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4 ideas
20660 | At one level maths and nature are very similar, suggesting some deeper origin [Wolfram] |
Full Idea: At some rather abstract level one can immediately recognise one basic similarity between nature and mathematics ...this suggests that the overall similarity between mathematics and nature must have a deeper origin. | |
From: Stephen Wolfram (A New Kind of Science [2002], p.772), quoted by Peter Watson - Convergence 17 'Philosophy' | |
A reaction: Personally I think mathematics has been derived by abstracting from the patterns in nature, and then further extrapolating from those abstractions. So the puzzle in nature is not the correspondence with mathematics, but the patterns. |
17890 | There are at least eleven types of large cardinal, of increasing logical strength [Koellner] |
Full Idea: Some of the standard large cardinals (in order of increasing (logical) strength) are: inaccessible, Mahlo, weakly compact, indescribable, Erdös, measurable, strong, Wodin, supercompact, huge etc. (...and ineffable). | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) | |
A reaction: [I don't understand how cardinals can have 'logical strength', but I pass it on anyway] |
17887 | PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner] |
Full Idea: To the extent that we are justified in accepting Peano Arithmetic we are justified in accepting its consistency, and so we know how to expand the axiom system so as to overcome the limitation [of Gödel's Second Theorem]. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.1) | |
A reaction: Each expansion brings a limitation, but then you can expand again. |
17891 | Arithmetical undecidability is always settled at the next stage up [Koellner] |
Full Idea: The arithmetical instances of undecidability that arise at one stage of the hierarchy are settled at the next. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) |