This thread isn't about how Schopenhauer applies a corrective to Kant and it would entail quite a long and off-topic discussion to explain. I don't see why my statements irk you so much. You could read the Appendix to WWR Volume I though if you are interested for a critique of Kant, the categories, and Kant's own misunderstanding of the thing-in-itself (for example Kant didn't understand the nature of the thing-in-itself - he didn't identify that noumena is nonsense - only noumenon makes sense (thing-in-itself CANNOT be plural) - Kant didn't understand what he was talking about, that's why even today Kantians talk about shit like noumenal space, or space in-itself.....................)

The thing-in-itself may remain unknown to Schopenhauer - unknown philosophically, but not mystically. Many occurrences reveal the thing-in-itself - romantic love being one of them - but these experiences resist conceptualisation, and hence remain forever out of the grasp of philosophy.

And this is where we find your mysticism, romantic love, character, etc.. — Metaphysician Undercover

No this isn't where you find them at all. They are just empty concepts in there. Bones with no meat on them.

Are you sure that Schopenhauer is not arguing that these ideals are analytic? — Metaphysician Undercover

Yes. But perhaps I wasn't clear. I didn't mean that space, time and causality are synthetic a prioris, but rather judgements involving them. Space, time and causality are forms of the mind. The reason why, for example, the propositions of geometry are synthetic a prioris is that they involve a synthesis of perception a priori through the form of space given by our cognitive faculty. This explains the certainty we have in geometry (ignoring for now Euclid's Fifth Postulate). The propositions are synthetic because the subject does not include/contain the predicate. "The shortest distance from a line to a point not on the line is the perpendicular on the line from that point" There's nothing in the perpendicular which contains it being the shortest distance - the two concepts are obviously related, but one does not entail the other. Hence these judgements must be synthetic. What Kant/Schopenhauer do, is that they go further and claim that, instead of being a posteriori to experience, they are a priori - hence why they are certain. If they were a posteriori, they wouldn't be certain. And they can't be analytic judgements because the subject "perpendicular" does not contain the predicate "shortest distance".

I guess you'd say that the judgements of mathematics are synthetic a posterioris granted that they have been changing with experience and over time. That may be your position, but it certainly isn't Schopenhauer's, and if you do adopt that position, then the statements of mathematics lose their certainty - they could have been otherwise.

Are you sure that Schopenhauer is not arguing that these ideals are analytic? — Metaphysician Undercover

Space, time and causality are neither analytic nor synthetic - only judgements are one of the two.

I would not class metaphysics with mathematics here, because being concerned with ontology, metaphysics must have respect for empirical reality as well. — Metaphysician Undercover

Yes, but put yourself in Schopenhauer's shoes. Euclidean geometry is capable to perfectly represent your reality in spatial terms. How is that possible? It's because the form that our mind imposes on experience (space) ensures that this is so. There is nothing to wonder about - they are certain because they are of subjective origin - they are forms through which experience itself is possible. In fact, remove those forms, and our experience itself becomes impossible. The world as representation is impossible if there is no space, time and causality. Why? Because any representation is a representation by virtue of being situated in space, time and causality. And these three are ideal - they are the structures of the mind - the forms provided by the mind.

But, as l explained earlier, mathematics itself changes and evolves in relation to what is practical, so even mathematics cannot guarantee certainty. — Metaphysician Undercover

But I'm not sure. Some parts of mathematics evolve - BUT, not all. For example, "the shortest distance from a line to a point is the perpendicular" is a judgement that is certain. How come it is certain?!

The point is, that from the perspective which I described, even the absolute certainty of mathematics is revealed as an illusion. — Metaphysician Undercover

I disagree - there are mathematical judgements that are clearly certain. In fact, non-Euclideanness is a higher viewpoint, which includes and accepts Euclidean geometry as merely one of its subsets.

This is why Plato is forced to posit "the good" in The Republic. The good is what makes all intelligible objects intelligible, and there is no exception here, not even mathematical objects. They are only intelligible in so far as they have a relation to the good. — Metaphysician Undercover

What do you mean to say with this?

