Gottlob Frege

German mathematician

• Born: November 8, 1848
• Birthplace: Wismar, Mecklenburg-Schwerin (now in Germany)
• Died: July 26, 1925
• Place of death: Bad Kleinen, Germany

Frege’s writings were never widely read or appreciated during his own time, and his complex system of symbols and functions was forbidding even to the best minds in mathematics. Nevertheless, he is recognized as the founder of modern symbolic logic and the creator of the first system of notations and quantifiers of modern logic.

Early Life

Friedrich Ludwig Gottlob Frege (FREH-gah) was the son of the principal of a private girls’ high school. While Frege was in high school in Wismar, his father died. Frege was devoted to his mother, who was a teacher and later principal of the girls’ school. He may have had a brother, Arnold Frege, who was born in Wismar in 1852. Nothing further is known about Frege until he entered the university at the age of twenty-one. From 1869 to 1871, he attended the University of Jena and proceeded to the University of Göttingen, where he took courses in mathematics, physics, chemistry, and philosophy. By 1873, Frege had completed his thesis and had received his doctorate from the university. Frege returned to the University of Jena and applied for an unsalaried position. His mother wrote to the university that she would support him until he acquired regular employment. In 1874, as a result of publication of his dissertation on mathematical functions, he was placed on the staff of the university. He spent the rest of his life at Jena, where he investigated the foundations of mathematics and produced seminal works in logic.

Frege’s early years at Jena were probably the happiest period in his life. He was highly regarded by the faculty and attracted some of the best students in mathematics. During these years, he taught an extra load as he assumed the courses of a professor who had become ill. He also worked on a volume on logic and mathematics. Frege’s lectures were thoughtful and clearly organized, and were greatly appreciated by his students. Much of Frege’s personal life, however, was beset by tragedies. Not only did his father die while he was a young man but his children also died young, as did his wife. He dedicated twenty-five years to developing a formal system, in which all of mathematics could be derived from logic, only to learn that a fatal paradox destroyed the system. During his life, he received little formal recognition of his monumental work and, with his death in 1925, passed virtually unnoticed by the academic world.

Life’s Work

Frege’s first major work in logic was published in 1879. Although this was a short book of only eighty-eight pages, it has remained one of the most important single works ever written in the field. Begriffsschrift: Eine der Arithmetischen Nachgebildete Formelsprache des reinen Denkens (conceptual treatise: a formal language, modeled upon that of arithmetic, for pure thought) presented for the first time a formal system of modern logic. He created a system of formal symbols that could be used more regularly than ordinary language for the purposes of deductive logic. Frege was by no means the first person to use symbols as representations of words, because Aristotle had used this device and was followed by others throughout the history of deductive logic.

Earlier logicians, however, had thought that in order to make a judgment on the validity of sentences, a distinction was necessary between subject and predicate. For the purposes of rhetoric, there is a difference between the statements “The North defeated the South in the Civil War” and “The South was defeated by the North in the Civil War.” For Frege, however, the content of both sentences conveyed the same concept and hence must be given the same judgment. In this work, Frege achieved the ideal of nineteenth century mathematics: that if proofs were completely formal and no intuition was required to judge the correctness of the proofs, then there could be complete certainty that these proofs were the result of explicitly stated assumptions. During this period, Frege began to use universal quantifiers in his logic, which cover statements that contain “some” or “every.” Consequently, it was now possible to cover a range of objects rather than a single object in a statement.

In 1884, Frege published Die Grundlagen der Arithmetik (The Foundations of Arithmetic , 1950), which followed his attempt to apply similar principles to arithmetic as his earlier application to logic. In this work, he first reviewed the works of his predecessors and then raised a number of fundamental questions on the nature of numbers and arithmetic truth. This work was more philosophical than mathematical.

