Thank you for contributing an actual quote from Kant to the discussion. As is usual with me and Kant, I'm not very sure about what he means by it, but I do notice one thing, which is that although he sounds very definite - almost dogmatic - about the primacy of the notion of space, and does connect it to axioms - presumably axioms of geometry - he does not mention Euclid's parallel postulate, which is what distinguishes Euclidean from non-Euclidean geometries.

Before Gauss, Lobachewski, Riemann and others developed the notion of non-Euclidean geometry in the 19th century, people tended to regard Euclid's axioms as a job lot, which you accepted or rejected all together, not picking and choosing. Nobody could see a way of accepting some but not others and still coming up with workable, useful notions.

But now, thanks to those 19th and then 20th century mathematicians we have a good understanding of many different possible axiomatisations, all producing workable, useful spaces. The mathematics is called Differential Geometry and the spaces are called Riemannian Manifolds.

What is common between the different spaces/manifolds - Euclidean and Non-Euclidean - is that they all satisfy the axioms of a Riemannian Manifold, the key ones of which are:

- the space is connected, so that from any starting point you can get to any other part of the space via a continuous path
- the space is continuous, so going along a path you won't suddenly find yourself in a completely different, faraway part of space
- the space has a constant dimensionality that is a positive integer n. For our space n=3.
- between any two points there is a measurable 'distance', which is the length of the shortest path from one to the other, and these distances must:
* be non-negative
* are zero iff the two points are the same
* are symmetric, so that the distance from A to B is the same as the distance from B to A
* obey the triangle inequality, so that the distance from A to C does not exceed the distance from A to B plus the distance from B to C

The Riemannian Manifold axioms are enough to give a very strong notion of space as a three-dimensional area in which things are located and can move about. Further, it accords well with our intuitions of space - well, with mine at least!

My hypothesis, which may be dispelled by further direct Kant quotes, is that perhaps it's this more general notion of space notion that Kant was insistent on, not on a notion that added additional axioms to make the space Euclidean. The test would be whether Kant actually directly mentions Euclid's parallel postulate, or something equivalent to it.

Note that mentions of Euclid or Euclidean do not count, because in Kant's time those terms only indicated a reference to geometry generally, not to something that is distinct from Non-Euclidean space - a meaning that only arose in the 19th century.

In the Kant quote that Agustino kindly provided for us (link above), Kant only mentions two specific aspects of space, which are:

(1) the triangle inequality for distances; and
(2) that things can be 'inside, outside or alongside one another'

These are properties that are satisfied by any Riemannian Manifold, not just Euclidean ones. Perhaps he mentions the parallel postulate somewhere else, but he certainly does not do so in the above quote.

One last thing. The parallel postulate says that there exist pairs of straight lines that never meet, and that pairs that do meet only do so at one place. I, and generations of mathematicians before me, do not find that particularly intuitive, whereas Euclid's other axioms do seem intuitive. That's why people wondered for centuries whether that aximo was necessary in order to do geometry at all. Gauss's brilliance was to show that it wasn't.

Another aspect of the parallel postulate is that the three internal angles of a triangle must add to 180 degrees (or to 'two right angles' as Euclid put it). Again, I do not find this at all intuitive. In non-Euclidean geometry the sum of angles can differ - it is more than 180 for elliptical manifolds and less than 180 for hyperbolic ones.

Anyway, TLDR, sorry about that, My question is: did Kant ever specifically insist that Euclid's parallel postulate was part of our a priori processing of intuitions?

In Kant, intuition is something closer to what we mean in common language by perception. — Agustino

Still the same issue, sensation is necessary for perception, so space, as a perception cannot provide the condition for the possibility of sensation. The ;logical order is reversed.

So pure intuition refers to the perception of space and time. — Agustino

"Perception of space and time" implies that there are these things, "space" and "time" which are being perceived. Either they are perceived through the senses (sense objects), or they are perceived directly by the mind (intuitions). If the latter, then the problem I indicated in my last post stands. These intuitions cannot provide the condition for the possibility of sensation because sensation is prior to intuition.

"The pure form of sensibility I shall call pure intuition" — Agustino

So this is meaningless nonsense, it's unintelligible, incomprehensible.

Kant goes through the thought experiment of taking all sensations away, "heavy", "red", etc. and finds that he cannot get rid of space. Even when one imagines nothing, one imagines that nothing in space. — Agustino

This thought experiment is faulty, because he has already had these sensations. So he cannot put himself in a place of never having had these sensations, therefore he cannot make any determinations about the conditions for the possibility of sensations in this way.

So, apparently you don't have a point of disagreement, then.

I think you are misreading Kant, though.

