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Summary Fictionalism denies the existence of abstract, aspatial and atemporal mathematical objects, at the same time claiming that mathematical theories are not true because there are no mathematical objects that those theories are supposed to be about. While this allows a fictionalist to avoid difficult questions about human knowledge of abstract objects, they have to handle a different problem. The applicability of mathematics and mathematicians' (usual) agreement suggest that there are some objective standards of correctness (if not truth) of mathematical theories and a fictionalist should explain what these standards are and how they are motivated.
Key works Freely accessible Balaguer 2008 contains an excellent list of key works in this field.
Introductions Nicely paced introductory surveys are Eklund 2010, Balaguer 2008 and Colyvan 2011
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  1. B. Armour-Garb (2011). The Implausibility of Hermeneutic Non-Assertivism. Philosophia Mathematica 19 (3):349-353.
    In a recent paper, Mark Balaguer has responded to the argument that I launched against Hermeneutic Non-Assertivism, claiming that, as a matter of empirical fact, ‘when typical mathematicians utter mathematical sentences, they are doing something that differs from asserting in a pretty subtle way, so that the difference between [asserting] and this other kind of speech act is not obvious’. In this paper, I show the implausibility of this empirical hypothesis.
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  2. B. Armour-Garb (2011). Understanding and Mathematical Fictionalism. Philosophia Mathematica 19 (3):335-344.
    In a recent paper in this journal, Mark Balaguer develops and defends a new version of mathematical fictionalism, what he calls ‘Hermeneutic non-assertivism’, and responds to some recent objections to mathematical fictionalism that were launched by John Burgess and others. In this paper I provide some fairly compelling reasons for rejecting Hermeneutic non-assertivism — ones that highlight an important feature of what understanding mathematics involves (or, as we shall see, does not involve).
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  3. Bradley Armour-Garb & James A. Woodbridge (2010). Why Deflationists Should Be Pretense Theorists (and Perhaps Already Are). In Cory D. Wright & Nikolaj J. L. L. Pedersen (eds.), New Waves in Truth. Palgrave Macmillan.
    In this paper, we do two things. First, we clarify the notion of deflationism, with special attention to deflationary accounts of truth. Seocnd, we argue that one who endorses a deflationary account of truth (or of semantic notions, generally) should be, or perhaps already is, a pretense theorist regarding truth-talk. In §1 we discuss mathematical fictionalism, where we focus on Yablo’s pretense account of mathematical discourse. §2 briefly introduces the key elements of deflationism and explains deflationism about truth in particular. (...)
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  4. Bradley Armour‐Garb & James A. Woodbridge (2014). From Mathematical Fictionalism to Truth‐Theoretic Fictionalism. Philosophy and Phenomenological Research 88 (1):93-118.
    We argue that if Stephen Yablo (2005) is right that philosophers of mathematics ought to endorse a fictionalist view of number-talk, then there is a compelling reason for deflationists about truth to endorse a fictionalist view of truth-talk. More specifically, our claim will be that, for deflationists about truth, Yablo’s argument for mathematical fictionalism can be employed and mounted as an argument for truth-theoretic fictionalism.
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  5. Jody Azzouni (2010). Talking About Nothing: Numbers, Hallucinations, and Fictions. Oxford University Press.
    Numbers -- Hallucinations -- Fictions -- Scientific languages, ontology, and truth -- Truth conditions and semantics.
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  6. Jody Azzouni (2004). The Derivation-Indicator View of Mathematical Practice. Philosophia Mathematica 12 (2):81-106.
    The form of nominalism known as 'mathematical fictionalism' is examined and found wanting, mainly on grounds that go back to an early antinominalist work of Rudolf Carnap that has unfortunately not been paid sufficient attention by more recent writers.
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  7. Mark Balaguer (2009). Fictionalism, Theft, and the Story of Mathematics. Philosophia Mathematica 17 (2):131-162.
    This paper develops a novel version of mathematical fictionalism and defends it against three objections or worries, viz., (i) an objection based on the fact that there are obvious disanalogies between mathematics and fiction; (ii) a worry about whether fictionalism is consistent with the fact that certain mathematical sentences are objectively correct whereas others are incorrect; and (iii) a recent objection due to John Burgess concerning “hermeneuticism” and “revolutionism”.
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  8. M. Balaguer (2011). Reply to Armour-Garb. Philosophia Mathematica 19 (3):345-348.
    Hermeneutic non-assertivism is a thesis that mathematical fictionalists might want to endorse in responding to a recent objection due to John Burgess. Brad Armour-Garb has argued that hermeneutic non-assertivism is false. A response is given here to Armour-Garb's argument.
