Structural analogies between mathematical and empirical theories. (English) Zbl 0874.00011

Ibarra, Andoni; Mormann, Thomas

Echeverria, Javier (ed.) et al., The space of mathematics. Philosophical, epistemological, and historical explorations. Revised papers from a symposium on structures in mathematical theories, Donostia/San Sebastian, Basque Country, Spain, September 1990. Berlin: Walter de Gruyter. Grundlagen der Kommunikation und Kognition. 31-46 (1992).

The authors propose a new concept for dealing with the existing analogies between mathematical and empirical sciences, especially physics. As an alternative to the “methodological analogy” of Lakatos’s methodology of scientific research programmes and the “functional analogy” of Quine’s holism they present a structural analogy, based in the representational character of cognition in general: “the representational character of cognition is not restricted to scientific knowledge but pervades all kinds of cognition, e.g. perception and measurement” (p. 32).
The authors discuss the representational structure of empirical theories following Henry Margenau’s distinction between two levels of physical conceptualization: the level of data and the level of symbolic constructs [cf. “Methodology of modern physics”, Philos. Sci. 2, 48-72, 164-187 (1935)]. They maintain that physical explanation usually starts in the range of data, swings over to the field of symbolic construction, and then returns to data, a movement which the authors call “swing” (p. 38).
A discussion of group theory is used to prove that the elements of data, symbolic constructs and swing also occur in mathematics. Category theory is presented as a tool for the reconstruction of the representational character of mathematical theories. This approach avoids the weaknesses of several positions in the philosophy of mathematics such as elementarism, fundamentalism and ontologism because it is much closer to the conceptions of the working scientists: “The relative, context-dependent characterization of mathematical entities as data and symbolic constructs \([\dots ]\) leads the representational approach to a distributed and variegated ontology of the objects of mathematical discourse” (p. 43).
For the entire collection see [Zbl 0839.00019].

Reviewer: V.Peckhaus (Erlangen)

MSC:

00A30	Philosophy of mathematics
00A79	Physics

Keywords:

Representational structure; cognition; data; symbolic constructs; swing; group theory; category theory; elementarism; fundamentalism; ontologism