Zigzag and Fregean Arithmetic
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In Frege’s logicism, numbers are logical objects in the sense that they are extensions of certain concepts. Frege’s logical system is inconsistent, but Richard Heck showed that its restriction to predicative (second-order) quantification is consistent. This predicative fragment is, nevertheless, too weak to develop arithmetic. In this paper, I will consider an extension of Heck’s system with impredicative quantifiers. In this extended system, both predicative and impredicative quantifiers co-exist but it is only permissible to take extensions of concepts formulated in the predicative fragment of the language. This system is consistent. Moreover, it proves the principle of reducibility applied to concepts true of only finitely many objects. With the aid of this form of reducibility, it is possible to develop arithmetic in a thoroughly Fregean way.
I would like to thank the invitation of Argument Clinic (an association of students of Philosophy at the University of Lisbon) for inviting me to give a presentation at the conference Principia Mathematica (1913–2013), held at the University of Lisbon in February 6–7, 2014. I came with the main idea of this paper while preparing a talk for this conference. Afterwards, I also had the chance to speak about the issues of this paper at the meeting “2014: Abstractionism/Neologicism” (University of Storrs, Connecticut, U.S.A.), at “Journée sur les Arithmétiques Faibles 33” (University of Gothenburg, Sweden) and at the conference “The Philosophers and Mathematics” (University of Lisbon, Portugal). I want to thank the organizers of these meetings (Marcus Rossberg, Ali Enayat and Hassan Tahiri, respectively), as well as the participants for their remarks (specially to Marco Panza, for detailed comments and questions). My final thanks are to an anonymous referee. In a preliminary version of this paper, Section 4.4 used a definition of finiteness due to Paul Stäckel in 1907 (for a reference see Parsons 1987). The referee complained about the artificiality of using this definition. We reformulated the section using a characterization of finiteness given by Frege in volume I of the Grundgesetze der Arithmetik (1893). I believe that this change made the paper more natural and tighter.
- Burgess, J. (2005). Fixing Frege. Princeton: Princeton University Press.Google Scholar
- Dummett, M. (1991). Frege. Philosophy of Mathematics. Cambridge, MA: Harvard University Press.Google Scholar
- Frege, G. (1967). Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought. In J. van Heijenoort (ed.), From Frege to Gödel (pp. 5–82). Harvard: Harvard University Press. A translation of Frege’s Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, which appeared in German in 1879. Translated by J. van Heijenoort.Google Scholar
- Frege, G. (1980). The Foundations of Arithmetic. Evanston: Northwestern University Press. A translation of Frege’s Die Grundlagen der Arithmetik, which appeared in German in 1884. Translated by J. L. Austin.Google Scholar
- Frege, G. (2013). Basic Laws of Arithmetic. Oxford: Oxford University Press. A translation of Frege’s two volumes of the Grundgesetze der Arithmetik, which appeared in German in 1893 and 1903. Translated and edited by P. A. Ebert and M. Rossberg with a foreword of Crispin Wright.Google Scholar
- Gödel, K. (2005). What is Cantor’s continuum problem. In P. Benacerraf and H. Putnam (eds.), Philosophy of Mathematics (selected readings) (pp. 470–485). Cambridge: Cambridge University Press (This is a revised and expanded version of a paper first published in 1947).Google Scholar
- Heck, R. (1998). The finite and the infinite in Frege’s Grundgesetze der Arithmetik. In M. Schirn (Ed.), The Philosophy of Mathematics Today (pp. 429–466). Oxford: Clarendon Press.Google Scholar
- Quine, W. O. (1963). Set Theory and its Logic. Harvard: Harvard University Press.Google Scholar
- Russell, B. (1973). On some difficulties in the theory of transfinite numbers and order types. In D. Lackey (Ed.), Essays in Analysis (pp. 135–164). Sydney: George Allen and Unwin Ltd. (This paper was first published in 1906.)Google Scholar
- Russell, B. (1993). Introduction to Mathematical Philosophy. New York: Dover Publications. (First published in 1919.)Google Scholar
- Russell, B., & Whitehead, A. N. (1927). Principia Mathematica (2nd ed.). Cambridge: Cambridge University Press.Google Scholar