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  9. Mark Balaguer, Fictionalism in the Philosophy of Mathematics. Stanford Encyclopedia of Philosophy.
    Mathematical fictionalism (or as I'll call it, fictionalism) is best thought of as a reaction to mathematical platonism. Platonism is the view that (a) there exist abstract mathematical objects (i.e., nonspatiotemporal mathematical objects), and (b) our mathematical sentences and theories provide true descriptions of such objects. So, for instance, on the platonist view, the sentence ‘3 is prime’ provides a straightforward description of a certain object—namely, the number 3—in much the same way that the sentence ‘Mars is red’ provides a (...)
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  10. Mark Balaguer (1998). Platonism and Anti-Platonism in Mathematics. Oxford University Press.
    In this book, Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He does this by establishing that both platonism and anti-platonism are defensible views. Introducing a form of platonism ("full-blooded platonism") that solves all problems traditionally associated with the view, he proceeds to defend anti-platonism (in particular, mathematical fictionalism) against various attacks, most notably the Quine-Putnam indispensability attack. He concludes by arguing that it is not simply that we do not currently have any good argument (...)
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  11. Mark Balaguer (1996). A Fictionalist Account of the Indispensable Applications of Mathematics. Philosophical Studies 83 (3):291 - 314.
  12. J. P. Burgess (2010). Mary Leng. Mathematics and Reality. Oxford: Oxford University Press, 2010. ISBN 978-0-19-928079-7. Pp. X + 278. Philosophia Mathematica 18 (3):337-344.
    (No abstract is available for this citation).
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  13. John P. Burgess (2004). Mathematics and Bleak House. Philosophia Mathematica 12 (1):18-36.
    The form of nominalism known as 'mathematical fictionalism' is examined and found wanting, mainly on grounds that go back to an early antinominalist work of Rudolf Carnap that has unfortunately not been paid sufficient attention by more recent writers.
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  14. John P. Burgess & Gideon A. Rosen (1997). A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics. Oxford University Press.
    Numbers and other mathematical objects are exceptional in having no locations in space or time or relations of cause and effect. This makes it difficult to account for the possibility of the knowledge of such objects, leading many philosophers to embrace nominalism, the doctrine that there are no such objects, and to embark on ambitious projects for interpreting mathematics so as to preserve the subject while eliminating its objects. This book cuts through a host of technicalities that have obscured previous (...)
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  15. Patrick Caldon & Aleksandar Ignjatović (2005). On Mathematical Instrumentalism. Journal of Symbolic Logic 70 (3):778 - 794.
    In this paper we devise some technical tools for dealing with problems connected with the philosophical view usually called mathematical instrumentalism. These tools are interesting in their own right, independently of their philosophical consequences. For example, we show that even though the fragment of Peano's Arithmetic known as IΣ₁ is a conservative extension of the equational theory of Primitive Recursive Arithmetic (PRA). IΣ₁ has a super-exponential speed-up over PRA. On the other hand, theories studied in the Program of Reverse Mathematics (...)
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  16. Jessica Carter (2013). Handling Mathematical Objects: Representations and Context. Synthese 190 (17):3983-3999.
    This article takes as a starting point the current popular anti realist position, Fictionalism, with the intent to compare it with actual mathematical practice. Fictionalism claims that mathematical statements do purport to be about mathematical objects, and that mathematical statements are not true. Considering these claims in the light of mathematical practice leads to questions about how mathematical objects are handled, and how we prove that certain statements hold. Based on a case study on Riemann’s work on complex functions, I (...)
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  17. Mark Colyvan (2011). Fictionalism in the Philosophy of Mathematics. In E. J. Craig (ed.), Routledge Encyclopedia of Philosophy.
    Fictionalism in the philosophy of mathematics is the view that mathematical statements, such as ‘8+5=13’ and ‘π is irrational’, are to be interpreted at face value and, thus interpreted, are false. Fictionalists are typically driven to reject the truth of such mathematical statements because these statements imply the existence of mathematical entities, and according to fictionalists there are no such entities. Fictionalism is a nominalist (or anti-realist) account of mathematics in that it denies the existence of a realm of abstract (...)
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  18. Mark Colyvan & Edward N. Zalta (1999). Mathematics: Truth and Fiction? Review of Mark Balaguer's. Philosophia Mathematica 7 (3):336-349.
    <span class='Hi'>Mark</span> Balaguer’s project in this book is extremely ambitious; he sets out to defend both platonism and fictionalism about mathematical entities. Moreover, Balaguer argues that at the end of the day, platonism and fictionalism are on an equal footing. Not content to leave the matter there, however, he advances the anti-metaphysical conclusion that there is no fact of the matter about the existence of mathematical objects.1 Despite the ambitious nature of this project, for the most part Balaguer does not (...)
