# Frege: Philosophy of Mathematics.

Frege is considered by many to be the founder of analytic
philosophy, but his importance for the philosophy of mathematics is
unrecognized. Despite his inability to deal with the paradoxes pointed
out by Russell, personal blind spots (fierce opposition to non-Euclidean
geometry, and failure to recognize the merit of the work of Hilbert,
Cantor, and Dedekind), and various serious philosophical shortcomings,
he is, says Dummett, "the greatest philosopher of mathematics yet
to have written" (p. 321). Not everything of interest on this
topic is given here; rather, Dummett means to go "to the heart of
Frege's philosophy of mathematics, setting aside everything not of
central importance for that purpose" (p. xi).

The initial focus is on Die Grundlagen der Arithmetik, "Frege's masterpiece" (p. 1), written as a prolegomenon to Frege's intended magnum opus, Grundgestze der Arithmetik. For Grundlagen--itself continuous with the work of the still earlier Begriffschrift--presents the philosophical underpinnings of Grundgestze, and so expresses Frege's mature philosophy of arithmetic. Frege's aim in Grundlagen is to show that the truths of arithmetic are analytic, and this requires identifying the principles and concepts basic to number theory; much of this work is taken up with analyzing numerical and arithmetical concepts. Frege's notion of "sense" has a prominent role here, for example, in his struggles with Husserl's psychologism. Interesting is Frege's view on analytic propositions, that they "encapuslate an inferential subroutine" (p. 36). Thus, dissection of a proposition reveals something new and is a process of concept formation; deductive reasoning requires the apprehension of such patterns and thus is creative. So Frege offers a solution to the problem of why deductive reasoning can reveal new knowledge--a problem rarely addressed.

A recurring theme is the status of numbers as objects. Dedekind and others hold that mathematical objects are "free creations of the human mind" (p. 49, quoting Dedekind). The resulting conceptions are not solipsistic because each creator will have performed analogous operations--a view rejected by Frege on the grounds that there is no means of comparing private mental contents. For him, judgeable mathematical contents are objective and communicable, the same for everyone. Dedekind believes that natural numbers have only the properties resulting from their positions in the order generated by the act of though creating them. But Frege holds that natural numbers are intimately connected with their applications and thus not solely identified by position: one gives a natural number in answer to the question, How many objects are there which satisfy a certain condition? Nevertheless, Frege is a staunch realist about numbers: given any domain of mathematical objects, statements resulting from quantification over it will be either true or false. Dummett objects to this; the paradoxes have revealed that some concepts are "indefinitely extensible" (p. 317) in not defining a definite totality. Thus we are forced to adopt a constructivist view of mathematical practice, a suggestion with which Dummett concludes his book.

There is much in this densely written but moderately sized volume: Frege's definition of a cardinal number and his claim that substantival uses of numerical expressions have primacy over adjectival ones; his adoption (in what is "arguably the most philosophically pregnant paragraph ever written" [p. 111] [Grundlagen, sec. 62]) of the linguistic turn in which numbers are defined contextually; the reason numbers have to be objects (because their domain must be infinite); Frege on the paradox and direction of analysis; and Frege's views on Kant, Mill, Dedekind, Hilbert, Cantor, Husserl, and Russell, and Whitehead. Dummett brings the discussion down to the present with comments on philosophers such as Waismann, Wittgenstein, Benacerraf, Quine, and Crispin Wright. Those wondering whether mathematics really needs the kind of foundation Frege attempts to provide will not be satisfied with Dummett's brief remarks here, but there is no doubt that he has provided an important exposition of the more distinctively mathematical side of Frege's thought.

The initial focus is on Die Grundlagen der Arithmetik, "Frege's masterpiece" (p. 1), written as a prolegomenon to Frege's intended magnum opus, Grundgestze der Arithmetik. For Grundlagen--itself continuous with the work of the still earlier Begriffschrift--presents the philosophical underpinnings of Grundgestze, and so expresses Frege's mature philosophy of arithmetic. Frege's aim in Grundlagen is to show that the truths of arithmetic are analytic, and this requires identifying the principles and concepts basic to number theory; much of this work is taken up with analyzing numerical and arithmetical concepts. Frege's notion of "sense" has a prominent role here, for example, in his struggles with Husserl's psychologism. Interesting is Frege's view on analytic propositions, that they "encapuslate an inferential subroutine" (p. 36). Thus, dissection of a proposition reveals something new and is a process of concept formation; deductive reasoning requires the apprehension of such patterns and thus is creative. So Frege offers a solution to the problem of why deductive reasoning can reveal new knowledge--a problem rarely addressed.

A recurring theme is the status of numbers as objects. Dedekind and others hold that mathematical objects are "free creations of the human mind" (p. 49, quoting Dedekind). The resulting conceptions are not solipsistic because each creator will have performed analogous operations--a view rejected by Frege on the grounds that there is no means of comparing private mental contents. For him, judgeable mathematical contents are objective and communicable, the same for everyone. Dedekind believes that natural numbers have only the properties resulting from their positions in the order generated by the act of though creating them. But Frege holds that natural numbers are intimately connected with their applications and thus not solely identified by position: one gives a natural number in answer to the question, How many objects are there which satisfy a certain condition? Nevertheless, Frege is a staunch realist about numbers: given any domain of mathematical objects, statements resulting from quantification over it will be either true or false. Dummett objects to this; the paradoxes have revealed that some concepts are "indefinitely extensible" (p. 317) in not defining a definite totality. Thus we are forced to adopt a constructivist view of mathematical practice, a suggestion with which Dummett concludes his book.

There is much in this densely written but moderately sized volume: Frege's definition of a cardinal number and his claim that substantival uses of numerical expressions have primacy over adjectival ones; his adoption (in what is "arguably the most philosophically pregnant paragraph ever written" [p. 111] [Grundlagen, sec. 62]) of the linguistic turn in which numbers are defined contextually; the reason numbers have to be objects (because their domain must be infinite); Frege on the paradox and direction of analysis; and Frege's views on Kant, Mill, Dedekind, Hilbert, Cantor, Husserl, and Russell, and Whitehead. Dummett brings the discussion down to the present with comments on philosophers such as Waismann, Wittgenstein, Benacerraf, Quine, and Crispin Wright. Those wondering whether mathematics really needs the kind of foundation Frege attempts to provide will not be satisfied with Dummett's brief remarks here, but there is no doubt that he has provided an important exposition of the more distinctively mathematical side of Frege's thought.

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Author: | Crittenden, Charles |
---|---|

Publication: | The Review of Metaphysics |

Article Type: | Book Review |

Date: | Mar 1, 1993 |

Words: | 639 |

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