Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-25T23:31:12.824Z Has data issue: false hasContentIssue false
  • English
  • Français

What is Neologicism?

Published online by Cambridge University Press:  15 January 2014

Bernard Linsky  and
Edward N. Zalta
Bernard Linsky
Affiliation:
Department of Philosophy, University of Alberta, Edmonton, Alberta T6G 2E5, Canada. E-mail: bernard.linsky@ualberta.ca
Edward N. Zalta
Affiliation:
CSLI, Ventura Hall, Stanford University, Stanford, CA 94305, USA. E-mail: zalta@stanford.edu
Get access Rights & Permissions [Opens in a new window]

Extract

Logicism is a thesis about the foundations of mathematics, roughly, that mathematics is derivable from logic alone. It is now widely accepted that the thesis is false and that the logicist program of the early 20th century was unsuccessful. Frege's [1893/1903] system was inconsistent and the Whitehead and Russell [1910–1913] system was not thought to be logic, given its axioms of infinity, reducibility, and choice. Moreover, both forms of logicism are in some sense non-starters, since each asserts the existence of objects (courses of values, propositional functions, etc.), something which many philosophers think logic is not supposed to do. Indeed, the tension in the idea underlying logicism, that the axioms and theorems of mathematics can be derived as theorems of logic, is obvious: on the one hand, there are numerous existence claims among the theorems of mathematics, while on the other, it is thought to be impossible to prove the existence of anything from logic alone. According to one well-received view, logicism was replaced by a very different account of the foundations of mathematics, in which mathematics was seen as the study of axioms and their consequences in models consisting of the sets described by Zermelo-Fraenkel set theory (ZF). Mathematics, on this view, is just applied set theory.

