The world appears to be mathematical because mathematics is part of the grammar - maybe a necessary part of the grammar - of the language we use to describe it. I'm saying "maybe" because I haven't said what mathematics is. This turns out to be important, I think.
I want to look at a hypothesis from Max Tegmark:
H: "The world is (just) mathematics"
This is a paraphrase, but I don't think it's wrong in any important way. Other ways of representing his position will share it's contradictions.
Since H is a statement about the world, it must (according to itself) be a kind of mathematical conjecture - perhaps even a hypothesis than can be proved or disproved. However, this can't be the case, because it uses the word "mathematics". We don't have a definition of this word that allows us to use it in this way (like a number or a symbol).
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