## Gottlob Frege

**-**

## Frege, Friedrich Ludwig Gottlob

# FREGE, FRIEDRICH LUDWIG GOTTLOB

(*b*. Wismar, Germany, 8 November 1848; *d*. Bad Kleinen, Germany, 26 July 1925),

*mathematics*, *logic*, *foundations of mathematics*. For the original article on Frege see *DSB*, vol. 5.

Although a mathematician, Gottlob Frege is regarded as one of the founding fathers of modern (analytical) philosophy. With his *Begriffsschrift* (concept script) of 1879 he created modern mathematical logic. He used it as a linguistic tool for a program of founding mathematical concepts exclusively on logical concepts (logicism). Frege was involved in controversies with representatives of the algebraic tradition in logic concerning the power of the different systems of symbolic logic, and with David Hilbert on the nature of mathematical axiom systems.

**Frege in Jena** . Frege spent most of his academic life at the University of Jena, except for five semesters of studies in Göttingen (1871–1873). He took courses in mathematics, physics, chemistry, and philosophy. Among his most important teachers in Jena were Karl Snell, who followed Jakob Friedrich Fries as chair of mathematics and physics, and Ernst Carl Abbe, at that time privatdozent for mathematics. Abbe became Frege’s mentor. He encouraged Frege to transfer to Göttingen in order to complete his university studies and later supported him in his career.

Back in Jena, Frege applied for the post of a privatdozent for mathematics, submitting as *Habilitationsschrift*“Methods of Calculation Based on an Extension of the Concept of Quality,” which contributed to the theory of functional equations, in particular iteration theory. Abbe initiated Frege’s promotion to außerordentlicher Professor (roughly equivalent to associate professor or reader) of mathematics in 1879. This early promotion was possible because Frege had published his first monograph, *Begriffsschrift*, in January of that year. In 1866 Abbe had become a scientific consultant for improving the construction of microscopes built by the Carl Zeiss optics industry. In 1875 he became an associate and limited partner of Zeiss. Abbe set up the Carl Zeiss Foundation (1889) by first establishing a fund for scientific purposes (Ministerialfond für wissenschaftliche Zwecke) in 1886, supporting teaching and research in mathematics and sciences at the University of Jena. Abbe’s foundation also made an improvement in Frege’s remuneration possible, and later financially supported his promotion to *ordentlicher Honararprofessor* (a payroll professorship in honor of the person) in 1896.

In 1907 Frege was awarded the prestigious title of *Hofrat*. His growing reputation is indicated by Ludwig Wittgenstein’s visit in 1911 (further visits took place in 1912 and 1913). In summer semester 1913 Rudolf Carnap attended Frege’s course Begriffsschrift II, and another course, Logic in Mathematics, in 1914. In 1918 Frege retired after having been on sick leave for a year. He moved to Bad Kleinen, a resort near Wismar. On the initiative of Heinrich Scholz, Frege’s *Nachlass* (literary estate) was transferred in 1935 to Münster, where it was purportedly destroyed during a bomb raid on Münster on 25 March 1945.

**Logic** . Frege’s publication of the *Begriffsschrift* is regarded in the early twenty-first century as “the single most important event in the development of modern logic” (Thiel and Beaney, p. 26). In this work, Frege created the first strict logical calculus in the modern sense, based on precise definitions of expressions and deduction rules arriving for the first time at an axiomatic development of classical quantification theory. Frege replaced the traditional analysis of elementary statements into subject and predicate with an analysis of a proposition into function and argument, which could be used to express the generality of a statement (and with this also existence statements) by bound variables and quantifiers.

It can be assumed that Frege took over the term *Begriffsschrift* from Friedrich Adolf Trendelenburg’s characterization of Gottfried Wilhelm Leibniz’s general characteristics (1854). The term had, however, already been used by Wilhelm von Humboldt in a treatise (1824) on the letter script and its influence on the construction of language.

In a lengthy review of the *Begriffsschrift* (1880), Ernst Schröder accused Frege of ignoring George Boole’s algebra of logic, first presented in *The Mathematical Analysis of Logic* (1847). Frege answered in articles (published only posthumously) by comparing Boole’s calculatory logic with his own, where he determined quantification theory as the main point of deviation (Frege, 1880–1881, 1882). It is indeed historically true that the Booleans had no quantification theory at that time, but this cannot be regarded as an essential difference between these variations of symbolic logic on a systematic level, because in 1883 the U.S. logician Charles S. Peirce and his student Oscar Howard Mitchell developed an almost equivalent quantification theory within the algebra of logic. The essential difference between the algebra of logic and Frege’s mathematical logic can thus be seen in different interpretations of the judgment. Another essential difference can be seen in the fact that Frege aimed at giving a logical structure of judgeable contents, which implied an inherent semantics. The Booleans, in contrast, were interested in logical structures themselves, which could be applied in different domains. Their systems allowed various interpretations. This required a supplementary external semantics.

**Logicism** . Frege’s work was above all devoted to investigations on the nature of number. It was, thus, essentially philosophical. There is evidence that he was influenced by the philosophy of his contemporaries, especially by neo-Kantian approaches. These influences found their way into Frege’s philosophy of mathematics with its metaphysical qualities.

Contrary to Immanuel Kant, who regarded mathematical (arithmetical and geometrical) propositions as examples for synthetic a priori propositions, that is, propositions that are not empirical, but enlarging knowledge, Frege wanted to prove that arithmetic could completely be founded on logic, that is, that each arithmetical concept, in particular the concept of number, could be derived from logical concepts. Arithmetic was, thus, analytical.

