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I originally posted this question on Math.SE and received some interesting comments but no answers. Now that some time has passed I thought that it might be appropriate to post here as well; perhaps it will incite some interesting discussion.

I have always felt moderately "skeptical" about the notion that Euclidean geometry as presented in a mathematics course (whether in a synthetic or an analytic form) actually corresponds to our intuitive, visual notion of flat (2d or 3d) space. By "skeptical" I of course do not mean that I actually doubt that it is true, I merely mean that I do not feel I have fully understood why it must be true, I have not fully convinced myself. Why, for instance, should a set like $\{(x,y)\in \mathbb{R}^2:y=mx+b\}$ for some $m,b \in \mathbb{R}$ correspond to the visual notion of a straight line? Why should something like $\arccos(\frac{v\cdot u}{|v||u|})$ (or however we end up defining angle formally) correspond to the visual notion of the angle between two vectors? Etc. I have never felt that I had totally satisfactory answers to these questions.

Some time ago I came across this answer on MO, explaining why topological spaces are defined in terms of open sets. The author demonstrates in a semi-rigorous way that the properties of open sets follow directly from certain intuitions we might have about "generalized rulers". Thus, from these intuitions about "generalized rulers", we can recover the formal definition of a topological space via open sets. I have been idly contemplating how one might do something similar for Euclidean geometry—demonstrate in a semi-formal way that the basic geometrical structure of $\mathbb{R}^n$ really does capture enough of our core intuitions about flat space that the picture we all have of $\mathbb{R}^n$ in our heads is really justified. But I am not sure how to do it. Certainly, since $\mathbb{R}^n$ is a model of any axiomatization of Euclidean geometry, showing that analytic geometry captures our intuitions about flat space is enough to show that synthetic geometry captures at least a subset of our intuitions about flat space.

I suspect that the "proof" here will be somewhat more complicated than the one in the linked question, as there are more intuitions to check. So I am not necessarily expecting a complete answer. But if anyone could suggest a direction in which to begin looking/thinking in order to convince myself of this fact, I would greatly appreciate it.

Edit: In light of a comment by Timothy Chow and the answer by Carlo Beenakker, I think it's perhaps worthwhile to disambiguate my terminology somewhat. Timothy rightly points out that visual intuition is not synonymous with spatial intuition, as is clear if one considers, for instance, the spatial intuitions that a person who has been blind since birth might have. To this I would also like to add that there is presumably a distinction between monocular and binocular visual intuition, being that the latter includes an element of depth perception and the former does not.

Carlo in his answer posts an image of train tracks running off to the horizon, and notes that despite the tracks being parallel, they visually appear to converge. I would argue that this may mix and match visual and spatial intuitions. It is our spatial intuition (and, presumably, spatial fact) which says that the tracks are parallel in 3d space. And in 2d, (monocular) visual space, the tracks do appear to converge at a point. But it seems to me that the tracks do not appear parallel in (monocular) visual space to begin with, and so Euclid's fifth postulate is not (necessarily) violated. When I look at the image of the tracks, I do not feel that I am seeing two parallel lines, rather I feel that I am seeing two non-parallel lines in 2d space whose preimages under projection happen to have been parallel in 3d space.

I suppose that in the narrowest sense, by my (now much more tentative) statement that "visual intuition is Euclidean", I meant that monocular visual space—the 2d projection of 3d space which we experience when we close one eye—appears to be flat. But now I am much less sure that this is so, and I suppose it does come down to an empirical question in the end. I haven't yet gotten a chance to look at the sources that everyone linked, so perhaps these distinctions are already addressed therein.

