Yes, I mean what ordinary people mean by intuition, not what Kant means . He uses words too weirdly for me. — andrewk

(1) What ordinary people mean by intuition cannot be used to defend Kant, who uses that word differently, and thus means different things by it than ordinary people.
(2) The way ordinary people use "intuition" and others words that are derived from it is extremely vague. In ordinary language, an "intuition" is just when I throw up my hands and tell you "I know it is this way, but I can't say why". Very often, habit can entrench thoughts, principles, and the like in people's mind, and they easily recall them, and feel very certain in them, but are unable to give justification for them.

Yes. It may be, as you say, cos I'm a mathematician. Or maybe I'm a mathematician cos I look at things that way. — andrewk

Which way?

As far as I understand it "intuition" for Kant means something pretty close to what we would call 'perception'. — Janus

I disagree with this, but I'll touch on it in replying to your third paragraph. Probably gets to the crux of our disagreement though.

You say "we are able to have synethic a priori knowledge about space due to our knowledge of geometry" but if this were true then it would not be "synthetic a priori knowledge" at all but synthetic a posteriori knowledge. I think it is more to the point that we are able to have knowledge of geometry due to our synthetic a priori knowledge of space. I think that is certainly what Kant thought. — Janus

I don't disagree with that interpretation of Kant here. This is why I think non-Euclidean geometry is problematic, just not destructive to the aesthetic. It can be "saved", that is -- and still feel reasonable rather than ludicrous.

So, following my second strategy, Euclidean geometry could be interpreted as synthetic a posteriori knowledge while non-Euclidean geometry could be interpreted as syntehtic a priori -- and the same would apply to any other geometry which predicts the events of the phenomenal world.

I don't think it makes sense to say that Euclidean or non-Euclidean geometries are "wrong"; both are intuitively obvious in their contexts. This is not say that it is, or even can be, intuitively obvious that spacetime is curved, because, to repeat myself, I don't think we have any reason to think that spacetime is the same thing as perceptual space, for the simple reason that we cannot perceive, or even visualize, the curvature of spacetime. Is there any reason you can think of why we must believe they are the same? — Janus

It's not our perception of space that's at issue, I'd say. The propositions of geometry are closely tied to physics, by my reading. Because our intuition follows mathematical laws we are also able to apply those mathematical laws to objects, which are themselves within our intuition.

Strictly speaking it's not perception which intuition is trying to explain, but rather intuition is one half of the elements of cognition which explains how knowledge of objects is possible. Clearly there are relations between perception and cognition, and granted the intuition's description relies heavily upon visual imagery (like a lot of Western philosophy), but the reason why mathematical laws are able to be posited and discovered in the phenomenal world is because our cognition relies upon this form. It sort of explains why we are able to make predictions which are actually caused -- meaning the "necessary connection" between two events -- in the first place, rather than merely the constant conjunction of non-related events believed by force of habit.

So if it turns out that Euclidean geometry is not the form of intuition it would seem to upend the notion that we have synthetic a priori knowledge of the form of intuition. Same goes for the physics based upon that synthetic a priori knowledge. However, if Euclidean geometry were merely empirical, an approximation of our cognitive faculties as Newton was an approximation, then I'd say that the aesthetic is saved.

But in either case, it's not how we perceive that's at issue. It's how we are able to know math and why it applies to the objects of our perception in the first place. Kind of a hair-thin distinction, but I'd say it's important because in one case we are dealing with phenomenology and psychology, and in the other we are dealing with the possibility of knowledge which seems to fit more in line with the whole Critique.

It's not our perception of space that's at issue, I'd say. The propositions of geometry are closely tied to physics, by my reading. Because our intuition follows mathematical laws we are also able to apply those mathematical laws to objects, which are themselves within our intuition.

Strictly speaking it's not perception which intuition is trying to explain, but rather intuition is one half of the elements of cognition which explains how knowledge of objects is possible. Clearly there are relations between perception and cognition, and granted the intuition's description relies heavily upon visual imagery (like a lot of Western philosophy), but the reason why mathematical laws are able to be posited and discovered in the phenomenal world is because our cognition relies upon this form. It sort of explains why we are able to make predictions which are actually caused -- meaning the "necessary connection" between two events -- in the first place, rather than merely the constant conjunction of non-related events believed by force of habit.

