The contemporary Platonists in the philosophy of mathematics argue that mathematical objects exist. One of the arguments by which they support this standpoint is the so-called Enhanced Indispensability Argument (EIA). This paper aims at pointing out the difficulties inherent to the EIA. The first is contained in the vague formulation of the Argument, which is the reason why not even an approximate scope of the set objects whose existence is stated by the Argument can be established. The second problem is (...) reflected in the vagueness of the very term indispensability, which is essential to the Argument. The paper will remind of a recent definition of the concept of indispensability of a mathematical object, reveal its deficiency and propose an improvement of this definition. Following this, we will deal with one of the consequences of the arbitrary employment of the concept of indispensability of a mathematical theory. We will propose a definition of this concept as well, in accordance with the common intuition about it. Eventually, on the basis of these two definitions, the paper will describe the relation between these two concepts, in the attempt to clarify the conceptual apparatus of the EIA. (shrink)

This discussion note points to some verbal imprecisions in the formulation of the Enhanced Indispensability Argument. The examination of the plausibility of alternative interpretations reveals that the argument’s minor premise should be understood as a particular, not a universal, statement. Interpretations of the major premise and the conclusion oscillate between de re and de dicto readings. The attempt to find an appropriate interpretation for the EIA leads to undesirable results. If assumed to be valid and sound, the argument warrants the (...) rationality of the belief in an unusual variant of Platonism. On the other hand, if taken as it stands, the argument is either invalid or is unsound or does not support the mathematical Platonism. Thus, the EIA in its present form cannot serve as a useful device for the Platonist. (shrink)

Platonists in mathematics endeavour to prove the truthfulness of the proposal about the existence of mathematical objects. However, there have not been many explicit proofs of this proposal. One of the explicit ones is doubtlessly Baker’s Enhanced Indispensability Argument, formulated as a sort of modal syllogism. We aim at showing that the purpose of its creation – the defence of Platonist viewpoint – was not accomplished. Namely, the second premise of the Argument was imprecisely formulated, which gave space for various (...) interpretations of the EIA. Moreover, it is not easy to perceive which of the more precise formulations of the above-mentioned premise would be acceptable. For all these reasons, it is disputable whether the EIA can be used to defend Platonist outlook. At the beginning of this century, Baker has shown that the so-called Quine-Putnam Indispensability Argument can not provide “full” platonism - a guarantee of the existence of all mathematical objects. It turns out, however, that the EIA has a similar disadvantage. (shrink)

A new trend in the philosophical literature on scientific explanation is that of starting from a case that has been somehow identified as an explanation and then proceed to bringing to light its characteristic features and to constructing an account for the type of explanation it exemplifies. A type of this approach to thinking about explanation – the piecemeal approach, as I will call it – is used, among others, by Lange (2013) and Pincock (2015) in the context of their (...) treatment of the problem of mathematical explanations of physical phenomena. This problem is of central importance in two different recent philosophical disputes: the dispute about the existence on non-causal scientific explanations and the dispute between realists and antirealists in the philosophy of mathematics. My aim in this paper is twofold. I will first argue that Lange (2013) and Pincock (2015) fail to make a significant contribution to these disputes. They fail to contribute to the dispute in the philosophy of mathematics because, in this context, their approach can be seen as question begging. They also fail to contribute to the dispute in the general philosophy of science because, as I will argue, there are important problems with the cases discussed by Lange and Pincock. I will then argue that the source of the problems with these two papers has to do with the fact that the piecemeal approach used to account for mathematical explanation is problematic. (shrink)

Marc Lange. Because Without Cause: Non-Causal Explanations in Science and Mathematics. Oxford Studies in the Philosophy of Science. Oxford University Press.

The author of “Evidence, Explanation, Enhanced Indispensability” advances a criticism to the Enhanced Indispensability Argument and the use of Inference to the Best Explanation in order to draw ontological conclusions from mathematical explanations in science. His argument relies on the availability of equivalent though competing explanations, and a pluralist stance on explanation. I discuss whether pluralism emerges as a stable position, and focus here on two main points: whether cases of equivalent explanations have been actually offered, and which ontological consequences (...) should follow from these. (shrink)

© The Authors [2018]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected] article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model...In his Moby Dick, Herman Melville writes that “to produce a mighty book you must choose a mighty theme”. Marc Lange’s Because Without Cause is definitely an impressive book that deals with a mighty theme, that of non-causal explanations in the empirical sciences and in mathematics. Blending a (...) variety of insights originating within the philosophy of science and the philosophy of mathematics, the book has an ambitious goal: capturing what features, in the many examples of putative explanations offered by mathematicians and scientists, make them genuinely explanatory.The architecture of Because Without Cause reflects its mighty character. The book is divided into four parts and eleven chapters, preceded by... (shrink)