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In 1879, Gottlob Frege, a German mathematician and logician, published a slim volume entitled Begriffsschrift: Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (“Begriffsschrift, a Formula Language, Modeled upon That of Arithmetic, for Pure Thought,” 1967; better known as Conceptual Notation, 1972), in which, for the first time in the history of logic, the fundamental ideas and principles of mathematical logic were set out. In this remarkable book, Frege achieved a much deeper analysis of deductive inference than had previous logicians; for example, the problem of deductive inferences involving multiply embedded expressions of generality (for example, “everyone loves someone”) was finally solved. Frege realized that the formal system of logic he had devised was precisely the instrument needed for realizing philosopher Gottfried Wilhelm Leibniz’s dream of reducing arithmetic to logic. Frege believed that before attempting this daunting task, an informal overview and defense of the project should be given; The Foundations of Arithmetic was written for that purpose. Frege later attempted the rigorous reduction of the concepts and truths of arithmetic to those of logic in Grundgesetze der Arithmetik, 1893-1903 (two volumes; partial translation, The Basic Laws of Arithmetic, 1964).

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Others’ Notions Refuted

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Frege’s concern in The Foundations of Arithmetic is with basic philosophical issues concerning the nature of numbers and the truths of arithmetic: What are numbers? How do we apprehend them? Why are the truths of arithmetic true? How do we know these truths? Adequate answers to these simple and basic questions would seem necessary for understanding this elementary branch of mathematics, but Frege came to believe that the answers given by previous writers were shallow and confused. Frege accordingly devotes the first half of The Foundations of Arithmetic to an examination and criticism of the views of his predecessors and contemporaries on the nature of numbers and arithmetical truth.

One prominent approach to foundational issues in mathematics is the empiricism of John Stuart Mill. Empiricism in general is the philosophical theory that all substantive knowledge is acquired by experience. Numbers, in Mill’s view, are properties of physical objects or collections of physical objects and therefore are conveyed to people through sense perception. The truths of arithmetic, according to Mill’s empiricism, are high-level inductive generalizations: Because, for example, when two objects are added to three objects the result is almost always a collection of five objects, we can conclude that 2 + 3 = 5. Frege points out, first of all, that numbers are not simply collections of physical objects or properties of physical objects. The same physical situation can be described as one deck, four suits, or fifty-two cards depending on the concept employed. Frege concludes that it is concepts, not physical things themselves, to which numbers are primarily ascribed. As far as the truths of arithmetic being inductive generalizations, Frege points out that induction presupposes probability theory, which in turn presupposes arithmetic. Furthermore, it is simply quite implausible to think that the process of adding a quart of water to a quart of popcorn should shake our confidence that 1 + 1 = 2.

Another approach to the foundations of mathematics is psychologism. This view holds that numbers are subjective mental entities and that the laws of arithmetic are descriptions of how humans as a matter of fact calculate. Psychologism views arithmetic as ultimately resting on naturalistic facts about human beings—facts about their ideas and mental processes. Frege believes that psychologism completely misconceives the nature of mathematics and logic. Because subjective ideas are private and can be apprehended only by their owners, psychologism implies, for example, that all humans think about a different object when they think about the number six. However, if each of us has our own number six, there would be nothing to prevent someone from legitimately asserting that six is a prime number—for how could this claim be rebutted? If psychologism is true, only that person has access to the number being discussed. It is obvious that arithmetic as a body of objective truths cannot rest on a view such as this. The claim that the laws of arithmetic are merely descriptions of how humans think implies that in the situation in which, because of some sort of mass hallucination, most people came to believe that 2 + 2 = 5, it would be true that 2 + 2 = 5. However, are the truths of arithmetic so dependent on what people happen to believe? This view ignores the normative character of mathematics and logic. These subjects are not concerned, as psychology and anthropology are, with how people reason and calculate; they are concerned with how we should reason and calculate.