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  19. Chris Daly (2006). Mathematical Fictionalism – No Comedy of Errors. Analysis 66 (291):208–216.
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  20. Chris John Daly (2008). Fictionalism and the Attitudes. Philosophical Studies 139 (3):423 - 440.
    This paper distinguishes revolutionary fictionalism from other forms of fictionalism and also from other philosophical views. The paper takes fictionalism about mathematical objects and fictionalism about scientific unobservables as illustrations. The paper evaluates arguments that purport to show that this form of fictionalism is incoherent on the grounds that there is no tenable distinction between believing a sentence and taking the fictionalist's distinctive attitude to that sentence. The argument that fictionalism about mathematics is ‘comically immodest’ is also evaluated. In place (...)
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  21. Antony Eagle (2008). Mathematics and Conceptual Analysis. Synthese 161 (1):67–88.
    Gödel argued that intuition has an important role to play in mathematical epistemology, and despite the infamy of his own position, this opinion still has much to recommend it. Intuitions and folk platitudes play a central role in philosophical enquiry too, and have recently been elevated to a central position in one project for understanding philosophical methodology: the so-called ‘Canberra Plan’. This philosophical role for intuitions suggests an analogous epistemology for some fundamental parts of mathematics, which casts a number of (...)
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  22. Hartry Field (1982). Realism and Anti-Realism About Mathematics. Philosophical Topics 13 (1):45-69.
  23. Gábor Forrai (2010). What Mathematicians' Claims Mean : In Defense of Hermeneutic Fictionalism. Hungarian Philosophical Review 54 (4):191-203.
    Hermeneutic fictionalism about mathematics maintains that mathematics is not committed to the existence of abstract objects such as numbers. Mathematical sentences are true, but they should not be construed literally. Numbers are just fictions in terms of which we can conveniently describe things which exist. The paper defends Stephen Yablo’s hermeneutic fictionalism against an objection proposed by John Burgess and Gideon Rosen. The objection, directed against all forms of nominalism, goes as follows. Nominalism can take either a hermeneutic form and (...)
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  24. J. Frans (2012). The Game of Fictional Mathematics: Review of M. Leng, Mathematics and Reality. [REVIEW] Constructivist Foundations 8 (1):126-128.
    Upshot: Leng attacks the indispensability argument for the existence of mathematical objects. She offers an account that treats the role of mathematics in science as an indispensable and useful part of theories, but retains nonetheless a fictionalist position towards mathematics. The result is an account of mathematics that is interesting for constructivists. Her view towards the nominalistic part of science is, however, more in conflict with radical constructivism.
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  25. James Hawthorne (1996). Mathematical Instrumentalism Meets the Conjunction Objection. Journal of Philosophical Logic 25 (4):363-397.
    Scientific realists often appeal to some version of the conjunction objection to argue that scientific instrumentalism fails to do justice to the full empirical import of scientific theories. Whereas the conjunction objection provides a powerful critique of scientific instrumentalism, I will show that mathematical instnrunentalism escapes the conjunction objection unscathed.
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  26. Sarah Hoffman, You Can't Mean That: Yablo's Figuralist Account of Mathematics.
    Burgess and Rosen argue that Yablo’s figuralist account of mathematics fails because it says mathematical claims are really only metaphorical. They suggest Yablo’s view is implausible as an account of what mathematicians say and confused about literal language. I show their argument isn’t decisive, briefly exploring some questions in the philosophy of language it raises, and argue Yablo’s view may be amended to a kind of revolutionary fictionalism not refuted by Burgess and Rosen.
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  27. Sarah Hoffman (2004). Kitcher, Ideal Agents, and Fictionalism. Philosophia Mathematica 12 (1):3-17.
    Kitcher urges us to think of mathematics as an idealized science of human operations, rather than a theory describing abstract mathematical objects. I argue that Kitcher's invocation of idealization cannot save mathematical truth and avoid platonism. Nevertheless, what is left of Kitcher's view is worth holding onto. I propose that Kitcher's account should be fictionalized, making use of Walton's and Currie's make-believe theory of fiction, and argue that the resulting ideal-agent fictionalism has advantages over mathematical-object fictionalism.
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  28. L. Horsten (2011). Review of M. Leng, Mathematics and Reality. [REVIEW] Analysis 71 (4):798-799.
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  29. Christopher Kennedy & Jason Stanley (2009). On 'Average'. Mind 118 (471):583 - 646.
