Truth, Existence and Explanation: Filmat 2016 Studies in the Philosophy of Mathematics

Cham, Switzerland: Springer Verlag (2018)
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Abstract

This book contains more than 15 essays that explore issues in truth, existence, and explanation. It features cutting-edge research in the philosophy of mathematics and logic. Renowned philosophers, mathematicians, and younger scholars provide an insightful contribution to the lively debate in this interdisciplinary field of inquiry. The essays look at realism vs. anti-realism as well as inflationary vs. deflationary theories of truth. The contributors also consider mathematical fictionalism, structuralism, the nature and role of axioms, constructive existence, and generality. In addition, coverage also looks at the explanatory role of mathematics and the philosophical relevance of mathematical explanation. The book will appeal to a broad mathematical and philosophical audience. It contains work from FilMat, the Italian Network for the Philosophy of Mathematics. These papers collected here were also presented at their second international conference, held at the University of Chieti-Pescara, May 2016.

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Chapters

Making Sense of Deflationism from a Formal Perspective: Conservativity and Relative Interpretability

The contemporary study of the notion of truth divides into two main traditions: a philosophical tradition concerned with the nature of truth and a logical one focused on formal solutions to truth-theoretic paradoxes. The logical results obtained in the latter are rich and profound but often hard to ... see more

Towards a Better Understanding of Mathematical Understanding

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Deflationary Truth Is a Logical Notion

The thesis that truth is a logical notion has been stated repeatedly by deflationists in philosophical discussions on the nature of truth. However, to prove the point, one would need to show that the truth predicate does classify as logical according to some reasonable criterion of logicality. Follo... see more

Parsimony, Ontological Commitment and the Import of Mathematics

In a recent paper Alan Baker has argued for the thesis that the use of a stronger mathematical apparatus in optimization explanations can reduce our concrete ontological commitment, and this results in an increase of explanatory power. The import of this thesis in the context of the Enhanced Indispe... see more

Existence vs. Conceivability in Aristotle: Are Straight Lines Infinitely Extendible?

Aristotle is committed to finitism in mathematics. But there are certain uses of infinity in mathematics which are indispensable, even in the mathematics of his day. So we had better understand Aristotle’s finitism in a way that is compatible with the mathematics of his time .In particular, we have ... see more

On Expressive Power Over Arithmetic

The paper is concerned with the fine boundary between expressive power and reducibility of semantic and intensional notions in the context of arithmetical theories. I will consider three notions of reduction of a theory characterizing a semantic or a modal notion to the underlying arithmetical base ... see more

The Explanatory Power of a New Proof: Henkin’s Completeness Proof

Mancosu writes But explanations in mathematics do not only come in the form of proofs. In some cases explanations are sought in a major recasting of an entire discipline. This paper takes up both halves of that statement. On the one hand we provide a case study of the explanatory value of a particul... see more

Some Remarks on True Undecidable Sentences

In this paper I try to discuss the question of the truth-value of Gödel-type undecidable sentences in a framework which keeps into due account the idea that mathematical inquiry develops in a three-level framework: informal mathematics, theories, and formal theories. Moreover, it is to be stressed t... see more

Applicability Problems Generalized

In this paper, I will do preparatory work for a generalized account of applicability, that is, for an account which works for math-to-physics, math-to-math, and physics-to-math application. I am going to present and discuss some examples of these three kinds of application, and I will confront them ... see more

Penrose’s New Argument and Paradox

In this paper we take a closer look at Penrose’s New Argument for the claim that the human mind cannot be mechanized and investigate whether the argument can be formalized in a sound and coherent way using a theory of truth and absolute provability. Our findings are negative; we can show that there ... see more

Structure and Structures

In this paper we critically evaluate the notion of the structure of the natural numbers with respect to the question how the internal structure of such a structure might be specified.

Can Proofs by Mathematical Induction Be Explanatory?

In this paper I discuss Marc Lange’s argument for the claim that inductive proofs can never be explanatory. I show that several of the assumptions on which Lange’s argument relies are problematic, and I argue that there are cases of explanatory inductive proof, providing a number of examples to back... see more

Church-Turing Thesis, in Practice

We aim at providing a philosophical analysis of the notion of “proof by Church’s Thesis”, which is – in a nutshell – the conceptual device that permits to rely on informal methods when working in Computability Theory. This notion allows, in most cases, to not specify the background model of computat... see more

Intensionality in Mathematics

Do mathematical expressions have intensions, or merely extensions? If we accept the standard account of intensions based on possible worlds, it would seem that the latter is the case – there is no room for nontrivial intensions in the case of non-empirical expressions. However, some vexing mathemati... see more

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Gabriele Pulcini
University of Campinas

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