Edmund Husserl on the applicability of formal geometry. (English) Zbl 1158.00005

Carson, Emily (ed.) et al., Intuition and the axiomatic method. Dordrecht: Springer (ISBN 978-1-4020-4039-9/hbk). Western Ontario Series in Philosophy of Science 70, 67-85 (2006).
The paper deals with E. Husserl’s philosophy of geometry. Husserl combined a phenomenological approach with D. Hilbert’s formal axiomatic foundation of Euclidean geometry. According to the author, for Husserl “one of the most important questions arising from Hilbert’s approach to geometry was whether such purely formal enquiry had any relevance beyond a pure theory of deductive systems” (p. 67). This question was closely connected with two applicability problems: (1) How can a given formal concept be applied in the physical sciences? (2) How does formal inquiry apply in pre-scientific experience? (p. 68)
Husserl distinguished between different concepts of space: the space of intuition (“space of everyday-life”), the space of pure geometry (geometric space), the space of applied geometry (“space of nature science”) and, after having become acquainted with Hilbert’s theory, formal space. The applicability problem can be seen as the problem to determine the relevance of formal geometry for the intuitive space conception. Husserl regards the formal inquiry as a tool for mathematical discovery intended for replacing material inquiry (p. 73). The result of a formal inquiry can be applied to intuitive space via geometric space.
In the core of paper the author discusses Husserl’s concept of a definite manifold which is the fundamental device for Husserl’s solution. A definite manifold is a domain of sentences which can all be derived logically from of finite number of axioms. Husserl elaborated this concept in connection with the foundations of arithmetic, but also applied it to geometry. “Husserl was looking for a condition that would make a formal theory perfect to such a degree that it could only be extended conservatively” (p. 77). Husserl’s main result is “Every extension of a formal axiomatic theory, which defines a manifold tat is definite in the absolute sense, is a conservative extension” (p. 78). Such conservative extensions could solve, as Husserl hoped, the applicability pro blem. This solution failed because of Gödel’s incompleteness results, but Husserl’s solution “should be valued as an attempt to give a criterion that secures the connection between concepts of space at different levels of abstraction”(p. 80).
For the entire collection see [Zbl 1140.03003].


00A30 Philosophy of mathematics
01A60 History of mathematics in the 20th century