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
So I submit, your honour, that the parallel postulate is not intuitive. — andrewk
So I submit, your honour, that the parallel postulate is not intuitive. — andrewk
The parallel postulate does not say what, based on your post, you appear to think it says. — andrewk
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, 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 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.
They always look parallel, and that's what matters to our intuitions. — andrewk
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
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
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
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