The logicist program is only sketched in the *Begriffsschrift*, where Frege gave purely logical definitions of equinumerousity and the successor relation. In his next book, the *Grundlagen der Arithmetik* (1884). Frege formulated the classical logicistic definition of number, according to which the number *n* is defined as the extension of the concept “equinumerous to the concept F_{n}” with F_{n}standing for a concept with exactly *n* objects falling under it. The series of F_{n}s starts with F_{0} = ¬x = x. F_{n+1}can be reconstructed recursively from F_{n}. The number 0 is defined as the extension of the concept “equinumerous to the concept ‘different from itself,’” and the number 1 as the extension of the concept “equinumerous to the concept ‘equals 0.’” The purely logical foundation of mathematics should not only disprove the Kantian paradigm, but also refute empiricist approaches to mathematics such as the one advocated by John Stuart Mill, and with this psychological interpretations of numbers as mental constructions. Frege pointedly expressed this criticism of psychologism in his harsh review of Edmund Husserl’s *Philosophie der Arithmetik* in 1894, as a result of which Husserl was brought to revise the foundational program of phenomenology and convert to antipsychologism.

Frege elaborated the logicistic program in the two volumes of the *Grundgesetze der Arithmetik* (1893/1903). In this last of his monographs Frege also presented the mature version of his ontology, developed earlier in the three papers “Function and Concept” (1891), “On *Sinn* und *Bedeutung*” (1892), and “Concept and Object” (1892), all three currently regarded as classic texts of analytical philosophy. There he further elaborated his earlier distinction between concept and object. In particular he introduced in the *Grundgesetze* value-ranges considered as a special kind of objects. The identity criterion is given in Basic Law V, according to which the value-ranges of two functions are identical if the functions coincide in their values for every argument, with this giving the modern abstraction schema. In terms of concepts the law says that whatever falls under the concept *F* falls under the concept *G* and vice versa, if and only if the concepts *F* and *G* have the same extension.

Frege’s conception of logicism failed, as Frege himself diagnosed, because of Basic Law V. Frege suggested an ad hoc solution forbidding that the extension of a concept may fall under the concept itself. This solution was proved to be insufficient by Stanis≠aw LeŚniewski (1939, unpublished) and Willard Van Orman Quine in 1955, but it indicates that the logical form of Basic Law V may be innocent of the emergence of the paradox and that the formation rules for function names may be too liberal in allowing impredicative function names.

In his latest publications Frege gave up logicism. He abandoned the talk of extensions of concepts and value-ranges, and the idea of numbers as logical objects, although he still held that they are objects of some other kind, based on the source of “geometrische Erkenntnisquelle,” that is, pure intuition.

Frege’s logicist program was later revived by the proponents of Frege-arithmetic and neologicism. In some of these directions Basic Law V is replaced by Hume’s principle, according to which two concepts *F* and *G* have the same number if and only if they are equinumerous, that is, if there is a one-to-one correspondence between the *F*’s and *G*'s.

**The Nature of Axiomatics** . After the failure of his logicistic program, Frege focused his research on geometry as the foundational discipline of mathematics. He kept the traditional understanding of geometry as an intuitive discipline, thereby opposing David Hilbert’s new formalistic approach to geometry that came along with a new kind of axiomatics. Frege’s opposition had its prehistory in his criticism of the older arithmetical formalism presented in a paper “Über formale Theorien der Arithmetik” (1885), taken up again in papers against his Jena colleague in mathematics, Carl Johannes Thomae (1906/1908). In these papers Frege opposes the understanding of arithmetic as a purely formal game with calculations bare of any contents. The older formalism regards arithmetic as a game like chess. It starts from certain initial formulas, then derives new formulas using a fixed set of transformation rules. But, neither the initial formulas nor the transition rules are justified, so the derived formulas are not justified either. Therefore, Frege concludes, these approaches could not provide any contribution to the foundations of arithmetic.

Hilbert overcame the traditional conception of axiomatics according to the model of Euclid’s *Elements* by giving an example. In his *Grundlagen der Geometrie* of 1899 he gave an axiomatic presentation of Euclidean geometry. Hilbert’s system proceeds from “thought things” in the Kantian sense, products of the human mind, but empty concepts because of lacking any element of (empirical) intuition. The geometrical concepts were not directly defined, but implicitly gained as concepts obeying the features set by some group of axioms, and justified by proving the independence of the axioms from one another, the completeness of the system, and its consistency. The formalistic approach aims at a theory of structures. This is pointedly expressed in Hilbert’s letter to Frege of 29 December 1899, in which Hilbert claimed that every theory is only a half-timbering or schema of concepts and implications with arbitrary basic elements. If instead of the system of points some system love, law, chimney sweep is thought, and if all axioms are regarded as relations between these elements, then all theorems, for example Pythagoras’s theorem, would be valid for them.

In a letter sent two days earlier, Frege had correctly criticized Hilbert’s use of implicit definitions arguing that he had blurred the differences between axioms and definitions. It became clear that Frege stuck to the traditional (Aristotelian) understanding of axioms in geometry, calling axioms sentences that are true but not proved, because they have emerged from a source of knowledge completely different from the logical, a source that can be called spatial intuition. From the truth of the axioms follows that they do not contradict each other, so no consistency of proof was needed. Hilbert answered that if the arbitrarily set axioms do not contradict each other with all their implications, then they are true and the defined objects exist. For Hilbert consistency (logical possibility) is, thus, the criterion of truth and existence.

Hilbert rejected Frege’s suggestion to publish the exchange of letters, so Frege took up his criticism in a series of papers published in 1903 and 1906 on the foundations of geometry, where he argued for his antiformalistic position. He demanded that after having defined the concept “point,” it should be possible to determine whether a certain object, for example, his pocket watch, was a point or not.

