We infer that space is curved; we do not experience it as curved. — John
We cannot perceive what it would be, because that would entail having 4D eyes. But - we can perceive what it is by analogy to other dimensions (and hence we can conceive it) — Agustino
I disagree with this. There can be no situation where measurement would indicate that the perpendicular from a line to a point isn't the shortest distance from the line to the point. If you think there can be, please conceive of and give me such an example. — Agustino
The OP is "can Schopenhauer's transcendental idealism survive the challenge posed to it by non-euclidean geometry?" — Agustino
Non-Euclidean geometry came along, and it turns out that we have empirical proof that Euclid's Fifth Postulate is actually false, with regards to space as investigated by physics. Now the curvature of space cannot be perceived - we perceive objects in space - things in space curve - but how can space itself curve - that is anathema to our perception. What does this mean for Schopenhauer? — Agustino
Nothing says that our spatial intuition has to be "right," either in the metaphysical thing-in-itself sense, or in the Kantian sense of always being confirmed by higher level theoretical judgments of the understanding, as in physics. And the following remains a fact: from the standpoint of cognitive science, we perceive the world in Euclidean terms. No discovery outside of cognitive science could change the fact that this is how we intuit the world. All of this is a way of saying that the space of physics is not the same thing as the space of ordinary, every-day experience (the former is conceptual and a posteriori, the latter is intuitive and a priori). If non-Euclidean geometry is useful for physics and is better at modeling "space," this means not that Kant must revise his concept of intuition as being of Euclidean space; rather, he must revise his conception of the relationship between understanding and intuition (accounting for the possibility of conflict and a posteriori, non-intuitive conceptions of space).
...ultimately, his system doesn't depend on the absence of such a conflict any more than our discovery of the non-Euclidean nature of space requires that we intuit our living room in non-Euclidean terms.
I don't see how either the mathematical discovery of non-euclidean geometries or the physical discovery of non-euclidean geometry of space-time invalidates Kant's reasoning. Physically, in general relativity it is the large-scale geometry that is non-Euclidean; and in the small-scale, that is locally - the scale appropriate to direct human perception (that is not magnified by extra-sensory instruments) - it is Euclidean. But this is besides the point; even were we to park ourselves close to somewhere where gravitational forces appreciably altered the curvature of spacetime - I think our direct understanding of space and time would remain euclidean. That is we would see for example a ball following a curved geodesic in spacetime as curved in space and through time and not a straight line.
Only if you replace Schopenhauer's conception of the thing-in-itself for Kant's (in other words, only if there is no possibility for a space in-itself)I think the same general objection applies to Kant as it does to Schopenhauer's metaphysics. — Wayfarer
Non-Euclidean geometry includes Euclidean as merely a subset of it, when the curvature of space is 0.But one question is, does non-Euclidean geometry invalidate Euclid's fifth postulate, or does it simply show that it's applicability is limited, in a way analoguous to how Newton's laws of motion were shown to have limited scope with the advent of relativity? — Wayfarer
The opinion is divided over Kant's transcendental idealism, because some people postulate a space in-itself, just like the partially examined life you yourself link to.Likewise, from what I am reading about the issue, opinion is divided as to whether non-Euclidean geometry actually invalidates 'transcendental idealism' — Wayfarer
This must be just false, since it assumes that our space is not curved - if space itself is curved, then you'd actually see it as a straight line.That is we would see for example a ball following a curved geodesic in spacetime as curved in space and through time and not a straight line
Yes indeed. How is it possible for there to be a conflict? Because there is a space-in-itself whose effects we notice, despite our inability to perceive this space-in-itself. This is a materialist re-appropriation of Kant, which is very common in today's world, but Kant (and Schopenhauer) would never ever agree to such interpretations. For them, space is intuited - any geometry always involve some a priori perception. You cannot even have a geometry formed of principles which are not synthetic - any conception of space must make a reference to perception (Sensation), and not just Understanding (concepts). So the only thing that can save Kant is noumenal space. If you admit noumenal space, you're a materialist, end of story. So Kant's project as he conceived it, is all but dead.he must revise his conception of the relationship between understanding and intuition (accounting for the possibility of conflict and a posteriori, non-intuitive conceptions of space).
