Combining Philosophers

Ideas for Pythagoras, Michael Hallett and Stephen Hetherington

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2 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / m. One

7485	For Pythagoreans 'one' is not a number, but the foundation of numbers [Pythagoras, by Watson]
	Full Idea: For Pythagoreans, one, 1, is not a true number but the 'essence' of number, out of which the number system emerges.
	From: report of Pythagoras (reports [c.530 BCE], Ch.8) by Peter Watson - Ideas Ch.8
	A reaction: I think this is right! Counting and numbers only arise once the concept of individuality and identity have arisen. Counting to one is no more than observing the law of identity. 'Two' is the big adventure.

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis

17836	The General Continuum Hypothesis and its negation are both consistent with ZF [Hallett,M]
	Full Idea: In 1938, Gödel showed that ZF plus the General Continuum Hypothesis is consistent if ZF is. Cohen showed that ZF and not-GCH is also consistent if ZF is, which finally shows that neither GCH nor ¬GCH can be proved from ZF itself.
	From: Michael Hallett (Introduction to Zermelo's 1930 paper [1996], p.1217)