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What are “foundations”?

Inference rules, axioms, language?

I was referring to axioms when I used the term "foundations" above. Inference rules are largely agreed upon across both maths and philosophy (and they're the same ones in both disciplines).

A tangential but related thought.

I’ve heard people say things like, “The world is math”, but it never seemed particularly coherent to me. Sometimes I’d assume they meant, “Inference systems with mathematical languages make useful predictions about my empirical experience.”

But now I’m favoring the interpretation that experience is purely formal/syntactic with no semantic component. There is no additional meaning beyond (or behind) appearance.

There is a notion of meaning in that it encodes possibility. A situation with lots of future possibilities has meaning. A dead-end situation has little or no meaning: stagnation, death, ...

If the formal/syntactic appearance is how leads to new possibilities involving more formal/syntactic appearances, then that is its meaning.

Are you suggesting that some people hold the normative position, “One ought prefer situations with more possibilities to those with fewer, ceteris paribus.”

But isn’t that just some hidden metaphysical structure? That is, it’s unavailable as formal appearance?

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