Realism and Logicism

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Frege’s criticisms of empiricism and psychologism prepare the way for his own view about the nature and status of numbers and the truths of arithmetic. Frege is a realist (or Platonist) about numbers; he maintains that they exist independently of our knowledge of them and that they have properties that are independent of our knowledge. The truths of arithmetic are, therefore, independent of our recognition of these truths. Numbers are not physical entities, but it does not follow from this that they are subjective mental entities. They are, rather, abstract logical objects. Numbers are nonactual in the sense that they cannot causally act on our sense organs. However, while not actual, they are objective: The way they are is independent of the way we might happen to think they are; and they are public entities in that all rational beings can apprehend the same numbers and arithmetical truths. Although it may seem that nothing could be both nonactual and objective, Frege points out that we think of things such as the equator in this way. It is not a physical or perceivable line, but for any point on Earth, it is an objective fact whether or not it is on the equator.

In addition to realism, the other main strand of Frege’s philosophy of arithmetic is the thesis that arithmetic can be reduced to logic, a view that has come to be known as logicism. Frege expresses his view using German philosopher Immanuel Kant’s distinction between analytic and synthetic propositions. According to Frege, a proposition is analytic if it is either a truth of logic or follows deductively from truths of logic; it is synthetic otherwise. Frege’s logicism is, then, the thesis that the truths of arithmetic are analytic. While expressing his view using Kantian terminology, Frege disagrees with Kant, who held that the propositions of arithmetic are synthetic. Kant’s reason for denying that the truths of arithmetic are analytic is that they are informative and constitute substantive knowledge, but the stock examples of analytic truths (for example, “all bachelors are unmarried”) hardly seem to have this status. Frege argues that Kant’s view is mistaken because it depends on a simplistic conception of analyticity and fails to recognize that knowledge is extended when we come to see analytic connections we did not see before. Frege suggests that the analytic consequences of a concept are not contained in it like the beams in a house, apparent to us through casual observation, but as plants are contained in their seeds. By becoming aware of these consequences through deductive reasoning, we acquire knowledge we did not have by merely knowing the meanings of the terms.

Frege’s logicism can be thought of as having two main stages: Numbers and arithmetical operations are first defined in terms of logical concepts, then the truths of arithmetic are deduced from axioms of a purely logical character. The first step in defining numbers in purely logical terms is the observation that an ascription of number is an assertion about a concept. For Frege, the expression “the number belonging to the concept F” refers to the number of objects falling under F. For example, the number belonging to the concept “natural satellite of Mars” is two because Mars has two moons. He next notes that a fundamental type of assertion involving numbers is when two concepts are said to have the same number belonging to them: “The number belonging to the concept F is the same as the number belonging to the concept G.” What Frege wants to do is explain how the content of this proposition can be expressed without mentioning numbers. His basic insight is that two concepts have the same number belonging to them if the objects falling under the concepts can be placed in a one-to-one correspondence. For example, the number of Gospels is the same as the number of suits in bridge because each Gospel can be paired with one of the suits: Matthew with hearts, Mark with spades, and so on. Thus, the content of a proposition asserting that two concepts have the same number can be expressed without mentioning numbers: “The objects falling under F can be put into a one-to-one correspondence with the objects falling under G.” The notion of a one-to-one correspondence is not a mathematical concept; it is a logical concept. A waiter does not have to know how to count in order to make sure the table has the same number of plates and knives; all he has to do is make sure there is a knife beside each plate. Thus, Frege has devised a way of capturing the content of a proposition asserting that the number of F’s is the same as the number of G’s in purely logical terms. This definition provided Frege with the basis for the rest of his definitions of arithmetical concepts and the ultimate deduction of the truths of arithmetic.

A Paradox

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In 1902, shortly before the second volume of The Basic Laws of Arithmetic was to be published, philosopher Bertrand Russell wrote Frege informing him of a contradiction, or paradox, in his system of logic. Frege’s system assumes that every concept determines a set; the concept “natural satellite of Earth,” for example, determines a set with the Moon as its only member. Two types of sets can be distinguished: sets that are members of themselves (the set of abstract objects is a member of itself because it itself is an abstract object) and sets that are not members of themselves (the set of mammals is not a mammal and so is not a member of itself). The contradiction can be generated by asking whether the set of all sets that are not members of themselves is itself a member of itself: If it is, then it is not; if it is not, then it is. Frege attempted to amend his system so that the contradiction would be blocked, but he soon came to the conclusion that his version of logicism could not be repaired.

