Combining Texts

Ideas for 'On the Question of Absolute Undecidability', 'The Statesman' and 'A Survey of Metaphysics'

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5 ideas

6. Mathematics / A. Nature of Mathematics / 1. Mathematics
4240 It might be argued that mathematics does not, or should not, aim at truth [Lowe]
     Full Idea: It might be argued that mathematics does not, or should not, aim at truth.
     From: E.J. Lowe (A Survey of Metaphysics [2002], p.375)
     A reaction: Intriguing. Sounds wrong to me. At least maths seems to need the idea of the 'correct' answer. If, however, maths is a huge pattern, there is no correctness, just the pattern. We can be wrong, but maths can't be wrong. Ah, I see…!
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
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]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
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.
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
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)
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
4241 If there are infinite numbers and finite concrete objects, this implies that numbers are abstract objects [Lowe]
     Full Idea: The Peano postulates imply an infinity of numbers, but there are probably not infinitely many concrete objects in existence, so natural numbers must be abstract objects.
     From: E.J. Lowe (A Survey of Metaphysics [2002], p.375)
     A reaction: Presumably they are abstract objects even if they aren't universals. 'Abstract' is an essential term in our ontological vocabulary to cover such cases. Perhaps possible concrete objects are infinite.