Because there is a space-in-itself whose effects we notice, despite our inability to perceive this space-in-itself. — Agustino
Yes it is >:O (which is the point I've been making all along) Now let me address your other postLet's suppose there is a "space-in-itself". Isn't this contradictory to Schopenhauer? — Metaphysician Undercover
This is irrelevant though. A geodesic appears as a straight line to observation - in fact, it actually is a straight line in a curved space. Non-Euclidean geometry includes Euclidean geometry - Euclidean geometry occurs when space simply has 0 curvature. But that the perpendicular is the shortest distance between a point and a line holds true in either Euclidean or Non-Euclidean space. In Non-Euclidean space, the perpendicular straight line (or geodesic) is still the shortest distance.I am not a physicist, but I understand that this is the case with 4d space-time. Space-time is curved according to the principles of general relativity. Do you know what a geodesic is? — Metaphysician Undercover
This is irrelevant though. A geodesic appears as a straight line to observation - in fact, it actually is a straight line in a curved space. Non-Euclidean geometry includes Euclidean geometry - Euclidean geometry occurs when space simply has 0 curvature. But that the perpendicular is the shortest distance between a point and a line holds true in either Euclidean or Non-Euclidean space. In Non-Euclidean space, the perpendicular straight line (or geodesic) is still the shortest distance. — Agustino
If you look here — Agustino
No it's really no illusion at all. If you are a two dimensional creature living your live on a two dimensional piece of paper which is curved to form a cylinder, when you're walking around the cylinder on a curved line, you yourself necessarily perceive it to be a straight line, and cannot perceive it as curved. The only way you can infer the curvature of your space, is if you find a way to alter it. We have found a way to alter it in our case - when the sun is between the earth and certain stars, it alters the curvature of the space between earth and those stars, and hence alters our measurement of their position, which we compare to when the sun isn't between the earth and those stars. If you cannot alter the curvature of your space, you cannot even know that it exists, except obviously by other signs such as you walk in a straight line and return to where you started from.To say that it "appears as a straight line", indicates that you recognize this as an illusion, which is not the true reality of the situation. — Metaphysician Undercover
No it's really no illusion at all. If you are a two dimensional creature living your live on a two dimensional piece of paper which is curved to form a cylinder, when you're walking around the cylinder on a curved line, you yourself necessarily perceive it to be a straight line, and cannot perceive it as curved. — Agustino
The only way you can infer the curvature of your space, is if you find a way to alter it. — Agustino
You are the one using a fiction. You rely on seeing those lines being curved in a Euclidean analogy to non-Euclidean geometry to say that they are curved in non-Euclidean geometry which is patently false.You are proceeding in the exact opposite way of reality, away from reality instead of toward reality. You base your unreal claim that two dimensional geometry is true by referring to a fictitious scenario. Then, you claim that you can only understand the true nature of space by altering it. But that's only because you are starting from your fictitious 2d assumptions, then claiming that the only way to make the reality of space compatible with you fictitious assumptions is to alter it. You fail to realize that the proper procedure is to alter your fictitious assumptions, because you cannot alter the reality of space. — Metaphysician Undercover
experience occurs on a stage which is ideal and not real. — Agustino
You are the one using a fiction. You rely on seeing those lines being curved in a Euclidean analogy to non-Euclidean geometry to say that they are curved in non-Euclidean geometry which is patently false.
The straightness of a line is governed by its intrinsic curvature. Non-Euclidean curvature is an extrinsic curvature - space itself is curving. This has nothing to do with the straightness of the line - with its intrinsic curvature. — Agustino
... No we're definitely not saying the same thing.It has an "intrinsic curvature", so its straightness is just an illusion — Metaphysician Undercover
Now Schopenhauer's ontological idealism — Agustino
If part of the stage is empirically real, then Schopenhauer's ontological idealism falls apart. — Agustino
And there is no quadruple aspect theory. Will is the ground of the phenomenon. Platonic Ideas are encounters with and glimpses of the thing-in-itself through art, or mystical experiences. The thing-in-itself is the unknown ground or source of the Platonic Ideas and of the Will. So it's still double aspect - Phenomenon composed hirearchically of Will and then the other Representations, and Thing-in-itself. — Agustino
Nonsense, for Kant the thing in itself is recognized as being logically necessary. He says that for there to be representation it follows logically that there must be something that is represented. It is thought by Kant as noumenal only in the sense that it utterly escapes, by its very definition, empirical investigation. — John
If the thing in itself is the noumenal and Will is not it, but rather merely "close to it", then is Will phenomenal? Obviously it cannot be part of the noumenal according to Schopenhauer, because the noumenal cannot have parts (according to both Schopenhauer and Kant). — John
And what about the platonic ideas? are they noumenal? If they are then how can there be more than one idea. And if all four the noumenal (timeless) the ideas ( timeless) the Will ( temporal only) and the phenomenal ( temporal and spatial) are different form one another, then how are there not four ontological categories? — John
I would say part of the principle of individuation because time for example also individuates.. First, I think we can say that space is the principle of individuation, i.e. it is that part of my cognition that makes what I perceive a plurality of distinct objects. — Thorongil
This is problematic. Space being inseparable from our cognition means that our knowledge is spatially mediated. Knowledge must have both perceptual and conceptual content according to Kant/Schopenhauer, because remember all knowledge must be ultimately reducible to some perception. If space is inseparable from our cognition, and non-Euclideanness is not perceivable in perception a priori, that means that non-Euclidean geometry cannot be knowledge, since it has no perceptual referent. That is obviously absurd.However, because space is inseparable from our cognition generally, the question is technically based on a category mistake, because it's asking for knowledge of that which conditions all knowledge — Thorongil
Yes but I meant it in a different way. I meant it in the sense that the Will isn't material. In that sense it is a form of ontological idealism - the Will is closer to an idea or a subject than to matter.No. He's an ontological voluntarist, in that the being of the world is will, as opposed to mind, a la Berkeley. — Thorongil
I don't mean that by stage. I mean space, time and causality by stage.The stage, assuming by that you mean the mental picture appearing before a conscious subject, is both empirically real and transcendentally ideal. Our experience of objects is not false. — Thorongil
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