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    $\begingroup$ this is the MSE question mentioned in the post: math.stackexchange.com/q/4865934/87355 $\endgroup$
    – Carlo Beenakker
    Commented Mar 20 at 10:06
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    $\begingroup$ Your question seems related to the “problem of space”, also known as the Riemann-Helmholtz (or Helmholtz-Lie) problem which approximately speaking consists of identifying (hopefully “visually obvious”) properties of the group of motions of space that characterize Euclidean, spherical and/or hyperbolic geometry. Various solutions have been proposed, notably by Lie and Engel, by Kolmogorov and by Tits (for the latter: see “Sur certaines classes d'espaces homogènes de groupes de Lie” (1955; =item [27] in vol. 1 of his collected works), section IV.E). $\endgroup$
    – Gro-Tsen
    Commented Mar 20 at 21:29
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    $\begingroup$ Shapes, such as triangles appear to be preserved under translation, rotation and reflection. This leads to a group-theoretic axiom system for Euclidean geometry. This comment could be expanded into a full answer. $\endgroup$
    – Kapil
    Commented Mar 21 at 3:04
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    $\begingroup$ I want to point out that spatial intuition is not necessarily identical with visual intuition (put yourself in the shoes of someone who has been totally blind since birth, for example). It seems to me that Euclidean geometry is intended to model spatial intuition and not visual intuition. The reason I intuitively feel that parallel train tracks do not meet isn't that they visually appear not to meet, but rather that if I imagine myself walking along indefinitely long train tracks, I do not expect them to ever meet. $\endgroup$
    – Timothy Chow
    Commented Mar 21 at 11:22

2 Answers 2

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Patrick Suppes asked Is Visual Space Euclidean? (1977). A more recent contribution that answers the question in the negative: Is perceptual space inherently non-Euclidean? (2009).

It is often assumed that the space we perceive is Euclidean, although this idea has been challenged by many authors. Here we show that if spatial cues are combined as described by Maximum Likelihood Estimation, Bayesian, or equivalent models, as appears to be the case, then Euclidean geometry cannot describe our perceptual experience. Rather, our perceptual spatial structure would be better described as belonging to an arbitrarily curved Riemannian space.

Concerning the Euclidean axiom that parallel lines do not cross, my own visual intuition is more like this:

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    $\begingroup$ Related (from 1947): mathoverflow.net/questions/178021/… $\endgroup$
    – Francois Ziegler
    Commented Mar 20 at 17:24
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    $\begingroup$ Well, they don't look to me like they cross …. I don't mean that completely facetiously: I'm OK with adding an ideal point at infinity where the lines meet, but nothing in this picture suggests to me continuing the lines past that point at infinity. $\endgroup$
    – LSpice
    Commented Mar 20 at 20:42
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    $\begingroup$ If you're referring to the two metallic things, they don't look parallel to me, at all. In fact, from this photograph alone, I don't know their geometry at all. Maybe they are two straight, tapering metallic sticks that have been arranged vertically so that they make an angle of around $30^\circ$–$40^\circ$. (Hopefully those who have been born on this planet can explain to me the meaning of these metallic objects, and why they are supposed to be parallel, but since I'm not from here, nothing directly tells me anything about Euclidean space from this picture.) $\endgroup$
    – printf
    Commented Mar 21 at 10:59
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To the sources mentioned by @Carlo Beenakker, I would like to add the works of Soviet physicist Boris Rauschenbach, most of them are in Russian, but there are some in English:

Perspective Pictures and Visual Perception
Boris V. Rauschenbach
Leonardo, Vol. 18, No. 1 (1985), pp. 45–49.

On My Concept of Perceptual Perspective That Accounts for Parallel and Inverted Perspective in Pictorial Art
Boris V. Rauschenbach
Leonardo, Vol. 16, No. 1 (Winter, 1983), pp. 28–30.

Perceptual Perspective and Cezanne's Landscapes
Boris V. Rauschenbach
Leonardo, Vol. 15, No. 1 (Winter, 1982), pp. 28–33.

He studied the question seriously, and analyzed many medieval and modern paintings. His main conclusion is that our visual perception is really not Euclidean.

Concerning Euclidean geometry, its axioms are based not only on the immediate visual perception but also on certain facts of optics, most of them can be experimentally verified. (A straight line is an abstraction of a light ray.) A notable exception is the 5th postulate, and already in antiquity there were doubts that it is really based on our experience, and this stimulated multiple attempts to prove it, until eventually a whole family of non-Euclidean geometries were discovered.

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    $\begingroup$ Of course, Einstein taught us that light rays, in the presence of massive objects, do things that Euclidean lines never do. $\endgroup$
    – Gerry Myerson
    Commented Mar 21 at 1:57

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