The two controversies mentioned show that Frege followed that traditional understanding of philosophy as all-embracing fundamental discipline that formed, along with logic, logical ontology, and epistemology the foundation for mathematics and sciences. He did not share the pragmatic attitude of some influential contemporaries in mathematics and sciences (like Hilbert) to keep philosophy away from their mathematical and scientific practice by simply fading out philosophical problems. Nevertheless, Frege opened the way for directions like philosophy of science, which aimed at bridging the gap between philosophy on the one hand and mathematics and science on the other, and which became successful in the twentieth century. His influence on Bertrand Russell’s logicism, codified in *Principia Mathematica*, Russell’s joint work with Alfred North Whitehead (1910/13), is well known. Through Carnap, Frege gained influence on logic and foundational research in the neopositivist movement of the Vienna Circle, which constituted the context of Kurt Gödel’s shaping of modern logic and foundational studies.

## SUPPLEMENTARY BIBLIOGRAPHY

### WORKS BY FREGE

“Methods of Calculation Based on an Extension of the Concept of Quantity” (1874). In McGuiness, *Collected Papers on Mathematics, Logic, and Philosophy*, 56–92.

*Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens*. Halle, Germany: L. Nebert, 1879. In *Conceptual Notation and Related Articles*, translated and edited with a biography and introduction by Terence Ward Bynum, 101–203. Oxford: Oxford University Press, 1972.

“Boole’s Logical Calculus and the Concept-Script.” (1880–1881). In Hermes et al., *Posthumous Writings*, 9–46.

“Boole’s Logical Formula-Language and My Concept-Script.”(1882). In Hermes et al., *Posthumous Writings*, 47–52.

*Die Grundlagen der Arithmetik, eine logisch mathematische Untersuchung über den Begriff der Zahl*. Breslau, Germany: W. Koebner, 1884. Translated by John L. Austin as *The Foundations of Arithmetic*. 2nd ed. Oxford: Blackwell, 1953.

“On Formal Theories of Arithmetic” (1885). In McGuiness, *Collected Papers on Mathematics, Logic, and Philosophy*, 112–121.

“Function and Concept” (1891). In Beany, *The Frege Reader*, 130–148.

“On *Sinn* und *Bedeutung*” (1892). In Beany, *The Frege Reader*, 151–171.

“On Concept and Object.” (1892). In Beany, *The Frege Reader*, 181–193.

“Grundgesetze der Arithmetik.” 2 Vols. Jena: H. Pohle, 1893 (Vol. 1); 1903 (Vol. 2).

Review of E. H. Husserl, *Philosophie der Arithmatik* I. (1894). In McGuiness, *Collected Papers on Mathematics, Logic, and Philosophy*, 195–209.

“On the Foundations of Geometry. First Series.” (1903). In McGuiness, *Collected Papers on Mathematics, Logic, and Philosophy*, 273–284.

“On the Foundations of Geometry. Second Series.” (1906). In McGuiness, *Collected Papers on Mathematics, Logic, and Philosophy*, 293–340./bibcit.composed>

“Reply to Mr. Thomae’s Holiday *Causerie*.” (1906). In McGuiness, *Collected Papers on Mathematics, Logic, and Philosophy*, 341–345.

“Renewed Proof of the Impossiblity of Mr. Thomae’s Formal Arithmetic.” (1908). In McGuiness, *Collected Papers on Mathematics, Logic, and Philosophy*, 346–350.

*Posthumous Writings*. Edited by Hans Hermes, Friedrich Kambartel, and Friedrich Kaulbach, with the assistance of Gottfried Gabriel and Walburga Rödding. Translated by Peter Long and Roger White, with the assistance of Raymond Hargreaves. Chicago: University of Chicago Press, 1979.

*Philosophical and Mathematical Correspondence*. Edited by Gottfried Gabriel, et al. Abridged for the English edition by Brian McGuinness. Translated by Hans Kaal. Chicago: University of Chicago Press, 1980.

*Collected Papers on Mathematics, Logic, and Philosophy*. Edited by Brian McGuinness. Translated by Max Black et al. Oxford: Blackwell, 1984.

*Kleine Schriften*. 2nd ed. Edited by Ignacio Angelelli. Hildesheim, Germany: G. Olms, 1990. Collection of Frege’s published papers and reviews, together with some letters and notes on Frege’s works by H. Scholz and E. Husserl.

“Diary: Written by Professor Dr. Gottlob Frege in the Time from 10 March to 9 April 1924.” Edited by Gottfried Gabriel and Wolfgang Kienzler. *Inquiry* 39 (1996): 303–342.

*The Frege Reader*. Edited by Michael Beaney. Oxford: Blackwell, 1997. The volume brings together all of Frege’s seminal writings, substantial parts of his three books, and selections from his posthumous writings and correspondence.

*Frege’s Lectures on Logic: Carnap’s Student Notes, 1910–1914*. Translated and edited by Erich H. Reck and Steve Awodey. Chicago: Open Court, 2004. Gives evidence for Frege’s late positions in logic and foundations. With an introductory essay by Gottfried Gabriel and a paper by the editors on Frege’s influence on the development of logic.

### OTHER SOURCES

Beaney, Michael. *Frege: Making Sense*. London: Duckworth, 1996. Frege’s concept of sense within his philosophy as a whole.

Beaney, Michael, and Erich H. Reck, eds. *Gottlob Frege: Critical Assessments of Leading Philosophers*. 4 vols. London: Routledge, 2005. Comprehensive collection of papers on Frege.

Brady, Geraldine. *From Peirce to Skolem: A Neglected Chapter in the History of Logic*. Amsterdam: Elsevier, 2000. Stresses the significance of the algebraic tradition for the development of logic.

