Combining Texts

Ideas for 'The Logic of Boundaryless Concepts', 'Mathematical Intuition' and 'Set Theory and the Continuum Hypothesis'

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2 ideas

6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism

18201	General principles can be obvious in mathematics, but bold speculations in empirical science [Parsons,C]
	Full Idea: The existence of very general principles in mathematics are universally regarded as obvious, where on an empiricist view one would expect them to be bold hypotheses, about which a prudent scientist would maintain reserve.
	From: Charles Parsons (Mathematical Intuition [1980], p.152), quoted by Penelope Maddy - Naturalism in Mathematics
	A reaction: This is mainly aimed at Quine's and Putnam's indispensability (to science) argument about mathematics.

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism

14248	We could accept the integers as primitive, then use sets to construct the rest [Cohen]
	Full Idea: A very reasonable position would be to accept the integers as primitive entities and then use sets to form higher entities.
	From: Paul J. Cohen (Set Theory and the Continuum Hypothesis [1966], 5.4), quoted by Oliver,A/Smiley,T - What are Sets and What are they For?
	A reaction: I find this very appealing, and the authority of this major mathematician adds support. I would say, though, that the integers are not 'primitive', but pick out (in abstraction) consistent features of the natural world.