There's matter in the physical world, and matter as content, subject matter. They both appear to be the same thing, looked at from two distinct perspectives, two sides of the same coin. In one case we look outward into the physical world, and we find it necessary to assume "matter" to substantiate our empirical observations. In the other case, we look inward, and must assume content, subject matter, to substantiate the existence of intelligible objects, ideas — Metaphysician Undercover

Fine, what is the distinction between the two? Which one underlies the other? For Schopenhauer, the thing-in-itself underlies the phenomena - the thing-in-itself is the source so to speak. Phenomena are merely its manifestation.

Take character - there is both empirical character, and noumenal character. What is the difference? Empirical character is character as it shows itself in the world - in particular situations governed by the forms. BUT - noumenal character is what instantiates itself through empirical character - indeed it is noumenal character that ultimately underlies empirical character. Why does this matter? Noumenal character is unchanging - character is in this sense destiny. If someone has an ugly character, empirically this may not show - he or she may never have the opportunity to show the ugliness of their character - but this doesn't change their noumenal character - it doesn't change who they are in their heart of hearts, and who they would be only if they had the chance

To me, it seems that "matter" as used by the Platonists is unnecessary. There just is no matter, end of story.

So the "two sides" are not actually two sides at all, they are two distinct things which bear the same name, "good". — Metaphysician Undercover

Yeah well said... which is tragic. It makes reality unintelligible. There is no controlling factor at all.

This is the same in the case of the assumed matter in the sensible world (thing-in -itself), and the internal subject matter, the mystical content. — Metaphysician Undercover

No the mystical is the thing-in-itself. The sensible world is merely the representation of the thing-in-itself mediated/individuated through the forms of the intellect (space, time/causality). That is why the sensible world is ultimately an illusion - the veil of Maya - merely the phenomenon of the thing-in-itself.

The fact that we do not have here two sides of the same thing, rather two distinct things, is why we adopt a dualist metaphysics. — Metaphysician Undercover

Disastrous! Dualism is really a disaster! It makes reality completely incoherent. How one side affects the other becomes impossible to explain. What am I really? Am I this or that? How am I both matter and subject matter? That just makes no sense. What is the necessary connection between matter and subject matter? None? Is that connection itself matter or subject matter? How is it possible that the connection is one and not the other?

This is interesting, so fine, let's have this discussion.

Let me first provide more detail on Schopenhauer. Now why is it important that space, time and causality are ideal? Well, their ideality explains both why they are (1) a priori, (2) synthetic, and (3) apodeictic, and hence certain. The Scholastics had trouble in proving the certainty of their principles. Why is the principle of causality certain for example? Schopenhauer answers that it is certain because it is an ideal form applied by our own mind - that also explains why the mind intuitively knows it.

So empirical reality is the domain of science. It is empirically real, and therefore worth studying. It is the external aspect of reality - one side of the coin (the other being thing-in-itself). The forms of the intellect (space, time and causality) are the study of metaphysics and theoretical mathematics (which is why metaphysics and mathematics are guaranteed certainty). The Will is the study of ethics (and metaphysics) and approaching the thing-in-itself. The thing-in-itself is the study of mysticism. The beauty of Schopenhauer's system is that it renders to all these sciences or areas of study their proper place, and also explains how it is that they are possible in the first place.

In addition, the thing-in-itself, and the mystical becomes capable of explaining different phenomena including, for example, romantic love, people's characters (character is destiny), etc. So all these elements form a gigantic explanatory framework for everything, from property to genius, to pretty much any other subject imaginable. I haven't seen any other philosophy do this, or even attempt it.

f we take the Platonic route, we resolve this problem by making the ideal the real. Now the stage, which is the ideal, is what is real, and the sense world is illusionary. — Metaphysician Undercover

If we do this, then how do we explain the certainty of mathematics? How do we explain its a priori nature and also it's non-logical, synthetic character?

Under this model then, the source of the content is the mind itself, matter is an assumption, it is purely theoretical, produced by the mind itself. Matter, content, is what the mind creates. — Metaphysician Undercover

What is the mind?