So if it turns out that Euclidean geometry is not the form of intuition it would seem to upend the notion that we have synthetic a priori knowledge of the form of intuition. Same goes for the physics based upon that synthetic a priori knowledge. However, if Euclidean geometry were merely empirical, an approximation of our cognitive faculties as Newton was an approximation, then I'd say that the aesthetic is saved.

But in either case, it's not how we perceive that's at issue. It's how we are able to know math and why it applies to the objects of our perception in the first place. Kind of a hair-thin distinction, but I'd say it's important because in one case we are dealing with phenomenology and psychology, and in the other we are dealing with the possibility of knowledge which seems to fit more in line with the whole Critique. — Moliere

Ptolemization, I see X-) - when it doesn't work, we'll add new fudge factors to make it work... Kind of ironic, given that this was supposed to be a Copernican revolution >:O

Well, for Kant, there is only one space and mathematics (geometry) describes it with apodictic certainty. — Agustino

Yes and that is the space of human perception. Spacetime, whatever it is, is not that space; that has been my point all along.

Parallel lines do meet, in our perception, at the horizon. So if you want to argue for this point (that our natural intuition of space is Euclidean), with which I actually agree, you cannot appeal to the "nature of visual perception". (2), there is no "perceptual" space as differentiated from "physical" space (the space we encounter when we do our physical experiments) in Kant - there is only one space. — Agustino

Parallel lines in perceptual space do not meet, otherwise trains could not operate. They only appear to meet, and it very well understood why that happens. We can build rail lines extending thousands of kilometers and the rails are (not perfectly, but on average, parallel). Can you think of any reason, other than practical limitations, why rail lines could not extend indefinitely?

(2) There is no "physical space" in Kant, as you have already acknowledged; so it seems you have fallen back into confusion again. Spacetime is a hypothetical construct; there is no actual spacetime that we can intuitively understand, as we can our perceptual spaces.

This is incoherent. Can you perceive non-euclidean geometries? — Agustino

Of course we can; we can intuitively understand them when they are visually represented on two dimensional curved surfaces. The analogy from curving or warping of two dimensional surfaces into the third dimension (which we can visualize) to curving of three dimensional space into a hypothetical fourth dimension is the only way we can get any notional sense of it; it is not something we can directly represent visually to ourselves at all.

But in Kant's system I can tell you for certain that it can be no other way. — Agustino

If you think anything can be no other way in Kant's system then I would conclude that you have not read Kant, or if you have, have not understood him. Kant scholar's have been arguing over just what he meant for centuries.

I'm sorry to do this, but I have little time at the moment, so I will direct you to my response to Agustino as I think it deals with some aspects of what constitutes our ongoing disagreement. I'll try to return to address your post more fully latter. :)

(1) What ordinary people mean by intuition cannot be used to defend Kant, who uses that word differently, and thus means different things by it than ordinary people. — Agustino

My intention is to defend not Kant, with whom I disagree on many important things (although I do have enormous admiration for him), but what I see as the amazing insight and usefulness of his notion of the Transcendental Aesthetic (TA). In the discussion over whether you and I find non-Euclidean geometries unintuitive, I see that as just a reflection on your and my particular cognitive processes, rather than about the TA, which is suggested to be universal to autonomous humans.

My interpretation of the TA, which has evolved in the course of this discussion (thank you everybody - this forum can be such a learning experience), is that humans process sensory input in a framework consisting of two Riemannian manifolds: a 3D one that we call 'space' and a 1D one that we call 'time'. That Kant did not describe it this way I ascribe to the fact that the language necessary to express that did not exist in his time.

Space as a 3D Riemannian manifold gives us points, lines, shapes, volumes, angles, directions, relative positions, insides and outsides, and distance.

As I see it, that, together with time, is enough for us to navigate, imagine and discuss the world. At most I would add a requirement that any curvature not be too extreme, because if that were the case we might find ourselves back where we started if we walked one metre (if the space were elliptic), That requirement is completely consistent with the region of the universe in which we evolved, and which we now inhabit.

It may well be the case that for some people the space manifold is also perfectly flat (ie no curvature, not even if unmeasurably small), as you report to be the case for you. But I suspect that is an individual variation, rather than a universal feature. For my own case, It is not necessary in order to obtain all the concepts listed in bold text above.

