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Ideas for 'Classical Cosmology (frags)', 'Introduction to the Theory of Logic' and 'Science and Hypothesis'

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2 ideas

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction

10891	If a set is defined by induction, then proof by induction can be applied to it [Zalabardo]
	Full Idea: Defining a set by induction enables us to use the method of proof by induction to establish that all the elements of the set have a certain property.
	From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.3)

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism

10180	Mathematicians do not study objects, but relations between objects [Poincaré]
	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.