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Mathematical descriptions

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Abstract

In this paper, the authors briefly summarize how object theory uses definite descriptions to identify the denotations of the individual terms of theoretical mathematics and then further develop their object-theoretic philosophy of mathematics by showing how it has the resources to address some objections recently raised against the theory. Certain ‘canonical’ descriptions of object theory, which are guaranteed to denote, correctly identify mathematical objects for each mathematical theory T, independently of how well someone understands the descriptive condition. And to have a false belief about some particular mathematical object is not to have a true belief about some different mathematical object.

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Notes

  1. This view has been defended elsewhere as well, though not exactly in the form put forward in Linsky and Zalta (1995), see Zalta (1983) (Chapter VI), Zalta (2000) and Nodelman and Zalta (2014).

  2. A somewhat more refined definition was put forward in Nodelman and Zalta (2014), to avoid a concern about indiscernibles: x is an object of a theory T just in case x is distinguishable in T, i.e., \( T\models \forall y(y \not =_{T} x \rightarrow \exists F(Fx \, \& \, \lnot Fy))\).

  3. At the end of the present paper, we offer one final point in response to anyone who objects that our analysis requires a technical understanding of object theory to successfully refer using a mathematical term.

  4. Formally, the semantics of descriptions can be spelled out as follows. Where \(\mathcal {I}\) is an interpretation of the language, f is an assignment function, \({\varvec{d}}_{\mathcal {I},f}(\tau )\) is the denotation of term \(\tau \) w.r.t \(\mathcal {I}\) and f, and \([\phi ]_{\mathcal {I},f}\) asserts \(\phi \) is true w.r.t. \(\mathcal {I}\) and f, we simply add a recursive clause that says:

    • \( {\varvec{d}}_{\mathcal {I},f}(\imath x\phi ) = \left\{ \begin{array}{ll} {\varvec{o}} & \quad \mathrm {if} \;\, f(x) = {\varvec{o}} \; \& \; [\phi ]_{\mathcal {I},f} \; \& \; \forall {\varvec{o}}'\forall f' (f'(x) = {\varvec{o}}' \, \& \, [\phi ]_{\mathcal {I},{f'}} \rightarrow {\varvec{o}}' = {\varvec{o}})\\ \mathrm {undefined}, &{}\quad \mathrm {otherwise}\end{array}\right. \)

    This says that the denotation of \(\imath x\phi \) is the object \({\varvec{o}}\) if and only if there is an assignment f such (a) that f assigns the object o to the variable x, (b) \(\phi \) is true w.r.t. \(\mathcal {I}\) and f, and (c) for all objects \(o'\) and assignments \(f'\), if \(f'\) assigns \({\varvec{o}}'\) to x and \(\phi \) is true w.r.t. \(\mathcal {I}\) and \(f'\), then \({\varvec{o}}'\) is identical to \({\varvec{o}}\). This captures Russell’s (1905) analysis of descriptions semantically.

  5. We recognize that this passage comes in the context where Benacerraf is presenting the “myth he learned as a youth”, but this bit is not the mythical part!

References

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Authors and Affiliations

  1. Department of Philosophy, University of Alberta, Edmonton, Canada

    Bernard Linsky

  2. Center for the Study of Language and Information, Stanford University, Stanford, CA, USA

    Edward N. Zalta

Authors
  1. Bernard Linsky
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  2. Edward N. Zalta
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Correspondence to Edward N. Zalta.

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Linsky, B., Zalta, E.N. Mathematical descriptions. Philos Stud 176, 473–481 (2019). https://doi.org/10.1007/s11098-017-1024-0

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  • DOI: https://doi.org/10.1007/s11098-017-1024-0

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