Which way? — Agustino

In a way that does not require the space manifold I use to be perfectly flat.

We can build rail lines extending thousands of kilometers and the rails are (not perfectly, but on average, parallel). — Janus

This got me thinking. How would we build a rail line to circumnavigate the equator, if there were a 5m wide land bridge all the way that followed the great circle of the equator? Say the land bridge is perfectly level (constant altitude above mean sea level) and extends at least 2.5m to either side of the equator at every point.

I'm pretty sure that the answer is that the rails would always be parallel and equidistant, but what we'd have to give up is the requirement that they be 'straight' - what's called a 'geodesic' in tech terminology. Say the gauge is standard and the centre of each rail is always 717.5mm away from the equator - one in the Northern and one in the Southern hemisphere. Then neither rail can follow a great circle but instead is constantly curving away from the equator at an incredibly small, constant rate.

So the lines would be parallel and a constant distance apart, but they would not be perfectly 'straight'. However a train could run along them with no difficulty at all.

Why can't the rails both be straight? Because a straight line on the surface of a sphere is a great circle, and any two great circles will intersect at two antipodal points. It would however be possible to make one of the rails a great circle and the other one not - eg if one rail followed the equator and the other were in the Southern hemisphere..

Yes and that is the space of human perception. — Janus

What is "human perception"? Is this not the same space as the space in which our bodies act and live? Before you said visual perception - that's not correct. We can have a notion of space through touch alone, for example.

Parallel lines in perceptual space do not meet, otherwise trains could not operate. — Janus

So then this is not visual space - what you see in front of your eyes, but rather something else. You admit that in visual perception, the lines appear to meet at the horizon.

Spacetime, whatever it is, is not that space; that has been my point all along. — Janus

What is spacetime? And how does it relate to the space we intuit?

Spacetime is a hypothetical construct; there is no actual spacetime that we can intuitively understand, as we can our perceptual spaces. — Janus

So what about light rays travelling in straight lines but bending around planets? We cannot perceive that or?

Of course we can; we can intuitively understand them when they are visually represented on two dimensional curved surfaces. — Janus

But we cannot intuitively understand them in three-dimensions, except by analogy, no?

The analogy from curving or warping of two dimensional surfaces into the third dimension (which we can visualize) to curving of three dimensional space into a hypothetical fourth dimension is the only way we can get any notional sense of it; it is not something we can directly represent visually to ourselves at all. — Janus

So then we really don't have an intuitive understanding of it? We have an understanding by proxy of 2D objects curved in the 3rd dimension. Furthermore, I think in mathematics, @andrewk should correct me, the notion of intrinsic curvature does not require the existence of another higher dimension for the space to curve into. So the 2D objects curving in another dimension - that's extrinsic curvature, and we can have an intuition of it. But we can't have an intuition of intrinsic curvature - in the Kantian sense of intuition.

If you think anything can be no other way in Kant's system then I would conclude that you have not read Kant, or if you have, have not understood him. Kant scholar's have been arguing over just what he meant for centuries. — Janus

I am aware there are Kant scholars who disagree - they are free to do so. But those who disagree, do such violence to Kant's system, that it is essentially unrecognisable, or otherwise a Ptolematization. I've seen and read scholars who don't take Kant's transcendental idealism seriously enough, and who buy into Kant's confused idea of the noumenon, and there being a real space out there (that physics figures out), and adapt Kant's ideas to take into account their naturalism, etc. - that's not philosophy if you ask me, that's nonsense. Schopenhauer understood Kant rightly, and at least set the noumenon bit straight, and avoided the pitfalls of naturalism.

If you have any Kant scholar who follows in the footsteps of Schopenhauer and deals with the issue of non-Euclidean geometry, feel free to let me know, and I will look into them.

In a way that does not require the space manifold I use to be perfectly flat. — andrewk

How do you imagine a 3D, non-flat space? How do you imagine intrinsic curvature? Hopefully, you won't say that you do via analogy to extrinsic curvature.