Type
Articles
Information
Bulletin of Symbolic Logic , Volume 12 , Issue 1 , March 2006 , pp. 60 - 99
Copyright
Copyright © Association for Symbolic Logic 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Anderson, D. and Zalta, E. [2004], Frege, Boolos, and logical objects, Journal of Philosophical Logic, vol. 33, no. 1, pp. 1–26.Google Scholar
Barwise, J. [1979], On branching quantifiers in English, Journal of Philosophical Logic, vol. 8, pp. 47–80.CrossRefGoogle Scholar
Batitsky, V. [2001], Some measurement-theoretic concerns about Hale's “Reals by abstraction”, Philosophia Mathematica, vol. 10, pp. 286–303.Google Scholar
Bell, J. L. [1994], Fregean extensions of first-order theories, Mathematical Logic Quarterly, vol. 40, pp. 27–30.Google Scholar
Benacerraf, P. [1965], What numbers could not be, The Philosophical Review, vol. 74, pp. 47–73.Google Scholar
Benacerraf, P. [1981], Frege: The last logicist, Midwest studies in philosophy: VI (French, P. et al., editors), University of Minnesota Press, Minneapolis, page reference is to the reprint in Demopoulos [1995].Google Scholar
Benacerraf, P. and Putnam, H. [1964], Philosophy of mathematics: Selected readings, Prentice Hall, Englewood Cliffs, NJ.Google Scholar
Boolos, G. [1986], Saving Frege from contradiction, Proceedings of the Aristotelian Society, vol. 87, pp. 137–151, reprinted in Boolos [1998, pp. 1998–182].Google Scholar
Boolos, G. [1987], The consistency of Frege's Foundations of Arithmetic, On being and saying (Thomson, J., editor), MIT Press, Cambridge, MA, reprinted in Boolos [1998, pp. 1998–201].Google Scholar
Boolos, G. [1990], The standard of equality of numbers, Meaning and method: Essays in honor of Hilary Putnam (Boolos, G., editor), Cambridge University Press, Cambridge, pp. 261–277; page reference is to the reprint in Boolos [1998, pp. 1998–219].Google Scholar
Boolos, G. [1994], The advantages of theft over honest toil, Mathematics and mind (George, A., editor), Oxford University Press, Oxford, page reference is to the reprint in Boolos [1998, pp. 255–274], pp. 22–44.Google Scholar
Boolos, G. [1997], Is Hume's principle analytic?, Logic, language, and thought (Heck, R., editor), Oxford University Press, Oxford, page reference is to the reprint in Boolos [1998, pp. 1998–261].Google Scholar
Boolos, G. [1998], Logic, logic, and logic, Harvard University Press, Cambridge, MA.Google Scholar
Burgess, J. [1998], On a consistent subsystemof Frege's Grundgesetze, Notre Dame Journal of Formal Logic, vol. 39, pp. 274–278.Google Scholar
Burgess, J. [2003], Review of Kit Fine, The Limits of Abstraction, Notre Dame Journal of Formal Logic, vol. 44, no. 4, pp. 227–251.Google Scholar
Burgess, J. [2005], Fixing Frege, Princeton University Press, Princeton, forthcoming; page reference is to the manuscript.Google Scholar
Cook, R. [2001], The state of the economy: Neologicism and inflation, Philosophia Mathematicas, vol. 10, no. 3, pp. 43–66.Google Scholar
Cook, R. [2003], Iteration one more time, Notre Dame Journal of Formal Logic, vol. 44, no. 2, pp. 63–92.Google Scholar
Cook, R. and Ebert, P. [2004], Discussion note: Kit Fine, Limits of abstraction, British Journal for the Philosophy of Science, vol. 55, pp. 791–800.CrossRefGoogle Scholar
Cook, R. and Ebert, P. [2005], Abstraction and identity, Dialectica, vol. 59, no. 2, pp. 1–19.Google Scholar
Demopoulos, W. [1995], Frege's Philosophy of mathematics, Harvard University Press, Cambridge, MA.Google Scholar
Dummett, M. [1991], Frege: Philosophy of mathematics, Harvard University Press, Cambridge, MA.Google Scholar
Etchemendy, J. [1990], The concept of logical consequence, Harvard University Press, Cambridge, MA.Google Scholar
Ferreira, F. and Wehmeier, K. [2002], On the consistency of the -CA fragment of Frege's Grundgesetze, Journal of Philosophical Logic, vol. 31, no. 4, pp. 301–312.Google Scholar
Field, H. [1989], Realism, mathematics, and modality, Blackwell, Oxford.Google Scholar
Fine, K. [2002], The limits of abstraction, Clarendon Press, Oxford.Google Scholar
Frege, G. [1893/1903], Grundgesetze der Arithmetik, vol. I/II, Verlag Hermann Pohle, Jena.Google Scholar
Goldfarb, W. [2001a], First-order Frege theory is undecidable, Journal of Philosophical Logic, vol. 30, pp. 613–616.Google Scholar
Goldfarb, W. [2001b], Frege's Conception of logic, Future pasts: The analytic tradition in twentieth century philosophy (Floyd, Juliet and Shieh, Sanford, editors), Oxford University Press, Oxford.Google Scholar