Demopoulos, William, ed. *Frege’s Philosophy of Mathematics*. Cambridge, MA: Harvard University Press, 1995.

Dummett, Michael. *Frege: Philosophy of Language*. 2nd ed. London: Duckworth, 1981.

———. *Frege and Other Philosophers*. Oxford: Clarendon Press; 1991.

———. *Frege: Philosophy of Mathematics*. Cambridge, MA: Harvard University Press, 1991.

Gabriel, Gottfried, and Uwe Dathe, eds. *Gottlob Frege: Werk und Wirkung; Mit den unveröffentlichten Vorschlägen für ein Wahlgesetz von Gottlob Frege*. Paderborn, Germany: Mentis, 2000.

Grattan-Guinness, Ivor. *The Search for Mathematical Roots*, *1870–1940: Logics, Set Theories, and the Foundations of Mathematics from Cantor through Russell to Gödel*. Princeton, NJ: Princeton University Press, 2000. See pp. 177–199. Frege’s role within the development of research in logic and the foundations of mathematics.

Gronau, Detlef. “Gottlob Frege’s Beiträge zur Iterationstheorie und zur Theorie der Funktionalgleichungen.” In *Gottlob Frege: Werk und Wirkung; Mit den unveröffentlichten Vorschlägen für ein Wahlgesetz von Gottlob Frege*, edited by Gottfried Gabriel and Uwe Dathe, 151–169. Paderborn, Germany: Mentis, 2000.

Hale, Bob, and Crispin Wright. *The Reason’s Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics*. Oxford: Clarendon Press, 2001. Collection of papers neo-Fregeanism and Frege arithmetic.

Kreiser, Lothar. *Gottlob Frege: Leben—Werk—Zeit*. Hamburg, Germany: Meiner, 2001. Full-scale biography of Frege.

Newen, Albert, Ulrich Nortmann, and Rainer Stuhlmann-Laeisz, eds. *Building on Frege: New Essays on Sense, Content, and Concept*. Stanford, CA: CSLI Publications, 2001. Collection of essays on Frege’s significance and heritage.

Quine, Willard Van Orman. “On Frege’s Way Out.” *Mind*, n.s., 64 (1955): 145–159. Proof that Frege’s solution of Russell’s paradox was not feasible.

Schirn, Matthias, ed. *Frege: Importance and Legacy*. Berlin: de Gruyter, 1996. Collection of essays on Frege’s significance and heritage.

Schröder, Ernst. “Review of Frege, 1879.” *Zeitschrift für Mathematik und Physik. Historisch-litterarische Abteilung* 25 (1880): 81–94.

Sluga, Hans D. *Gottlob Frege*. London: Routledge and Kegan Paul, 1980.

Sullivan, Peter M. “Frege’s Logic.” In *Handbook of the History of Logic*, vol. 3, edited by Dov M. Gabbay and John Woods, 659–750. Amsterdam: Elsevier North Holland, 2004.

Thiel, Christian. “Zur Inkonsistenz der Fregeschen Mengenlehre.” In *Frege und die moderne Grundlagenforschung: Symposium, gehalten in Bad Homburg im Dezember 1973*, edited by Christian Thiel, 134–159. Meisenheim am Glan, Germany: Verlag Anton Hain, 1975.

———. “‘Not Arbitrarily and Out of a Craze for Novelty’: The Begriffsschrift 1879 and 1893.’’ In *Gottlob Frege: Critical Assessments of Leading Philosophers*, 4 vols., edited by Michael Beaney and Erich H. Reck, vol. 2, 13–28. London: Routledge, 2005. Surveys the nature of Frege’s logical notation and its development.

Thiel, Christian, and Michael Beaney. “Frege’s Life and Work: Chronology and Bibliography.” In *Gottlob Frege: Critical Assessments of Leading Philosophers*, 4 vols., edited by Michael Beaney and Erich H. Reck, vol. 1, 23–39. London: Routledge, 2005. Brief chronology on the basis of information now available, and comprehensive bibliography.

Weiner, Joan. *Frege Explained: From Arithmetic to Analytic Philosophy*. Chicago: Open Court, 2004. Compact explanation of Frege’s work.

*Volker Peckhaus*

## Frege, Friedrich Ludwig Gottlob

# Frege, Friedrich Ludwig Gottlob

(*b*. Wismar, Germany, 8 November 1848; *d*. Bad Kleinen, Germany, 26 July 1925)

*logic, foundations of mathematics*.

Gottlob Frege was a son of Alexander Frege, principal of a girl’s high school, and of Auguste Bialloblotzky. He attended the Gymnasium in Wismar, and from 1869 to 1871 he was a student at Jena. He then went to Göttingen and took courses in mathematics, physics, chemistry, and philosophy for five semesters. In 1873 Frege received his doctorate in philosophy at Göttingen with the thesis, *Ueber eine geometrische Darstellung der imäginaren Gebilde in der Ebene*. The following year at Jena he obtained the *venia docendi* in the Faculty of Philosophy with a dissertation entitled “Rechungsmethoden, die sich auf eine Erweitung des Grössenbegriffes gründen,” which concerns one-parameter groups of functions and was motivated by his intention to give such a definition of quantity as gives maximal extension to the applicability of the arithmetic based upon it. The idea presented in the dissertation of viewing the system of an operation *f* and its iterates as a system of quantities, which in the introduction to his *Grundlagen der Arithmetik* (1884) Frege essentially ascribes to Herbart, hints at the notion of *f-* sequence expounded in his *Begriffsschrift* (1879).