So the lines would be parallel and a constant distance apart, but they would not be perfectly 'straight'. However a train could run along them with no difficulty at all. — andrewk

That's true; they would not be straight in the vertical plane, because they would curve to remain parallel to the curvature of the Earth. What if we could build a rail line into space; it could be straight and parallel in both planes I think.

What is spacetime? And how does it relate to the space we intuit? — Agustino

I already said it is a hypothetical construct.

So what about light rays travelling in straight lines but bending around planets? We cannot perceive that or? — Agustino

We cannot directly perceive light rays at all. On account of our explanatory theories about what we do observe we can infer that they are bending. We can further infer that the bending is caused by curvature of spacetime in accordance with other theories.

I keep getting sucked back into these discussions and sometimes they just take up too much time, I don't have much time right now, so...really gotta go...

I already said it is a hypothetical construct. — Janus

What does it mean that it is a hypothetical construct?

I keep getting sucked back into these discussions and sometimes they just take up too much time, I don't have much time right now, so...really gotta go... — Janus

Okay, answer when you have time then :P

We cannot directly perceive light rays at all. On account of our explanatory theories about what we do observe we can infer that they are bending. We can further infer that the bending is caused by curvature of spacetime in accordance with other theories. — Janus

We perceive them via instruments, that is still perception. It's like looking at a cell with a microscope - still counts as percieving, even though not directly (I don't see how that is relevant though). Even so called direct perception is mediated through our eyes - if we're color blind, we perceive things differently. So... Whether mediated through eyes, or telescopes or whatever - makes no difference as far as I see it. We basically see that they are bending.

Anyway, apart from that point, do you believe atoms exist? We also infer the existence of atoms from related evidence. So spacetime and its curvature isn't just a theory, it really exists.

How do you imagine a 3D, non-flat space? How do you imagine intrinsic curvature? Hopefully, you won't say that you do via analogy to extrinsic curvature. — Agustino

I'll describe below how I imagine it, but that's beside my point, which is that I think we don't need to imagine it. I think all we need to cognise the world is the bolded list of items in my previous post, and we get that from any Riemannian Manifold, whether flat or curved. When we use those things in navigating the world we can remain uncommitted as to whether the space is slightly curved or perfectly flat.

Now to reply to your specific question. You are right, it is weird to imagine. Here's a couple of ways:

1. Two spaceships set off on a journey, travelling initially parallel and starting 1km apart, going at the same speed and steering straight ahead. If the space is flat they will remain 1km apart. If not, they will subsequently measure that they are getting further apart if the space is hyperbolic, or closer if it is elliptic.

3. Set up three space stations 1, 2, 3 in deep space, each firing a laser beam at the next: 1 to 2, 2 to 3, 3 to 1. Each measures the angles between the incoming and outgoing beam. The stations are floating freely, not firing rockets to accelerate. The three angles will add to 180 degrees if the space is flat, less than that if hyperbolic and more than that if elliptic.

It's cool. Take as much time as you need. It does seem, based on what you've said to Agustino, that you prefer the first strategy I proposed. I prefer the 2nd, or at least some modification of the 2nd, since I still think about this stuff and am not settled on it. So we'll see where the conversation takes us.

I'll describe below how I imagine it, but that's beside my point, which is that I think we don't need to imagine it. I think all we need to cognise the world is the bolded list of items in my previous post, and we get that from any Riemannian Manifold, whether flat or curved. When we use those things in navigating the world we can remain uncommitted as to whether the space is slightly curved or perfectly flat. — andrewk

I will address this later when I have more time.

Now to reply to your specific question. You are right, it is weird to imagine. Here's a couple of ways:

1. Two spaceships set off on a journey, travelling initially parallel and starting 1km apart, going at the same speed and steering straight ahead. If the space is flat they will remain 1km apart. If not, they will subsequently measure that they are getting further apart if the space is hyperbolic, or closer if it is elliptic.

3. Set up three space stations 1, 2, 3 in deep space, each firing a laser beam at the next: 1 to 2, 2 to 3, 3 to 1. Each measures the angles between the incoming and outgoing beam. The stations are floating freely, not firing rockets to accelerate. The three angles will add to 180 degrees if the space is flat, less than that if hyperbolic and more than that if elliptic. — andrewk

I don't see how you're imagining anything. To imagine is to create a visual, tactile, or in any case sensory picture or image of intrinsic curvature in your mind. To imagine isn't to come up with some experiments that would prove or disprove the hypothesis.

