The eleatic and the indispensabilist

The debate over whether we should believe that mathematical objects exist quickly leads to the question of how to determine what we should believe.  Indispensabilists claim that we should believe in the existence of mathematical objects because of their ineliminable roles in scientific theory.  Eleatics argue that only objects with causal properties exist.  Mark Colyvan’s recent defenses of Quine’s indispensability argument against some contemporary eleatics attempt to provide reasons to favor the indispensabilist’s criterion.  I show that Colyvan’s argument is not decisive against the eleatic and sketch a way to capture the important intuitions behind both views

In Defense of Mathematical Inferentialism

I defend a new position in philosophy of mathematics that I call mathematical inferentialism. It holds that a mathematical sentence can perform the function of facilitating deductive inferences from some concrete sentences to other concrete sentences, that a mathematical sentence is true if and only if all of its concrete consequences are true, that the abstract world does not exist, and that we acquire mathematical knowledge by confirming concrete sentences. Mathematical inferentialism has several advantages over mathematical realism and fictionalism

Maurinian Truths : Essays in Honour of Anna-Sofia Maurin on her 50th Birthday

This book is in honour of Professor Anna-Sofia Maurin on her 50th birthday. It consists of eighteen essays on metaphysical issues written by Swedish and international scholars

What we talk about when we talk about numbers

In this paper, I describe and motivate a new species of mathematical structuralism, which I call Instrumental Nominalism about Set-Theoretic Structuralism. As the name suggests, this approach takes standard Set-Theoretic Structuralism of the sort championed by Bourbaki and removes its ontological commitments by taking an instrumental nominalist approach to that ontology of the sort described by Joseph Melia and Gideon Rosen. I argue that this avoids all of the problems that plague other versions of structuralism

Guest editor’s introduction

Guest Editor’s introduction to the Monographic Section

Ficcionalismo e explicações matemáticas

In this paper, I place Mary Leng’s version of mathematical instrumentalism within the context of the debate in mathematical realism/anti-realism as well as within the context of the platonism/nominalism debate. I maintain that although her position is able to show how the conjunction of Quinean naturalism and confirmational holism does not necessarily lead to the conclusion that mathematical objects must necessarily exist for they are indispensable in our best scientific theories; her usage of both theses still leads to platonism. Such is the case for her characterization of scientific theories as akin to a set-theory that accommodates fictitious objects and statements within it is untenable due to the dependence of fictions on a realist ontology.Keywords: fictionalism, mathematical instrumentalism, indispensability argument, Mary Leng, platonism, nominalism.Neste artigo, situo a versão de Mary Leng do instrumentalismo matemático no contexto do debate do realismo/antirrealismo matemático, bem como no contexto do debate do platonismo/nominalismo. Sustento que, embora sua posição seja capaz de mostrar como a conjunção do naturalismo quineano e do holismo confirmatório não leva necessariamente à conclusão de que os objetos matemáticos devem necessariamente existir, pois são indispensáveis em nossas melhores teorias científicas, seu uso de ambas as teses ainda leva ao platonismo. Esse é o caso de sua caracterização das teorias científicas, semelhante a uma teoria dos conjuntos que acomoda objetos e declarações fictícias dentro dela, sendo insustentável devido à dependência de ficções em relação a uma ontologia realista.Palavras-chave: ficcionalismo, instrumentalismo matemático, argumento da indispensabilidade, Mary Leng, platonismo, nominalismo

Mathematical Explanation and Ontology: An Analysis of Applied Mathematics and Mathematical Proofs

The present work aims at providing an account of mathematical explanation in two different areas: scientific explanation and within mathematics. The research is addressed from two different perspectives: the one arising from an ontological concern about mathematical entities, and the other originating from a methodological choice: to study our chosen problems (mathematical explanation in science and in mathematics itself) in mathematical practice, that is to say, looking at the way mathematicians understand and perform their work in these diverse areas, including a case study for the context of intra-mathematical explanation. The central target is the analysis of the role that mathematical explanation plays in science and its relevance to the success or failure of scientific theories. The ontological question of whether the explanatory role of abstract objects, mathematical objects in particular, is enough to postulate their existence will be one of the issues to be addressed. Moreover, the possibility of a unified theory of explanation which can accommodate both external and internal mathematical explanation will also be considered. In order to go deeper into these issues, the research includes: (1) an analysis how the question of what is involved in internal mathematical explanation has been addressed in the literature, an analysis of the role of mathematical proof and the reasons why it makes sense to search for more explanatory proofs of already known results, and (2) an analysis of the relation between the use of mathematics in scientific explanation and the ontological commitment that arises from these explanatory tools in science. Part of the present work consists of an analysis of the explanatory role of mathematics through the study of cases reflecting this role. Case studies is one of the main sources of data in order to clarify the role mathematical entities play, among other methodological resources

Nominalizations

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