After the publication of the *Begriffsschrift*, Frege was appointed extraordinary professor at Jena in 1879 and honorary professor in 1896. His stubborn work toward his goal—the logical foundation of arithmetic—resulted in his two-volume *Grundgesetze der Arithmetik* (1893–1903). Shortly before publication of the second volume Bertrand Russell pointed out in 1902, in a letter to Frege, that his system involved a contradiction. This observation by Russell destroyed Frege’s theory of arithmetic, and he saw no way out. Frege’s scientific activity in the period after 1903 cannot be compared with that before 1903 and was mainly in reaction to the new developments in mathematics and its foundations, especially to Hilbert’s axiomatics. In 1917 he retired. His *Logische Untersuchungen*, written in the period 1918–1923, is an extension of his earlier work.

In his attempt to give a satisfactory definition of number and a rigorous foundation to arithmetic, Frege found ordinary language insufficient. To overcome the difficulties involved, he devised his *Begriffsschrift* as a tool for analyzing and representing mathematical proofs completely and adequately. This tool has gradually developed into modern mathematical logic, of which Frege may justly be considered the creator.

The *Begriffsschrift* was intended to be a formula language for pure thought, written with specific symbols and modeled upon that of arithmetic (i.e., it develops according to definite rules). This is an essential difference between Frege’s calculus and, for example, Boole’s or Peano’s, which do not formalize mathematical proofs but are more flexible in expressing the logical structure of concepts.

One of Frege’s special symbols is the assertion sign
(properly only the vertical stroke), which is in
terpreted if followed by a symbol with judgeable content. The interpretation of
*A* is “*A* is a fact.” Another symbol is the conditional
, and
is to be read as “*B* implies *A*.” The assertion
is justified in the following cases: (1) *A* and *B* are true; (2) *A* is true and *B* is false; (3) *A* and *B* are false.

Frege uses only one deduction rule, which consists in passing from
and
to
. The assertion that *A* is not a fact is expressed by
i.e., the small vertical stroke is used for negation. Frege showed that the other propositional connectives, “and” and “or,” are expressible by means of negation and implication, and in fact developed propositional logic on the basis of a few axioms, some of which have been preserved in modern presentations of logic. Yet he did not stop at propositional logic but also developed quantification theory, which was possible because of his general notion of function. If in an expression a symbol was considered to be replaceable, in all or in some of its occurrences, then Frege calls the invariant part of the expression a function and the replaceable part its argument. He chose the expression Φ(*A*) for a function and, for functions of more than one argument, Ψ(*A,B*). Since the φ in φ(*A*) also may be considered to be the replaceable part, φ(*A*) may be viewed as a function of the argument φ. This stipulation proved to be the weak point in Frege’s system, as Russell showed in 1902.

Generality was expressed by

which means that φ(*a*) is a fact, whatever may be chosen for the argument. Frege explains the notion of the scope of a quantifier and notes the allowable transition from
to
, where a occurs only as argument of *X*(*a*), and from
to
, where *a* does not occur in *A* and in φ (*a*) occurs only in the argument places.

Existence was expressed by (a). There was no explicitly stated rule of substitution.

It should be observed that Frege did not construct his system for expressing pure thought as a formal system and therefore did not raise questions of completeness or consistency. Frege applied his *Begriffsschrift* to a general theory of sequences, and in part III he defines the ancestral relation on which he founded mathematical induction. This relation was afterward introduced informally by Dedekind and formally by Whitehead and Russell in *Pricipia mathematica*.

The *Begriffsschrift* essentially underlies Frege’s definition of number in *Grundlagen der Arithmetik* (1884), although it was not used explicitly. The greater part of this work is devoted to a severe and effective criticism of existing theories of number. Frege argues that number is something connected with an assertion concerning a concept; and essential for the notion of number is that of equality of number (i.e., he has to explain the sentence “The number which belongs to the concept *F* is the same as that which belongs to G.”). He settled on the definition “The number which belongs to the concept *F* is the extence of the concepts of beaing eqal to the concept *F*” where eqality of concepts is understood as the extension of a one-to-one correspondence between their extensions. The number zero is that belonging to a concept with void extension, and the number one is that which belongs to the concept equal to zero. Using the notion of *f* sequence, natural numbers are defined, with ∞ the number belonging to the notion of being a natural number.^{1}

Frege’s theories, as well as his criticisms in the *Begriffsschrift* and the *Grundlagen*, were extended and refined in his *Grundgesetze*, in which he incorporated the essential improvements on his Begriffsschrift that had been expounded in the three important papers “Funktion und Begriff” (1891), “Über Sinn und Bedeutung” (1892), and “Über Begriff und Gegensstand” (1892). In particular, “Über Sinn und Bedeutung” is an essential complement to his *Begriffsschrift*. In addition, it has had a great influence on philosophical discussion, specifically on the development of Wittgenstein’s philosophy. Nevertheless, the philosophical implications of the acceptance of Frege’s doctrine have proved troublesome.

An analysis of the identity relation led Frege to the distinction between the sense of an expression and its denotation. If *a* and *b* are different names of the same object (refer to or denote the same object), we can legitimately express this by *a* = *b*, but = cannot be considered to be a relation between the objects themselves.

Frege therefore distinguishes two aspects of an expression: its denotation, which is the object to which it refers, and its sense, which is roughly the thought expressed by it. Every expression expresses its sense. An unsaturated expression (a function) has no denotation.

These considerations led Frege to the conviction that a sentence denotes its truth-value; all true sentences denote the True and all false sentences denote
the False—in other words, are names of the True and the False, respectively. The True and the False are to be treated as objects. The consequences of this distinction are further investigated in “Über Begriff und Gegenstand.” There Frege admits that he has not given a definition of concept and doubts whether this can be done, but he emphasizes that concept has to be kept carefully apart from object. More interesting developments are contained in his “Funktion und Begriff.” First, there is the general notion of f-sequence already briefly mentioned in the *Grundlagen*, and second, with every function there is associated an object, the so-called *Wertverlauf*, which he used essentially in his *Grundgesetze der Arithmetik*.

