Mathematical explanation: problems and prospects.

*(English)*Zbl 1141.00007The author deals with the vague concept of mathematical explanation in its relation to mathematical proof and explanation in science. The paper is divided into three parts.

The first part is devoted to conceptual clarifications, in the second part different positions concerning explanations in mathematics are presented. The focus is on an important tradition in the philosophy of mathematics which the author calls “hypothetico-inductivist”. The third part gives an example: Alfred Pringsheim’s explanatory approach to the foundations of complex analysis.

Part 1 starts with an example taken over from G. Bouligand, three proofs of Pythagoras’ theorem: Euclid’s own proof, an explanatory proof, and an intuitive, but not explanatory proof. The author considers it as problematic to give an clear definition of “explanation” covering all the different uses of this term. He hints at D. Sandborg’s observation that in mathematical practice mathematical phenomena are explained by reference to case studies from contemporary mathematics [cf. Br. J. Philos. Sci. 49, No. 4, 603–624 (1998; Zbl 0945.03504)]. To complement this approach, he wants to pay attention to forms of explanation in mathematics “in which a particular presentation of a theory provides the natural explanation for its results” (p. 101).

The author sees a connection between scientific and mathematical explanation in the concept of hypothetico-inductivism (h-inductivism), discussed in part 2. “H-inductivism is, roughly, a conception of mathematics which asserts that the acceptance of axioms for a mathematical discipline might be motivated not by criteria of evidence and certainty but rather, like hypotheses in physics, by their success in deriving and systematizing a certain number of familiar consequences. In this sense the consequences are often more evident than the axioms we are appealing to in deriving them” (p. 103–04). Typical examples for h-inductivism are J. St. Mill’s claim that mathematics presents explanations just as any empirical science does, a consequence of his empiricism not to be confused with it. B. Russell [especially in his paper “The Regressive method of discovery the premises of mathematics” (1907)] and K. Gödel are presented as h-inductivists such as I. Lakatos with his “quasi-empiricism” and R. Hersh’s fallibilist approach to the philosophy of mathematics.

The paper is closed in part 3 with a discussion of A. Pringsheim’s approach to complex analysis as presented in his Vorlesungen über Funktionenlehre [1916–1932; cf. JFM 46.0221.05; JFM 46.0318.05; JFM 46.0318.06; JFM 51.0237.01; JFM 58.0296.18] which is based on the claim “that only according to his method it is possible to ‘explain’ a great number of results, which in previous approaches, in particular Cauchy’s, remain mysterious and unexplained” (p. 108). The main conceptual tool is Pringsheim’s notion of mean value of a function.

The first part is devoted to conceptual clarifications, in the second part different positions concerning explanations in mathematics are presented. The focus is on an important tradition in the philosophy of mathematics which the author calls “hypothetico-inductivist”. The third part gives an example: Alfred Pringsheim’s explanatory approach to the foundations of complex analysis.

Part 1 starts with an example taken over from G. Bouligand, three proofs of Pythagoras’ theorem: Euclid’s own proof, an explanatory proof, and an intuitive, but not explanatory proof. The author considers it as problematic to give an clear definition of “explanation” covering all the different uses of this term. He hints at D. Sandborg’s observation that in mathematical practice mathematical phenomena are explained by reference to case studies from contemporary mathematics [cf. Br. J. Philos. Sci. 49, No. 4, 603–624 (1998; Zbl 0945.03504)]. To complement this approach, he wants to pay attention to forms of explanation in mathematics “in which a particular presentation of a theory provides the natural explanation for its results” (p. 101).

The author sees a connection between scientific and mathematical explanation in the concept of hypothetico-inductivism (h-inductivism), discussed in part 2. “H-inductivism is, roughly, a conception of mathematics which asserts that the acceptance of axioms for a mathematical discipline might be motivated not by criteria of evidence and certainty but rather, like hypotheses in physics, by their success in deriving and systematizing a certain number of familiar consequences. In this sense the consequences are often more evident than the axioms we are appealing to in deriving them” (p. 103–04). Typical examples for h-inductivism are J. St. Mill’s claim that mathematics presents explanations just as any empirical science does, a consequence of his empiricism not to be confused with it. B. Russell [especially in his paper “The Regressive method of discovery the premises of mathematics” (1907)] and K. Gödel are presented as h-inductivists such as I. Lakatos with his “quasi-empiricism” and R. Hersh’s fallibilist approach to the philosophy of mathematics.

The paper is closed in part 3 with a discussion of A. Pringsheim’s approach to complex analysis as presented in his Vorlesungen über Funktionenlehre [1916–1932; cf. JFM 46.0221.05; JFM 46.0318.05; JFM 46.0318.06; JFM 51.0237.01; JFM 58.0296.18] which is based on the claim “that only according to his method it is possible to ‘explain’ a great number of results, which in previous approaches, in particular Cauchy’s, remain mysterious and unexplained” (p. 108). The main conceptual tool is Pringsheim’s notion of mean value of a function.

Reviewer: Volker Peckhaus (Paderborn)

##### MSC:

00A30 | Philosophy of mathematics |

03A05 | Philosophical and critical aspects of logic and foundations |