(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.Yes, I mean what ordinary people mean by intuition, not what Kant means . He uses words too weirdly for me. — andrewk
Which way?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
As far as I understand it "intuition" for Kant means something pretty close to what we would call 'perception'. — Janus
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 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
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 >:OIt'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
Well, for Kant, there is only one space and mathematics (geometry) describes it with apodictic certainty. — Agustino
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
This is incoherent. Can you perceive non-euclidean geometries? — Agustino
But in Kant's system I can tell you for certain that it can be no other way. — 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.(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
In a way that does not require the space manifold I use to be perfectly flat.Which way? — Agustino
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.We can build rail lines extending thousands of kilometers and the rails are (not perfectly, but on average, parallel). — 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.Yes and that is the space of human perception. — 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.Parallel lines in perceptual space do not meet, otherwise trains could not operate. — Janus
What is spacetime? And how does it relate to the space we intuit?Spacetime, whatever it is, is not that space; that has been my point all along. — Janus
So what about light rays travelling in straight lines but bending around planets? We cannot perceive that or?Spacetime is a hypothetical construct; there is no actual spacetime that we can intuitively understand, as we can our perceptual spaces. — Janus
But we cannot intuitively understand them in three-dimensions, except by analogy, no?Of course we can; we can intuitively understand them when they are visually represented on two dimensional curved surfaces. — 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.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
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 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
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
What is spacetime? And how does it relate to the space we intuit? — Agustino
So what about light rays travelling in straight lines but bending around planets? We cannot perceive that or? — Agustino
Okay, answer when you have time then :PI 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
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.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
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.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 will address this later when I have more time.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 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.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
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.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
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.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
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.Now show me that you can imagine intrinsic curvature in the same way. — Agustino
See, I am tired of "reinterpretations" of Kant such as:Kant scholar's — Janus
From here.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
Actually... I misread your solution initially. At least you seem to understand what the problem is. So here are my comments again:You don't disagree that my solution "works" then, though? — Moliere
On what grounds do we judge a geometrical proposition to be a synthetic a priori?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
(1) Why is it sensible that we could be wrong about the form of the intuition?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
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 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
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?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
So if we don't have synthetic a priori knowledge of the form of intuition there are two main questions: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.
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.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.
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
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
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
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