Since a function is expressed by an unsaturated expression *f(x)*, which denotes an object if x in it is replaced by an object, there arises the possibility of extending the notion of function because sentences denote objects (the True [*T*] and the False [*F*]), and one arrives at the conclusion that, e.g., (x^{2} = 4) = (x >1) is a function. If one replaces x by 1, then, because 1^{2} = 4 denotes F, as does 1 > 1, it follows that (1^{2} = 4) = (1 > 1) denotes *T*.

Frege distinguishes between first-level functions, with objects as argument, and second-level functions, with first-level functions as arguments, and notes that there are more possibilities. For Frege an object is anything which is not a function, but he admits that the notion of object cannot be logically defined. It is characteristic of Frege that he could not take the step of simply postulating a class of objects without entering into the question of their nature. This would have taken him in the direction of a formalistic attitude, to which he was fiercely opposed. In fact, at that time formalism was in a bad state and rather incoherently maintained. Besides, Frege was not creating objects but was concerned mainly with logical characterizations. This in a certain sense also holds true for Frege’s introduction of the *Wertverlauf*, which he believed to be something already there and which had to be characterized logically.

In considering two functions, e.g., x^{2} –4x and x(x–4), one may observe that they have the same value for the same argument. Therefore their graphs are the same. This situation is expressed by Frege true for Frege’s introduction of the *Wertverlauf*, as x^{2}–4x”. Without any further ado he goes on to speak of the *Wertverlauf* of a function as being something already there, and introduced a name for it. The *Wertverläufe* of the above mentioned functions x(x–4) and X^{2}-4x are denoted by ὰ(α–4) and ὲ(ε^{2} — 4ε) respectively, and in general ὲ*f*(ε) is used to denote the *Wertverlauf* of function *f* (ξ). This *wertuerlauf* is taken to be an object, and Frege assumes the basic logical law characterizing equality of *Wertverläufe:*

Frege extends this to logical functions (i.e., concepts), which are conceived of as functions whose values are truth-values, and thus extension of a concept may be identified with the *Wertverlauf* of a function assuming only truth-values. Therefore, e.g.,

In the appendix to volume II of his *Grundgesetze*, Frege derives Russell’s paradox in his system with the help of the above basic logical law. Russell later succeeded in eliminating his paradox by assuming the theory of types.

It is curious that the man who laid the most suitable foundation for formal logic was so strongly opposed to formalism. In volume II of the *Grundgesetze*, where he discusses formal arithmetic at length, Frege proves to have a far better insight than its exponents and justly emphasizes the necessity of a consistency proof to justify creative definitions. He is aware that because of the introduction of the *Wertverläufe* he may be accused of doing what he is criticizing. Nevertheless, he argues that he is not, because of his logical law concerning *Wertverläufe* (which proved untenable).

When Hilbert took the axiomatic method a decisive step further, Frege failed to grasp his point and attacked him for his imprecise terminology. Frege insisted on definitions in the classic sense and rejected Hilbert’s “definition” of a betweenness relation and his use of the term “point.” For Frege geometry was still the theory of space. But even before 1814 Bolzano had already reached the conclusion that for an abstract theory of space, one may be obliged to assume the term point as a primitive notion capable of various interpretations. Hilbert’s answer to Frege’s objections was quite satisfactory, although it did not convince Frege.

## BIBLIOGRAPHY

I. Original Works. Frege’s writing includes *Ueber eine geometrische Darstellung der imaginären Gebilde in der Ebene*, his inaugural diss. to the Faculty of Philosophy at Göttingen (Jena, 1873); *Begriffsschrift, eine der arithme- tischen nachgebildete Formelsprache des reinen Denkens* (Halle, 1879), 2nd ed., I. Angelelli, ed. (Hildesheim, 1964), English trans, in J. van Heijenoort, *From Ferge to Godel* (Cambridge, 1967), pp.1–82; *Die Grundlagen der Arithmetik* (Breslau, 1884), trans. with German text by J. L. Austin (Oxford, 1950; 2nd rev. ed. 1953), repr. as *The Foundations of Arithmetic* (Oxford, 1959); *Function und Begriff* (Jena, 1891), English trans. in P. Geach and M. Black, pp. 21–41 (see below); “Uber Sinn und Bedeutung,” in *Zeitschrift
für Philosophie und philosophische Kritik*, n. s. **100** (1892) 25–50, English trans, in P. Geach and M. Black, pp. 56–78 (see below); “Über Begriff und Geogenstand” in *Viertel jahrschrift für wissenschaftliche philosophie*, **16** (1892) 192–205, English trans, in P. Geach and M. Black, pp. 42–55 (see below): *Grundgesetze der Arithmetik*, 2 vols. (Jena, 1893–1903), repr. in 1 vol. (Hildesheim, 1962); and “Über die Grundlagen der Geometrie in *Jaheresbericht der Dutschen Mathematikervereinigung***12** (1903) 319–324, 368–375; **15** (1906) 293–309, 377–403, 423–430

