Frege, Gottlob

▪ German mathematician and philosopher

Introduction

born November 8, 1848, Wismar, Mecklenburg-Schwerin

died July 26, 1925, Bad Kleinen, Germany

 German mathematician and logician (logic, history of), who founded modern mathematical logic (formal logic). Working on the borderline between philosophy and mathematics—viz., in the philosophy of mathematics and mathematical logic (in which no intellectual precedents existed)—Frege discovered, on his own, the fundamental ideas that have made possible the whole modern development of logic and thereby invented an entire discipline.

      Frege was the son of Alexander Frege, a principal of a girls' high school in Wismar. His mother, Auguste Frege, née Bialloblotzky, who was perhaps of Polish origin, outlived her husband, who died in 1866. Frege entered the University of Jena in 1869, where he studied for two years, and then went to the University of Göttingen for a further two—in mathematics, physics, chemistry, and philosophy. Frege spent the whole of his working life as a teacher of mathematics at Jena: he became a Privatdozent in May 1871, was made an ausserordentlicher Professor (associate professor) in July 1879, and became statutory professor of mathematics in May 1896. He lectured in all branches of mathematics (though his mathematical publications outside the field of logic are extremely few) and also on his own logical system. A great many of his publications, however, were expressly philosophical in character: he himself once said, “Every good mathematician is at least half a philosopher, and every good philosopher at least half a mathematician.” He kept aloof from his students and even more aloof from his colleagues.

      Though Frege was married, his wife died during World War I, leaving him no children of his own. There was an adopted son, Alfred, however, who became an engineer.

      Frege was, in religion, a liberal Lutheran and, in politics, a reactionary. He had a great love for the monarchy and for the royal house of Mecklenburg, and during World War I he developed an intense hatred of socialism and of democracy, to which he came to ascribe the loss of the war and the shame of the Treaty of Versailles. A diary kept at the end of his life reveals, as well, a loathing of the French and of Catholics and an anti-Semitism extending to a belief that the Jews must be expelled from Germany.

      Frege had a vivid awareness of his own genius and a belief that it would one day be recognized; but he became increasingly embittered at the failure of scholars to recognize it during his lifetime. He delighted in controversy and polemic; but the originality of his own work, the almost total independence of his own ideas from other influences, past or present, was quite exceptional and, indeed, astonishing.

System of mathematical logic.

      In 1879 Frege published his Begriffsschrift (“Conceptscript”), in which, for the first time, a system of mathematical logic in the modern sense was presented. No one at the time, however—philosopher or mathematician—comprehended clearly what Frege had done, and when, some decades later, the subject began to get under way, his ideas reached others mostly as filtered through the minds of other men, such as Peano (Peano, Giuseppe); in his lifetime there were very few—one was Bertrand Russell (Russell, Bertrand)—to give Frege the credit due to him. He was not yet too downcast by the failure of the learned world to appreciate the Begriffsschrift, which, after all, discourages the reader by the use of a complex and unfamiliar symbolism to express unfamiliar ideas. He resolved, however, to compose his next book without the use of any symbols at all.

      There followed a period of intensive work on the philosophy of logic and of mathematics, embodied initially in his first book, Die Grundlagen der Arithmetik (1884; The Foundations of Arithmetic). The Grundlagen was a work that must on any count stand as a masterpiece of philosophical writing. The only review that the book received, however, was a devastatingly hostile one by Georg Cantor (Cantor, Georg), the mathematician whose ideas were the closest to Frege's, who had not bothered to understand Frege's book before subjecting it to totally unmerited scorn.

      Wounded by the reception of his second book, Frege nevertheless devoted the next decade to producing a series of brilliant philosophical articles in which he elaborated his philosophy of logic. These articles contain many deep insights, although, as Frege systematized his theories, there appeared a certain hardening into a kind of scholasticism. There followed a return to the philosophy of mathematics with the first volume of Grundgesetze der Arithmetik (1893; partial Eng. trans., Basic Laws of Arithmetic), in which Frege presented, in a modified version of the symbolic system of the Begriffsschrift, a rigorous development of the theory of Grundlagen. This, too, received only a single review (by Peano). The neglect of what was to have been his chef d'oeuvre finally embittered Frege, who had complained, in the preface, of the apparent ignorance of his work on the part of writers working in allied fields. The resulting bitterness shows in the style of Frege's controversial writing. Seldom has criticism of previous writers been more deadly than in his Grundlagen; but it is expressed with a lightness of touch and is never unfair. In volume 2 of the Grundgesetze (1903), however, the attacks became heavyhanded and abusive—a means of getting back at the world that had ignored him.

