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Tyrrell_McAllister comments on Open Thread: May 2010, Part 2 - Less Wrong
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That's not a problem for logicism per se. Logicism isn't really a claim about what it takes to prove mathematical claims. So it doesn't fail if you can't prove some mathematics by a certain means. Rather, logicism is a claim about what mathematical assertions mean. According to logicism, mathematical claims ultimately boil down to assertions about whether certain abstract relationships among predicates entail other abstract relationships among predicates, where this entailment holds completely regardless of the meaning of the predicates. That is, mathematical claims boil down to claims of pure logical entailment.
So, if you discover that your particular mathematical system is incomplete, then what you've really done is discover that you had missed some principles of logic. It's as though you'd known that P ∧ P entails P, but you just hadn't noticed that P ∨ P entails P as well.
(But you were right about why logicism ultimately failed to convince everyone: Mathematics seems to have ontological commitments, where pure logic does not.)
I didn't downvote either comment. Your comment was probably downvoted because some readers considered its arguments to be wrong or unclear. zero call's comment was probably downvoted because it smacks of the mind projection fallacy, especially here:
The organization of facts into axioms, rules of inference, proofs, and theorems doesn't seem to be an ontologically fundamental one. We superimpose this structure when we form mental models of things. That is, the logical structure of things exists in the map, not the territory.
I wish you would have made this last comment on the post directly, so that I could reply to that there. Anyways, the point I was offering was that the logical structure does exist in the territory, not just the map. Our maps are merely reflecting this property of the territory. The fundamental signature of this is the observation that physical systems, when viewed in a map which exists only as a re-representation or translation (as opposed to an interpretation) amenable to logical analysis, are shown to prohibit logical contradiction. (For example, the two statements (if A, then B) and (if A, then not B) cannot both be true, where A and B are statements in some re-representation of the physical system.)
I'll move that part of my comment there, with my apologies.
That's quite alright -- thank you for your discussion.