II. Secondary Literatrure. On Frege and his work, see I. Angelelli, *Studies on Gottlob Frege and Traditional Philosophy* (Dordrecht, 1967); P. Geach and M. Black, *Translation of the Philosophical Writings of Gottlob Frege*, 2nd ed. (oxford, 1960); H. Hermes, *et al., Gottlob frege, Nachgelassene Schriften* (Hamburg,1969); J. van Heijenfoort, *From Frege to Godel* (Cambridge, 1967): P. E. B. Jourdain, “The Development of the Theories of Mathematical Logic and the Principles of Mathematics. Gottlob Frege,” In *Quarterly Journal of Pure and Applied Mathematics*, **43** (1912), 237–269; W. Kneale and Martha Kneale, *The Development of Logic* (Oxford, 1962) pp. 435–512; J. Largeault, *Logique et philosophie chez Frege* (Paris–Louvain, 1970); C. Parsons, “Frege’s Theory of Number,” in M. Black, ed., *Philosophy in America* (London, 1965), pp. 180-203; G. Patzig, *Gottlob Frege, Funktion, Begriff, Bedeutung* (Göttingem, 1966); and *Gottlob Frege, Logische Untersuchungen* (Göttingen, 1966); B. Russell, *The Principles of Mathematics* (Cambridge, 1903), Appendix A, “The Logical and Arithmetical Doctrines of Frege” M. Steck “Ein unbeakannter Brief von Gottlob Frege über Hilberts erste Vorlesung über die Grundlagen der Geometrie, in *Sitzungsberichte der Heidelberger Akademie der Wissenschaften*, Abhandlung 6 (1940); and “Unbekannte Briefe Frege’s über die Grundlagen der Geometrie und Antwort brief Hilbert’s and Frege *ibid* Abhandlung 2 (1941); H. G. Steiner, “Frege und die Grundlagen der Geometric I. II’ in *Mathematische-physikalishce Semesterberichte*, n. s. **10** (1963), 175–186 and **11** (1964), 35–47; and J. D. B. Walker, *A Study of Frege* (Oxford, 1965).

B. Van Rootselaar

## Frege, Gottlob

# FREGE, GOTTLOB

mathematical backgroundsystematization of science and sources of knowledge

the history of frege's project

frege's legacy

bibliography

**FREGE, GOTTLOB** (1848–1925), German mathematician, logician, and philosopher.

Friedrich Ludwig Gottlob Frege devoted most of his career to a single project: the attempt to provide foundations for arithmetic. What is it to provide foundations for arithmetic? To provide foundations for a science, in Frege's sense, is to systematize it: to list its primitive truths and concepts and to show how these primitive truths and concepts figure in the justification of the truths of the science. The resulting systematization of a science is meant to exhibit the source of the knowledge of its truths. Mathematical work provided him with one model of this sort of systematization, the arithmeticization of analysis.

## mathematical background

Analysis originated in the seventeenth century as a response to the needs of physics and astronomy. Its proofs originally exploited techniques of geometry. In the mid-nineteenth century, however, it became apparent that many of the geometrical proofs were not as secure as they seemed. Some apparently good proofs were identified as fallacious. The difficulties were attributable, in part, to confusions about some basic notions of analysis, including those of limit and continuity. The response was to try to show that the foundations of analysis lay in arithmetic: to offer a systematization of analysis in which only terms of arithmetic were used. Arithmetic was also systematized. Julius Wilhelm Richard Dedekind (1831–1916) provided an easily surveyable list of axioms from which, it was thought, all truths of arithmetic were derivable. But Frege was not content with Dedekind's systematization. For Dedekind's axiomatization did not, according to Frege, exhibit the source of our knowledge of the truths of arithmetic. The problem was that the terms of arithmetic (for example, "0") appeared in Dedekind's axioms, and that his axioms included familiar truths of arithmetic (for example, that 0 does not equal 1). Such terms and truths are not primitive, in Frege's sense, because it is not clear what the source of knowledge about numbers is. To see why, it is necessary to look at Frege's views about the sources of knowledge.

## systematization of science and sources of knowledge

Sense experience is indisputably a source of some knowledge. The empiricist view can be characterized as the view that sense experience is the source of *all* knowledge. Frege thought that the German philosopher Immanuel Kant (1724–1804) had refuted this view by showing that some knowledge, including knowledge of Euclidean geometry, does not have the senses as its source. The source of this knowledge is, rather, pure intuition—a faculty underlying perception of objects in space. But Kant erred, according to Frege, in failing to recognize that some substantive knowledge requires neither sense experience nor pure intuition for its justification. Knowledge of arithmetic, Frege believed, is more basic than that of either the special sciences or geometry. The source of this knowledge is something that underlies all knowledge: reason alone. The science of reason, the science of the general laws that underlie all correct inference, is logic. Frege was convinced that the truths of arithmetic and, indeed, of all mathematics other than Euclidean geometry were logical truths. But many truths of arithmetic (e.g., that 0 does not equal 1) seem to be truths about particular objects, not general laws of logic. Frege thought he could show that all arithmetical truths are truths of logic by defining the numbers and the concept of number from purely logical concepts and proving the basic truths of arithmetic using only these definitions and logical laws. The result would be logical foundations for arithmetic: foundations showing that reason alone is the source of the knowledge of the truths of arithmetic.

## the history of frege's project

Although Frege was convinced that the truths of arithmetic were logical truths, he was aware that they were not derivable using the logical systems generally accepted in his time. But he was also convinced that these logical systems were inadequate. Thus he began by constructing a new logic. In the 1879 monograph, *Begriffsschrift,* he set out the first logical system adequate for the expression and evaluation of the arguments that are regarded as logically valid. His next major work, *Die Grundlagen der Arithmetik* (1884; *The Foundations of Arithmetic,* 1950), was an informal description of his project, its motivations, and Frege's strategy for accomplishing his goals. The project was to have been completed in his *Grundgesetze der Arithmetik* (*Basic Laws of Arithmetic*). The first volume of *Basic Laws* was published in 1893. In 1902, when the second volume was in press, Frege received a now-famous letter from the mathematician and philosopher Bertrand Russell (1872–1970), demonstrating that the logical system was inconsistent. Frege ultimately concluded that the project, as he had envisioned it, was doomed to failure.