Now we are nearly consistent with Schopenhauer, the thing-in-itself, matter, content, is what is created by the mind. — Metaphysician Undercover

No you're not, because for Schopenhauer the thing-in-itself is one side of the coin, and the other is the phenomenon. Hence "World as Will and Representation". It doesn't seem like you have two sides here. That's a problem, because how will you account for mysticism, romantic love, character, etc?

Ironically it is the existence of the pure a priori which necessitates the assumption of the noumenon (noumena) — Metaphysician Undercover

Yes kind of. For Schopenhauer, what necessitates the assumption of the noumenon is the fact that experience occurs on a stage which is ideal and not real. So the question arises as to what lies behind the curtain so to speak. If space, time and causality are ideal, then all of experience, which is structured by these, becomes mere phenomenon. So the structure of experience is ideal, but the content must be real. Therefore what is the source of this content? The thing-in-itself as it manifests within the categories of space, time and causality. The Will is the closest we get to thing-in-itself, because the Will is outside of space and causality, but not outside of time. We find ourselves willing so and so; nothing causes it. And neither is our will as we experience it subjectively in space. But our will is in time - one movement of the will occurs after another. These are obviously paraphrased for what Schopenhauer says, which is a lot more nuanced, than my brief and very downgraded remarks here.

Also the thing-in-itself is not a plural - this is indeed incoherent, as the categories of space, time and causality which together form the principle of individuation (hence plurality) do not apply to the thing-in-itself.

No, that's odd; I can control it. — John

Oh yeah, I'm sure you can John. Granted by the fact that you can't even control your mouth, I have high doubts about your capacity to control something else.

Kant seems far better and not intrinsically different when it comes to what matters. — John

Kant is thoroughly confused and muddled up. Schopenhauer clarifies and redeems Kant for the most part, while also in some way maybe also critically deforming the Kantian project. In either case, Schopenhauer is encyclopaedic in the way K will never hope to be.

Can you explain what your own view of mathematics which we've talked about before by the way, has to do with Schopenhauer's transcendental idealism? :s

Right, so you should be able to present a list of, say, about ten of Schopenhauer's original and unique insights then. Can't wait to read them. — John

First you should watch for that prickly and arrogant attitude of yours. I'm under no obligation to provide you with anything, especially while you lack respect.

Just two - (because I don't feel like indulging in your childish nonsense) - I open the WWR Volume I randomly and look at just that page.

"For property, that is not taken from a person without wrong, can, in view of our explanation of wrong, be only what is made by his own powers. Therefore by taking this, we take the powers of his body from the will objectified in it, in order to make them serve the will objectified in another body. For only in this way does the wrongdoer, by seizing not another's body, but an inanimate thing entirely different from it, break into the sphere of another's affirmation of will, since the powers, the work of another's body, are, so to speak, incorporated in, and identified with, this thing [...] 'Wise men who know olden times declare that a cultivated field is the property of him who cut down the wood and cleared and ploughed the land, just as an antelope belongs to the first hunter who mortally wounds her'" (page 334)

"Therefore the man of genius requires imagination, in order to see in things not what nature has actually formed, but what she endeavoured to form, yet did not bring about, because of the conflict of her forms with one another, which was referred to in the previous book [...] The imagination extends the mental horizon of the genius beyond the objects that actually present themselves to his person, as regards both quality and quantity" (page 185)

Literarily every page has markings and notings. Literarily. Schopenhauer has the greatest and at least for certain the most ambitious philosophical system ever attempted, even more ambitious than Spinoza. If it wasn't for my doubts with regards to transcendental idealism as S expounds it, then definitely his achievement would be unparalleled in the whole history of philosophy.

Saying you have read works and displaying understanding of them are two very >:O different things. I admit I am not very familiar with Schopenhauer, so I can't really judge your understanding of his philosophy, but you have previously claimed to understand Hegel, with whom I am very familiar, and yet displayed poor understanding of his philosophy, and that fact, coupled with the general paucity of intellectual sophistication displayed by your posts does not inspire much confidence in me as to your general philosophical acumen. — John

>:O I never claimed to have thorough knowledge of Hegel. Hegel is not Aristotle, Spinoza, Wittgenstein, Pascal, Schopenhauer, Hamann, Kant, Aquinas, Eric Voegelin, or any other philosopher that I have a thorough understanding of. I've read parts of the Pheno and secondary works about Hegel. Never devoted much time to studying the man because his philosophy doesn't interest me that much. Much ado about nothing. If you're so keen maybe I lend you my copies of WWR with notes on every page and highlightings - Schopenhauer is infinitely rich and eminently worth studying - i have to read 50 pages of Hegel to find even one insight. With Schopenhauer they are on every page.

