Ideas from 'Philosophy of Mathematics' by Leon Horsten [2007], by Theme Structure

[found in 'Stanford Online Encyclopaedia of Philosophy' (ed/tr Stanford University) [plato.stanford.edu ,-]].

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2. Reason / D. Definition / 8. Impredicative Definition

10882

Predicative definitions only refer to entities outside the defined collection

			Full Idea: Definitions are called 'predicative', and are considered sound, if they only refer to entities which exist independently from the defined collection.
			From: Leon Horsten (Philosophy of Mathematics [2007], §2.4)

5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms

10884

A theory is 'categorical' if it has just one model up to isomorphism

			Full Idea: If a theory has, up to isomorphism, exactly one model, then it is said to be 'categorical'.
			From: Leon Horsten (Philosophy of Mathematics [2007], §5.2)

6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics

10885

Computer proofs don't provide explanations

			Full Idea: Mathematicians are uncomfortable with computerised proofs because a 'good' proof should do more than convince us that a certain statement is true. It should also explain why the statement in question holds.
			From: Leon Horsten (Philosophy of Mathematics [2007], §5.3)

6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory

10881

The concept of 'ordinal number' is set-theoretic, not arithmetical

			Full Idea: The notion of an ordinal number is a set-theoretic, and hence non-arithmetical, concept.
			From: Leon Horsten (Philosophy of Mathematics [2007], §2.3)