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All the ideas for 'fragments/reports', 'Philosophy of Mathematics' and 'Intro to I: Classical Logic'

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6 ideas

1. Philosophy / D. Nature of Philosophy / 3. Philosophy Defined

23367	Even pointing a finger should only be done for a reason [Epictetus]
	Full Idea: Philosophy says it is not right even to stretch out a finger without some reason.
	From: Epictetus (fragments/reports [c.57], 15)
	A reaction: The key point here is that philosophy concerns action, an idea on which Epictetus is very keen. He rather despise theory. This idea perfectly sums up the concept of the wholly rational life (which no rational person would actually want to live!).

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

10882	Predicative definitions only refer to entities outside the defined collection [Horsten]
	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 / A. Overview of Logic / 6. Classical Logic

9463	Classical logic is bivalent, has excluded middle, and only quantifies over existent objects [Jacquette]
	Full Idea: Classical logic (of Whitehead, Russell, Gödel, Church) is a two-valued system of propositional and predicate logic, in which all propositions are exclusively true or false, and quantification and predication are over existent objects only.
	From: Dale Jacquette (Intro to I: Classical Logic [2002], p.9)
	A reaction: All of these get challenged at some point, though the existence requirement is the one I find dubious.

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 [Horsten]
	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 [Horsten]
	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 [Horsten]
	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)