Throughout the work, Frege enunciated three basic positions concerning the world of philosophical logic. Mental images of a word as perceived by the speaker are irrelevant to the meaning of a word in a sentence in terms of its truth or falsity. The word “grass” in the sentence “the grass is green” does not depend on the mental image of “grass” but on the way in which the word is used in the sentence. Thus the meaning of a word was found in its usage. A second idea was that words have meaning only in the context of a sentence. Rather than depending on the precise definition of a word, the sentence determined the truth-value of the word. If “all grubs are green,” then it is possible to understand this sentence without necessarily knowing anything about “grubs.” Also, it is possible to make a judgment about a sentence that contains “blue grubs” as false, because “all grubs are green.”

Frege’s third idea deals with the distinction between concepts and objects. This distinction raises serious questions concerning the nature of proper names, identity, universals, and predicates, all of which were historically troublesome philosophical and linguistic problems.

After the publication of The Foundations of Arithmetic, Frege became known not only as a logician and a mathematician but also as a linguistic philosopher. Although the notion of proper name is important for his system of logic, it also extends far beyond those concerns. There had existed an extended debate as to whether numbers such as “1,2,3,…” or directions such as “north” were proper names. Frege argued that it was not appropriate to determine what can be known about these words and then see if they can be classified as objects. Rather, like his theory of meaning, in which the meaning of a word is determined by its use in a sentence, if numbers are used as objects they are proper names.

Frege’s insistence on the usage of words extended to the problem of universals. According to tradition, something that can be named is a particular, while a universal is predicated on a particular. For example, “red rose” comprises a universal “red” and a particular “rose.” Question arose as to whether universals existed in the sense that the “red” of the “red rose” existed independently of the “rose.” Frege had suggested that universals are used as proper names in such sentences as “The rose is red.”

Between 1893 and 1903, Frege published two volumes of his unfinished work Grundgesetze der Arithmetik (the basic laws of arithmetic). These volumes contained both his greatest contribution to philosophy and logic and the greatest weaknesses of his logical system.

Frege made a distinction between sense and reference, in that words frequently had the same reference, but may imply a different sense. Words such as “lad,” “boy,” and “youth” all have the same reference or meaning, but not in the same sense. As a result, two statements may be logically identical, yet have a different sense. Hence, 2 + 2 = 4 involves two proper names of a number, namely “2 + 2” and “4,” but are used in different senses. Extending this idea to a logical system, the meaning or reference of the proper names and the truth-value of the sentence depend only on the reference of the object and not its sense. Thus, a sentence such as “The boy wore a hat” is identical to the sentence “The lad wore a hat.” Because the logical truth-value of a sentence depends on the meaning of the sentence, the inclusion of a sentence without any meaning within a complex statement means that the entire statement lacks any truth-value. This proved to be a problem that Frege could not resolve and became a roadblock to his later work.

A further problem that existed in Grundgesetze der Arithmetik, which was written as a formal system of logic including the use of terms, symbols, and derived proofs, was the theory of classes. Frege wanted to use logic to derive the entire structure of mathematics to include all real numbers. To achieve this, Frege included, as part of his axioms, a primitive theory of sets or classes.

While the second volume of Grundgesetze der Arithmetik was being prepared for publication, Frege received a letter from Bertrand Russell describing a contradiction that became known as the Russell Paradox. This paradox, sometimes known as the Stranger Loop, asks, is “the class of all classes that are not members of itself” a member of itself or not? For example, the “class of all dogs” is not a dog; the “class of all animals” is not an animal. If the class of all classes is a member of itself, then it is one of those classes that are not members of themselves. However, if it is not a member of itself, then it must be a member of all classes that are members of themselves, and the loop goes on forever. Frege replaced the class axiom with a modified and weaker axiom, but his formal system was weakened, and he never completed the third volume of the work.

Between 1904 and 1917, Frege added few contributions to his earlier works. During these years, he attempted to work through those contradictions that arose in his attempt to derive all of mathematics from logic. By 1918, he had begun to write a new book on logic, but he completed only three chapters. In 1923, he seemed to have broken through his intellectual dilemma and no longer believed that it was possible to create a foundation of mathematics based on logic. He began work in a new direction, beginning with geometry, but completed little of this work before his death.

