedekind e do grupo francês Nicholas Bourbaki. / [en] Frege’s reflections on the nature of numbers, presented in his Grundlagen der Arithmetik (1884), provide the guiding theme for this dissertation. His characterization of numbers as logical objects was sharply criticized by Paul Benacerraf (1965), who suggested that since mathematicians (as mathematicians) work within a structural framework, philosophical research on mathematics should preserve this structural aspect of mathematics. Benacerraf’s arguments, as well as earlier works by the Bourbaki group, gave raise to several structuralist research projects, one of the most interesting of which is developed by Stewart Shapiro. We present the ontological and epistemological basis of Shapiro’s socalled ante rem structuralism, as well as some of the debates from the late 19th and early 20th centuries which throw some light on the conceptual presuppositions of Shapiro and Benacerraf. In this connection we emphasize the work of the German mathematician Richard Dedekind and of the French group Nicholas Bourbaki.

<p>This paper studies the philosopher Ludwig Wittgenstein during his years (1920–26) as an elementary school teacher in remote Niederösterreich, Austria. The paper gives a survey of his life, and also a brief account of three of his main works: Tractatus Logico-Philosophicus, Philosophical Investigations and Remarks on the Foundations of Mathematics. Attention is given to his alphabetical word list, Wörterbuch für Volksschulen, published for educational use in elementary schools. The study is focused on Wittgenstein’s educational practise, and establishes a connection between his experience as a teacher and his late philosophy.</p>

This paper studies the philosopher Ludwig Wittgenstein during his years (1920–26) as an elementary school teacher in remote Niederösterreich, Austria. The paper gives a survey of his life, and also a brief account of three of his main works: Tractatus Logico-Philosophicus, Philosophical Investigations and Remarks on the Foundations of Mathematics. Attention is given to his alphabetical word list, Wörterbuch für Volksschulen, published for educational use in elementary schools. The study is focused on Wittgenstein’s educational practise, and establishes a connection between his experience as a teacher and his late philosophy.

Current reform in mathematics teaching and learning is rooted in a changing vision of school mathematics, one that includes constructivist learning, student-centered pedagogy, and the use of worthwhile tasks (National Council of Teachers of Mathematics, 1989, 1991, 2000). This changing vision not only challenges teachers’ beliefs about mathematics instruction but their philosophies of mathematics as well (Dossey, 1992). This study investigates the processes that four teachers’ go through as they implement a new task-based mathematics curriculum while exploring their personal philosophies of mathematics. The participants were part of a graduate-level course that examined, through the writings of Davis and Hersh (1981), Lakatos (1976), Polya (1945/1973), and others, a humanist/fallibilist philosophy of mathematics. These participants shared, through reflective writings and interviews, their struggles to, first, define mathematics and its purpose in society and in schools, and second, to reconcile their views of mathematics with their instructional practices. The study took place as the participants, two classroom teachers and two instructional coaches, engaged in the initial implementation of a reform mathematics curriculum, a reform based in social constructivist learning theories. Using narrative analysis, this study focuses on the unique mathematical stories of four experienced educators. Each of the participants initially expressed a traditional, a priori view of mathematics, seeing mathematics as right/wrong, black/white, a subject outside of human construction. The participants’ expressed views of mathematics changed as they attempted to align their personal philosophies of mathematics with their (changing) classroom practices. They shared their personal struggles to redefine themselves as mathematics teachers through a process of experimentation, reflection, and adaptation. This process was echoed in their changing philosophies of mathematics. These participants came to see mathematics as fluid and a human construct; they also came to see their philosophies of mathematics as fluid and ever-changing, a process more than a product.

