Archive for the ‘somewhat philsophical’ Category

Annoyingly brilliant comment to the last post

January 8, 2009

I can see absolutely no flaw in the following argument by Daryl McCullough.

My complaint about rephrasing Godel’s result as saying that the Godel sentence G can neither be proved nor disproved is that, to me, the ontological status of “G cannot be proved” is *exactly* the same as the status of “G is true”. To say that G is true is to say that there does not exist a natural number satisfying such and such a property. To say that G cannot be proved is to say that there does not exist a proof satisfying such and such a property. I don’t see how the latter makes any less ontological commitment than the former.

GARGH! Thus (unless someone can furnish me with a reason why it may be wrong) I probably going to have to accept that there isn’t really a good reason not to say “true”. After all there is no alternative rendering that avoids the same potential confusions and we don’t expect popularisations to define things perfectly – just to give the “shape” of the idea, so any confusion between a technical use of “true” and an everyday one is acceptable (they are different but close enough to get some understanding – it’s not like “group”).

It still seems squicky to me but I guess that’s my problem.

The truth about mathematics.

January 5, 2009

I often read paraphrases of one or other of Gödel’s theorems that talk about true, unprovable statements. I’ve said before that I’m a formalist of sorts. Talk of undecidable statements in a system being true gives me headaches. And I’m an analyst so I work in ZFC. If I say “statement X is true” I’m telling you that there exists a proof of statement X in ZFC. If you ask me if I think the continuum hypothesis is true, I’ll explain to you that it’s known to be undecidable. If you tell me you know it’s undecidable but still want to know if I think it’s true, I’ll look at you as if you asked me what colour integrity is.
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