departmental bulletin paper

Departmental Bulletin Paper

Text

Osaka Human Sciences. 2025, 11, p. 31-43

From the Axiomatic Method of Hilbert’s Mathematics to Cavaillès’s Philosophy of the Concept

Kondo, Kazunori

Graduate School of Human Sciences, Osaka University

English

Text

2025-03

We support the thesis of the fruitfulness of the rationalist philosophy of science, as opposed to the naturalist philosophy of science that is recognised as dominant today. Our position, as far as rationalist philosophy is concerned, is that presented by Cavaillès under the title of “philosophy of the concept”, inspired by the Hilbertian axiomatic method. Here are a few points of clarification concerning the assertions in our article: 1. Hilbert does not see the axiomatic system as a mere convention. Thus, he thinks that the axiomatic system has a content of its own in itself that is distinct from the particular object. The latter is relative to the axiomatic system (formal system) that formalizes it. 2. The Hilbertian method makes the existence of this formal content possible. Using this method, Hilbert tried to justify infinitist mathematics, such as set theory. For Hilbert, the condition for this justification is to show that it does not entail contradiction (non-contradiction requirement). 3. One of Hilbert’s greatest discoveries is that this proper formal, or abstract structure, is relative to the axiomatic system, and we can make this structure analysable through it. 4. The philosophy of the concept comes from the Hilbertian perspective on the axiomatic method. These features are: 1. the abandonment of the absolute foundation (nonfoundationalism), 2. the abstract structure relative to the external formal system, 3. the powers of the expansion of the abstract structure, expanding and modifying the external formal system, 4. Relative truth concerning the formal system that specifies it, 5. Progressive revelation of truth with the thematization and expansion of the abstract structure, 6. This thematization, which presupposes the earlier structure, and then contains it in the later structure, is therefore not arbitrary (non-relativism).

10.18910/100847

100847

From the Axiomatic Method of Hilbert’s Mathematics to Cavaillès’s Philosophy of the Concept

Kondo, Kazunori

Graduate School of Human Sciences, Osaka University

English

2025-03

Osaka Human Sciences

11

31

43

https://hdl.handle.net/11094/100847

10.18910/100847

https://doi.org/10.18910/100847