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Metascience (2009) 18:433–436 Ó Springer 2009 DOI 10.1007/s11016-009-9300-z REVIEW NEW MODEL NATURALISM John P. Burgess, Mathematics, Models, and Modality: Selected Philosophical Essays. Cambridge: Cambridge University Press, 2008. Pp. xiv+301. US$99.00 HB. By Øystein Linnebo This is a collection of...

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Ó Springer 2009

Metascience (2009) 18:433–436 DOI 10.1007/s11016-009-9300-z

REVIEW

NEW MODEL NATURALISM

John P. Burgess, Mathematics, Models, and Modality: Selected Philosophical Essays. Cambridge: Cambridge University Press, 2008. Pp. xiv+301. US$99.00 HB.

By Øystein Linnebo This is a collection of twelve previously published and two new articles on topics in the philosophy of mathematics, ontology, modality, and the theory of reference, accompanied by a useful introduction. Written by a respected logician and philosopher, the articles are technically and historically well informed, philosophically interesting, and at times provocative. Readers acquainted with Burgess’s work will recognise some familiar themes, such as his radical naturalism, anti-revisionism vis-a`-vis scientific practice, and careful logical analyses. Anyone working in the relevant areas will find much to learn in this book. Good philosophy should according to Burgess be Ônaturalistic’ – in Quine’s sense that it ‘‘abstains from imposing philosophical constraints on science’’ (p. 48). Thus one central theme in the book is naturalism in the philosophy of mathematics. Burgess’s naturalistic approach was first laid out in his influential ÔWhy I am not a Nominalist’ (1983), which developed an early version of an important argument later refined in his 1997 book with Gideon Rosen: A Subject Without an Object. Although science makes extensive use of mathematical objects, nominalists deny that there are any such objects. Burgess distinguishes two forms of nominalism. Hermeneutic nominalists claim that, properly interpreted, science isn’t committed to mathematical objects. Revolutionary nominalists claim there are scientific reasons to Ônominalise’ physics. The former claim belongs to linguistics and should be evaluated as such. The latter belongs to physics and should be

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evaluated accordingly. Since both claims are without merit when thus evaluated, Burgess concludes we should reject nominalism. In the introduction to the present volume, Burgess is unapologetic about the anti-revisionism he espoused in 1983, which denied philosophy any autonomy vis-a`-vis actual scientific practice. Philosophy must leave science precisely as it is. But it may be responded that the boundary between philosophy and science is often quite indistinct (think of debates about the nature of space–time, the interpretation of quantum mechanics, the levels of Darwinian selection, and the foundations of economics) and that philosophers are sometimes better equipped than scientists to identify tensions within science. Burgess’s anti-revisionism is also operative in his highly critical ÔDummett’s Case for Intuitionism’, which rejects Dummett’s case as being dependent on an outdated behaviourism and at odds with current linguistic practice. This is a fair challenge. But Dummett would reject Burgess’s dichotomy between staying within current science and being anti-scientific. Whether this suffices to salvage Dummett’s argument is another matter. ÔMathematics and Bleak House’ attacks fictionalist accounts of mathematics. The comparison of mathematics with fiction is dismissed as unhelpful. Then follows a discussion of an argument (due to Chihara and Yablo) that mathematicians’ reluctance to endorse a Platonistic ontology indicates that they don’t intend any ontological commitment to abstract mathematical objects. Burgess correctly observes that the mentioned reluctance admits many alternative explanations (for instance mathematicians’ desire to avoid trouble with philosophers). The article ends by endorsing the Carnapian claim that many of the ontological questions under debate presuppose an untenable form of metaphysical realism. The self-styled Ôsequel’ ÔQuine, Analyticity, and Philosophy of Mathematics’ develops this Carnapian view of ontology further. Although both Carnap and Quine reject Ôalienated’ conceptions of ontology, they disagreed about the distinction between internal and external questions. Burgess argues, in defence of Carnap, that the distinction can do important work, such as explaining the obviousness of elementary mathematics. Quine’s criticism that the distinction presupposes an untenable analytic-synthetic distinction is countered by outlining a Ôpragmatic’ notion of analyticity which Burgess commends to Quinians and non-Quinians alike.

