Combining Texts

Ideas for 'Mahaprajnaparamitashastra', 'Guidebook to Wittgenstein's Tractatus' and 'Introduction to Zermelo's 1930 paper'

unexpand these ideas     |    start again     |     choose another area for these texts

display all the ideas for this combination of texts


4 ideas

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
Counting needs to distinguish things, and also needs the concept of a successor in a series [Morris,M]
     Full Idea: Just distinguishing things is not enough for counting (and hence arithmetic). We need the crucial extra notion of the successor in a series of some kind.
     From: Michael Morris (Guidebook to Wittgenstein's Tractatus [2008], Intro.5)
     A reaction: This is a step towards the Peano Axioms of arithmetic. The successors could be fingers and toes, taken in a conventional order, and matched one-to-one to the objects. 'My right big toe of cows' means 16 cows (but non-verbally).
To count, we must distinguish things, and have a series with successors in it [Morris,M]
     Full Idea: Distinguishing between things is not enough for counting. …We need the crucial extra notion of a successor in a series of a certain kind.
     From: Michael Morris (Guidebook to Wittgenstein's Tractatus [2008], Intro)
     A reaction: This is the thinking that led to the Dedekind-Peano axioms for arithmetic. E.g. each series member can only have one successor. There is an unformalisable assumption that the series can then be applied to the things.
Discriminating things for counting implies concepts of identity and distinctness [Morris,M]
     Full Idea: The discrimination of things for counting needs to bring with it the notion of identity (and, correlatively, distinctness).
     From: Michael Morris (Guidebook to Wittgenstein's Tractatus [2008], Intro.5)
     A reaction: Morris is exploring how practices like counting might reveal necessary truths about the world.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
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)