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Mallah comments on The mathematical universe: the map that is the territory - Less Wrong
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For continuous functions, we do. See "abstract stone duality".
Interesting. Do you know of place on the net where I can see what other (independent, mathematically knowledgeable) people have to say about its implications? It's asking for a lot maybe, but I think that would be the most efficient way for me to gain info about it, if there is.