Friedrich Ludwig Gottlob Frege

Gottlob Frege (b. 1848, d. 1925) was a German mathematician, logician, and philosopher who worked at the University of Jena. He wrote philosophical works about logic, mathematics, and language.

Principal Works

• Begriffsschrift (‘Concept Notation’), eine der arithmetischen nachgebildete Formelsprache des reinen Denkens , Halle a. S., 1879
• Die Grundlagen der Arithmetik (“The Foundations of Arithmetic”): eine logisch-mathematische Untersuchung über den Begriff der Zahl, Breslau, 1884
• Funktion und Begriff (“Function and Concept”): Vortrag, gehalten in der Sitzung vom 9. Januar 1891 der Jenaischen Gesellschaft für Medizin und Naturwissenschaft, Jena, 1891
• “Über Sinn und Bedeutung” (“On Sense and Denotation”), in Zeitschrift für Philosophie und philosophische Kritik, C (1892): 25–50
• “Über Begriff und Gegenstand” (“On Concept and Object”), in Vierteljahresschrift für wissenschaftliche Philosophie, XVI (1892): 192–205
• Grundgesetze der Arithmetik (“Basic Laws of Arithmetic”), Jena: Verlag Hermann Pohle, Band I (1893), Band II (1903)
• “Was ist eine Funktion?” (“What is a Function?”), in Festschrift Ludwig Boltzmann gewidmet zum sechzigsten Geburtstage, 20. Februar 1904, S. Meyer (ed.), Leipzig, 1904, pp. 656–666
• “Der Gedanke” (“The Thought”). Eine logische Untersuchung”, in Beiträge zur Philosophie des deutschen Idealismus I (1918): 58–77

Frege’s Life

• Born, November 8, 1848, in Wismar (Mecklenburg-Schwerin)
• 1869, entered the University of Jena
• 1871, entered the University of Göttingen
• 1873, Ph. D. in Mathematics (Geometry), University of Göttingen
• 1874, Habilitation in Mathematics, University of Jena
• 1874, Privatdozent, University of Jena
• 1879, Professor Extraordinarius, University of Jena
• 1896–1917, ordenlicher Honorarprofessor, University of Jena
• Died, July 26, 1925, in Bad Kleinen (now in Mecklenburg-Vorpommern)

Frege’s Advances in Logic

Frege virtually founded the modern discipline of mathematical logic. He developed a system of conceptual notation (inspired by Leibniz’s conception of a rational calculus), and though we no longer use his notation, his system constituted the first predicate calculus. Frege’s second-order predicate calculus was based on the ‘function-argument’ analysis of propositions and it freed logicians from the limitations of the ‘subject-predicate’ analysis of Aristotelian logic. Frege’s formal system made it possible for logicians to develop a strict definition of a proof. Unfortunately, Frege employed a principle (Basic Law V) in his later system (Grundgesetze) which turned out to be inconsistent. Despite the fact that a contradiction invalidated his system, Frege validly derived the Peano Axioms governing the natural numbers from a powerful and consistent principle now known as Hume’s Principle (some philosophers have proposed that the derivation of the Peano Axioms from Hume’s Principle should be called ‘Frege’s Theorem’). Frege is most well-known among philosophers, however, for suggesting that the expressions of language have both a sense and a denotation (i.e., that at least two semantic relations are required to explain the significance of linguistic expressions). This seminal idea in the philosophy of language has inspired research in the field for over a century.

Further Reading

General

• Currie, G., Frege: An Introduction to His Philosophy, Sussex, The Harvester Press (Totowa, NJ: Barnes and Noble), 1982
• Dummett, M., “Gottlob Frege”, in Encyclopedia of Philosophy (Volume 3), New York: MacMillan, 1967
• Zalta, E., “Gottlob Frege”, in Stanford Encyclopedia of Philosophy
• Zalta, E., “Frege’s Theorem and Foundations for Arithmetic”, in Stanford Encyclopedia of Philosophy

Specific

• Anderson, David, and E. Zalta, “Frege, Boolos, and Logical Objects”, Journal of Philosophical Logic, 33(1) (February 2004): 1–26
• Boolos, G., “Saving Frege From Contradiction”, Proceedings of the Aristotelian Society, 87 (1986/87): 137–151
• Boolos, G., “The Consistency of Frege’s Foundations of Arithmetic”, in J. Thomson (ed.), On Being and Saying, Cambridge, MA: The MIT Press, 1987, pp. 3–20
• Demopoulos, W., (ed.), Frege’s Philosophy of Mathematics, Cambridge, MA: Harvard University Press, 1995
• Heck, R., “The Development of Arithmetic in Frege’s Grundgesetze Der Arithmetik”, Journal of Symbolic Logic, 58(2) (June 1993): 579–601
• Nodelman, U., and E. Zalta, “Number Theory and Infinity Without Mathematics”, Journal of Philosophical Logic, 53 (2024): 1161–1197. https://doi.org/10.1007/s10992-024-09762-7
• Parsons, T., “On the Consistency of the the First-Order Portion of Frege’s Logical System”, Notre Dame Journal of Formal Logic 28(1) (January 1987): 161–168
• Resnik, M., Frege and the Philosophy of Mathematics, Ithaca, NY: Cornell University Press, 1980
• Sluga, H., Gottlob Frege, London: Routledge and Kegan Paul, 1980
• Wright, C., Frege’s Conception of Numbers as Objects, Aberdeen: Aberdeen University Press, 1983
• Zalta, E., “Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege’s Grundgesetze in Object Theory”, Journal of Philosophical Logic, 28(6) (1999): 619–660

Copyright © 2025, by Edward N. Zalta. All rights reserved.