Ideas from 'Russell' by A.C. Grayling [1996], by Theme Structure

[found in 'Russell' by Grayling,A.C. [OUP 1996,0-19-287683-x]].

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6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism

6408	Russell needed three extra axioms to reduce maths to logic: infinity, choice and reducibility

			Full Idea: In order to deduce the theorems of mathematics from purely logical axioms, Russell had to add three new axioms to those of standards logic, which were: the axiom of infinity, the axiom of choice, and the axiom of reducibility.
			From: A.C. Grayling (Russell [1996], Ch.2)
			A reaction: The third one was adopted to avoid his 'barber' paradox, but many thinkers do not accept it. The interesting question is why anyone would 'accept' or 'reject' an axiom.

12. Knowledge Sources / A. A Priori Knowledge / 2. Self-Evidence

6414	Two propositions might seem self-evident, but contradict one another

			Full Idea: Two propositions might contradict each other despite appearing self-evident when considered separately.
			From: A.C. Grayling (Russell [1996], Ch.2)
			A reaction: Russell's proposal (Idea 5416) is important here, that self-evidence comes in degrees. If self-evidence was all-or-nothing, Grayling's point would be a major problem, but it isn't. Bonjour explores the idea more fully (e.g. Idea 3704)