As for intellectual paucity - you should have a look in the mirror - you claim to have studied Spinoza for years and your knowledge of him is piss poor >:O

A comment on Euclidean geometry and relativity by philosopher Kelly Ross: — Wayfarer

Again why does that matter? You post things which are really irrelevant to the thread that is going on. Yes, the fact that non-Euclidean geometries are conceivable logically has never been under discussion. There is a reason why I said I'm talking about Schopenhauer's transcendental idealism and not Kant's. You can save Kant by saying that perceptual space is Euclidean (just like John says) but actual space can be different. For Schopenhauer, there is no space apart from perceptual space. Space doesn't apply to the thing-in-itself. So only the phenomenon is structured by the a priori form of space - the a priori form of space - read that again. So if perceptual space is Euclidean, and perceptual space is all there is - where do we get non-Euclideanism from? Indeed it should be impossible for non-Euclideanism to exist - at least seemingly - Schopenhauer himself states:

But that eleventh axiom [11th axiom is equivalent in the context of Euclidean geometry with Euclid's Fifth Postulate] regarding parallel lines is a synthetic proposition a priori, and as such has the guarantee of pure, not empirical, perception; this perception is just as immediate and certain as is the principle of contradiction itself, from which all proofs originally derive their certainty. At bottom this holds good of every geometrical theorem — Schopenhauer

Really Wayfarer, engage with the actual discussion. We're not talking about Kant's transcendental idealism at all. Schopenhauer says that denying the parallel postulate is like denying the principle of contradiction. Right there - I quoted it for you. See that? Now contribute to the discussion, and get off the ivory tower from which you throw unrelated links and the like.

Maggotsino — John

:-} I have read Schopenhauer's WWR Vol I and II, On the Fourfold Root, and numerous works on him like Bryan Magee's, etc.

You should stop with this retardedness - it's getting below the minimum level of decency expected of someone.

The other interesting feature is that non-Euclideanness always presupposes a larger, n+1 dimensional space. So presumably spaces cannot be non-Euclidean ad infinitum. A 4D space could curve in 5D, and a 5D in 6D and so forth, but ultimately we must reach an Euclidean space, which seems to be fundamental to reaching an end-point to the series. So perhaps we are not mistaken about the Euclideanness of space - we are mistaken about its dimensionality?

appear to converge due to perspective effect, with the falsehood that they are they are perceived to actually converge. — John

You are making an appearance/reality distinction again here. So is this a pheonemon/noumenon distinction then? You are using your terms in a very strange way it seems to me at least. We perceive that they actually converge, just by looking at them. We need to move and change our position etc. to understand that our vision is "tricking" us due to our perspective. Then we conceive of how space really is, behind the appearances. But these distinctions don't fit in Schopenhauer's project at all.

I would say the "non-Euclideanness" as an ontological assertion, is coming from, for one example at least, the fact that we interpret some actual observations, such as the gravitational lensing effect, that was predicted by Einstein's adaptations of Riemannian geometry (if I remember right) to be confirmations that space can be 'warped' by mass. But I would say that we really have no idea what that could mean, because we cannot visualize such a thing. And even if we could visualize such a thing we could not be at all certain in extending it to be any kind of absolute or transcendental ontological claim. — 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). Imagine you are a 2D creature, just width and length, living on a flat piece of paper. You have only two degrees of freedom - left/right and forward/backward. So you move on the piece of paper - the flat paper is an Euclidean surface. Now imagine that your flat piece of paper is folded in the 3rd dimension, such that it forms a cylinder. You drop a rectangle at your starting position, and then you walk in a straight line in the direction in which the paper was folded. If you keep walking, you will return and stumble over the rectangle which you initially dropped. This is a direct effect of the non-Euclideanness of your space, which you can perceive - by walking in a straight, not curved line, you return to your starting position. So there is nothing inconceivable about non-Euclideanness.

