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Ideas for 'talk', 'Remarks on the Foundations of Mathematics' and 'On Second-Order Logic'

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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order

10833	Many concepts can only be expressed by second-order logic [Boolos]
	Full Idea: The notions of infinity and countability can be characterized by second-order sentences, though not by first-order sentences (as compactness and Skolem-Löwenheim theorems show), .. as well as well-ordering, progression, ancestral and identity.
	From: George Boolos (On Second-Order Logic [1975], p.48)

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

11073	Two and one making three has the necessity of logical inference [Wittgenstein]
	Full Idea: "But doesn't it follow with logical necessity that you get two when you add one to one, and three when you add one to two? and isn't this inexorability the same as that of logical inference? - Yes! it is the same.
	From: Ludwig Wittgenstein (Remarks on the Foundations of Mathematics [1938], p.38), quoted by Robert Hanna - Rationality and Logic 6
	A reaction: This need not be a full commitment to logicism - only to the fact that the inferential procedures in mathematics are the same as those of logic. Mathematics could still have further non-logical ingredients. Indeed, I think it probably does.