The point is asking "for the nature of it" (in the sense you mean) is incoherent becasue it is defined by not having one at all. It's a "mystery" in empirical terms because there is no empirical account that grasps it.

This is it's nature. We do know what we know. The thing-in-itself is a logical object without empirical form or account. The myth we do not understand this comes out of expecting it to have an empirical form. We mistaken think we need to define it empirically to understand what it is.

We infer that space is curved; we do not experience it as curved. — John

So what are we experiencing when we employ gravitational lensing to see distant planets circling distant stars? - https://en.wikipedia.org/wiki/Gravitational_microlensing

When John mentioned this on p2, you dismissed this rather too glibly - talking about spacetime curvature in a way that was out-of-date since Gauss first defined the notion of intrinsic curvature.

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

3D space doesn't require a fourth spatial dimension in which to curve. The Universe could be a hypersphere - a compact curved spherical 3D space - but it wouldn't mathematically need to float in a larger space like a planet in a void.

So if we are talking about actual modern physical concepts, then even watching a falling rock accelerating in a gravitational field counts as "seeing spacetime curvature". A frame of reference is only "flat" to the degree that energy potential differences have been globally constrained to make that the case.

The non-Euclidean argument really ought to be reversed.

The early assumption was that flatness was natural as the simplest state. And that did mathematically appear to accord with the world as experienced at a "classical scale" - a scale of a temperature and extent convivial to human existence. You could build things using Euclidean geometry and experience would seem to say that you could achieve a perfect carpentered fit. A picture really could be absolutely level to some idea of a horizontal plane.

But nowadays, the more natural presumption would be that flatness is the surprising answer to the ontic question of "how much is everything curved?". Naively, spatial extent could have any curvature value. What's stopping it not? And yet our Universe has this remarkably fine-tuned balance where there is this incredibly tiny deviation from perfect flatness - the hyperbolic curvature of a universal "dark energy" - that means there is in fact something rather than nothing.

A Universe that was actually flat couldn't even exist because it would be too unstable. It would shrivel up under its own weight with the slightest nudge of any local fluctuation.

I'm not sure what this says about TI - as the whole issue of synthetic a-prioris makes sense to me only as saying something about the fact that we find mathematical abstractions to be a way to grasp the symmetry principles reality must employ to organise itself.

So you can see why mathematical abstractions seem a special kind of deal - a way to transcend the epistemic and grasp the ontic. But also just as clearly, for humans the grasping of the relevant maths has been a work in progress.

We started by glimpsing the principles in the near at hand, classical scale of empirical experience. Euclid showed how flat spaces could be constructed from straight lines and stationary points. And from there we have reversed around to see existence from its other end - the view of spacetime (and energy even) as the product of top-down organising constraint.

Again, naively (in this more sophisticated ontic view) the Universe could have, so should have, every kind of spatiotemporal curvature. That means a theory of everything has to aim at discovering a mathematical-strength reason it is instead the case that the Universe is - almost - absolutely flat and classical over at least 140 orders of magnitude of empirically-observable scale.

Then how does it differ from the 'unknowable X' that Kant postulated? Kant did not deny we know this is our situation with regard to our thoughts about the limits of knowledge and reason; that this where we find ourselves placed logically, so to speak, in relation to what we conceive as the 'in itself'.

We know it. It is conceivable. Rather than an absence of knowledge (i.e. a thing we don't know, as we don't have access to its empirical forms), it is a presence of knowledge (i.e. the timeless understood or conceived). In this understanding, we fully grasp the thing-in-itself. Rather than an "unknowable X," it is a knowable X.

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

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? You can research this on Wikipedia and other sources. Euclidian geometry assumes a static space, as if there is a static "present time" at which measurements can be made. But once we account for the fact that nothing is static, everything is moving in time, so that the present time is indefinite, our concept of space must be adjusted to allow for this. That is why a straight line cannot be consider to be the shortest distance between two points. Physicists now take into account that time is passing when they make such measurements. The entirety of Euclidean geometry can be dismissed because it assumes a static space without the influence of passing time. The concept of space which Euclidian geometry presupposes, has been demonstrated to be inaccurate. Therefore Euclidian geometry itself is inaccurate.

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

I think the same general objection applies to Kant as it does to Schopenhauer's metaphysics.

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?

Likewise, from what I am reading about the issue, opinion is divided as to whether non-Euclidean geometry actually invalidates 'transcendental idealism':

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.

Wes Alwan Partially-Examined Life

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.

