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Ideas for 'fragments/reports', 'Semantic Necessity' and 'Philosophy of Mathematics'

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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic

23446	You can't prove consistency using a weaker theory, but you can use a consistent theory [Linnebo]
	Full Idea: If the 2nd Incompleteness Theorem undermines Hilbert's attempt to use a weak theory to prove the consistency of a strong one, it is still possible to prove the consistency of one theory, assuming the consistency of another theory.
	From: Øystein Linnebo (Philosophy of Mathematics [2017], 4.6)
	A reaction: Note that this concerns consistency, not completeness.

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism

23448	Mathematics is the study of all possible patterns, and is thus bound to describe the world [Linnebo]
	Full Idea: Philosophical structuralism holds that mathematics is the study of abstract structures, or 'patterns'. If mathematics is the study of all possible patterns, then it is inevitable that the world is described by mathematics.
	From: Øystein Linnebo (Philosophy of Mathematics [2017], 11.1)
	A reaction: [He cites the physicist John Barrow (2010) for this] For me this is a major idea, because the concept of a pattern gives a link between the natural physical world and the abstract world of mathematics. No platonism is needed.