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Logicism is the idea that all mathematics is reducible to logic. This is commonly supposed to have been dealt a death blow by Gödel's incompleteness theorem, though it's not clear why.

What is mathematics? It is the construction of mathematical theories. A mathematical theory starts with a set of axioms, and then proceeds to derive interesting consequences of those axioms through logical inference.

Now, Gödel's theorem is a theorem about mathematics, not about logic. It uses an ingenious self-reference argument to impose limits on the possible scopes of mathematical theories. However, it is possible to use the same kind of self-reference argument to show that logic itself is not subject to such limits. For consider: assume that there is a logical argument that demonstrates that there are areas of human thinking or knowledge where logic is not valid. But that argument assumes the validity of logic, does it not? Therefore, logic must still apply even in reasoning about those areas.

The difference between logic and mathematics is that mathematics is always built on axioms, while logic is not. Logic must be independent of any axioms, otherwise it couldn't be used freely to reason about the consequences of arbitrary axioms.

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