Hale, B. [1987], Abstract objects, Blackwell, Oxford.Google Scholar
Hale, B. [2000], Reals by abstraction, Philosophia Mathematica, vol. 8, pp. 100–123.Google Scholar
Hale, B. [2002], Real numbers, quantities, and measurement, Philosophia Mathematica, vol. 10, pp. 304–323.CrossRefGoogle Scholar
Hale, B. and Wright, C. [2001], The reason's proper study, Clarendon, Oxford.Google Scholar
Heck, R. [1992], On the consistency of second-order contextual definitions, Noûs, vol. 26, pp. 491–494.Google Scholar
Heck, R. [1996], The consistency of predicative fragments of Frege's Grundgesetze der Arithmetik, History and Philosophy of Logic, vol. 17, pp. 209–220.Google Scholar
Hempel, C. [1945], On the nature of mathematical truth, The American Mathematical Monthly, vol. 52, pp. 543–556, page reference is to the reprint in Benacerraf and Putnam [1964].Google Scholar
Henkin, L. [1961], Some remarks on infinitely long formulas, Infinitistic methods, Pergamon Press, New York, pp. 167–183.Google Scholar
Hodes, H. [1984], Logicism and the ontological commitments of arithmetic, Journal of Philosophy, vol. lxxxi, no. 3, pp. 123–149.Google Scholar
Hodes, H. [1991], Where do sets come from?, The Journal of Symbolic Logic, vol. 56, no. 1, pp. 151–175.Google Scholar
Kneale, W. and Kneale, M. [1962], The development of logic, Clarendon Press, Oxford.Google Scholar
Linsky, B. [1990], Was the axiom of reducibility a principle of logic? Russell, The Journal of the Bertrand Russell Archives, vol. 10, pp. 125–140.Google Scholar
Linsky, B. and Zalta, E. [1994], In defense of the simplest quantified modal logic, Philosophical Perspectives, vol. 8, pp. 431–458.Google Scholar
Linsky, B. and Zalta, E. [1995], Naturalized Platonism vs. Platonized Naturalism, The Journal of Philosophy, vol. xcii, no. 10, pp. 525–555.Google Scholar
MacFarlane, J. [2002], Frege, Kant, and the logic in logicism, The Philosophical Review, vol. 111, pp. 25–65.Google Scholar
Martin-Löf, P. [1984], Intuitionistic type theory, Bibliopolis, Naples.Google Scholar
Parsons, T. [1987], The consistency of the first-order portion of Frege's logical system, Notre Dame Journal of Formal Logic, vol. 28, no. 1, pp. 161–168.Google Scholar
Quine, W. V. O. [1937a], Logic based on inclusion and abstraction, The Journal of Symbolic Logic, vol. 2, pp. 145–152.Google Scholar
Quine, W. V. O. [1937b], New foundations for mathematical logic, American Mathematical Monthly, vol. 44, pp. 70–80.Google Scholar
Quine, W. V. O. [1970], Philosophy of logic, Prentice-Hall, Englewood Cliffs.Google Scholar
Schoeder-Heister, P. [1987], A model-theoretic reconstruction of Frege's permutation argument, Notre Dame Journal of Formal Logic, vol. 28, no. 1, pp. 69–79.Google Scholar
Shapiro, S. [1991], Foundations without foundationalism: A case for second-order logic, Clarendon, Oxford.Google Scholar
Shapiro, S. [2004], Foundations of mathematics: Metaphysics, epistemology, and structure, The Philosophical Quarterly, vol. 54, no. 214, pp. 16–37.Google Scholar
Shapiro, S. and Weir, A. [1999], New V, ZF, and abstraction, Philosophia Mathematica, vol. 7, pp. 293–321.Google Scholar
Sieg, W. [1999], Hilbert's programs: 1917–1922, this Bulletin, vol. 5, no. 1, pp. 1–44.Google Scholar
Tarski, A. [1935], Der Wahrheitsbegriff in den formalisierten Sprachen, Studia Philosophica, vol. 1, pp. 261–405.Google Scholar
Tennant, N. [2004], A general theory of abstraction operators, The Philosophical Quarterly, vol. 54, no. 214, pp. 105–133.Google Scholar
Wehmeier, K. [1999], Consistent fragments of Grundgesetze and the existence of nonlogical objects , Synthese, vol. 121, pp. 309–328.Google Scholar
Weir, A. [2003], Neo-Fregeanism: An embarrassment of riches, Notre Dame Journal of Formal Logic, vol. 44, no. 1, pp. 13–48.Google Scholar
Whitehead, A. and Russell, B. [1910–1913], Principia mathematica, Cambridge University Press, Cambridge, second edition, 1925–1927.Google Scholar
Wright, C. [1983], Frege's Conception of numbers as objects, Scots Philosophical Monographs, vol. 2, Aberdeen University Press, Aberdeen.Google Scholar
Zalta, E. [1999], Natural numbers and natural cardinals as abstract objects: A partial reconstruction of Frege's Grundgesetze in object theory , Journal of Philosophical Logic, vol. 28, no. 6, pp. 619–660.Google Scholar
Zalta, E. [2000], Neo-logicism? an ontological reduction of mathematics to metaphysics, Erkenntnis, vol. 53, no. 1–2, pp. 219–265.Google Scholar