Talking about numbers. Easy arguments for mathematical realism (Q2825439)

The book discusses so-called ``easy arguments'' for mathematical realism. A paradigmatic example of easy argument, and the central focus of the entire book, is the following. Consider the sentences: (1) ``Mars has two moons'' and (2)``The number of moons of Mars is two''. It seems uncontroversial that if sentence (1) is true, then also sentence (2) is true. The truth of sentence (2), however, seems to demand of the world that it contain numbers. The argument makes use of the latter claim and of the implication from (1) to (2), to conclude for the existence of numbers, from the seemingly innocent assumption (1). What makes arguments of this kind problematic is the observation that sentences (1) and (2) seem to differ significantly for what concern their ontological commitment. Sentence (1) is usually perceived as not committing us to mathematical entities, as it only says something about concrete things (Mars and its moons). Sentence (2), instead, is typically considered to be ontologically loaded, as it says something about numbers. The apparent facility with which we go from assuming the truth of sentence (1), with supposedly no ontological commitment, to the truth of a seemingly ontologically loaded one as (2), has prompted the designation ``easy arguments for mathematical realism''. Arguments of this kind and strategies proposed to counter them are the subject of this book. The author argues that easy arguments do not succeed. She observes that such arguments rely on the following crucial premises: (i) if sentence (1) is true, then sentence (2) is true; (ii) the truth of sentence (2) demands of the world that it contains numbers. NEWLINENEWLINENEWLINEAfter an introductory chapter that runs through basic linguistic theory, the first part of the book analyses the first premise above, and reaches the conclusion that prominent attempts to deny (i) do not succeed. The second part of the book discusses premise (ii), and critically reviews a number of attempts to counter it. In a final chapter, the author offers her own argument against (ii). NEWLINENEWLINENEWLINEChapter 1 is a general introduction, clearly setting out the problem and summarising the contents of the book. Chapter 2 introduces preliminary notions. In particular, the author reviews the main features of a type theory whose purpose is to offer a detailed analysis of the semantic function of expressions, and applies it to an analysis of sentences (1) and (2). Chapter 3 is a brief introduction to Part 1, in which the author outlines the subsequent analysis of two prominent attempts to deny the first premise. In Chapter 4, Felka discusses fictionalism and in Chapter 5 indifferentialism. (Chapter 6 is a brief summary of these two chapters). NEWLINENEWLINENEWLINEFictionalism and indifferentialism are characterised as sharing a common core, as their proponents maintain that sentence (1) is true, but sentence (2) is untrue, i.e., they deny premise (i). According to Felka, the principal challenge these positions face, is to clarify why the implication from the truth of (1) to the truth of (2) still appears as obvious to us. A further common aspect of these approaches, as surveyed by Felka, is that they offer a similar explanation of this fact, as according to proponents of both positions speakers assert something true in uttering (2), even if (2) itself is untrue. The principal differences between fictionalism and indifferentialism lies in how they spell out the details of what it is that is asserted truly in asserting (2). Chapters 4 and 5 offer a detailed analysis of attempts made to defend fictionalist and indifferentialist accounts. The author argues that both strategies do not succeed. In the case of fictionalism, the main difficulty lies in explaining what vindicates the claim that the number of moons of Mars is true according to the relevant fiction. The author maintains that appeals to either the explicit or the implicit part of the relevant fiction do not succeed. As to indifferentialism, this is characterised as the thesis that speakers assert something true in uttering the sentence ``The number of moons of Mars is two'', and this makes the sentence appear true. However, this sentence lacks truth value as it has a false presupposition, the presupposition that numbers exist. According to Felka, for indifferentialism if from the sentence above ``we subtract the content of the sentence that arises due to its presuppositions, then its remaining content is that Mars has two moons.'' (p. 123) Felka argues that prominent indifferentialist strategies ``cannot be used to establish that speakers who utter the sentence `The number of moons of Mars is two' assert something true.'' (p. 84) Having argued that prominent strategies to deny premise (i) do not succeed, Felka discusses the alternative that arises if we deny premise (ii). The second premise is typically justified by claiming that (2) is an identity statement, in which the verb ``is'' is flanked by two singular terms: ``the number of moons of Mars'' and ``two''. The thought is that singular terms refer, and therefore the truth of sentence (2) requires that number two exists. NEWLINENEWLINENEWLINEThe centre of the second part of the book, from Chapter 7 to 10, are strategies that make use of linguistic analysis to counter the claim that (2) is an identity statement. First, the author discusses a prominent analysis that denies that (2) is an identity statement, claiming instead that it is a so-called focus construction. This, in turn, is used to argue that (2) is ontologically innocent after all. The author contends that this argument is not convincing. However, she supports the claim that (2) is not an identity statement, and rather argues that (2) is a disguised ``question-answer'' pair. Felka maintains that disguised question-answer pairs are ontologically innocent. She therefore concludes that the existence of numbers is not as easily provable as the easy arguments suggest.