What you're failing to grasp is that I don't disagree that we know how we think the in itself (and that we know it in this way is all that what you are saying really amounts to). The point is , though, that the in itself is thought as that which logically must 'be there' (even the terms 'be' or 'be there' are not really appropriate here, and in fact nor is any of our language), but of which we can know nothing in any of the kinds of senses that we know the empirical. All we know of it is given in apophatic terms as the negation of the ways in which empirical things are conceived. Non-spatial, non-temporal, non-causal and so on.

Sigh... the point is that's mistaken. The thing-in-itself is not merely an absence of emprical form, but rather its own thing, understood and concieved itself. Rather than merely a negation of empirical forms, the thing-in-itself is something in its own right.

Spacial, causal, temporal, etc. are NOT emprical forms. When we speak about them, we aren't describing the emprical. They are logical expressions.

This means that not only can we not know the thing-in-itself emprically, but also that the emprical is irrelevant to knowledge of the thing-in-itself. We don't lack any knowledge of the thing-in-itself because we can't give it in empirical terms.

I think it is I who should be sighing! You really haven't said anything here. Except for your disagreements with what you think I or Kant have claimed there is literally no content here. And the disagreements themselves are mistaken. Please quote where I said that space, time and causality are "empirical forms".

If you think we know something, anything, positive about the in itself that is given in terms completely independent of any negative reference to the empirical then tell us what that item of knowledge is.

If you think we can't know anything about the in itself, but you want to say that we don't lack any knowledge of the in itself on account of the fact that we can't couch such knowledge in empirical terms, then on account of what do you say we lack any such knowledge?

I really doubt you will try to honestly, and in a spirit of good will, address the questions I'm asking here, but if you don't, I won't respond to you again, unless I see some genuine attempt to address others on common terms in your future posts. Your style of 'engaging' really is abysmal, Willow, seriously.

actually have to say as an amateur Kantian I think Willows post is OK

I think the same general objection applies to Kant as it does to Schopenhauer's metaphysics. — Wayfarer

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)

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

Non-Euclidean geometry includes Euclidean as merely a subset of it, when the curvature of space is 0.

Likewise, from what I am reading about the issue, opinion is divided as to whether non-Euclidean geometry actually invalidates 'transcendental idealism' — 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.

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

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.

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).

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.

Because there is a space-in-itself whose effects we notice, despite our inability to perceive this space-in-itself. — Agustino

Let's suppose there is a "space-in-itself". Isn't this contradictory to Schopenhauer? Space cannot be a pure intuition, nor an idea. And this validates what I have said, none of our spatial principles, Euclidian or otherwise, can be certain. They are all susceptible to doubt.

Let's suppose there is a "space-in-itself". Isn't this contradictory to Schopenhauer? — Metaphysician Undercover

Yes it is >:O (which is the point I've been making all along) Now let me address your other post

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.

If you look here
https://en.wikipedia.org/wiki/Geodesic#/media/File:Spherical_triangle.svg

You see straight line AC (actually a curve from our point of view, but a straight line from the point of view of being embedded in the curved space)

From point B to AC, the shortest distance is still the perpendicular little a (also a curve from a point of view external to that space, but from the point of view internal to it, it is a straight line)

Either the thing in itself gives rise to the world we experience or it is utterly disconnected from it; which would make it irrelevant to us. — John

No it doesn't give rise to the world in a causal (empirical) sense.

3D space doesn't require a fourth spatial dimension in which to curve. The Universe could be a hypersphere - a compact curved spherical 3D space - but it wouldn't mathematically need to float in a larger space like a planet in a void. — apokrisis

This is correct.

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

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. The straight line is one dimensional. The perpendicular employs a second dimension. So it is just an extremely simplistic way of looking at the space around us. The fact that we can establish compatibility between the simplistic way of seeing things, and the more complicated (but more accurate) way of seeing things, does not indicate that the simplistic way is true. It is just the case that in order to convert one's mind from seeing things in the simplistic way, to seeing things in the more accurate, but complicated way, compatibility must be established. This is evident when we went from a geocentric to a heliocentric way of looking at the solar system. Despite the fact that the movements of all the planets, sun and moon, could be predicted from the geocentric model, and these movements had to be made consistent with the heliocentric model, it would be wrong to argue that the geocentric model is still a true way of seeing things.

If you look here — Agustino

So all that is being done here, is that the more complicated, and more accurate conception, "curved space", is being made to be consistent with "a straight line" that you may see. But you only actually see a straight line on a 2d surface, or a 1d string line. And this does not account for the space which exists between 2d surfaces, or all around the string line. It's an extremely simplistic, artificial and manufactured way of "seeing" things. Even the 3d representation does not account for the fact that time is passing, so it is still way too simplistic, artificial and manufactured. It is not a true representation of "space-in-itself" because it is too simplistic. Space-in-itself does not exist independently of time passing, so any representation which cannot account for time passing is inaccurate.

