4 ideas
| 15927 | Definition just needs negation, known variables, conjunction, disjunction, substitution and quantification [Weyl, by Lavine] |
| Full Idea: For mathematics, Weyl arrived (by 1917) at a satisfactory list of definition principles: negation, identification of variables, conjunction, disjunction, substitution of constants, and existential quantification over the domain. | |
| From: report of Hermann Weyl (works [1917]) by Shaughan Lavine - Understanding the Infinite V.3 | |
| A reaction: Lavine summarises this as 'first-order logic with parameters'. |
| 20653 | Six reduction levels: groups, lives, cells, molecules, atoms, particles [Putnam/Oppenheim, by Watson] |
| Full Idea: There are six 'reductive levels' in science: social groups, (multicellular) living things, cells, molecules, atoms, and elementary particles. | |
| From: report of H.Putnam/P.Oppenheim (Unity of Science as a Working Hypothesis [1958]) by Peter Watson - Convergence 10 'Intro' | |
| A reaction: I have the impression that fields are seen as more fundamental that elementary particles. What is the status of the 'laws' that are supposed to govern these things? What is the status of space and time within this picture? |
| 9567 | Maths deals with quantities of physical significance, ignoring irrelevant features [Geroch] |
| Full Idea: Mathematics can serve to provide a framework within which one deals only with quantities of physical significance, ignoring other, irrelevant things. | |
| From: Robert Geroch (Mathematical Physics [1985], p.1), quoted by Charles Chihara - A Structural Account of Mathematics 9.8 | |
| A reaction: This is a modern physicist espousing abstractionism, as derided and dismissed by Frege and Geach. It's common sense, really. |
| 10246 | The limit of science is isomorphism of theories, with essences a matter of indifference [Weyl] |
| Full Idea: A science can determine its domain of investigation up to an isomorphic mapping. It remains quite indifferent as to the 'essence' of its objects. The idea of isomorphism demarcates the self-evident boundary of cognition. | |
| From: Hermann Weyl (Phil of Mathematics and Natural Science [1949], 25-7), quoted by Stewart Shapiro - Philosophy of Mathematics | |
| A reaction: Shapiro quotes this in support of his structuralism, but it is a striking expression of the idea that if there are such things as essences, they are beyond science. I take Weyl to be wrong. Best explanation reaches out beyond models to essences. |