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JoshuaZ comments on Train Philosophers with Pearl and Kahneman, not Plato and Kant - Less Wrong
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He doesn't discuss it that much, but there's a strong argument that it is operating the background 1 (pdf). The same author as linked wrote an essay about this, but I can't find it right now.
This is strange, because your link is about Kant disagreeing with other philosophers on the nature of Euclid's parallel postulate. I took your claim to be that because Kant was seemingly only aware of Euclidean geometry, he used properties specific to only Euclidean geometry in his discussion of geometry.
Show me explicitly where this "operating in the background" is, and I'd be more convinced.
Hmm, ok. Rereading the link and thinking about this more, it looks like I'm either strongly misremembering what it said or am just hopelessly confused. I'll need to think about this more.
Thinking about this less and something else more is also a good option.