    This article investigates the semantics of sentences that express numerical averages, focusing initially on cases such as 'The average American has 2.3 children'. Such sentences have been used both by linguists and philosophers to argue for a disjuncture between semantics and ontology. For example, Noam Chomsky and Norbert Hornstein have used them to provide evidence against the hypothesis that natural language semantics includes a reference relation holding between words and objects in the world, whereas metaphysicians such as Joseph Melia and (...)
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  30. Glenn Kessler (1978). Mathematics and Modality. Noûs 12 (4):421-441.
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  31. Mary Leng (2010). Mathematics and Reality. OUP Oxford.
    Mary Leng offers a defense of mathematical fictionalism, according to which we have no reason to believe that there are any mathematical objects. Perhaps the most pressing challenge to mathematical fictionalism is the indispensability argument for the truth of our mathematical theories (and therefore for the existence of the mathematical objects posited by those theories). According to this argument, if we have reason to believe anything, we have reason to believe that the claims of our best empirical theories are (at (...)
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  32. Mary Leng, Structuralism, Fictionalism, and Applied Mathematics.
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  33. Mary Leng (2005). Revolutionary Fictionalism: A Call to Arms. Philosophia Mathematica 13 (3):277-293.
    This paper responds to John Burgess's ‘Mathematics and Bleak House’. While Burgess's rejection of hermeneutic fictionalism is accepted, it is argued that his two main attacks on revolutionary fictionalism fail to meet their target. Firstly, ‘philosophical modesty’ should not prevent philosophers from questioning the truth of claims made within successful practices, provided that the utility of those practices as they stand can be explained. Secondly, Carnapian scepticism concerning the meaningfulness of metaphysical existence claims has no force against a naturalized version (...)
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  34. Mary Leng, Alexander Paseau & Michael D. Potter (eds.) (2007). Mathematical Knowledge. Oxford University Press.
    What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions. Written by some of the world's leading philosophers of mathematics, psychologists, and mathematicians, Mathematical Knowledge gives a lively sense of the current state of debate in this fascinating field. Contents 1. (...)
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  35. David Liggins (2014). Abstract Expressionism and the Communication Problem. British Journal for the Philosophy of Science 65 (3):599-620.
    Some philosophers have recently suggested that the reason mathematics is useful in science is that it expands our expressive capacities. Of these philosophers, only Stephen Yablo has put forward a detailed account of how mathematics brings this advantage. In this article, I set out Yablo’s view and argue that it is implausible. Then, I introduce a simpler account and show it is a serious rival to Yablo’s. 1 Introduction2 Yablo’s Expressionism3 Psychological Objections to Yablo’s Expressionism4 Introducing Belief Expressionism5 Objections and (...)
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  36. David Liggins (2012). Weaseling and the Content of Science. Mind 121 (484):997-1005.
    I defend Joseph Melia’s nominalist account of mathematics from an objection raised by Mark Colyvan.
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  37. David Liggins (2010). The Autism Objection to Pretence Theories. Philosophical Quarterly 60 (241):764-782.
    A pretence theory of a discourse is one which claims that we do not believe or assert the propositions expressed by the sentences we utter when taking part in the discourse: instead, we are speaking from within a pretence. Jason Stanley argues that if a pretence account of a discourse is correct, people with autism should be incapable of successful participation in it; but since people with autism are capable of participiating successfully in the discourses which pretence theorists aim to (...)
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  38. M. Liston (2013). Christopher Pincock. Mathematics and Scientific Representation. Oxford University Press, 2012. ISBN 978-0-19-975710-7. Pp. Xv + 330. [REVIEW] Philosophia Mathematica 21 (3):371-385.
  39. Michael Liston (1993). Taking Mathematical Fictions Seriously. Synthese 95 (3):433 - 458.
    I argue on the basis of an example, Fourier theory applied to the problem of vibration, that Field's program for nominalizing science is unlikely to succeed generally, since no nominalistic variant will provide us with the kind of physical insight into the phenomena that the standard theory supplies. Consideration of the same example also shows, I argue, that some of the motivation for mathematical fictionalism, particularly the alleged problem of cognitive access, is more apparent than real.
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  40. Jarosław Mrozek (1996). Matematyka - narzędzie czy opis? Instrumentalistyczna i realistyczna interpretacja zastosowań matematyki. Filozofia Nauki 2.
    In the paper there are presented two proposals of the interpretations of the applications of mathematics in the natural sciences - realistic and instrumentalistic. The realistic conception, in accordance with the successes of science, maintains that there exists some kind of correspondence between the mathematical structures and the internal structure of the world. It is expressed in the thesis of the mathematicality of nature. The instrumentalistic approach separates the cognitive content of the scientific theory from the mathematical means of expression (...)
