Frege, Gottlob

Gottlob Frege (1848-1925) was a professional mathematician who, together with Bertrand Russell, is considered to be one of the two grandfathers of modern analytic philosophy. However, the importance of his work extends far beyond the field of philosophy. Frege first introduced the concepts of modern quantificational LOGIC (1879). Indeed, with apologies to C. S. Pierce, it is no exaggeration to call modern quantificational logic Frege's discovery. Frege was also the first to present a formal system in the modern sense in which it was possible to carry out complex mathematical investigations (cf. FORMAL SYSTEMS, PROPERTIES OF).

In addition to contemporary, second-order quantificational logic, a host of logical and semantical techniques occur explicitly for the first time in his work. In Part III of his 1879 work, he introduced the notion of the ancestral of a relation, which yields a logical characterization of one important notion of mathematical sequence; for example, the ancestral can be used to define the notion of natural number. Indeed, the ancestral provides a general technique for transforming an inductive definition of a concept into an explicit one (Dedekind was the codiscoverer of this notion). Frege's later work was also of logico-semantical significance. The "smooth-breathing" operator of his Grund-gesetze der Arithmetik, a variable binding device for the formation of complex names for extensions of functional expressions, is the inspiration for lambda abstraction. The brilliant semantic discussion in Part I, though hindered by the lack of an analysis of the consequence relation, nonetheless anticipated many future developments in logic and semantics. For instance, Frege's hierarchy of functions (see Dummett 1973: chap. 3) could be taken as the catalyst for CATEGORIAL GRAMMAR. Even the influential technique of treating two-place functional expressions as denoting functions from objects to one-place functions described in Schoenfinkel (1924) is anticipated by Frege in his discussion of the extensions of two-place functional expressions (1893: §36).

Other ideas of Frege have also had a tremendous impact on research in the cognitive sciences. Perhaps the most important of these is the distinction between sense and reference, which occurs in his 1892 paper, "On Sense and Reference," the defining article of the analytic tradition in philosophy (see SENSE AND REFERENCE for an extended discussion). In that paper, he also gave the first modern informal semantical analysis of PROPOSITIONAL ATTITUDES and introduced the notion of presupposition into the literature (though he introduced the negation test for presupposition, it is clear that he was unaware of the Projection Problem; see PRESUPPOSITION). His 1918 essay "Thoughts" contains a sophisticated discussion of indexicality (cf. INDEXICALS AND DEMONSTRATIVES). Though some of the students of BRENTANO also made distinctions like the one between sense and reference, and even had interesting discussions of indexicals and demonstratives (e.g., Husserl 1903: Book VI), none of them achieved the conceptual clarity of Frege on these topics. Furthermore, Frege's conception of thoughts as structured in a way similar to sentences is a precursor to one aspect of Fodor's (1975) LANGUAGE OF THOUGHT, though Frege's conception of the ontology of thoughts as abstract, mind independent entities, much like numbers and sets, is incompatible with a Fodorian construal of them, and indeed with much of what is said on the matter in the philosophy of mind today.

Though Frege's ideas and discoveries have clearly had a profound effect on subsequent research in philosophy, computer science, and linguistics, his life's project, logicism, is usually considered to be a failure (for an influential defense of part of Frege's version of logicism, see Wright 1983). Logicism is the doctrine that arithmetic is reducible to logic. Frege announced this project in his Foundations of Arithmetic, which contained the most sophisticated discussion of the concept of number in the history of philosophy, together with an informal description of how the logicist program could be carried out. In his Magnum Opus, Grundgesetze der Arithmetik (Basic Laws of Arithmetic) (1893, 1903), Frege tried to carry out the logicist program in full detail, attempting to derive the basic laws of arithmetic, and indeed analysis, within a formal system whose axioms he believed expressed laws of logic. Unfortunately, the theory was inconsistent. This discovery, by Bertrand Russell, devastated Frege, and essentially ended his career as a mathematician. Recent research has shown, however, that there is a great deal of interest that is salvageable from his mathematical work (Wright 1983; and the essays in Demopoulos 1995).

Frege is not merely of historical interest for the student of cognitive science. Rather than being interested in how we in fact reason, he is interested in how we ought to reason, and rather than being interested in the biological component of mentality, he is interested in the abstract structure of thought. Studying his works provides a useful curative for those who need to be reminded about the public and normative aspects of the notions that concern cognitive science.

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-- Jason C. Stanley

References

Demopoulos, W., Ed. (1995). Frege's Philosophy of Mathematics. Cambridge, MA: Harvard University Press.

Dummett, M. (1973). Frege: Philosophy of Language. London: Duckworth.

Fodor, J. (1975). The Language of Thought. New York: Thomas Crowell.

Frege, G. (1879). Begriffsschrift: a formula language, modeled upon that of arithmetic, for pure thought. In Jean Van Heihenoort, Ed., 1967, From Frege to Goedel: A Source Book in Mathematical Logic, 1879-1931. Cambridge: Harvard University Press, pp. 5 - 82.

Frege, G. (1884/1980). The Foundations of Arithmetic. Evanston, IL: Northwestern University Press.

Frege, G. (1891). On Sense and Reference. In Peter, Geach, and Max Black, Eds., (1993). Translations from the Philosophical Writings of G Frege. Oxford: Blackwell, pp. 56-78 (there translated as "On Sense and Meaning").

Frege, G. (1893, 1903/1966). Grundgesetze der Arithmetik. Hildesheim: Georg Olms Verlag.

Frege, G. (1918). Thoughts. In Brian McGuiness, Ed., 1984, Collected Papers. Oxford: Blackwell, pp. 351-372.

Frege, G. (1984). Collected Papers. Brian McGuiness, Ed. Oxford: Blackwell.

Husserl, E. (1903/1980). Logische Untersuchungen. Tuebingen: Max Niemeyer.

Schoenfinkel, M. (1924). On the building blocks of mathematical logic. In J. Van Heihenoort, Ed., 1967, From Frege to Goedel: A Source Book in Mathematical Logic, 1879-1931. Cambridge, MA: Harvard University Press, pp. 357 - 366.

Wright, C. (1983). Frege's Conception of Numbers as Objects. Aberdeen: Aberdeen University Press .