Contents: Acknowledgements. Conventions. Preface. Biographical sketch. 1 Introduction. 2 The Contributions. 3 Early work in analysis. 4 The Theory of Science . 5. Later mathematical studies. A On Kantian Intuitions. B The Bolzano-Cauchy Theorem.

This paper is a fresh attempt to articulate the role of a theory of truthmakers. We argue that truthmaker theory constitutes a cornerstone of good methodology in metaphysics, but that a conflation of truthmaker theory with the theory of truth has been responsible for certain excesses associated with truthmaker-based approaches in the recent literature. If truthmaker theory is not a component of a theory of truth, then truthmaker maximalism – the view that every truth has a truthmaker – loses its (...) primary motivation. More generally, if the task of truthmaker theory is not to provide a definition or account of truth in truthmaker terms, there is no pressing need for hard, a priori principles stating which truths have truthmakers and which do not. (shrink)

The philosophical analysis of mathematical explanations concerns itself with two different, although connected, areas of investigation. The first area addresses the problem of whether mathematics can play an explanatory role in the natural and social sciences. The second deals with the problem of whether mathematical explanations occur within mathematics itself. Accordingly, this entry surveys the contributions to both areas, it shows their relevance to the history of philosophy and science, it articulates their connection, and points to the philosophical pay-offs to (...) be expected by deepening our understanding of the topic. (shrink)

There is a long line of thinkers in the philosophy of mathematics who have sought to base an account of proof on what might be called a 'metaphysical ordering' of the truths of mathematics. Use the term 'metaphysical' to describe these orderings is intended to call attention to the fact that they are regarded as objective and not subjective and that they are conceived primarily as orderings of truths and only secondarily as orderings of beliefs. -/- I describe and consider (...) two models for such orderings, (i) an implicational model and something I call (ii) a mosaic model. -/- Both models fail to do what such orderings should do. Moreover, in both cases, the trouble arises because the class of truths to be ordered is taken to be closed under various weakening implications. This is a result of their employing a global conception of logic. (shrink)

Truth is one of the most debated topics in philosophy; Wolfgang Kunne presents a comprehensive critical examination of all major theories, from Aristotle to the present day. He argues that it is possible to give a satisfactory 'modest' account of truth without invoking problematic notions like correspondence, fact, or meaning. The clarity of exposition and the wealth of examples will make Conceptions of Truth an invaluable and stimulating guide for advanced students and scholars.

We take as our starting point a thesis to the effect that, at least for true judgments of many varieties, there are parts of reality which make such judgments are true. We argue that two distinct components are involved in this truthmaker relation. On the one hand is the relation of necessitation, which holds between an object x and a judgment p when the existence of x entails the truth of p. On the other hand is the dual notion of (...) projection, which holds between a judgment p and an object x when the truth of p entails the existence of x. A truthmaker for a judgment p is then a necessitator for p which satisfies the further constraint that it is part of p’s projection. We offer a formal theory of the truthmaker relation thus defined, exploiting ontological tools of basic mereology and the theory of dependence. We then apply the theory to a range of problems connected with generic expressions, ellipsis, vagueness, and indexical and perceptual judgments. (shrink)

The main question of this paper is how to understand the notion of a truth-maker. In section 1, I show that the identification of truth-making with necessitation cannot capture the pretheoretic understanding of notions such as ‘x makes something true’. In section 2, I examine Barry Smith’s reaction to this problem: he defines truth-making as the combination of necessitation and projection. I focus on the formal part of Smith’s account, which is shown to yield undesired results. However, in section 3, (...) I present an alternative account of projection, which fares better and can fruitfully be employed to circumvent the problems raised in section 1. Unfortunately, the account still has to face some troublemakers, as I show in the final section. I conclude, therefore, with a pessimistic view on the project of defining truth-making via necessitation and projection. (shrink)

