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).

Others’ Notions Refuted

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...

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Realism and Logicism

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...

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A Paradox

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...

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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...

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