Yes, hence visual/sensory fields — Agustino

OK, but that still doesn't change the fact that you were conflating the truth that parallel lines are perceived to appear to converge due to perspective effect, with the falsehood that they are they are perceived to actually converge.

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. — Agustino

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.

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...

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.

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.

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.

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?

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

A key traditional misunderstanding is that the very existence of non-Euclidean geometry refutes Kant's theory of mathematics. What is often seen stated, e.g. by the great French mathematician Poincaré, is that non-Euclidean geometry is impossible if the postulates of geometry are synthetic a priori propositions. However, it was recognized by Leonard Nelson that Kant's theory in fact allows for a prediction of the existence of non-Euclidean geometry. That is a big difference. The confusion occurs because people forget the basic definition of "synthetic": that any synthetic proposition can be denied without contradiction and thus that the contradictory of any synthetic proposition is conceivable, just as Hume would have said that the contradictory of any "matter of fact" is conceivable. That is true of any synthetic proposition, whether a priori or a posteriori. But, if the postulates of Euclid are axiomatically independent, and if the contradictories of the postulates of Euclid are conceivable and involve no contradiction, then a non-Euclidean geometry built with them would be just as consistent as that of Euclid. The construction of non-Euclidean geometries thus vindicated Kant rather than refuted him. It was Hume and Hegel, who thought that the postulates were analytic, who were refuted.

http://www.friesian.com/penrose.htm

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.

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.

Seemed related to me:

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.

If it's not, I'll but out and go back to earning a living. X-)

What about Maggotsino and Shyster Eggfart? If you don't have a good understanding of Schopenhauer yourself how could you possibly tell whether others do or not? — John

Well, one of them is named after him, darth seems to bring him in a lot, so he's at least read him, and Thoro seems to have a hardon for him, so I dunno. If I'm wrong, then they can let me know >:O

If HE can come up with spoofy names for others then why should such names not be invented for himself and you?

Saying you have read works and displaying understanding of them are two very 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.

Of course that's just my opinion, but it is my honest opinion, and I'm neither trying to offend you nor spare your feelings. If such critique helps you, then good; if not, what can I do? If it does no more than offend you, then taking into account your generally obnoxious style, so full of straw dogs and ad hominems, then I simply couldn't care less.

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. — Agustino

Start from the proposition that space is purely conceptual. It doesn't have to be Euclidean, it doesn't have to be any particular way at all, it is however we as the conceptualizers, make it to be. For example, the circle doesn't have to be 360 degrees, it doesn't have to be divided into four right angles, or anything like that. Human beings chose to do this. They chose to do this because they had a system of numbers, and they wanted to apply the number system to the sensible world, in measurement. So they produced geometry in order to apply numbers in measurement.

Now we have a concept of "space" which has been constructed through the use of geometry. You talk about the "non-Euclideanness of space", but all this means is that you opt for a conception of space which is non-Euclidean. Why would you opt for such a conception? Well, if Euclidean geometry proved to be inadequate for certain activities of applying mathematics to the sensible world, in measurement, then we would have to produce a Non-Euclidean geometry which was adequate. This geometry would produce a non-Euclidean concept of space, and so, the non-Euclideanness of space.

So the question of where does the non-Euclideanness of space come from, is answered with "it comes from the inadequacies of the Euclidean concept of space to fulfil our purposes of measurement". We do not need to make any judgements about whether or not space is "empirically real", all we need to judge is whether or not our geometry (and therefore concept of space), is adequate for measuring the aspects of the world which we judge to be empirically real.

Hardon!!!...in an avowed celibate...now that's inappropriate... but funny!

You do realize that you can't control an erection, correct? >:O

No it's really simple: some of our immediate perceptions tell us that parallel lines appear to converge, and in their broader, more inclusive, scope our perceptions tells us that they do not actually converge, or even appear to converge when we are properly placed in relation to them. It's somewhat like the situation with the bent stick in water. The stick appears bent, so you can say that our visual perception tells us that, but our broader perceptual experience (when we feel the stick or take it out of the water, for example) tells us that the stick is not bent.

No, that's odd; I can control it. Perhaps that's too much information... but you made the erroneous claim...
O:)

Besides, to say that someone has a hardon for something suggests much more than a merely transient randomly appearing tumescence (which they perhaps truly cannot totally control) and suggests something more like a cultivated yearning for something (which they certainly should be able to control, particularly if they wish to be, or become, celibate).