This thesis addresses an issue in the philosophy of Mathematics which is little discussed, and indeed little recognised. This issue is the phenomenon of a ‘change of setting’. Changes of setting are events which involve a change in a scientific framework which is fruitful for answering questions which were, under an old framework, intractable. The formulation of the new setting usually involves a conceptual re-orientation to the subject matter. In the natural sciences, such re-orientations are arguably unremarkable, inasmuch as it is possible that within the former setting for one’s thinking one was merely in error, and under the new orientation one is merely getting closer to the truth of the matter. However, when the subject matter is pure mathematics, a problem arises in that mathematical truth is (in appearance) timelessly immutable. The conceptions that had been settled upon previously seem not the sort of thing that could be vitiated. Yet within a change of setting that is just what seems to happen. Changes of setting, in particular in their effects on the truth of individual propositions, pose a problem for how to understand mathematical truth. Thus this thesis aims to give a philosophical analysis of the phenomenon of changes of setting, in the spirit of the investigations performed in Wilson (1992) and Manders (1987) and (1989). It does so in three stages, each of which occupies a chapter of the thesis: 1. An analysis of the relationship between mathematical truth and settingchanges, in terms of how the former must be viewed to allow for the latter. This results in a conception of truth in the mathematical sciences which gives a large role to the notion that a mathematical setting must ‘explain itself’ in terms of the problems it is intended to address. 2. In light of (1), I begin an analysis of the change of setting engendered in mathematical logic by Gottlob Frege. In particular, this chapter will address the question of whether Frege’s innovation constitutes a change of setting, by asking the question of whether he is seeking to answer questions which were present in the frameworks which preceded his innovations. I argue that the answer is yes, in that he is addressing the Kantian question of whether alternative systems of arithmetic are possible. This question arises because it had been shown earlier in the 19th century that Kant’s conclusion, that Euclid’s is the only possible description of space, was incorrect. 3. I conclude with an in-depth look at a specific aspect of the logical system constructed in Frege’s Grundgesetze der Arithmetik. The purpose of this chapter is to find evidence for the conclusions of chapter two in Frege’s technical work (as opposed to the philosophical). This is necessitated by chapter one’s conclusions regarding the epistemic interdependence of formal systems and informal views of those frameworks. The overall goal is to give a contemporary account of the possibility of setting-changes; it will turn out that an epistemic grasp of a mathematical system requires that one understand it within a broader, somewhat historical context.

Traditionally, mathematics and metaphor have been thought of as disparate: the former rigorous, objective, universal, eternal, and fundamental; the latter imprecise, derivative, nearly - if not patently - false, and therefore of merely aesthetic value, at best. A growing amount of contemporary scholarship argues that both of these characterizations are flawed. This dissertation shows that there are important connexions between mathematics and metaphor that benefit our understanding of both. A historically structured overview of traditional theories of metaphor reveals it to be a notion that is complicated, controversial, and inadequately understood; this motivates a non-traditional approach. Paradigmatically shifting the locus of metaphor from the linguistic to the conceptual - as George Lakoff, Mark Johnson, and many other contemporary metaphor scholars do - overcomes problems plaguing traditional theories and promisingly advances our understanding of both metaphor and of concepts. It is argued that conceptual metaphor plays a key role in explaining how mathematics is grounded, and simultaneously provides a mechanism for reconciling and integrating the strengths of traditional theories of mathematics usually understood as mutually incompatible. Conversely, it is shown that metaphor can be usefully and consistently understood in terms of mathematics. However, instead of developing a rigorous mathematical model of metaphor, the unorthodox approach of applying mathematical concepts metaphorically is defended.

In this dissertation on Hilary Putnam's philosophy, I investigate his development regarding meaning and necessity, in particular mathematical necessity. Putnam has been a leading American philosopher since the end of the 1950s, becoming famous in the 1960s within the school of analytic philosophy, associated in particular with the philosophy of science and the philosophy of language. Under the influence of W.V. Quine, Putnam challenged the logical positivism/empiricism that had become strong in America after World War II, with influential exponents such as Rudolf Carnap and Hans Reichenbach. Putnam agreed with Quine that there are no absolute a priori truths. In particular, he was critical of the notion of truth by convention. Instead he developed a notion of relative a priori truth, that is, a notion of necessary truth with respect to a body of knowledge, or a conceptual scheme. Putnam's position on necessity has developed over the years and has always been connected to his important contributions to the philosophy of meaning. I study Hilary Putnam's development through an early phase of scientific realism, a middle phase of internal realism, and his later position of a natural or commonsense realism. I challenge some of Putnam’s ideas on mathematical necessity, although I have largely defended his views against some other contemporary major philosophers; for instance, I defend his conceptual relativism, his conceptual pluralism, as well as his analysis of the realism/anti-realism debate.