Ideas from 'Mathematics and the Metaphysicians' by Bertrand Russell [1901], by Theme Structure

[found in 'Mysticism and Logic' by Russell,Bertrand [Unwin 1989,0-04-824021-4]].

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5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / a. Achilles paradox

7557	To solve Zeno's paradox, reject the axiom that the whole has more terms than the parts

			Full Idea: Presumably Zeno appealed to the axiom that the whole has more terms than the parts; so if Achilles were to overtake the tortoise, he would have been in more places than the tortoise, which he can't be; but the conclusion is absurd, so reject the axiom.
			From: Bertrand Russell (Mathematics and the Metaphysicians [1901], p.89)
			A reaction: The point is that the axiom is normally acceptable (a statue contains more particles than the arm of the statue), but it breaks down when discussing infinity (Idea 7556). Modern theories of infinity are needed to solve Zeno's Paradoxes.

6. Mathematics / A. Nature of Mathematics / 1. Mathematics

10059

In mathematic we are ignorant of both subject-matter and truth

			Full Idea: Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
			From: Bertrand Russell (Mathematics and the Metaphysicians [1901], p.76)
			A reaction: A famous remark, though Musgrave is rather disparaging about Russell's underlying reasoning here.

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / b. Mark of the infinite

7556	A collection is infinite if you can remove some terms without diminishing its number

			Full Idea: A collection of terms is infinite if it contains as parts other collections which have as many terms as it has; that is, you can take away some terms of the collection without diminishing its number; there are as many even numbers as numbers all together.
			From: Bertrand Russell (Mathematics and the Metaphysicians [1901], p.86)
			A reaction: He cites Dedekind and Cantor as source for these ideas. If it won't obey the rule that subtraction makes it smaller, then it clearly isn't a number, and really it should be banned from all mathematics.

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

7554	Self-evidence is often a mere will-o'-the-wisp

			Full Idea: Self-evidence is often a mere will-o'-the-wisp, which is sure to lead us astray if we take it as our guide.
			From: Bertrand Russell (Mathematics and the Metaphysicians [1901], p.78)
			A reaction: The sort of nice crisp remark you would expect from a good empiricist philosopher. Compare Idea 4948. However Russell qualifies it with the word 'often', and all philosophers eventually realise that you have to start somewhere.