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Ideas for 'Mathematical Methods in Philosophy', 'What are Sets and What are they For?' and 'Mathematics and Philosophy: grand and little'

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4 ideas

6. Mathematics / A. Nature of Mathematics / 1. Mathematics
9812 In mathematics, if a problem can be formulated, it will eventually be solved [Badiou]
     Full Idea: Only in mathematics can one unequivocally maintain that if thought can formulate a problem, it can and will solve it, regardless of how long it takes.
     From: Alain Badiou (Mathematics and Philosophy: grand and little [2004], p.17)
     A reaction: I hope this includes proving the Continuum Hypothesis, and Goldbach's Conjecture. It doesn't seem quite true, but it shows why philosophers of a rationalist persuasion are drawn to mathematics.
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
14246 If mathematics purely concerned mathematical objects, there would be no applied mathematics [Oliver/Smiley]
     Full Idea: If mathematics was purely concerned with mathematical objects, there would be no room for applied mathematics.
     From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.1)
     A reaction: Love it! Of course, they are using 'objects' in the rather Fregean sense of genuine abstract entities. I don't see why fictionalism shouldn't allow maths to be wholly 'pure', although we have invented fictions which actually have application.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
9813 Mathematics shows that thinking is not confined to the finite [Badiou]
     Full Idea: Mathematics teaches us that there is no reason whatsoever to confne thinking within the ambit of finitude.
     From: Alain Badiou (Mathematics and Philosophy: grand and little [2004], p.19)
     A reaction: This would perhaps make Cantor the greatest thinker who ever lived. It is an exhilarating idea, but we should ward the reader against romping of into unrestrained philosophical thought about infinities. You may be jumping without your Cantorian parachute.
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
14247 Sets might either represent the numbers, or be the numbers, or replace the numbers [Oliver/Smiley]
     Full Idea: Identifying numbers with sets may mean one of three quite different things: 1) the sets represent the numbers, or ii) they are the numbers, or iii) they replace the numbers.
     From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.2)
     A reaction: Option one sounds the most plausible to me. I will take numbers to be patterns embedded in nature, and sets are one way of presenting them in shorthand form, in order to bring out what is repeated.