Ideas of Henri Poincaré, by Theme

[French, 1854 - 1912, Professor of Mathematics at the Sorbonne.]

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6. Mathematics / A. Nature of Mathematics / 2. Geometry
One geometry cannot be more true than another
     Full Idea: One geometry cannot be more true than another; it can only be more convenient.
     From: Henri Poincaré (Science and Method [1908], p.65), quoted by Stewart Shapiro - Philosophy of Mathematics
     A reaction: This is the culminating view after new geometries were developed by tinkering with Euclid's parallels postulate.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
Poincaré rejected the actual infinite, claiming definitions gave apparent infinity to finite objects
     Full Idea: Poincaré rejected the actual infinite. He viewed mathematics that is apparently concerned with the actual infinite as actually concerning the finite linguistic definitions the putatively describe actually infinite objects.
     From: report of Henri Poincaré (On the Nature of Mathematical Reasoning [1894]) by Shaughan Lavine - Understanding the Infinite
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
Mathematicians do not study objects, but relations between objects
     Full Idea: Mathematicians do not study objects, but relations between objects; it is a matter of indifference if the objects are replaced by others, provided the relations do not change. They are interested in form alone, not matter.
     From: Henri Poincaré (Science and Hypothesis [1902], p.20), quoted by E Reck / M Price - Structures and Structuralism in Phil of Maths §6
     A reaction: This connects modern structuralism with Aritotle's interest in the 'form' of things. Contrary to the views of the likes of Frege, it is hard to see that the number '7' has any properties at all, apart from its relations. A daffodil would do just as well.
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
Convention, yes! Arbitrary, no!
     Full Idea: Poincaré once exclaimed, 'Convention, yes! Arbitrary, no!'.
     From: report of Henri Poincaré (talk [1901]) by Hilary Putnam - Models and Reality
     A reaction: An interesting view. It mustn't be assumed that conventions are not rooted in something. Maybe a sort of pragmatism is implied.
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
Avoid non-predicative classifications and definitions
     Full Idea: Never consider any objects but those capable of being defined in a finite number of word ...Avoid non-predicative classifications and definitions.
     From: Henri Poincaré (The Logic of Infinity [1909], p.63), quoted by Penelope Maddy - Naturalism in Mathematics II.4
26. Natural Theory / D. Laws of Nature / 11. Against Laws of Nature
The aim of science is just to create a comprehensive, elegant language to describe brute facts
     Full Idea: In Poincaré's view, we try to construct a language within which the brute facts of experience are expressed as comprehensively and as elegantly as possible. The job of science is the forging of a language precisely suited to that purpose.
     From: report of Henri Poincaré (The Value of Science [1906], Pt III) by Rom Harré - Laws of Nature 2
     A reaction: I'm often struck by how obscure and difficult our accounts of self-evident facts can be. Chairs are easy, and the metaphysics of chairs is hideous. Why is that? I'm a robust realist, but I like Poincaré's idea. He permits facts.