Notes about a comment on “Kantian space”

“How did Kantians in the 1800s and afterward address non-Euclidean Geometry? I have strong issues with Kant’s reasoning, since it implicitly relies on the assumption that Euclidean Geometry is the fundamental basis of spatial reasoning, where triangles inherently must all have angles which sum to 180 degrees. In non-Euclidean Geometry, there are triangles which ARE triangles yet do not have angle sums equal to 180 degrees. Same with mathematical reasoning of 5+7=12, which assumes Base 10 and some underlying logic axioms, but does not apply, say, to other number bases such as Base 9, or to clouds (5 clouds + 7 clouds may combine into 1 cloud). Did Kantians address these assumptions of Kant versus later mathematical developments? Your suggestion that it is about the structure of geometry goes to Kant’s thinking and other philosophers’ thinking that you only need logic and some structural reasoning to reach universal truths. But logic and reasoning are flawed due to their implicit assumptions, such as focusing on one type of geometry or counting system as opposed to other types. When one considers the possibility that 5+7 does not always equal 12 (unless one lays out, in detail, the implicit assumptions being used), one should see that Kant’s arguments are flawed.”

(Edit: Should have started the whole post with “In my opinion:”) The particular geometry of Euclid is just an example, the Kantian arguments are not about specific geometries but rather about the structure of any kind of geometry itself, so it doesn’t rely on those assumptions. It can be understood just as abstract as the notion of space evolved over time in linear algebra and physics. Even if we don’t count specific properties of fields, rings and spaces, there is always an idea of “in-ness” and you can’t get rid of it this easily by mentioning non-euclidean geometry. It was hinted in the comment, that every system is “flawed” by having a perspective, because their truths are inherently bound by the frames of that given system. Maybe I’m defending the coherence theory of truth way too much here, but the comment itself presupposes a similar theory of truth, so I’m trying to reply in the same frame of reference. This is not an universal truth, at least we don’t know from just this, may be just a human bias even, this really depends on your further views.

The thing is, that we are talking about geometries and a priori principles, not about some universal truth (that seems like a quite strange idea in itself). What the a priori notion of space means, is that there is something general about them (all of spaces, any space), and it is the same generalization that mathematics is formalizing by discovering all the possible spaces and relations in them.  (Type theory should be especially interesting for philosophy.)
Kant’s examples may be outdated, but the arguments themselves are not affected by that because there is some kind of “space-ness” in everything we think about, it’s not just a semantical problem of using phrases like “in my head” or “out of question”. My personal opinion is that the categories of a priori and a posteriori would need a rework themselves but there is still a sharp distinction between at least two types of ideas. (Here’s an analogy within set theories, you could think of sets as a priori and relations or attributes like being well-ordered as a posteriori.) The very  crucial nature of space is certainly mysterious, but nevertheless it seems to be a fundamental “base” to interpret, imagine or understand anything at all. I think an interesting addition is how modern physics seems to be in agreement with this view in a way, because according to the current models where there is anything (forces), there is space as well. Could it be any other way?
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