Furthermore, I'm not saying that it says something about the ontological nature of space. I'm simply pointing a fact that Schopenhauer doesn't seem able to account for the non-Euclideanness of space as it is not an a priori perception generated synthetically through a form our cognitive faculties impose on us - this only raises the question, where does it come from? And at least one part of space is not transcendentally ideal, and therefore, it can only be empirically real.

Non-Euclidean spaces are nonetheless, insofar as they are spaces at all, and not merely mathematical models, spaces imagined (however limitedly) by us, and so speak no more to what is in itself than our 'normal' perceptions and intuitions do. So, I'm still not really seeing your point. — John

Sure. But how do we have such knowledge? Is it synthetic a priori as per Schopenhauer/Kant? Do we perceive non-Euclideanness in a priori perception? If we don't, as you have already said we don't, then there is a big problem. Where is the non-Euclideanness coming from? It's clearly not a form imposed by our cognitive faculties, because if it was, then we'd be able to see it in perception a priori...

Perception encompasses all the sensory modalities, including the somato-sensory and proprioception; it is not restricted to the merely visual, obviously. — John

Is perception what appears in my visual/sensory fields? — Agustino

Yes, hence visual/sensory fields

The reason behind this thread is that Schopenhauer makes it clear in WWR that space, time and causality cannot apply to the thing-in-itself - indeed space time and causality apply only to the phenomenon/representation which is constituted through the principium individuationis. This for me is the only interesting transcendental idealism - Kant's is too shy, because Kant allows for a noumenal space, Schopenhauer doesn't. But if non-Euclidean geometry is the case, then there appears to be something to space that is not given directly in perception a priori - there appears that there is a noumenal space - this is catastrophic for Schopenhauer.

This is nonsense. We know via perception that parallel lines never really meet. Think of railway lines; they never meet and it would be disastrous if they did. You are conflating the idea that due to perspective effects parallel lines are perceived to appear to meet at the horizon, with the very different and erroneous idea that they are actually perceived to meet. — John

Okay, then I seemingly misunderstand the way you are using perception. Would you like to define it more clearly, so that I can make future arguments based on it? Is perception what appears in my visual/sensory fields? If not, then what is perception in the way you use it?

Eckhart Tolle? O:) >:O

Or Eckhart Meister? >:O In either case, I have no doubt about that lol - I never read either one.

Thoronkill, Schopenpower1, darthbarrasputum to name a few, I think. — Heister Eggcart

In addition to those three, @The Great Whatever also deserves a mention. While he has moved beyond Schopenhauer, he has read him and has good understanding.

Curved relative to what? — Mongrel

Curved in a 4th dimension, obviously.

This is a curve. What are we using the x-y axes for? — Mongrel

Y is time and X is space :-O - but what happens if the x, y, z axis are space, and time is the m axis ;) ? Can you see that? A system with four axes.

That humans can't conceptualize curved space. — Question

But we can conceptualise it very easily. We can't imagine it perceptually, but that's another story.

Thoronkill — Heister Eggcart

Yes that is true, but I've already been discussing with him. He is indeed the only one I'm aware of who is highly knowledgeable in the metaphysics of Schopenhauer

Schopenpower1, darthbarrasputum — Heister Eggcart

These two may be knowledgeable in Schopenhauer, but they are more pessimists, than they are transcendental idealists :P

This is obviously a false assumption to make. — Question

Which one?

I think the OP is off-target with the idea that non-Euclidean geometries have any bearing on the validity of human a priori intuitions about the nature of space. The latter are certainly valid when it comes to perceptual space. — John

I should add that for Kant, at least (I cannot speak for Schopenhauer) time, space and the twelve categories apply only to the empirical world (the world of perception). — John

I hate dealing with such misrepresentations as these - how is space as given in perception even Euclidean to begin with? Parallel lines in perception do meet at the horizon - just like parallel lines meet at the vanishing point in a painting. So perceptual space isn't even Euclidean to begin with. Our intuition of space - which isn't the same thing as perceptual space - is Euclidean.

