On Two Conceptions of Formality in 19Th Century Symbolic Logic

CLE 30/ XV EBL/ XIV SLALM Section: Foundational and Philosophical Aspects of Logic

On Two Conceptions of Formality in 19th Century Symbolic Logic

Javier Legris University of Buenos Aires and CEF, ANCBA / CONICET E-mail: [email protected]

Extended Abstract

This paper deals with two conceptions of formality underlying 19th Century symbolic logic. In both conceptions the idea of symbolic system played an important role. The discussion is guided by a distinction devised by Gottlob Frege between two sorts of formal theories. In the paper both conceptions are connected to the attempts of constructing universal scientific languages at that time. As a conclusion it will be shown that each of the two conceptions of formality places logic in different levels and determines different conceptions of universality. Along the 19th Century, different notions of formality coexisted. The nature of formal objects and, consequently, of formal theories was understood in various ways. According to these different ideas, logic and mathematic had different domains. As a consequence, the very formal nature of logic received different interpretations. In a paper from 1885, Frege maintained a particular conception of formal theories in Arithmetic. This conception was introduced by Frege in the context of his criticism of formalist positions in the mathematics of the time, like those proposed by Heine and Thomae. According to Frege, a theory for arithmetic is formal if every law of the theory is logically derived only from logical notions via definitions. Frege wrote:

“I here want to consider two views, both of which bear the name ‘formal theory’. I shall agree with the first; the second I shall attempt to confute. The first has it that all arithmetic propositions can be derived from definitions alone using purely logical means, and consequently that they also must be derived in this way”. (Frege 1885, p. 94, engl. transl., p. 141)

Frege connected this characterization explicitly with his idea of universality: The concept of number is universally applicable. It can be applied to entities of every kind. He refers in this case to a “general applicability” of this concept (see Frege 1885/1886). Some years later, in Grundgesetze, Frege characterized his conception of formal theories as contentual. For the meaning of the symbols are the real objects of the theory, the symbols being only a medium to express this meaning (see Frege 1903, p. 154). This is the sense according to which Frege constructed arithmetic in his conceptual script. Frege opposed this conception of formal theory to the idea of symbolic systems whose elements had no

1 meaning at all. They are, as Frege stated, ‘empty signs’ (see Frege 1885/1886, p. 97, engl. transl. p. 144). Now, Ernst Schröder with his program of an abstract algebra had just advocated for the latter idea of formal theory: Formal algebra could be applied to different domains, and the operations received diverse interpretations depending on the domain considered (see Schröder 1873, p233). This general applicability of formal algebra suggested a new idea of universality: Formal algebra is universal in the sense that it can be applied to every domain. So, following John Corcoran´s notion of scheme (see Corcoran 2006), in a formal theory every symbol is schematic, so that formal is not equivalent to logical. On the contrary, for Frege in a formal theory non symbol is schematic. Schröder expanded later his algebraic program by introducing the algebra of relatives, and he envisaged the idea of a universal scientific language, a pasigraphy as he called it (see Schröder 1898, p. 149). Thus, the symbolic system of the algebra of relatives turns into a language, which had the pragmatic aim of expressing every mathematical theory - and prima facie every scientific theory. The two cases of Frege and Schröder suggest that to different conceptions of formal theories correspond different conceptions of universality. For example, Frege´s characterization of formal theories as theories with a logical content is in agreement with his conceptual script as a universal language in which every expression has a fixed presupposed meaning. So, Formalism and universality turn to be strongly related notions at the end of the 19th Century.

References

Corcoran, John. 2006. “Schemata: The Concept of Schema in The History of Logic”. Bulletin of Symbolic Logic 12, 2, pp. 219-240.

Frege, Gottlob. 1885/1886. “Über formale Theorien der Arithmetik”. En Sitzungsberichte der Jeaneischen Gesellschaft für Medizin und Naturwissenschaft 12, pp. 94-104. English trans. by Eike-Henner W. Kluge in Frege, Gottlob: On the Foundations of Geometry and Formal Theories of Arithmetic. New haven – London, Yale University Press, 1971, pp. 141- 153.

Frege, Gottlob. 1903. Grundgesetze der Arithmetik Bd. III. Jena. Reprinted Hildesheim, Olms, 1966.

Schröder, Ernst. 1873. Lehrbuch der Arithmetik und Algebra. Lepizig, B. G. Teubner.

Schröder, Ernst. 1898. "Über Pasigraphie, ihren gegenwärtige Stand und die pasigraphische Bewegung in Italien". In Verhandlungen der Ersten Internationales Mathematiker-Kongresses in Zürich vom 9. bis 11. August 1897, ed. by Ferdinand Rudio. Leipzig, Teubner, pp. 147-162. Reprint: Nendeln, Kraus, 1967. English translation “On Pasigraphy. Its Present State and The Pasigraphic Movement in Italy” en The Monist 9 (1899), pp. 44-62 (corrigenda, p. 320).