Kant was trying to show why mathematics is so effective at describing physical space - if mathematics is just a human construct, its effectiveness cannot be accounted. — Agustino

I would say there is no "physical space" for Kant; space is not a physical object. Can you provide a citation that supports your idea that Kant was specifically concerned with showing "why mathematics is so effective at describing physical space"? Of course geometry is effective at describing perceptual space, because it just consists in formulations of our intuitions of the nature of our visual perception.

For Kant space is a pure form of intuition, it is not given empirically, rather it gives the empirical. Euclidean geometry is the direct intuition of the characteristics of perceptual space. Non-Euclidean geometries are not empirically given either but are intuitively derived models of how geometrical principles would diverge form the Euclidean on curved two-dimensional planes. The curvature of space-time is also not empirically given, but is a hypothetical construct, whose predictions have been very precisely confirmed and measured. The point is, though, that spacetime is not the same as space and time understood as pure forms of intuition; it is something else, we know not what, something that we cannot even visualize.

due to its a priority, and the role the pure form of sensation (space) has in constituting all (spatial) experiences, and hence any possible experimental result.

Can you quote a passage from Kant where he clearly claims that all experimental results must be in accord with our synthetic a priori conceptions of the pure forms of intuition? I have read Kant pretty extensively and I don't recall any such claim. The mistake you are making consists in thinking that spacetime is a "(spatial) experience", and that's wrong; spacetime is not perceptual space.

I don't see how the parallel postulate in particular must be mentioned by Kant to count against his apparent view. Kant didn't mark out that postulate as being any less certain that the others, so I don't think one can say he was still correct in insisting EG as metaphysically certain. Interestingly - and it's a sadly not much explored topic outside 1 book and a few papers - you can have inconsistent geometries, dealing with things like Escher pictures and such.

did Kant ever specifically insist that Euclid's parallel postulate was part of our a priori processing of intuitions? — andrewk

There is a remark about something similar:

Suppose that the conception of a triangle is given to a philosopher and that he is required to discover, by the philosophical method, what relation the sum of its angles bears to a right angle. He has nothing before him but the conception of a figure enclosed within three right lines, and, consequently, with the same number of angles. He may analyse the conception of a right line, of an angle, or of the number three as long as he pleases, but he will not discover any properties not contained in these conceptions. But, if this question is proposed to a geometrician, he at once begins by constructing a triangle. He knows that two right angles are equal to the sum of all the contiguous angles which proceed from one point in a straight line; and he goes on to produce one side of his triangle, thus forming two adjacent angles which are together equal to two right angles. He then divides the exterior of these angles, by drawing a line parallel with the opposite side of the triangle, and immediately perceives that be has thus got an exterior adjacent angle which is equal to the interior. Proceeding in this way, through a chain of inferences, and always on the ground of intuition, he arrives at a clear and universally valid solution of the question. — Kant

In addition, there are the remarks of Kant's student, Schopenhauer, more clearly about the same subject:

In fact, it seems to me that the logical method is in this way reduced to an absurdity. But it is precisely through the controversies over this, together with the futile attempts to demonstrate the directly certain as merely indirectly certain, that the independence and clearness of intuitive evidence appear in contrast with the uselessness and difficulty of logical proof, a contrast as instructive as it is amusing. The direct certainty will not be admitted here, just because it is no merely logical certainty following from the concept, and thus resting solely on the relation of predicate to subject, according to the principle of contradiction. 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 WWR Vol II §8

So all evidence available seems to point to the fact that Kant (& Schopenhauer) did consider Euclid's parallel postulate to be a synthetic a priori.

One last thing. The parallel postulate says that there exist pairs of straight lines that never meet, and that pairs that do meet only do so at one place. I, and generations of mathematicians before me, do not find that particularly intuitive, whereas Euclid's other axioms do seem intuitive. That's why people wondered for centuries whether that aximo was necessary in order to do geometry at all. Gauss's brilliance was to show that it wasn't. — andrewk

Why don't you find it intuitive? When you imagine space, isn't this how you necessarily would imagine it? I lean towards saying that my intuition is thoroughly Euclidean, and non-Euclidean geometry wasn't discovered for so long precisely because we don't have an intuition / direct perception of it. Otherwise, why did it take non-Euclidean geometry so long to be discovered?

I don't think one can say he was still correct in insisting EG as metaphysically certain — MindForged

What I'm pointing out is that the words 'Euclidean Geometry' have a different meaning now from what they had in the 18th century. In the 18th century they just meant Geometry simpliciter, because Euclid was seen as the father of geometry and was considered synonymous with it, and because no other sort of Geometry was known and people imagined no other sort was possible.

