What would Kant have made of non-Euclidan geomety?

• 11k
Yes, I mean what ordinary people mean by intuition, not what Kant means . He uses words too weirdly for me.
(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.
Which way?
• 1.1k
As far as I understand it "intuition" for Kant means something pretty close to what we would call 'perception'.

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.

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?

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.
• 11k
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.
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
• 5k
Well, for Kant, there is only one space and mathematics (geometry) describes it with apodictic certainty.

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.

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?

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.

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.
• 5k

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. :)
• 1.2k
(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.
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?
In a way that does not require the space manifold I use to be perfectly flat.
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We can build rail lines extending thousands of kilometers and the rails are (not perfectly, but on average, parallel).
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..
• 11k
Yes and that is the space of human perception.
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.
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.
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.
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.
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.
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.
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.
• 11k
In a way that does not require the space manifold I use to be perfectly flat.
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.
• 5k
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.

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.
• 5k
What is spacetime? And how does it relate to the space we intuit?

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?

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...
• 11k
I already said it is a hypothetical construct.
What does it mean that it is a hypothetical construct?
• 11k
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...
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.
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.
• 1.2k
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.
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.
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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.
• 11k
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.
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.
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.
• 11k
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?
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@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.
• 1.2k
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.
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.
• 11k
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
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