Abstract
This paper deals with conceptions of formality underlying 19th century symbolic logic, where notations and manipulation of signs played an important role. It is devoted specifically to the case of Ernst Schröder’s “formal algebra”, which extended with the algebra of relatives (as developed by C.S. Peirce) constituted the basis for a Pasigraphy as a universal notation system. The discussion will begin with the well-known distinction devised by Gottlob Frege between two sorts of formal theories. In the paper, both conceptions of formality will be connected with the corresponding attempts of constructing universal scientific notations (Schröder’s Pasigraphy and Frege’s Begriffsschrift). It will be shown that the Pasigraphy was an interpretation of that formal algebra. As a further conclusion, it will be suggested that each of the two conceptions of formality places logic in different levels and determines different conceptions of universality.
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Notes
- 1.
Thommae, together with Heinrich Eduard Heine and Hermann Hankel among others, would represent the school of “old formalism”, different from the later formalism of Hilbert’s school.
- 2.
Ignacio Angelelli called this view “philosophical logicism”: Arithmetic would deal with the more general features of reality, that is, these features that are in every domain. In this sense it could be compared with ontology or logic (see [1, p. 244]).
- 3.
In fact, Frege mainly refers to Alwin R. Korselt and his paper “Über die Grundlagen der Geometrie” (see [7, p. 281]). Korselt had worked on Schröder’s algebra of logic.
- 4.
This is essentially what Detlefsen called “Frege’s Problem” (see [3, pp. 9 ff]).
- 5.
In the Pasigraphy-paper, the algebra of relatives has a privileged interpretation in set theory and Schröder proposed a set-theoretic reconstruction of arithmetic. Therefore, it can be suggested that the formal algebra has an intended interpretation, being then a language in a strict sense. However, the distinction between formal system and possible interpretations remains. For a further discussion of this problem, see [13, pp. 597 ff].
- 6.
Thommae also referred to “schemes” in his aforementioned book: real numbers should be understood as “pure schemes without content”.
- 7.
Frege made use of the notion of schema in his criticism on Korselt’s idea of formal system (see [7, p. 304]).
References
Angelelli, I.: Studies on Gottlob Frege and Traditional Philosophy. Dordrecht, Reidel (1967)
Corcoran, J.: Schemata: the concept of schema in the history of logic. Bull. Symb. Log. 12, 219–240 (2006)
Detlefsen, M.: Hilbert’s Program. An Essay on Mathematical Instrumentalism. Dordrecht, Reidel (1986)
Dutilh Novaes, C.: The different ways in which logic is (said to be) formal. Hist. Philos. Logic 32, 303–332 (2011)
Frege, G.: Über formale Theorien der Arithmetik. Sitz.ber. Jeaneischen Ges. Med. Nat.wiss. 12, 94–104 (1885/1886). Reprinted in Frege G.: Kleine Schriften, 2nd edn. (1990) Edited by Ignacio Angelelli. Olms, Hildesheim–Zürich–New York, Olms, pp. 103–111
Frege, G.: Grundgesetze der Arithmetik, vol II. Jena (1903). Reprinted Hildesheim, Olms, 1966
Frege, G.: Über die Grundlagen der Geometrie. Jahresber. Dtsch. Math.-Ver. 15, 293–309 (1906). Reprinted in Frege 1990, pp. 281–323
Hodges, W.: A formality. CD Festschrift for the 50th birthday of Johan van Benthem, Vossiuspers AUP (1999). www.aup.nl
Kant, I.: KrV. Kritik der reinen Vernunft, 1st edn. 1781, 2nd edn. 1787. In: Weischedel, W. (ed.) Werke in zwölf Bänden, vol. III. Suhrkamp, Frankfurt (1968)
Legris, J.: Deux approches des relations logique-mathématiques Frege et Schröder. In: En Justifier en Mathématiques, comp. por Dominique Flament y Philippe Nabonnand, pp. 215–254. Editions de la Maison des sciences de l´homme, Paris (2011)
Legris, J.: Universale Sprache und Grundlagen der Mathematik bei Ernst Schröder. In: Mathematik—Logik—Philosophie. Ideen und ihre historischen Wechselwirkungen, comp. por Günther Löffladt. pp. 255–269. Harri Deutsch, Frankfurt am Main (2012)
MacFarlane, J.: What does it mean to say that logic is formal? PhD Dissertation, University of Pittsburgh (2000)
Peckhaus, V.: Schröder’s logic. In: Gabbay, D.M., Woods, J. (eds.) Handbook of the History of Logic. Vol. 3. The Rise of Modern Logic: From Leibniz to Frege, pp. 557–609. Elsevier-North Holland, Amsterdam (2004)
Schröder, E.: Lehrbuch der Arithmetik und Algebra. Teubner, Lepizig (1873)
Schröder, E.: Review of Begriffsschrift by Gottlob Frege. Z. Math. Phys. 25, 81–94 (1880)
Schröder, E.: Über Pasigraphie, ihren gegenwärtige Stand und die pasigraphische Bewegung in Italien. In: Rudio, F. (ed.) Verhandlungen der Ersten Internationales Mathematiker-Kongresses in Zürich vom 9. bis 11. August 1897. The Monist, vol. 9, pp. 147–162. Teubner, Leipzig (1898). Reprint Nendeln, Kraus, 1967, English translation: on Pasigraphy. Its Present State and The Pasigraphic Movement in Italy en The monist 9, 1898, pp. 44–62 (corrigenda, p. 320)
van Heijenoort: Logic as calculus and logic as language. Synthese 24, 324–330 (1967)
Acknowledgements
It is a truism that the idea of universality is essential to Jean-Yves Beziau’s work. In some respect, Jean-Yves devotes his life to this idea. The following historical comments can be interesting in connection with it. This paper is a result of a research project supported by the Consejo Nacional de Investigaciones Científicas y Técnicas from Argentina (PIP 11220080101334, CONICET). It also benefited from the additional financial support of the Alexander von Humboldt Foundation. I wish to thank Volker Peckhaus for helpful and valuable comments on an earlier draft of this paper.
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Legris, J. (2015). On Universality and Formality in 19th Century Symbolic Logic: The Case of Schröder’s “Absolute Algebra”. In: Koslow, A., Buchsbaum, A. (eds) The Road to Universal Logic. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-10193-4_16
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