## frege's legacy

Although Frege failed to accomplish the task he set himself, he made profound contributions to logic and philosophy. One of these is Frege's new logic. The inconsistency in the logic set out in *Basic Laws* is easily eliminated by omitting the two basic logical laws that were needed for the definitions of the numbers. The resulting logical system is not only more powerful than earlier logical systems, it is also formal in several important respects. In Frege's logic it is a mechanical task to determine, for any string of symbols in the logical language, whether it is a well-formed name or sentence of the language. It is also a mechanical task to determine whether a sentence is a basic law and whether or not a sentence follows immediately by one of the rules of inferences from other sentences. Thus there is a mechanical procedure for evaluating a purported gapless proof of the argument in the formal language. These formal features make it possible to regard the logical system itself as a mathematical entity. The field of mathematical logic thus has its origin in Frege's new logic.

Another important Fregean legacy comes from his approach to his philosophical problem. Frege believed that he could solve a philosophical problem about the nature of the truths of arithmetic by introducing definitions, using purely logical terms, that could replace numerals in all contexts. The justification of these definitions was provided by an analysis of how certain symbols (the numerals) are used and a demonstration that these symbols can be dispensed with by defining them from other terms. The philosophical question that Frege wanted to answer appears to have nothing in particular to do with language or meaning. Yet he answered the question by engaging in a linguistic investigation. The use of this strategy marks Frege as one of the first (perhaps the very first) to take the so-called linguistic turn that is characteristic of analytic philosophy, the dominant school in Anglo-American philosophy since the middle of the twentieth century.

Finally, many of Frege's writings on specific issues concerning language, logic, and mathematics remain immensely influential in the twenty-first century. A great deal of work in linguistics and the philosophy of language has its origin in his discussions of language. Indeed, a substantial number of early-twenty-first-century philosophers regard themselves as neo-Fregeans. Even the logicist project that Frege regarded as having been decisively refuted has been resurrected and forms an important strand of contemporary philosophical thought about arithmetic. Frege's work attracted only a small audience in his lifetime. But in the years since, his influence on contemporary philosophy, especially on thought about language and logic, has become ubiquitous.

*See also***Science and Technology.**

## bibliography

### Primary Sources

Frege, Gottlob. *Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number*. Translated by J. L. Austin. 2nd edition. Oxford, U.K., 1980. Translation of *Die Grundlagen der Arithmetik*.

Beaney, Michael, ed. *The Frege Reader.* Oxford, U.K., 1997. Collection containing Frege's most well known articles as well as excerpts from his most important books.

### Secondary Sources

Dummett, Michael. *Frege: Philosophy of Mathematics.* London, 1991. A scholarly work by one of Frege's most influential contemporary interpreters

Weiner, Joan. *Frege Explained: From Arithmetic to Analytic Philosophy.* Chicago, 2004. An introduction to Frege's works intended for nonspecialists.

Joan Weiner

## Gottlob Frege

# Gottlob Frege

The German mathematician and philosopher Gottlob Frege (1848-1925) is considered the founder of modern mathematical logic. His work was almost wholly ignored during his lifetime but now exerts a great influence on the philosophy of logic and language.

Gottlob Frege was born on Nov. 8, 1848, at Wismar. He began his university studies at Jena in 1869 but after 2 years moved to Göttingen. He studied mathematics, the natural sciences, and philosophy and took his degree in 1873. Thereafter he taught at Jena in the department of mathematics. He was made a professor in 1896 and retired in 1918. Frege was married to Margarete Lieseberg, and the couple had one adopted son. Frege died on July 26, 1925, in Bad Kleinen.

Frege invented the concept of a formal system of mathematical logic, and in his first major work, *Begriffsschrift* (1879), he presented the first example of such a system in his formulation of a propositional and predicate calculus. He introduced the mathematical notion of function and variable into logic and invented the idea of quantifiers. He was
also the first writer on axiomatic theory to make clear the distinction between an axiom and a rule of inference.

Further progress in this work convinced Frege that the basic ideas of arithmetic (but not of geometry) could be articulated solely in logical expressions. He expressed his new program first in a nonsymbolic work, *The Foundations of Arithmetic* (1884), which also featured a brilliant and devastating polemic against all previous attempts at the subject. The crown of his work was to be his *Basic Laws of Arithmetic.* The first volume of this work appeared in 1893; but in 1903, as Frege was about to issue the second volume, Bertrand Russell pointed out a contradiction in Frege's use of the concept of a "class," which undermined the proofs in the work. Frege hastily added an appendix that sought to remedy the defect (this effort was later proved defective), but thereafter he seemed to lose interest in the great project. Two decades later he regarded the whole enterprise as an error and fell back upon the Kantian interpretation of mathematical judgments as synthetic a priori.

Frege also made important contributions to the philosophy of logic. Concerning the old question: what is it for a proposition to have meaning?—he introduced a variety of distinctions that are being exploited by contemporary philosophers. Frege rejected epistemology as the starting point of philosophy and revived the classical view, dominant before René Descartes, that philosophical logic holds this place.

## Further Reading

There are no biographies of Frege. His work, however, has been extensively studied, especially since translations of it have become available, beginning in the 1950s. A convenient collection of most of the important critical essays is in E. D. Klemke, ed., *Essays on Frege* (1968), which also has a complete bibliography. Two difficult but rewarding full-length studies are Jeremy D. B. Walker, *A Study of Frege* (1965), and Robert Sternfeld, *Frege's Logical Theory* (1966). □

## Frege, Gottlob

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