×

Logicism as making arithmetic explicit. (English) Zbl 1346.03024

Summary: This paper aims to shed light on the broader significance of Frege’s logicism (and hence the phenomenon of modern logic) against the background of discussing and comparing Wittgenstein’s ‘showing/saying’-distinction with Brandom’s idiom of logic as the enterprise of making the implicit rules of our linguistic practices (something we do) explicit (by something we say). The main thesis of this paper is that the problem of Frege’s logicism lies deeper than in its inconsistency (which has since turned out to be reparable, as the neologicists have shown): it lies in the basic idea that in arithmetic (and prospectively in language in general) one can, and should, express everything that is implicitly presupposed so that nothing is left unsaid. This, in fact, is the target of Wittgenstein’s critique. Rather than the Tractatus, with its claim that logicism attempts to say something that can only be shown (e.g. what ‘object’, ‘function’ or ‘number’ are), it is the Philosophical Investigations, with its argument by regress against the thesis that every rule which one can follow must be of an explicit nature, that is of real significance here.

MSC:

03A05 Philosophical and critical aspects of logic and foundations
03-03 History of mathematical logic and foundations
01A55 History of mathematics in the 19th century
01A65 Development of contemporary mathematics
00A35 Methodology of mathematics
Full Text: DOI

References:

[1] Bolzano, B. (1851). Paradoxien des Unendlichen. Leipzig: Reclam. · Zbl 0134.24601
[2] Boolos, G.; Thompson, JJ (ed.), The consistency of Frege’s ‘Foundations of Arithmetic’, 3-20 (1987), Cambridge, MA
[3] Brandom, R. (1994). Making it explicit. Reasoning, representing, and discursive commitment. Cambridge, MA: Harvard University Press.
[4] Brandom, R. (2008a). Between saying and doing. Towards an analytic pragmatism. Oxford: Oxford University Press. · doi:10.1093/acprof:oso/9780199542871.001.0001
[5] Brandom, R.; Stekeler-Weithofer, P. (ed.), Responses, 209-230 (2008), Amsterdam · doi:10.1075/bct.15.14bra
[6] Brouwer, L. E. J. (1907). Over de grondslagen der wiskunde. Amsterdam: Universiteit Amsterdam. · JFM 38.0081.04
[7] Carroll, L. (1895). What the tortoise said to achilles. Mind,3, 278-280. · doi:10.1093/mind/IV.14.278
[8] Fine, K. (2002). The limits of abstraction. Oxford: Oxford University Press. · Zbl 1043.00005
[9] Frege, G. (1879). Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle: L. Nebert.
[10] Frege, G. (1884). Die Grundlagen der Arithmetik. Eine logisch mathematische Untersuchung über den Begriff der Zahl. Breslau: W. Koebner. · Zbl 0654.03005
[11] Frege, G. (1893/1903). Grundgesetze der Arithmetik. Begriffsschriftlich abgeleitet I-II. Jena: H. Pohle. · JFM 25.0101.02
[12] Frege, G. (1983). Nachgelassene Schriften (2nd ed.). Hamburg: Felix Meiner. · Zbl 0553.01026
[13] Hale, B., & Wright, C. (2001). The reason’s proper study. Essays towards a Neo-Fregean philosophy of mathematics. Oxford: Oxford University Press. · Zbl 1005.03006
[14] Kamlah, W., & Lorenzen, P. (1967). Logische Propädeutik. Vorschule des vernünftigen Redens. Mannheim: Bibliographisches Institut.
[15] Kolman, V. (2005). Lässt sich der Logizismus retten? Allgemeine Zeitschrift für Philosophie,30, 159-174.
[16] Kolman, V. (2010). Continuum, name, paradox. Synthese,175, 351-367. · Zbl 1208.03013 · doi:10.1007/s11229-009-9527-7
[17] Kripke, S. (1982). Wittgenstein on rules and private language. An elementary exposition. Cambridge, MA: Harvard University Press.
[18] Lakatos, I. (1978). Cauchy and the continuum: The significance of non-standard analysis for the history and philosophy of mathematics. Mathematical Intelligencer,1, 151-161. · Zbl 0398.01009 · doi:10.1007/BF03023263
[19] Lorenzen, P. (1955). Einführung in die operative Logik und Mathematik. Berlin: Springer. · Zbl 0066.24802 · doi:10.1007/978-3-662-01539-1
[20] Lorenzen, P., & Lorenz, K. (1978). Dialogische Logik. Darmstadt: Wissenschaftliche Buchgesellschaft. · Zbl 0435.03011
[21] McDowell, J.; Stekeler-Weithofer, P. (ed.), Motivating inferentialism: Comments on making it explicit (Ch. 2), 109-126 (2008), Amsterdam · doi:10.1075/bct.15.09mcd
[22] Odifreddi, P. (1989). Classical recursion theory. The theory of functions and sets of natural numbers. Amsterdam: North-Holland. · Zbl 0661.03029
[23] Poincaré, H. (1908). Science et méthode. Paris: Flammarion. · JFM 39.0095.03
[24] Potter, M. (2000). Reason’s nearest kin. Philosophies of arithmetic from Kant to Carnap. Oxford: Oxford University Press. · Zbl 1016.03003
[25] Potter, M. (2008). Wittgenstein’s notes on logic. Oxford: Oxford University Press. · doi:10.1093/acprof:oso/9780199215836.001.0001
[26] Quine, W. V. O. (1953). From a logical point of view. Cambridge, MA: Harvard University Press. · Zbl 0050.00501
[27] Ramsey, F. P. (1931). The foundations of mathematics and other logical essays. London: Routledge. · JFM 57.0047.06
[28] Russell, B. (1905). On denoting. Mind,14, 479-493. · doi:10.1093/mind/XIV.4.479
[29] Russell, B. (1920). Introduction to mathematical philosophy. London: George Allen & Unwin. · Zbl 0865.03001
[30] Russell, B., & Whitehead, A. N. (1910-1913). Principia Mathematica. Cambridge: Cambridge University Press. · JFM 41.0083.02
[31] Stekeler-Weithofer, P. (1986). Grundprobleme der Logik. Berlin: de Gryuter. · Zbl 1160.00001 · doi:10.1515/9783110855005
[32] Wittgenstein, L. (1922). Tractatus logico-philosophicus. London: Routledge & Kegan Paul. · JFM 48.1128.13
[33] Wittgenstein, L. (1953). Philosophical investigations. Oxford: Blackwell. · Zbl 1028.03003
[34] Wittgenstein, L. (1964). Philosophical remarks. Oxford: Blackwell. · Zbl 0976.01502
[35] Wittgenstein, L. (1974). Philosophical grammar. Oxford: Blackwell.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.