Are there so few people fluent in Schopenhauer on these boards? :s So far no one with any knowledge whatsoever of Schopenhauer has replied in this thread. This kind of annoys me, because I want to solve this problem. Please reply only if you have read at least one volume of World as Will and Representation, otherwise it seems that it's impossible to have sufficient knowledge.

Idols are easier to commit one's time and eyes to because they're manifested in material. Honesty, loyalty, love, contrition, and so on are all virtues that are beneath the surface. You can't walk into a statue of love like you can a golden cow. — Heister Eggcart

Yes, I think you are partly correct. But then, not all idols, as Wayfarer adds, are material - there are also immaterial idols. So what is the commonality between the two? The commonality is clearly in how the person relates to the object, whether this is a physical or a mental object. So why does a person relate to an object such that the object becomes an idol? How does the person make the object into an idol?

You can not serve too masters at once -- you can't serve God and mammon at the same time. But again, that assumes one believes in God. No god, no mammon. — Bitter Crank

No, all that it assumes is that one must have a master in the sense used Biblically. Even if there is no God, one must have a master.

Probably nothing practically, I don't know. Rather as a Christian is not a good person, but a sinner, and climbers can often be found at the foot of the mountain rather than the peak. — unenlightened

So if nothing practically makes your life about love, then it really isn't about love at all is it? It's one thing to try and fail, and another not to try at all. So if you did mean that you're trying and failing, what is it that you're doing that means that you're trying? If someone looks at your life from the outside, would they say "this man's life is about love"?

When I have given a ring, it was a commitment. — unenlightened

The ring was a commitment? What is the relationship between the ring and the commitment? When does the ring become by analogy an idol?

Yea...I long ago dismissed Agustino as a moral lunatic. — Arkady

"Yes, it happened when he told me that casual sex is immoral" X-)

>:O

If I take your meaning here, yes. Christ taught a change of heart. Jews were only getting the rituals, while they failed to live moral lives, which is why idolatry is a sensitive topic because a lot of Christians think it prudent to remove all symbols and such, which is going too far. — Heister Eggcart

Yes! It's not the action, but the heart.

This goes, again, with Christ's teaching of a turning over of the heart. — Heister Eggcart

Indeed!

The next thing to investigate is how an object, an idea, a person, or anything else can become an idol - what about the relationship of the person with it causes the person to use it as an invisible mirror reflecting their own self back to themselves while hiding the process and making it seem that God is reflected back? What makes the Pharisee blind - spiritually - to his idolatry? And what makes the penitent tax collector worship in truth - authentically?

Let's go back to the Kierkegaard quote with which this thread started. If I can bow down before a stone shaped idol, and still pray to the true God, that means that what makes something an idol depends on me - on my interiority - on my hidden inner life. Would you agree?

Likewise, someone can bow before the icon of the true God, and pray to an idol - because it's their interiority that makes the object an idol - they are making it an idol. How is this possible? How do they make it an idol? What about them makes it an idol? They pray in untruth ... What is untruth?

Is the Pharisee praying to God watching porn? Is he actually doing what the porn viewer is doing? He looks at the true God - but sees in the true God what he wants to see. The true God speaks what he wants him to speak - congratulating him for his righteousness. Who is the "true God"? It's the Pharisee himself - but the Pharisee does not see this. This remains hidden to him.

So what is an idol? An invisible mirror through which the self speaks to the self as if God were speaking...

I'd be careful here. Life is frequently a concept. It's a meaning we refer to and reason about all the time. Even living itself is a concept in this sense (I'm talking about it right now).

Life is not a concept. This means something different than "Life cannot be reduced to a concept," and is not mutually exclusive with it. — TheWillowOfDarkness

Well I stated my point clearly - so is it a true statement that "life cannot be reduced to a concept" or is it a false statement?