I do recommend abstaining from Schopenhauer, by the way. Kant seems far better and not intrinsically different when it comes to what matters. From the little reading I have done of and about his work, Schopenhauer's differences from Kant certainly seem to matter to him, but are really not of much significance. This is to say that the parts of his philosophy which differ substantially from Kant's are not actually philosophically interesting, or at least I have never come across any presentation or explanation of them that makes them seem interesting to me. I am open to being convinced otherwise, though.

But - we can perceive what it is by analogy to other dimensions (and hence we can conceive it). — Agustino

No, it should read: "But, we can conceive ('perceive' is quite simply the wrong term here) what it is by analogy to other dimensions (and hence we can conceive it). Unfortunately, this is a vacuous tautology.

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

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. — Agustino

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.

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.

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

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.

I would if I knew Schopenhauer's transcendental idealism, but I don't, so I can't. The existence of numbers and mathematics is difficult because it appears as purely ideal. Geometry is easier because it is clearly not purely ideal. Geometry surely gets shaped by its practical application just as much as it does by its free creation in theory. In the case of mathematics it is not so clear, but I believe that there are examples of the theoretical principles of mathematics changing according to practicality. Mathematical principles evolve. We've seen the zero come into existence. We've seen calculus and algebra come into existence. We now have imaginary numbers, and to my mind imaginary numbers allows for contradiction within the mathematical principles.

This implies that there may be no such thing as pure a priori. Perhaps only time will prove to be purely a priori. I think that Kant allows that geometry and spatial concepts are a priori. Schopenhauer may have wanted to limit the pure a priori to mathematics. Ironically it is the existence of the pure a priori which necessitates the assumption of the noumenon (noumena), because this is what transcends experience. By confining, limiting, that which transcends experience, we get closer and closer to the noumenon itself.

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.

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. — Agustino

If 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.

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. — Agustino

Next we have the issue of content. Under the Platonic scheme, the content is no longer real. This comes out as Aristotle's "matter". The form is what is real, and matter is something which we need to assume the existence of, in order to make change intelligible. So as much as "the structure of experience" is real, it is only intelligible through the assumption of matter, which is not real, it is only assumed. Content is assumed in order to make experience intelligible.

Therefore what is the source of this content? — Agustino

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.

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. — Agustino

Now we are nearly consistent with Schopenhauer, the thing-in-itself, matter, content, is what is created by the mind. Schopenhauer may represent it as the creative power of the mind, the Will itself. Or, as you say, the Will is "the closest we get to the thing-in-itself" We've removed Will from all spatial context, so what we can do next is to see if it has temporal context. By referring to Aristotle's concept of matter, which has now become consistent with mental content, as subject matter, we find that this concept refers to that which violates the law of excluded middle. It is what may or may not be, potential. We can give this "potential" temporal context at the present, the middle, between the past and future.

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?

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. — Agustino

Actually, I think you have contradiction here Agustino. Something which is purely ideal cannot be (2) synthetic. It's being analytic which makes it ideal, and this is what guarantees that it is apodeitic as well. Are you sure that Schopenhauer is not arguing that these ideals are analytic?

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). — Agustino

I would not class metaphysics with mathematics here, because being concerned with ontology, metaphysics must have respect for empirical reality as well. But, as l explained earlier, mathematics itself changes and evolves in relation to what is practical, so even mathematics cannot guarantee certainty. The fact that mathematical principles have changed through the passing of time, as we have progressed with our understanding of empirical reality, demonstrates that even mathematics does not guarantee 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. — Agustino

It may be the purpose of Schopenhauer's metaphysics to put all the areas of study in their proper place, but this itself is an empirical practise. It is a process which describes each existing science, and is therefore empirical.

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? — Agustino

You mean non-logical analytical character don't you? The point is, that from the perspective which I described, even the absolute certainty of mathematics is revealed as an illusion. It may be the case that there is something which is a priori, in an absolute sense, the intellect, the will, the soul, God, or some such thing, but even the principles of mathematics are tainted by empirical practicality. 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.

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? — Agustino

You don't see the two sides of the coin here? 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. And this is where we find your mysticism, romantic love, character, etc..

We have the same "two sides" in a lesser developed way with Plato's "the good". There is "the good" in the sense of what is intended, what is desired, wanted, and this is within your mind, an ideal. Then there is "the good" as it is brought into existence in the physical world. In your mind, the good is a desired state, something to be brought into existence, what is wanted. When we act we actually bring something into existence. The intent is to bring into existence the good as it is desired, as it exists in the mind. But the action does not necessarily bring into existence that very same good which was intended, and this is the source of uncertainty. So the "two sides" are not actually two sides at all, they are two distinct things which bear the same name, "good". 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. They are not really two sides of the same thing, they are completely distinct. 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.

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?

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.