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prase comments on Meanings of Mathematical Truths - Less Wrong
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We should maybe taboo the word "true", since for a mathematical theorem to be true is not exactly the same as for an interpreted sentence about the physical world. How would you then formulate the sentence "the math that's less tied to the physical world isn't less true"?
In this case, I mean something like "if you start off with consistent and true beliefs, adding more true beliefs won't lead to self contradiction." I can define self-contradiction formally, as asserting both a statement and its formal negation.
This may seem slightly circular, but I think it's still a useful definition that captures what I want. I also think some circularity is useful to capture what we mean by an axiomatic system.