I'm hardly being quick. Four or so years of your posting is quite a ripe age. — TheWillowOfDarkness

Even an entire life may not be sufficient to know someone :)

We are living. Take us away, there is no-one experiencing well-being. — TheWillowOfDarkness

Right. So then life cannot be reduced to concept, is that correct?

just as you were — TheWillowOfDarkness

Why so quick to judge? :)

I should add that for Kant, at least (I cannot speak for Schopenhauer) time, space and the twelve categories apply only to the empirical world (the world of perception). Mathematics and geometry thus would also apply only to the empirical world. If there is an anomaly between how we understand time and space intuitively and how empirical observations seem to suggest it 'really is': the question we are left with is What can that "how it really is" be independent of our perceptions and judgements?". Can it be anything for us? Can it be anything in itself? Can it be anything in itself, beyond what we might think it is in itself? It must always remain a speculative hypothesis, I would say. — John

This is not a thread for discussing Kant's transcendental idealism - that should have been clear. You do perceive non-Euclidean space. Not directly. But you perceive its effects. If space is shaped as a sphere, you could drop an egg here, walk in a straight line, and return to the egg. That is perceiving space to be non-Euclidean. What you mean is that you cannot see space itself curving. But that is obvious - you'd need 4D eyes to see your 3D space curving. All that tells us is that our perception is limited and we have blind spots.

The abstraction of "how" or "why" you love is just a dishonest reduction to particular concepts or images, where the lived well-being is reduced to some idea, standard or authority. — TheWillowOfDarkness

Why is this a problem? Why can't the lived well-being be reduced to some conception?

No. — Heister Eggcart

Greater, of course. — Heister Eggcart

For my love to be given, yes. — Heister Eggcart

Ok. So if Love is greater than you, it is not completely contained by you - it always exceeds you. And in order to know it, and be closer to it - you have to reach out of the prison of your self. So man is created in the image of God - destined to reach out of their own self - to point towards God.

To some who were confident of their own righteousness and looked down on everyone else, Jesus told this parable: “Two men went up to the temple to pray, one a Pharisee and the other a tax collector. The Pharisee stood by himself and prayed: ‘God, I thank you that I am not like other people—robbers, evildoers, adulterers—or even like this tax collector. I fast twice a week and give a tenth of all I get.’
But the tax collector stood at a distance. He would not even look up to heaven, but beat his breast and said, ‘God, have mercy on me, a sinner.’ — Luke 18:9-13

Who is the Pharisee really praying to? And who is the tax collector praying to?

If one who lives in a Christian culture goes up to God’s house, the house of the true God, with a true conception of God, with knowledge of God and prays—but prays in a false spirit; and one who lives in a idolatrous land prays with the total passion of the infinite, although his eyes rest on the image of an idol; where is there most truth? The one prays in truth to God, although he worships an idol. The other prays in untruth to the true God and therefore really worships an idol — Soren Kierkegaard

Why is the Christian, in the house of the true God, with a true conception and knowledge of God praying to an idol, while the one kneeling before the statue of the false God is praying in truth to the real God? What is the relationship between the object in front of which one prays and idolatry?

Love is the truth and it is in me - but unless it can reflect itself in the world, it is dead — Heister Eggcart

Is all of it in you? Or is your asceticism a way to reach out of yourself? Is Love greater than you or equal to you?

I think the OP is off-target with the idea that non-Euclidean geometries have any bearing on the validity of human a priori intuitions about the nature of space. — John

So it seems you see no problem with a priori intuitions being false. In trying to save Schopenhauer, you're already on the run - re-treating to saying something merely about human perception and not reality - which is what Schopenhauer and Kant have been attempting to do all along.

Did Schopenhauer mean that the truths of geometry are true only in-so-far as our perception goes? Or did he hold that, since our cognitive faculties structure the world via space, time and causality - these form the very structure of the world, and hence are apodeictic, and cannot be wrong or disproven empirically? Indeed it is this which makes them synthetic a priori judgements - their truth doesn't depend on experience.

I would also question the metaphor of the intuition of space as a 'stage' on which 'things occur'. — Wayfarer

The metaphor isn't central. Space, time and causality form the framework in which representation necessarily occurs, according to Schopenhauer. This framework is provided by the cognitive faculty, and is not empirically real.