• andrewk
    1.4k
    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.
  • andrewk
    1.4k
    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.
  • Agustino
    11.3k
    Kant scholar'sJanus
    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.
  • Moliere
    1.3k
    You don't disagree that my solution "works" then, though?
  • Agustino
    11.3k
    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.
  • Janus
    5.6k
    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 directlyAgustino

    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.
  • Janus
    5.6k
    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".
  • andrewk
    1.4k
    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.
  • andrewk
    1.4k
    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.
  • Janus
    5.6k
    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.
  • charleton
    1.2k


    "What would Kant have made of non-Euclidan geomety?"

    He'd have said it was obvious if you decide that triangles can exist across 3D space, or on the surface of spheres.
    He was smart enough to realise that Euclid assumed 2D.

    Move along now.... nothing to see here!
  • Metaphysician Undercover
    4.2k
    So I submit, your honour, that the parallel postulate is not intuitive.andrewk

    The parallel postulate is simply true by the law of non-contradiction. Lines which do not meet are parallel lines. Lines which are not parallel meet. Anything else would be contradictory.
  • andrewk
    1.4k

    The parallel postulate does not say what, based on your post, you appear to think it says.
  • Janus
    5.6k
    So I submit, your honour, that the parallel postulate is not intuitive.andrewk

    So, say two lines met a billion parsecs away. In that case do they start out truly parallel? If they do, then at what point do they cease to be parallel? If they don't, then does that rule out the possibility of two lines being parallel tout court? If so, on account of what exactly would that be the case? So, take the case of a Globe like the Earth; we have a great circle at the Equator; is a circle precisely one meter north or south of that not concentric, and if it is would not the two lines qualify as parallel against an imaginary plane 90 degrees to the horizontal?

    Of course there are no perfect circles, perfectly straight or parallel lines in nature, but that would seem to be an entirely separate issue.
  • Metaphysician Undercover
    4.2k
    The parallel postulate does not say what, based on your post, you appear to think it says.andrewk

    Yes, it says just what I thought it says, I looked it up before I posted to make sure. It's all a matter of definition, and non-contradiction.
  • andrewk
    1.4k
    So, say two lines met a billion parsecs away. In that case do they start out truly parallel? If they do, then at what point do they cease to be parallel?
    They always look parallel, and that's what matters to our intuitions. There is no part of the triangle we can look at in which the bit we can visualise doesn't either look like two parallel lines, or one line when they are so close together that we cannot distinguish them.

    So if we want an intuitive parallel postulate, I imagine it would have to be something like:

    If a line segment intersects two straight lines forming two interior angles on the same side that sum to an amount that is VISUALISABLY less than two right angles, then there is some VISUALISABLE distance such that, if the two lines are extended for that distance in the direction on which the angles sum to less than two right angles, they meet.

    and this postulate would be met by any space that is no more than very slightly curved, which would include our real world space.
  • Metaphysician Undercover
    4.2k
    They always look parallel, and that's what matters to our intuitions.andrewk

    A line exists by definition, it is a defined thing. You can't see what it looks like, nor can you even imagine it. I think this throws everyone off, you can only know what it is, by its definition. The principles of geometry are definitions which must be adhered to in creating geometric forms. In the case of the parallel postulate we have the definition of "line", and also some defined relations between lines. There is the defined relation of parallel, and the defined relation between parallel lines which intersect another line. The parallel postulate holds, (is valid) only if one respects the various definitions which support it.

    The idea of a "plane", and the relationship between planes is something created by definition, it cannot be sensed nor can it be imagined. In common use, "intuited" means directly apprehended by the mind. This is how these principles are understood. The key thing is to "understand" them, because whether or not they are "true", and how they relate to the physical world is irrelevant to understanding them. They just need to be understood to be used in construction in the physical world. But the fact that they are useful in the physical world justifies the claim that there is some sort of "truth" to them.
  • Perplexed
    70
    According to Kant, the a priori synthetic truths must be certain from the perspective of the phenomenon and our experience. One repercussion of this is that you could not do a physics experiment which did not obey the laws of geometry.Agustino

    Are you saying that Kant would have denied that any physics experiment could reveal that spacial geometry was not Euclidean?
  • charleton
    1.2k
    The parallel postulate is simply true by the law of non-contradiction. Lines which do not meet are parallel lines. Lines which are not parallel meet. Anything else would be contradictory.Metaphysician Undercover

    Since we do not have, and cannot have access to infinity then the premise can only work at a mundane level. Since we can never know if the universe is infinite, or tell if it might or might not coalesce into a pinprick then it might be the case that all lines, including apparently parallel ones might at some point meet.
  • Perplexed
    70
    One way of arguing is that our intuition is still Euclidean. So in spite of non-Euclidean geometry, our form is Euclidean.

    Another way of arguing: you could say that our intuition of space is actually non-Euclidean (or whatever happens to be the correct geometry of space, supposing non-Euclidean geometry is superseded), and Euclidean geometry was merely an empirical concept of that form.
    Moliere

    But if physical effects external to ourselves can be shown to influence the geometry of space, is this not fatal to the assumption that space is an a priori form of our intuition?
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