Suppose the curvature is very high, such that if you take 10 steps in one direction, you return to the same point where you started. This is a thought experiment, an unrealistic one, but it's useful. Suppose there are a series of poles, 1 step apart, in front of you, with the pole right next to you being red (so that you can keep track of when you return), while the others are some other colors. How would this visually look to you?

@andrewk

To further illustrate what imagination is, when I imagine a curved line, I imagine that line curving right in front of my eyes. Basically I see what anyone would see if they were to draw a curved line on a piece of paper. So if someone asks me about extrinsic curvature, that's what it is - that's how you imagine it. Now show me that you can imagine intrinsic curvature in the same way.

That's true; they would not be straight in the vertical plane, because they would curve to remain parallel to the curvature of the Earth. What if we could build a rail line into space; it could be straight and parallel in both planes I think. — Janus

If the rail line were stationary relative to Earth, the lines could not be both straight and parallel, because in that reference frame the spatial slices are curved. Since parallelness is necessary in order for the train to be able to run but straightness is not (trains can go around curves), we would have to give up straightness, rather than parallelness.

It may be useful to be clear what we mean by parallel. What I mean is that if we draw a straight line perpendicular to one track then it meets the other track at right angles.

Interestingly, if the track were in free fall towards Earth then it may be possible for the lines to be both straight and parallel. That's because, subject to a few other initial conditions being met, its reference frame could be the one I referred to earlier as one in which curved spacetime can have flat spatial slices. It would make the devil of a mess when it hit the Earth though.

This lecture is a good summary of the refutation of Kant's views:

http://faculty.poly.edu/~jbain/spacetime/lectures/13.Kant_and_Geometry.pdf

Suppose the curvature is very high, such that if you take 10 steps in one direction, you return to the same point where you started. This is a thought experiment, an unrealistic one, but it's useful. Suppose there are a series of poles, 1 step apart, in front of you, with the pole right next to you being red (so that you can keep track of when you return), while the others are some other colors. How would this visually look to you? — Agustino

Say there are three poles, coloured red, yellow, blue, at distances of 3 1/3 steps away from one another, in a straight line in the direction I'm looking. There can't be more than three because the fourth pole would be where the first one is.

As I look along the line I see an infinite series of poles: red, yellow, blue, red, yellow, blue, etc. Next to every red pole is an image of me, seen from the back. The images of poles and of me diminish in size as they move along the line of vision, just as a series of poles beside a long, straight road does.

It would be somewhat similar to what one gets when one stands between two opposing mirrors, except that the view of myself would always be from behind. You may be interested in this essay I wrote about something like this - what happens when we point a TV camera at its monitor, inspired by a comment Alan Watts made in one of his talks. There are some pictures and videos in it that I find quite cool.

Now show me that you can imagine intrinsic curvature in the same way. — Agustino

I can only repeat what I said above, that we don't need to imagine it. Cognising space as a Riemannian Manifold is not non-Euclidean, but aEuclidean (think of the difference between immoral and amoral). It is uncommitted as to whether the space may be curved, as long as it is not heavily curved.

I would call the experiments I described a way of 'imagining' a non-Euclidean 3D space. But I feel no need to argue if you don't consider that imagining.

Kant scholar's — Janus

See, I am tired of "reinterpretations" of Kant such as:

Under the understanding of a prioricity at issue pre-Two Dogmas of Empiricism, a priori truths were largely conflated with necessary truths. So, if you could recognize the possibility of the failure of the parallel postulate, that would constitute a falsification of its necessity and thus (given the conflation) a falsification of the claim that it was a priori.

Where Kant went wrong, if this was indeed what he held, was in thinking that our intuition of space and time represented the world as it actually is. Frege famously made the same mistake in one of his later articles, "Foundations of Geometry". — Dennis

From here.

If this interpretation is correct, Kantianism is dead anyway. Schopenhauer would rightly find the notion of a space and time beyond perceptual space and time (the Euclidean ones) abhorrent to Kant's doctrine, and rightly so. If you ask me, such reinterpretations are pathetic, and they exist because people can't abandon a dead doctrine, and try to change it to fit the facts, when it really should be let go of.