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. 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.

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

Show me a two dimensional creature living on a two dimensional surface. That's fictitious. Why do you need to refer to a fictitious scenario to demonstrate your claim unless your claim is itself fictitious?

The only way you can infer the curvature of your space, is if you find a way to alter it. — Agustino

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.

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

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.

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

OK, we both seem to be saying the same thing, the straight line is really curved. It has an "intrinsic curvature", so its straightness is just an illusion. If you want to say that its straightness is real, and the intrinsic curvature is an illusion, that's fine by me. But remember, you are the one who said "experience occurs on a stage which is ideal and not real". As far as I can tell, the straightness is ideal and not real.

In any case, the reason I wanted to separate geometry from mathematics is that I think all geometrical principles are dependent on empirical evidence, and therefore uncertain. I think this discussion only serves as proof of that point. I would prefer to move on to the principles of mathematics, in an attempt to determine whether there are any purely a priori principles.

It has an "intrinsic curvature", so its straightness is just an illusion — Metaphysician Undercover

... No we're definitely not saying the same thing.

http://mathworld.wolfram.com/IntrinsicCurvature.html

As I have said a million times, Non-Euclidean geometry does not refute the axiom that the shortest distance is the perpendicular - among many other axioms that aren't refuted. So you have to explain to me where does this axiom get its certainty from, because it seems that regardless how our space is, it can't be refuted.

I feel I should make an appearance. As I told Agustino, I freely admit to being fairly mathematically illiterate, which may hinder my ability to see the purported force of his objection. Agustino thinks that non-Euclidean geometry spells the death knell for transcendental idealism, while I have been pushing back against that claim in various ways. He drifts toward realism, whereas I, to the extent that I acknowledge the cogency of his objection, drift toward Berkeleyan idealism, for I say that I do not perceive space Euclideanly or in any other way. What I perceive are impressions, e.g. colors, shapes, sounds, etc, that my intellect fashions into distinct objects in space and time. Thus, I still regard space as an a priori form of knowledge.

Agustino wonders what space is if not its properties, Euclidean or otherwise. There are several ways to answer this question. 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. 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. Space can no more be known in itself than the eye can see itself or digestion can digest itself. It can still be known and perceived, but not in the way that the question assumes. Lastly, geometry tries to determine the properties of points, lines, surfaces, and so on, so it's technically not correct to say that it determines the properties of space itself, since points, lines, and surfaces are themselves in space. Any attempt to know what space is through experience, that is, a posteriori, necessarily presupposes it.

Now Schopenhauer's ontological idealism — Agustino

No. He's an ontological voluntarist, in that the being of the world is will, as opposed to mind, a la Berkeley.

If part of the stage is empirically real, then Schopenhauer's ontological idealism falls apart. — Agustino

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.

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

I think we see two bifurcations in Schopenhauer. There is first will and presentation, which is the world (hence his title). Then there is the world and the thing-in-itself.

Sure, except it doesn't address anything I've said although it purports to do so. I find that frustrating to have someone continually disagreeing with me on account of continually misunderstanding what I am saying.

OK, that's obviously true, but I haven't said it does, so I don't see the point of your post

That's right, and for Schopenhauer the world seems to be Will and Representation, and then over and above the world there seem to be the Platonic Ideas and the in itself. That's what I was trying to explain to 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

This is correct.

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

The will can be considered a weird sort of phenomenon, yes. But it is not composed of parts, given that it's not in space. It is what appears in appearance or what is presented in presentation, which makes it the thing-in-itself for us. But it is not the thing-in-itself in and by itself.

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

The Ideas are outside of space, time, and causality, but they still presuppose the subject/object relation and so are still phenomenal. I would break up Schopenhauer's ontology in the following way:

- The thing-in-itself, which is completely unknowable.
- The will as the thing-in-itself when the latter becomes conscious of itself, which is knowable in time as distinct acts of will identical to the movements of the body.
- The Platonic Ideas as the different grades of the will's objectifying itself, that is, the different degrees of what the will wills, which is life/existence, and knowable in aesthetic contemplation when willing has temporarily abated, wherein one is conscious solely of the Idea and not the movements of one's body or of individual objects in space and time.
- Individual objects as the Platonic Ideas come under and known in space, time, and causal relation to each other.

The last three are all technically phenomenal, but in very different ways.

. 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

I would say part of the principle of individuation because time for example also individuates.

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

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.

No. He's an ontological voluntarist, in that the being of the world is will, as opposed to mind, a la Berkeley. — 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.

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

I don't mean that by stage. I mean space, time and causality by stage.