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Gottlob Frege

Gottlob Frege

(1848-1925) was a German mathematician and logician as well as a philosopher. He is considered the major founder of modern . His uniquely profound research led him from pure mathematics, like the methodizing of natural numbers or “the proof,” to a highly philosophical approach to mathematical logic and the foundation of mathe­matics, and even to consequential findings in the field of linguistics. Applicable to most fields of logic, Frege’s research exerted an influence far beyond pure mathematical logic. He is considered to have advanced logic beyond Aristotle, despite the fact that he failed in constructing a consistent axiomatic foundation for logic. Consequently, Frege holds a significant place in the historical development of logic and mathematics from ancient Greece to our modern times.

Life and Work

F. L. Gottlob Frege was born on November 8, 1848, in Wismar, northern Germany, in the state of Mecklenburg. He was the son of Alexander Frege, the principal of a private high school in Wismar. It is probable that the influence of his teacher at the local gymnasium, Leo Sachse, motivated Frege to study after school. Subsequently he enrolled as a student in mathematics, chemistry, physics, and ; first from 1869 to 1871 at the University of Jena under the encouragement of the famous physicists Ernst Abbe and Karl Snell; next he expanded his studies for 2 more years at Göttingen University, where he eventually obtained a doctoral degree in 1873 with the valued geometrical thesis translated as On a Geometrical Representation of Imaginary Figures in a Plane. Just one year later, Frege received his second doctorate (habilitation) for his work Methods of Calculation Based Upon an Amplification of the Concept of Magnitude, comprehending some initial steps of his theory of higher (complex) mathematical functions.

Also in the year 1874, upon recommendation of his academic teacher Abbe, Frege became a lecturer in mathematics at the University of Jena, where he would remain all his professional life. He taught extensively in all mathematical disciplines. His research, however, concentrated on the philosophy of logic. Persistent dialogue—noteworthy because Frege was extremely reserved in general—with his Jena colleague and one of his few friends Rudolf Eucken, later a winner of the Nobel Prize in litera­ture, supported Frege’s philosophically mathematic thinking.

In 1879, Frege’s seminal first work Concept Script, a Formal Language of Pure Thought Modelled Upon That of Arithmetic () was published. He developed a principle for the construction of a logical language. In the after­math of the Begriffsschrift, Frege was made associ­ate professor at Jena University. Frege married Margarete Lieseberg, but unfortunately neither of their two children survived into adulthood, so they adopted a boy named Alfred.

Frege’s book (Grundlagen) appeared in 1884. It comprised for the first time a complete system for the foundation of arithmetic based on a set of mathematically logical axioms. To gain higher recognition for his work, this book was written in completely non­technical, natural language.

In conjunction with his profound research into the logical system of mathematics, Frege felt impelled to develop a . His major work on a linguistic system supporting the philosophy of logic is On (1892). In this book, Frege’s two famous linguistic puzzles were presented, distinguishing between sense and the denotation of terms in order to resolve the ambiguities of language.

In 1893, with the first volume of Basic Laws of Arithmetic (Grundgesetze), Frege’s major opus on the philosophy of mathematics was published. Frege used to say that “every good mathematician is at least half a philosopher, and every good phi­losopher is at least half a mathematician.” Because almost no colleague of his in the world was able to understand what Frege had managed to find, none of his publications achieved immediate success, and he even received some very poor reviews.

Nonetheless, Frege was promoted to honorary professor in Jena in 1896, giving him a regular income for the first time in his life. Years of bad luck followed. In 1902, Frege received a letter from Bertrand Russell who modestly pointed out that he had discovered a paradox caused by a severe inconsistency in Frege’s set of logical axi­oms. Also, the second volume of the Basic Laws was based on the misarranged axioms but was already finished and was (at that exact time) with the printer. Unfortunately, even the amendment to his axiomatic system that Frege added as an appendix proved to be inconsistent; as some say, Frege must have known this but was unable to accept his failure. Frege gave up writing the intended third volume of the Basic Laws and never again published any research. His late attempt to base logic on geometry instead of arithmetic could not be elaborated any further.

During the First World War, Frege retired and after the death of his wife he left Jena; he lived in increasing reclusiveness and died on July 26, 1925, in Bad Kleinen, Germany.

Influence

Frege left deep traces in two fields of philosophi­cal research: in the logic of mathematics and in the philosophy of language. In the first, he marks the beginning of modern science. His invention of quantified variables (predicate calculus) to replace the ambiguous meaning of natural language not suitable for the denotation of complex mathemat­ical terms was seminal to the development of mathematics. His influence arises not only from separate theorems Frege published himself, but also is founded on the diffusion of his very ele­mentary logic and philosophical findings into countless studies.

Frege’s analytical philosophy is connected with Locke’s and Hume’s empiricism. In a far-reaching empiricist manner and as a central point in his logic, which is founded on the sole relevance of the logical truth of an argument, Frege did not accept any non- falsifiable fact or instance; hence he was in deep doubt about transcendental phenomena, and epochs do not seem to have attracted his attention.

Personality

For a man in determined pursuit of new scientific findings, Frege was extremely conservative in his political attitude. Especially in his later years, the embittered Frege was an enthusiastic monarchist, anti-democrat, anti-French, anti-Catholic, and anti-Semite.

Frege displayed signs of immoderate self-regard and refused to accept or even to consider the criti­cism of his colleagues. To the contrary, he reacted with embitterment and polemic attacks on his crit­ics, which seems an inadequate response in light of Frege’s undoubtedly immense and lasting scientific achievements in the fields of logic and the philoso­phy of mathematics and language.

Matthias S. Hauser

See also Aristotle; Hume, David; Language

Further Readings

Beaney, M. (Ed.). (1997). The Frege reader. Oxford, UK: Blackwell.

Dummett, M. (1991). Frege: Philosophy of mathematics. Cambridge, MA: Harvard University Press.

Sluga, H. (1980). Gottlob Frege. London: Routledge & Kegan Paul.

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