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According to Kant, geometry is possible because of our intuition of space. But, this intuition is presumably 3-dimensional, as we experience the world 3-dimensionally. So, how would higher-dimensional (4 dimensions, 10 dimensions, 1000 dimensions?) geometry fit into this narrative? Humans are capable of this, but I don't see how this would fit into his view of geometry being possible because of our intuition of space.

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    Intuition is overrated.
    – Frank
    Apr 2 at 18:41
  • Higher-dimensional geometry would not fit into Kant's spatial intuition. However, he could accommodate its arithmetized version, i.e. analytic geometry with coordinates being tuples of real (as we would call them) numbers, the same way he accommodated arithmetic, through temporal intuition. Arguably, that is all that higher-dimensional "geometry" comes down to for us, sophisticated arithmetic in disguise. Of course, due to precise analogies between lower and higher dimensional analytic geometries, spatial intuition ends up being analogically useful even in the latter.
    – Conifold
    Apr 2 at 21:25
  • @Conifold Would you say that geometry/algebraic geometry has become "the same" as number theory/arithmetic? I think you can do algebraic geometry without doing any arithmetic.
    – Frank
    Apr 3 at 13:53
  • @Frank No more than probability theory has become "the same" as measure theory. The formal shell is shared, but motivations and intuitions that drive concepts and problems of interest in the fields are very different. And much of geometry is not even algebraic geometry.
    – Conifold
    Apr 3 at 14:04

2 Answers 2

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Kant's claim was that we have intuitive knowledge of things like "between any two points there is a straight line", and "two lines that cross, intersect in a point". He didn't foresee modern geometry and wasn't making any claims about it.

Modern geometry is arithmetic-based, meaning it's done with numbers and symbols instead of a compass and straight edge. This allows one to explore far beyond our intuitions of space, but truth is based on formal consistency rather than perceptible spatial relationships. No intuition of space is required, but consequently, the results don't need to conform to anything real in space either.

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  • "Measuring the Earth" became entirely abstract and unintelligible. Huh. We should quit while we're ahead.
    – Scott Rowe
    Apr 2 at 21:14
  • I think this is overly simplistic. A look at papers will show that most reasoning in modern geometry is not arithmetical/algebraic, it is wrapped into formalizations only after the fact, if at all, and surely not "based" on them. Geometric intuition, continuous with the "spatial" reasoning in 2D and 3D, is very much required to get anywhere. Hyperbolic geometry, as done by Bolyai and Lobachevsky, was not even formally arithmetic-based, they performed constructions mimicking those of Elements. Perhaps, our "spatial" intuition is more fragmentary and flexible than Kant thought.
    – Conifold
    Apr 3 at 9:57
  • @conifold, Yes, I should have said "hyper-geometry" instead of "modern geometry". Non-Euclidean geometry in three dimensions doesn't violate our Euclidean intuitions; it relies on them. Apr 3 at 15:30
  • Non-Euclidean geometry in two dimensions does not "violate" them either, Bolyai and Lobachevsky did not use 3D models, those were discovered much later. The same goes for fragments of "hyper-geometry", in 4D+ dimensions, as well. People perform "geometric constructions" even in Hilbert and Banach spaces, it is common in functional analysis. Our "spatial intuition", whatever that is, is not rigidly tied to 2-3D, or to Euclidean objects, although it is most comfortable there. The cleaned up processed redux that Kant took from Elements was not a first hand account of what he was after.
    – Conifold
    Apr 3 at 18:01
  • @Conifold, I wasn't contrasting 3 dimensions with 2 dimensions; I was contrasting it with more than 3 dimensions. I deny that we have spatial intuitions of multi-dimensional spaces. At best we have analogies. Can extremely bright mathematicians and physicists develop the ability to visualize 4-dimensional spaces? Possibly. Some of them claim to, at any rate, but that doesn't make it a Kantian intuition of space. Apr 3 at 20:09
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It is important to understand what one means by the term 'geometry' specifically. The Kantian implication clearly refers to a 'geometric' intuition, a spatial understanding of 'basic' concepts that are a priori, or in certain ways intrinsic, things that you are aware of naturally, and don't require an experience or stimulation from the so called 'real world'. There are two considerations when Kant makes such claims:

  1. The level of abstraction he is operating at, for instance he could talk about 'science' and not refer to the natural sciences(In this case he's clearly operating on a highly generalised plane), which brings me to the second point

  2. How Kant defines words; In the Critique of Pure Reason, Kant defines intuition as "that through which knowledge refers to its objects immediately" Now, pesky word choices, what does immediately mean in a temporal sense? Well, it is trying to communicate some sort of a spontaneous understanding. This clearly differs from the modern use of the word, which I've noticed is quite a common error made when one tends to interpret Kant.

Coming back to your question, what we mean when the refer to the term 'geometry' is usually the modern Mathematical treatment of the word, so, as pointed above, not what Kant had in mind.

Secondly, from a pure mathematical standpoint, we make generalisations in order to operate in higher dimensions. That is, our intuition is still restricted to 3-space, no human being is capable of imagining 3 vectors perpendicular to one another and magically adding a 4th that is also normal to all three of them. However that is the concept of a 'dimension' roughly. We saw how we moved from 1 space to 2 space to 3 space and therefore arrived at a rough idea of how higher evolutions might proceed. In the human brain we are still operating by means of 3-space analogies, or mathematical intuitions(here not in the Kantian sense, in the comprehension/internalisation of a concept sense) that is readily available to us as a consequence of our 'actual' reality.

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