Ideas from 'Maths as a Science of Patterns' by Michael D. Resnik [1997], by Theme Structure

[found in 'Mathematics as Science of Pattern Recognition' by Resnik,Michael D. [OUP 1999,0-19-825014-2]].

Click on the Idea Number for the full details    |     back to texts     |     expand these ideas


4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
Axioms are often affirmed simply because they produce results which have been accepted
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
Mathematical realism says that maths exists, is largely true, and is independent of proofs
6. Mathematics / A. Nature of Mathematics / 5. Geometry
Euclid's geometry is synthetic, but Descartes produced an analytic version of it
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / a. Structuralism
Sets are positions in patterns
Mathematical constants and quantifiers only exist as locations within structures or patterns
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / d. Platonist structuralism
Maths is pattern recognition and representation, and its truth and proofs are based on these
Congruence is the strongest relationship of patterns, equivalence comes next, and mutual occurrence is the weakest
Structuralism must explain why a triangle is a whole, and not a random set of points
There are too many mathematical objects for them all to be mental or physical