In this paper, I develop a "safety result" for applied mathematics. I show that whenever a theory in natural science entails some non-mathematical conclusion via an application of mathematics, there is a counterpart theory that carries no commitment to mathematical objects, entails the same conclusion, and the claims of which are true if the claims of the original theory are "correct": roughly, true given the assumption that mathematical objects exist. The framework used for proving the safety result has some advantages (...) over existing nominalistic accounts of applied mathematics. It also provides a nominalistic account of pure mathematics. (shrink)

Easy-road mathematical fictionalists grant for the sake of argument that quantification over mathematical entities is indispensable to some of our best scientific theories and explanations. Even so they maintain we can accept those theories and explanations, without believing their mathematical components, provided we believe the concrete world is intrinsically as it needs to be for those components to be true. Those I refer to as “mathematical surrealists” by contrast appeal to facts about the intrinsic character of the concrete world, not (...) to explain why our best mathematically imbued scientific theories and explanations are acceptable in spite of having false components, but in order to replace those theories and explanations with parasitic, nominalistically acceptable alternatives. I argue that easy-road fictionalism is viable only if mathematical surrealism is and that the latter constitutes a superior nominalist strategy. Two advantages of mathematical surrealism are that it neither begs the question concerning the explanatory role of mathematics in science nor requires rejecting the cogency of inference to the best explanation. (shrink)

FieldHartry. Science Without Numbers: A Defense of Nominalism 2nd ed.Oxford University Press, 2016. ISBN 978-0-19-877792-2. Pp. vi + 56 + vi + 111.

ABSTRACTIn the target article ‘If-Thenism’, Stephen Yablo develops a novel form of if-thenism, that appeals to the notion of logical subtraction. In this commentary, I explore the limits of Yablo's proposed subtraction procedure, by leaning on an analogy with photographic subtraction. In particular, I will argue that there will be cases when there's nothing interesting left after the subtraction and, as a consequence, there are serious limits to the applicability of the subtraction procedure.

ABSTRACTAccording to Yablo, in uttering sentences that imply the existence of abstract objects, we do not assert their literal content. Instead, we only make a weaker conditional claim that does not have the controversial implication. In this commentary I argue that the conditional claims Yablo suggests we are making are true only if abstract objects exist and, thus, also carry the controversial implication.

ABSTRACTStandard theories treat A→C and A→D as equivalent when C and D coincide on A. However, Yablo's incremental conditional does not behave in this way. Consider the following: a = b→Fa. a = b→Fb.According to Yablo, the real content of is Fb and the real content of is Fa. In worlds where a = b is true, both and have the same truth-value. However, ‘Fa can come apart truth-value-wise from Fb in worlds where a isn't b.’ In this paper, I (...) argue that this feature of Yablo's incremental conditional makes his if-thenism incompatible with fictionalism. (shrink)

ABSTRACTIn our paper, we mount a novel argument, which trades on recent work by Roy Sorensen [2016], following work by Saul Kripke, against Yablo's preferred reading of if-thenism, which is an attempt to read problematically ontologically committing sentences in a way that does not carry such ontological commitments. Although our argument is directed at Yablo's proposed reading of if-thenism, if the argument is successful, other versions of if-thenism may be affected. After reviewing Sorensen's recent work and presenting our argument, we (...) consider a possible way out and use this as a means for presenting some challenges to Yablo. (shrink)