Once Frege realized that it was impossible to construct a version of logicism employing only self-evident logical principles that would therefore provide a foundation for the knowledge of arithmetic, he simply lost interest in logicism. The program of logicism continued through the work of Russell and others, though because of the apparatus needed to block the contradiction, it could no longer be held to reduce arithmetic to self-evident axioms that are purely logical in character.

The Foundations of Arithmetic reached beyond the confines of logic and the philosophy of mathematics to general philosophy by influencing the approach philosophers take in dealing with problems. A striking example of this new approach occurs in section 62, in which the issue is how we can apprehend numbers given that they are not physical entities (and so are not perceivable) and given that they are not subjective mental entities available to us through introspection. Frege deals with this epistemological problem by recasting it as the problem of what propositions containing number terms mean. Also once we have an adequate answer to this question, perhaps the central question of The Foundations of Arithmetic, it is possible to maintain that we apprehend numbers by grasping the sense of propositions in which number terms occur. This strategy of translating metaphysical and epistemological issues into questions about language, meaning, and understanding is constitutive of what is known as the analytic tradition in philosophy. The Foundations of Arithmetic can be regarded, therefore, as the fountainhead of analytic philosophy.

Bibliography

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Additional Reading

Beaney, Michael. Frege: Making Sense. London: Duckworth, 1996. An analysis of Frege’s logic and resulting philosophy.

Currie, Gregory. Frege: An Introduction to His Philosophy. Brighton, England: Harvester Press; Totowa, N.J.: Barnes and Noble, 1982. Offers an introduction to the central points of Gottlob Frege’s philosophical program and traces the historical development of his views. It provides a very clear explanation of Frege’s formal system, philosophy of mathematics, and philosophical logic. Recommended for advanced undergraduates.

Dummett, Michael. Frege: Philosophy of Language. 2d ed. Cambridge, Mass.: Harvard University Press, 1981. Although Dummett’s interpretations are hotly contested, this magisterial study is the definitive work on Frege. A very long work, it presupposes knowledge of the main currents of analytic philosophy and of symbolic logic in places, but the summaries impart key arguments very clearly. For advanced undergraduates.

Dummett, Michael. The Interpretation of Frege’s Philosophy. Cambridge, Mass.: Harvard University Press, 1987. This book is a reply to criticisms of Dummett’s Frege: Philosophy of Language. It is also shorter, less technical, and places Frege in a broader historical setting, thus making it suitable as an introduction.

Grossman, Reinhardt. Reflections on Frege’s Philosophy. Evanston, Ill.: Northwestern University Press, 1969. The author sees “On Sense and Reference” as the solution—though not a faultless one—to a number of problems emerging from Frege’s earlier work. This is a superb introductory text for the advanced undergraduate.

Kenny, Anthony. Frege. New York: Penguin Books, 1995. This marvelous survey briefly explains and assesses the full range of Frege’s work. Accurate but nontechnical, this is perhaps the most accessible introduction to Frege for the student or general reader.

Kneale, William, and Martha Kneale. The Development of Logic. Oxford: Clarendon Press, 1962. Though lacking the benefit of recent scholarship, this classic and authoritative text contains a profound yet accessible analysis of Frege’s career, his relation to his predecessors, and his theory of number. It devotes a chapter to his general logic, including expositions of his Conceptual Notation, the theory of sense and reference, and The Basic Laws of Arithmetic.

Schirn, Matthias, ed. Frege: Importance and Legacy. New York: Walter de Gruyter, 1996. This book examines Frege’s analytical philosophy and its lasting legacy.

Sluga, Hans D. Gottlob Frege. Boston: Routledge and Kegan Paul, 1980. A readable account of Frege’s theories set against the philosophical concerns of late nineteenth century Germany. Traces the influences of Gottfried Wilhelm Leibniz, Immanuel Kant, and later philosophers on Frege and his contemporaries.

Walker, Jeremy D. B. A Study of Frege. Ithaca, N.Y.: Cornell University Press, 1965. A thoughtful exposition, especially strong on the philosophy of language of Frege’s middle and late periods. Suitable for advanced undergraduates.

Weiner, Joan. Frege in Perspective. Ithaca, N.Y.: Cornell University Press, 1990. Though not expansive enough to serve as an introduction, this study is not addressed solely to specialists. It discusses Frege’s writings on number theory and the laws of thought and seeks to discover Frege’s motivation for attempting to prove the truths of arithmetic.

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