Classical logic is explosive in the face of contradiction, yet we find ourselves using inconsistent theories. Mark Colyvan, one of the prominent advocates of the indispensability argument for realism about mathematical objects, suggests that such use can be garnered to develop an argument for commitment to inconsistent objects and, because of that, a paraconsistent underlying logic. I argue to the contrary that it is open to a classical logician to make distinctions, also needed by the paraconsistent logician, which allow a (...) more nuanced ranking of theories in which inconsistent theories can have different degrees of usefulness and productivity. Facing inconsistency does not force us to adopt an underlying paraconsistent logic. Moreover we will see that the argument to best explanation deployed by Colyvan in this context is unsuccessful. I suggest that Quinean approach which Colyvan champions will not lead to the revolutionary doctrines Colyvan endorses. (shrink)

According to standard accounts of mathematical representations of physical phenomena, positing structure-preserving mappings between a physical target system and the structure picked out by a mathematical theory is essential to such representations. In this paper, I argue that these accounts fail to give a satisfactory explanation of scientific representations that make use of inconsistent mathematical theories and present an alternative, robustly inferential account of mathematical representation that provides not just a better explanation of applications of inconsistent mathematics, but also a (...) compelling explanation of mathematical representations of physical phenomena in general. 1Inconsistent Mathematics and the Problem of Representation2The Early Calculus3Mapping Accounts and the Early Calculus3.1Partial structures3.2Inconsistent structures3.3Related total consistent structures4A Robustly Inferential Account of the Early Calculus in Applications 4.1The robustly inferential conception of mathematical representation4.2The robustly inferential conception and inconsistent mathematics4.3The robustly inferential conception and mapping accounts5Beyond Inconsistent Mathematics. (shrink)

In this paper I present an argument for belief in inconsistent objects. The argument relies on a particular, plausible version of scientific realism, and the fact that often our best scientific theories are inconsistent. It is not clear what to make of this argument. Is it a reductio of the version of scientific realism under consideration? If it is, what are the alternatives? Should we just accept the conclusion? I will argue (rather tentatively and suitably qualified) for a positive answer (...) to the last question: there are times when it is legitimate to believe in inconsistent objects. (shrink)

This paper scrutinizes the debate over logical pluralism. I hope to make this debate more tractable by addressing the question of motivating data: what would count as strong evidence in favor of logical pluralism? Any research program should be able to answer this question, but when faced with this task, many logical pluralists fall back on brute intuitions. This sets logical pluralism on a weak foundation and makes it seem as if nothing pressing is at stake in the debate. The (...) present paper aims to improve this situation by looking at a promising case study and drawing general lessons about the kind of evidence that would support logical pluralism. I argue that the best motivation for logical pluralism will ultimately be rooted in certain kinds of performative data. (shrink)

The work done so far on the understanding of mathematical (proof) texts focuses mostly on logical and heuristical aspects; a proof text is considered to be understood when the reader is able to justify inferential steps occurring in it, to defend it against objections, to give an account of the “main ideas”, to transfer the proof idea to other contexts etc. (see, e.g., Avigad in The philosophy of mathematical practice, Oxford University Press, Oxford, 2008). In contrast, there is a rich (...) philosophical tradition dealing with the concept of understanding and interpreting texts, namely philosophical hermeneutics, represented, e.g., by Schleiermacher, Dilthey, Heidegger or Gadamer. In this tradition, “understanding” generally refers to the integration in a comprehensive (historical, existential, life-worldly,...) context. In this article, we take some first steps towards exploring the question how the ideas from philosophical hermeneutics presented in Gadamer’s “Truth and Method” apply to mathematical texts and what (if anything) can be learned from these for the didactics and presentation of mathematics. (shrink)

This paper is about the underlying logical principles of scientific theories. In particular, it concerns ex contradictione quodlibet (ECQ) the principle that anything follows from a contradiction. ECQ is valid according to classical logic, but invalid according to paraconsistent logics. Some advocates of paraconsistency claim that there are ‘real’ inconsistent theories that do not erupt with completely indiscriminate, absurd commitments. They take this as evidence in favor of paraconsistency. Michael (2016) calls this the non-triviality strategy (NTS). He argues that this (...) strategy fails in its purpose. I will show that Michael's criticism significantly over-reaches. The fundamental problem is that he places more of a burden on the advocate of paraconsistency than on the advocate of classical logic. The weaknesses in Michael's argument are symptomatic of this preferential treatment of one viewpoint in the debate over another. He does, however, make important observations that allow us to clarify some of the complexities involved in giving a logical reconstruction of a theory. I will argue that there are abductive arguments deserving of further consideration for the claim that paraconsistent logic offers the best explanation of the practice of inconsistent science. In this sense, the debate is still very much open. (shrink)

There has always been interest in inconsistency in science, not least within science itself as scientists strive to devise a consistent picture of the universe. Some important early landmarks in this history are Copernicus’s criticism of the Ptolemaic picture of the heavens, Galileo’s claim that Aristotle’s theory of motion was inconsistent, and Berkeley’s claim that the early calculus was inconsistent. More recent landmarks include the classical theory of the electron, Bohr’s theory of the atom, and the on-going difficulty of reconciling (...) Einstein’s general relativity and quantum theory. But over the past few decades philosophers have taken a particular and increasing interest in inconsistency in science. In 2002 this culminated in the first collection of articles specifically dedicated to the topic: Inconsistency in Science, edited by Joke Meheus, published by Kluwer, and featuring twelve articles on a range of topics in the philosophy of science and mathematics.Since then philosophic. (shrink)

In this paper I sketch some lines of response to Mark Colyvan’s indispensability arguments for the existence of inconsistent objects, being mainly concerned with the indispens ability of inconsistent mathematical entities. My response will draw heavily on Jody Azzouni’s deflationary nominalism.

Scientific pluralism is the view according to which there is a plurality of scientific domains and of scientific theories, and these theories are empirically adequate relative to their own respective domains. Scientific monism is the view according to which there is a single domain to which all scientific theories apply. How are these views impacted by the presence of inconsistent scientific theories? There are consistency-preservation strategies and inconsistency-toleration strategies. Among the former, two prominent strategies can be articulated: Compartmentalization and Information (...) restriction. Among the inconsistency-toleration strategies, we have: Paraconsistent compartmentalization and Dialetheism. In this paper, I critically assess the adequacy of each of these four views. (shrink)

This thesis examines possible philosophies to account for the practice of mathematics, exploring the metaphysical, ontological, and epistemological outcomes of each possible theory. Through a study of the two most probable ideas, mathematical platonism and fictionalism, I focus on the compelling argument for platonism given by an appeal to the sciences. The Indispensability Argument establishes the power of explanation seen in the relationship between mathematics and empirical science. Cases of this explanatory power illustrate how we might have reason to believe (...) in the existence of mathematical entities present within our best scientific theories. The second half of this discussion surveys Newtonian Cosmology and other inconsistent theories as they pose issues that have received insignificant attention within the philosophy of mathematics. The application of these inconsistent theories raises questions about the effectiveness of mathematics to model physical systems. (shrink)