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Burgess’s ‘‘farewell to the issue of nominalism’’ (p. 6), in ÔBeing Explained Away’ is a wonderful example of how logic can shed light on the debate about nominalism. When two theories are mutually interpretable, logicians regard them as being of the same overall strength. This strength can manifest itself in either ontological or ideological (that is, conceptual) commitments. Nominalists go to great lengths to replace ontological commitments with ideological ones. Using a trick due to Quine’s ÔVariables Explained Away’, Burgess shows how all ontological commitments can be eliminated in favour of ideological ones, which he takes to show that the nominalist goal is absurd. Next follow two more technical pieces. ÔE pluribus unum’ uses George Boolos’ plural logic to reinterpret and improve on a motivation of the axioms of standard ZF set theory due to Paul Bernays. Although this supplies an elegant reinterpretation and an important discussion of the relation between plurals and sets, I have expressed misgivings about the improvements sought over Bernays’ theory in ÔBurgess on Plural Logic and Set Theory’ (Philosophia Mathematica, 2001). ÔLogicism: A New Look’ provides just that. Anyone who believes that logicism was buried by Russell’s Paradox more than a century ago will be introduced to some recent logicist approaches still worthy of attention. Burgess concentrates on two less well known approaches. One, due to Richard Heck, is to use only predicative second-order logic. Although this approach blocks the paradoxes and justifies some elementary mathematics, its mathematical strength is severely limited. Another approach, due to Richard Jeffrey, exploits the idea that mathematics stands to logic as physics stands to its empirical data: logic providing the data, which mathematics explains and Ôpredicts’. Another theme of the book is modality. First out is ÔTarski’s Tort’, which is an inventory of the ills allegedly caused by Tarski’s liberal use of the term Ôsemantics’. The term traditionally refers to the empirical study of linguistic meaning. But Tarski used it also for his purely mathematical theory of models and definability. This has encouraged two sort of confusion. Ontological commitments of a model theory for a language have unjustifiably been ascribed to the language itself. And the development of a model theory has unjustifiably been taken to provide a theory of meaning. More controversially, Burgess contends that Tarski’s tort has lent undeserved prestige to truth-conditional theories of meaning and that such

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theories might otherwise never have been associated with Tarski’s theory of truth. I believe, however, that this overstates matters. The difference between mathematical and empirical Ôsemantics’ seems more like that between differential geometry and the study of physical space: different but related. Which modal logic is the right one? An eponymous article clarifies two targets – semantic logical validity and demonstrability in principle – which are studied by validity logic and demonstrability logic, respectively. The article provides a superb overview of the issues, outlining key arguments and providing valuable references for the soundness and completeness of the modal logic S5 for validity logic, and the soundness of S4 for demonstrability logic. ÔCan Truth Out?’ is a useful examination of Fitch’s paradox of knowability in the light of its temporal analogue, which targets the view that every truth will become known. Burgess points out that the obvious tense-logical formulation of this view conflates different moments at which truths are considered. Since this formulation parallels the standard modal-logical formulation of verificationism, the latter too is called into question. ÔQuinus ab Omni Naevo Vindicatus’ is – as its title indicates – an attempt to free Quine from every blemish. More specifically, the article defends Quine’s critique of quantified modal logic against various pre-Kripkean attacks. This is meant not just to vindicate Quine but also to highlight the importance and novelty of Kripke’s contribution (which is challenged by Ruth Barcan Marcus, against whom the article forcefully polemicises). Quine’s target is a linguistic conception of modality that analyses necessity as truth in virtue of meaning. Quine is right that, on this conception, modality de re becomes impossible; for this linguistic modality applies only to sentences, not to objects. (And because objects don’t have unique names, there can be no reduction of de re modality to de dicto.) It took until Kripke to articulate an alternative, metaphysical conception of modality that sidesteps Quine’s attack – provided that one accepts its strong metaphysical assumptions. In short, this is a collection of useful and thought-provoking essays, warmly recommended to anyone with an interest in the relevant areas. Department of Philosophy University of Bristol Bristol, UK...