Significance

In Begriffsschrift, Gottlob Frege created the first comprehensive system of formal logic since the ancient Greeks. He provided some of the foundations of modern logic with the formulation of the principles of noncontradiction and excluded middle. Equally important, Frege introduced the use of quantifiers to bind variables, which distinguished modern symbolic logic from earlier systems.

Frege’s works were never widely read or appreciated. His system of symbols and functions was forbidding even to the best minds in mathematics. Russell, however, made a careful study of Frege and was clearly influenced by his system of logic. Also, Ludwig Wittgenstein incorporated a number of Frege’s linguistic ideas, such as the use of ordinary language, into his works. Frege’s distinction between sense and reference later generated a renewed interest in his work, and a number of important philosophical and linguistic studies are based on his original research.

Bibliography

Bynum, Terrell W. Introduction to Conceptual Notations, by Gottlob Frege. Oxford, England: Clarendon Press, 1972. An eighty-page introduction to the logic of Frege. Although sections of the text on the logic are not suited for the general reader, the introductory text is clear, concise, and highly accessible. The significant works by Frege are outlined in simple terms, and the commentary is useful.

Currie, Gregory. Frege: An Introduction to His Philosophy. Brighton, England: Harvester Press, 1982. Discusses all the major developments in Frege’s thought from a background chapter to the Begriffsschrift, theory of numbers, philosophical logic and methods, basic law, and the fatal paradox. Some parts of this text are accessible to the general reader; other parts require a deeper understanding of philosophical issues.

Dummett, Michael A. E. Frege: Philosophy of Language. London: Duckworth, 1973. One of the leading authorities on the philosophy of Frege. The advantage of this text over others is that Dummett is in part responsible for the idea that Frege is a linguistic philosopher. Somewhat difficult but good introduction to Frege.

Grossmann, Reinhardt. Reflections on Frege’s Philosophy. Evanston, Ill.: Northwestern University Press, 1969. Delineates three major areas of Frege’s thoughts as found in Begriffsschrift and The Foundations of Arithmetic, and describes the distinction between meaning, sense, and reference. Within these areas the author writes an exposition on a few selected problems that are of current interest.

Hill, Claire O. Rethinking Identity and Metaphysics: The Foundations of Analytic Philosophy. New Haven, Conn.: Yale University Press, 1997. The author provides a reassessment of twentieth century analytic philosophy by examining the writings of Frege, Bertrand Russell, and Willard Quine. Hill concludes that the lack of clarity inherent in the abstract issue of identity has implications for solid subjects, such as medical ethics.

Kneale, William, and Martha Kneale. The Development of Logic. Oxford, England: Clarendon Press, 1962. Three chapters in this work are useful. Chapter 7 covers Frege and his contemporaries, Frege’s criticism of his predecessors, and Frege’s definition of natural numbers. Chapter 8 covers Frege’s three major works and outlines his contributions to the world of logic. Chapter 9 covers formal developments in logic after Frege and reveals his pivotal position in the development of modern symbolic logic.

Noonan, Harold W. Frege: A Critical Introduction. Malden, Mass.: Blackwell, 2001. An overview of Frege’s ideas, emphasizing his logic, theory of meaning, distinctions between sense and reference, and ideas about object, concept, and function.

Reck, Erich H., ed. From Frege to Wittgenstein: Perspectives on Early Analytic Philosophy. New York: Oxford University Press, 2002. Considers how the two men developed an analytic philosophy, including a discussion of Wittgenstein’s debt to Frege, an explanation of the roots of analytic philosophy, and interpretations of some of Frege’s writings.

Salerno, Joseph. On Frege. Belmont, Calif.: Wadsworth/Thomson Learning, 2001. Brief, nontechnical overview of Frege’s ideas aimed at students seeking a better understanding of his philosophy.

Weiner, Joan. Frege Explained: From Arithmetic to Analytic Philosophy. Chicago: Open Court, 2004. A summary of Frege’s philosophy, tracing the development of his ideas about logic, sense, meaning, and other subjects. Includes a chapter about his life and character.