You don't disagree that my solution "works" then, though?

You don't disagree that my solution "works" then, though? — Moliere

Actually... I misread your solution initially. At least you seem to understand what the problem is. So here are my comments again:

So, following my second strategy, Euclidean geometry could be interpreted as synthetic a posteriori knowledge while non-Euclidean geometry could be interpreted as syntehtic a priori -- and the same would apply to any other geometry which predicts the events of the phenomenal world. — Moliere

On what grounds do we judge a geometrical proposition to be a synthetic a priori?

That's what I mean. Surely it's sensible that we could be wrong about the form of inuition. So, supposing non-Euclidean geometry is the true geometry of the space we experience it doesn't seem like a large step to say that we were simply wrong before about the form of intuition. At least not to me. If that were the case, then it would just be an empirical concept, though -- since a priori concepts of space are apodeictic. — Moliere

(1) Why is it sensible that we could be wrong about the form of the intuition?
(2) Does the form of intuition belong to our subjectivity? If so, is it possible to be wrong about our own subjectivity?
(3) Can we know whether a geometric statement really is a synthetic a priori with certainty? And if so, how?

I don't think "intuition" in Kant means the same thing as intuitive. Space isn't intuitively obvious to us. Others have been wrong about space -- like Leibniz and Newton, for instance. So while the examples Kant uses are from Euclidean geometry it seems to me that one could modify the philosophy without losing the core of the aesthetic. It's not that something is obvious, but rather that we are able to have synethic a priori knowledge about space due to our knowledge of geometry. If one geometry is wrong then, just like Newton could be wrong, we could understand such sciences as something which wasn't part of our cognitive faculties but was derived from them, and is therefore empirical in that sense (and not synthetic a priori knowledge, but instead rests upon that) — Moliere

I don't follow how "we are able to have synthetic a priori knowledge about space due to our knowledge of geometry". Our synthetic a priori knowledge of space is what we codify through geometry.

I also don't follow what you mean by "space isn't intuitively obvious to us". For example, it seems impossible to imagine 4D space. So is the three-dimensionality of space not something intuitively obvious to us? Could we be wrong about that too? And what would that even mean?

It's not our perception of space that's at issue, I'd say. The propositions of geometry are closely tied to physics, by my reading. Because our intuition follows mathematical laws we are also able to apply those mathematical laws to objects, which are themselves within our intuition.

Strictly speaking it's not perception which intuition is trying to explain, but rather intuition is one half of the elements of cognition which explains how knowledge of objects is possible. Clearly there are relations between perception and cognition, and granted the intuition's description relies heavily upon visual imagery (like a lot of Western philosophy), but the reason why mathematical laws are able to be posited and discovered in the phenomenal world is because our cognition relies upon this form. It sort of explains why we are able to make predictions which are actually caused -- meaning the "necessary connection" between two events -- in the first place, rather than merely the constant conjunction of non-related events believed by force of habit. — Moliere

If I follow you correctly, your point is the traditional Kantian one that the phenomenal world is organised through the a priori forms of space and time and the categories of the understanding - so in this specific case, space doesn't exist "out there", it is just how we represent the phenomenal world to ourselves. In other words, space continues to be transcendentally ideal per your view?

So if it turns out that Euclidean geometry is not the form of intuition it would seem to upend the notion that we have synthetic a priori knowledge of the form of intuition. Same goes for the physics based upon that synthetic a priori knowledge. However, if Euclidean geometry were merely empirical, an approximation of our cognitive faculties as Newton was an approximation, then I'd say that the aesthetic is saved.

So if we don't have synthetic a priori knowledge of the form of intuition there are two main questions:

(1) Since the form of intuition is subjective, why don't we have such knowledge? How does acting in the world (empiricism, scientific experiments, etc.) help us gain that knowledge? Aren't we ultimately gaining knowledge about ourselves then?
and
(2) How do we even know that synthetic a priori knowledge even exists if we do not know when we have it? How can we know if a piece of synthetic knowledge is a priori (Riemmann) or a posteriori (Euclidean)?

But in either case, it's not how we perceive that's at issue. It's how we are able to know math and why it applies to the objects of our perception in the first place. Kind of a hair-thin distinction, but I'd say it's important because in one case we are dealing with phenomenology and psychology, and in the other we are dealing with the possibility of knowledge which seems to fit more in line with the whole Critique.

