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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.
Gist of Idea
Mathematicians do not study objects, but relations between objects
Source
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.
| 18203 | Avoid non-predicative classifications and definitions [Poincaré] |
| 15923 | Poincaré rejected the actual infinite, claiming definitions gave apparent infinity to finite objects [Poincaré, by Lavine] |
| 10180 | Mathematicians do not study objects, but relations between objects [Poincaré] |
| 10245 | One geometry cannot be more true than another [Poincaré] |
| 9916 | Convention, yes! Arbitrary, no! [Poincaré, by Putnam] |
| 15877 | The aim of science is just to create a comprehensive, elegant language to describe brute facts [Poincaré, by Harré] |