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  41. Richard Pettigrew (2012). Indispensability Arguments and Instrumental Nominalism. Review of Symbolic Logic 5 (4):687-709.
    In the philosophy of mathematics, indispensability arguments aim to show that we are justified in believing that abstract mathematical objects exist. I wish to defend a particular objection to such arguments that has become increasingly popular recently. It is called instrumental nominalism. I consider the recent versions of this view and conclude that it has yet to be given an adequate formulation. I provide such a formulation and show that it can be used to answer the indispensability arguments. -/- There (...)
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  42. Hugly Philip & Sayward Charles (1989). Mathematical Relativism. History and Philosophy of Logic 10 (1):53-65.
    We set out a doctrine about truth for the statements of mathematics?a doctrine which we think is a worthy competitor to realist views in the philosophy of mathematics?and argue that this doctrine, which we shall call ?mathematical relativism?, withstands objections better than do other non-realist accounts.
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  43. AhtiVeikko Pietarinen (2008). Why Pragmaticism is Neither Mathematical Structuralism nor Fictionalism. Proceedings of the Xxii World Congress of Philosophy 41:19-25.
    Despite some surface similarities, Charles Peirce’s philosophy of mathematics, pragmaticism, is incompatible with both mathematical structuralism and fictionalism. Pragmaticism has to do with experimentation and observation concerning the forms of relations in diagrammatic and iconic representations ofmathematical entities. It does not presuppose mathematical foundations although it has these representations as its objects of study. But these objects do have a reality which structuralism and fictionalism deny.
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  44. Christopher Pincock, Alan Baker, Alexander Paseau & Mary Leng (2012). Science and Mathematics: The Scope and Limits of Mathematical Fictionalism. [REVIEW] Metascience 21 (2):269-294.
    Science and mathematics: the scope and limits of mathematical fictionalism Content Type Journal Article Category Book Symposium Pages 1-26 DOI 10.1007/s11016-011-9640-3 Authors Christopher Pincock, University of Missouri, 438 Strickland Hall, Columbia, MO 65211-4160, USA Alan Baker, Department of Philosophy, Swarthmore College, Swarthmore, PA 19081, USA Alexander Paseau, Wadham College, Oxford, OX1 3PN UK Mary Leng, Department of Philosophy, University of York, Heslington, York, YO10 5DD UK Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796.
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  45. Stephen Pollard (2010). 'As If' Reasoning in Vaihinger and Pasch. Erkenntnis 73 (1):83 - 95.
    Hans Vaihinger tried to explain how mathematical theories can be useful without being true or even coherent, arguing that mathematicians employ a special kind of fictional or "as if" reasoning that reliably extracts truths from absurdities. Moritz Pasch insisted that Vaihinger was wrong about the incoherence of core mathematical theories, but right about the utility of fictional discourse in mathematics. This essay explores this area of agreement between Pasch and Vaihinger. Pasch's position raises questions about structuralist interpretations of mathematics.
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  46. G. Priest (2011). Jody Azzouni. Talking About Nothing. New York: Oxford University Press, 2010. Isbn 978-0-19-973894-64. Pp. IV + 273. Philosophia Mathematica 19 (3):359-363.
  47. Davide Rizza (2011). Review of M. Leng, Mathematics and Reality. [REVIEW] Philosophical Quarterly 61 (244):655-657.
  48. Matthias Schirn (ed.) (1998). The Philosophy of Mathematics Today. Clarendon Press.
    This comprehensive volume gives a panorama of the best current work in this lively field, through twenty specially written essays by the leading figures in the field. All essays deal with foundational issues, from the nature of mathematical knowledge and mathematical existence to logical consequence, abstraction, and the notions of set and natural number. The contributors also represent and criticize a variety of prominent approaches to the philosophy of mathematics, including platonism, realism, nomalism, constructivism, and formalism.
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  49. Stewart Shapiro (1983). Conservativeness and Incompleteness. Journal of Philosophy 80 (9):521-531.
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  50. Jonathan Tallant (2013). Pretense, Mathematics, and Cognitive Neuroscience. British Journal for the Philosophy of Science 64 (4):axs013.
    A pretense theory of a given discourse is a theory that claims that we do not believe or assert the propositions expressed by the sentences we token (speak, write, and so on) when taking part in that discourse. Instead, according to pretense theory, we are speaking from within a pretense. According to pretense theories of mathematics, we engage with mathematics as we do a pretense. We do not use mathematical language to make claims that express propositions and, thus, we do (...)
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