To go back to your initial question, your solution doesn't appear like a cop-out, but there are a lot of things to flesh out.

We perceive them via instruments, that is still perception. It's like looking at a cell with a microscope - still counts as percieving, even though not directly — Agustino

We don't observe light rays or curvature of space in the way we see cells through a microscope, though. We observe other phenomena about which light rays and curvature of space are explanatory theories.

As far as I understand it "intuition" for Kant means something pretty close to what we would call 'perception'. — Janus


I disagree with this, but I'll touch on it in replying to your third paragraph. Probably gets to the crux of our disagreement though. — Moliere

I still don't have a lot of time, so since you think this might be the salient point of our disagreement, we probably should focus on defining our terms and thereby hopefully gaining enough confluence to progress the discussion.

"Thoughts without content are empty, intuitions without concepts are blind." This well-worn quotation form Kant I have always taken to be suitably paraphrased as " Conceptions without perceptual content are empty, perceptions without conceptual content are blind".

I've had another response to my question on measuring curvature of space. It makes the excellent point - which I had completely missed - that the path traced out by a laser beam in a constant-time spatial hypersurface is not necessarily a geodesic (straight line) of that hypersurface. Certainly there is no obvious mathematical reason why it should be so, even though our instincts expect it to be. Unless that traced-out path is a geodesic, even the experiment involving the three space stations and lasers may be unable to directly demonstrate a spatial curvature. One would need to do a very complex calculation that decomposed the deviation from 180 degrees of the triangle's angle sum into a component attributable to curvature and a component attributable to the laser paths not being geodesics.

So curvature of space, if it exists at all (recall the observation above that there exist coordinate systems (reference frames) within which entire regions of space are flat), cannot be directly observed even with extremely high tech equipment. One needs to be proficient in GR, and very patient and determined and have a lot of time on one's hands, even to do the calculations that might indicate a curvature.

I suggest that, if curvature of space cannot be directly observed, but only inferred from long, complex calculations that most people would not understand, it interferes with or invalidates our intuition of space in the TA not one whit.

I have also been thinking about what it would mean to have an intuition of the parallel postulate, if that means being able to visualise constructions that demonstrate it.

Consider a line segment AB of length 1cm, with a line L1 going through point A at right angles to AB and another line L2 going through point B at right angles. The parallel postulate says that L1 and L2 never meet.

Now replace L2 by a line L3 through point B, that is at an angle that differs from 90 degrees by such a tiny amount that it intersects L1 at point C, a distance of a billion megaparsecs from AB.

Can you imagine triangle ABC? I can't. If we look at the AB end, what we see looks like one end of a rectangle. If we look at the C end, what we see looks like a single straight line. I cannot hold the whole triangle in my mind's eye.

The parallel postulate says that L2 meets L1 but L3 does not. But I cannot distinguish in my mind's eye between AB with L1 and L2 attached and AB with L1 and L3 attached.

So, for me, the parallel postulate is not something that can be visualised.

Another way of saying that is that 'infinity is a very long way'. It is such a long way that the difference between 'these two lines will never meet' and 'these two lines will meet at a point a billion megaparsecs away' is meaningless - to me at least. Of course I can do calculations with it that have different consequences for the two cases. But that is not visualising it, and I suggest it is not intuiting it either.

So I submit, your honour, that the parallel postulate is not intuitive.

Where Kant went wrong, if this was indeed what he held, was in thinking that our intuition of space and time represented the world as it actually is. — Dennis

But this is not what Kant held. He held that our intuition of space and time represented the world as it is experienced by us. The Euclidean conception of space is the conception of space of our everyday perception of the world. We do not perceive the space of the world as curved. Admittedly we don't perceive it as "straight" either; it is simply neutral. But neutral is straight, and in fact spacetime overall is Euclidean, it is curved only in the proximity of massive objects.

In any case your example of perspective effects is not apt, because our perceptual space in the comprehensive sense is not merely a single view from the ground. All the perspective effects cancel themselves out; if we look at the rail from one end it diminishes to the other; if we look at it from the other, it diminishes to the first, so we know the lines are parallel. The salient point is straightness in any case.