Contradictions in Frege's system.

      A worse disaster than neglect, however, was in store for him. While volume 2 of the Grundgesetze was at the printer's, he received on June 16, 1902, a letter from one of the few contemporaries who had read and admired his works—Bertrand Russell. The latter pointed out, modestly but correctly, the possibility of deriving a contradiction in Frege's logical system—the celebrated Russell paradox (Russell's paradox). The two exchanged many letters; and, before the book was published, Frege had devised a modification of one of his axioms intended to restore consistency to the system. This he explained in an appendix to the book. After Frege's death, it would be shown by a Polish logician, Stanisław Leśniewski (Leśniewski, Stanisław), that Frege's modified axiom still leads to contradiction. Probably Frege never discovered this. Even a brief inspection, however, of the proofs of the theorems in volume 1 would have revealed that several crucial proofs would no longer go through, and this Frege must have found out.

      In any case, 1903 effectively marks the end of Frege's productive life. He never published the projected third volume of the Grundgesetze, and he took no part in the development of the subject, mathematical logic, that he had founded, though it had progressed considerably by the time of his death. He published a few polemical pieces; but, with the exception of three essays in the philosophy of logic produced after the end of the war, he did no further creative work. In 1912 he declined, in terms expressing deep depression, an invitation by Russell to address a mathematical congress in Cambridge.

      At the very end of Frege's life, he again started to work on the philosophy of mathematics, having arrived at the conclusion that one of the fundamental bases of his earlier work—the attempt to found arithmetic on logic—had been mistaken; but the work did not progress very far and was not published.

      Up to an advanced age, Frege hiked every summer in Mecklenburg, his native region. He finally retired during World War I and went to live in Bad Kleinen, in Mecklenburg.

Influence of Frege's work.

      Frege's work represents the beginning of modern logic because of his invention of the notation of quantifiers (quantification) and variables. (In natural language, generality is represented by inserting an expression like “everything” or “something” in the argument-place of the predicate; in the notation used in logic since Frege, the argument-place is filled by a variable letter, say x, and the resulting expression prefixed by a quantifier, “For every x” or “For some x,” said to “bind” that variable.) By means of this notation he solved the problem that had baffled the logicians of the Middle Ages and prevented the further advance of logic ever since, viz., the analysis of sentences involving multiple generality. In him there also appeared the first clear separation between the formal characterization of logical laws and their semantic justification. His philosophical work is of an importance far more general than the area to which he principally applied it, the philosophy of mathematics: he initiated a revolution, in fact, as profound as that of René Descartes in the 17th century. Whereas Descartes had made epistemology the starting point for all philosophy, Frege gave this place to the theory of meaning or the philosophy of language. His work has been influential because he made the restricted part of philosophy in which he worked basic to all the rest. The effect was imparted in the first place, however, through the work of others, particularly that of Wittgenstein (Wittgenstein, Ludwig), who visited him in 1914 and who revered him. But, since John Austin's translation of the Grundlagen into English in 1950, the direct influence of Frege's writing among English-speaking philosophers has been very great. No one supposes that Frege said the last word on any topic; but there is scarcely a live question in contemporary philosophy of language for whose examination Frege's views do not form at least the best starting point.

Michael A.E. Dummett

Additional Reading

Works.

The principal English translations of Frege's works are: J.L. Austin (trans.), The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, 2nd ed. rev. (1959), a two-language version; Peter Geach and Max Black (eds.), Translations from the Philosophical Writings of Gottlob Frege, 2nd ed. (1960); Montgomery Furth (ed. and trans.), The Basic Laws of Arithmetic (1964); J. van Heijenoort (ed. and trans.), “Begriffschrift,” in From Frege to Gödel (1967); and A. and M. Quinton (trans.), “The Thought,” in Peter F. Strawson (ed.), Philosophical Logic (1967).

Critical sources.

E.D. Klemke (ed.), Essays on Frege (1968), contains recent work on Frege. See also Ludwig Wittgenstein, Tractatus Logicophilosophicus (1922; Eng. trans. 1961) and Philosophische Untersuchungen (written 1936–49; Philosophical Investigations, 1953); Bertrand Russell, “On Denoting,” Mind, 14:479–493 (1905, reprinted in Logic and Knowledge, ed. by R.C. Marsh, 1956); and Michael Dummett, “Frege, Gottlob,” Encyclopedia of Philosophy, vol. 3 (1967) and The Interpretation of Frege's Philosophy (1980).

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