A realist theory of truth for a class of sentences holds that there are entities in virtue of which these sentences are true or false. We call such entities ‘truthmakers’ and contend that those for a wide range of sentences about the real world are moments (dependent particulars). Since moments are unfamiliar, we provide a definition and a brief philosophical history, anchoring them in our ontology by showing that they are objects of perception. The core of our theory is the (...) account of truthmaking for atomic sentences, in which we expose a pervasive ‘dogma of logical form’, which says that atomic sentences cannot have more than one truthmaker. In contrast to this, we uphold the mutual independence of logical and ontological complexity, and the authors outline formal principles of truthmaking taking account of both kinds of complexity. (shrink)

The article is primarily concerned with the notion of a truth-maker. An explication for this notion is offered, which relates it to other notions of making something such-and-such. In particular, it is shown that the notion of a truth-maker is a close relative of a concept employed by van Inwagen in the formulation of his Consequence Argument. This circumstance helps understanding the general mechanisms of the concepts involved. Thus, a schematic explication of a whole battery of related notions is offered. (...) It is based on an explanatory notion, introduced by the sentential connector “because”, whose function is examined in some detail. Finally, on the basis of the explication proposed, an argument is developed to the effect that the objects usually regarded as truth-makers are not apt to play this role. (shrink)

Die "Paradoxien des Unendlichen" sind ein Klassiker der Philosophie der Mathematik und zugleich eine gute Einführung in das Denken des "Urgroßvaters" der analytischen Philosophie. Das Unendliche - seit jeher ein Faszinosum für die philosophische Reflexion - wurde in der Zeit nach der Grundlegung der Analysis durch Leibniz und Newton in der Mathematik zunächst als Problem betrachtet, das sich nicht vollkommen widerspruchsfrei behandeln lässt. Bernard Bolzano, der heute als "Urgroßvater der analytischen Philosophie" (Michael Dummett) gilt, zeigt in diesem klassisch gewordenen Text (...) jedoch, dass es beim Nachdenken über das Unendliche, sofern man begrifflich differenziert argumentiert und klare Definitionen zugrunde legt, keine echten Widersprüche gibt und sich die "Paradoxien des Unendlichen" auflösen lassen. Bolzano setzt sich zunächst mit den Unendlichkeitsbegriffen der Mathematik und der (idealistisch geprägten) Philosophie seiner Zeit auseinander, bevor er sich Schritt für Schritt die analytischen Grundlagen einer konsistenten Theorie des Unendlichen erarbeitet. Bereits einige Jahrzehnte vor Georg Cantors bis heute maßgeblicher mathematischer Unendlichkeitsdefinition gelingt Bolzano mit diesem Werk, dessen logische Klarheit und dessen Grad an philosophischer Durchdachtheit bis heute Beispielcharakter haben, die Rehabilitation des Unendlichen für die Philosophie der Mathematik. Die "Paradoxien des Unendlichen" wurden zuerst 1851 aus dem Nachlass herausgegeben. Diese Neuausgabe beruht auf dem der Erstausgabe zugrundeliegenden Manuskript. (shrink)

Does mathematics ever play an explanatory role in science? If so then this opens the way for scientific realists to argue for the existence of mathematical entities using inference to the best explanation. Elsewhere I have argued, using a case study involving the prime-numbered life cycles of periodical cicadas, that there are examples of indispensable mathematical explanations of purely physical phenomena. In this paper I respond to objections to this claim that have been made by various philosophers, and I discuss (...) potential future directions of research for each side in the debate over the existence of abstract mathematical objects. (shrink)

Wesley Salmon is renowned for his seminal contributions to the philosophy of science. He has powerfully and permanently shaped discussion of such issues as lawlike and probabilistic explanation and the interrelation of explanatory notions to causal notions. This unique volume brings together twenty-six of his essays on subjects related to causality and explanation, written over the period 1971-1995. Six of the essays have never been published before and many others have only appeared in obscure venues. The volume includes a section (...) of accessible introductory pieces, as well as more advanced and technical pieces, and will make essential work in the philosophy of science readily available to both scholars and students. (shrink)