But now the term 'Euclidean Geometry' is used to refer to a subset of Geometry that excludes manifolds with curvature. To argue that Kant intended that meaning, without additional evidence, is to participate in an anachronism, using a meaning of the term that was not the meaning at the time it was used.

The only way to substantiate a claim that Kant was not just referring to Geometry generally (Riemannian Manifolds) is to find a quote where he specifically insists on the importance of the parallel postulate.

I see that @Agustino has just posted a new quote from Kant involving the sum of angles in a triangle, so I'll read that and see where it leads me. Since I find reading Kant really hard work, it'll probably be quite a while before I have anything coherent to say about it.

But the fact that Geometry was Euclidean Geometry at the time would suggest that he meant Euclidean Geometry, especially as (as you say) Euclid was considered synonymous with it. There was no other way of him to conceive of geometry at the time other than what we refer to as Euclidean Geometry. Granted, it's not as direct as his goof with syllogistic but I don't see how he could have referred to anything else.

What exactly is the criticism? That he didn't discover the existence of elliptic and hyperbolic geometries in 1781? That's a pretty harsh standard, since nobody else came up with them until at least 1813.

Or that he didn't at least understand that the parallel postulate was not necessary in order to have all the usual concepts of continuity, connectedness, insides, outsides, points, lines, angles, volumes, shapes? Again, nobody else knew that until 1813, so why should we have expected Kant to realise it?

The following is counterfactual, and hence unfalsifiable and otherwise empty, but no more so than the rest of the discussion:

I suggest that if a mathematician that Kant respected had discovered these things and had explained to Kant in 1781 that you can get all those things without the parallel postulate, Kant would have taken that on board and related his Transcendental Aesthetic to Riemann and his manifolds, rather than to Euclid.

One background factor in this discussion is provided by the statement of Schopenhauer (given by @Agustino above):

that eleventh axiom [11th axiom is equivalent in the context of Euclidean geometry with Euclid's Fifth Postulate] regarding parallel lines is a synthetic propositiona 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

I think the distinction 'pure and not empirical' is significant, as it refers to any principle which is immediately evident to intuition itself without reference to any empirical or sensory object. This reflects the Platonist distinction between the intellectual intuition which is able to grasp ideas directly, with sensory perception which is of a lower order in only grasping its objects mediately. Whereas, empiricism generally wants to start from sensory perception, and validate any proposition with respect to it. This, then, was the historical grounds that Euclidean principles were to provide insight a kind of higher truth. Whereas:

The discovery of the non-Euclidean geometries had a ripple effect which went far beyond the boundaries of mathematics and science. The philosopher Immanuel Kant's treatment of human knowledge had a special role for geometry. It was his prime example of synthetica priori knowledge; not derived from the senses nor deduced through logic — our knowledge of space was a truth that we were born with. Unfortunately for Kant, his concept of this unalterably true geometry was Euclidean. Theology was also affected by the change from absolute truth to relative truth in the way that mathematics is related to the world around it, that was a result of this paradigm shift.

The existence of non-Euclidean geometries impacted the intellectual life of Victorian England in many ways and in particular was one of the leading factors that caused a re-examination of the teaching of geometry based on Euclid's Elements. This curriculum issue was hotly debated at the time and was even the subject of a book, Euclid and his Modern Rivals, written by Charles Lutwidge Dodgson (1832–1898) better known as Lewis Carroll, the author of Alice in Wonderland. — Wikipedia

(I am sure the Cheshire Cat's grin is relevant here, but can't quite put my finger on how. Perhaps I should try and read his book.)

Looking up Euclid's axioms on wiki, I noticed something interesting. Here's wiki's translation of them:

Let the following be postulated":

1. "To draw a straight line from any point to any point."
2. "To produce [extend] a finite straight line continuously in a straight line."
3. "To describe a circle with any centre and distance [radius]."
4. "That all right angles are equal to one another."
5. The parallel postulate: "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles." — wiki version of Thomas Heath's translation of Euclid

Unless I'm missing something, the 5th postulate would also be true for an elliptic surface, such as the surface of a sphere. In order to exclude elliptic geometries, the words ',and not on the other side' would have to be added at the end of the sentence.

The same applies to the alternative version of the postulate given in the following section of the wiki article:

In a plane, through a point not on a given straight line, at most one line can be drawn that never meets the given line.

It seems to me that 'at most' needs to be changed to 'exactly' in order to exclude elliptic surfaces.

I feel I must be missing something. Can anybody help me find what it is?

I think the distinction 'pure and not empirical' is significant, as it refers to any principle which is immediately evident to intuition itself without reference to any empirical or sensory object. — Wayfarer

Yes, this is an important distinction. There no are truly Euclidean or non-Euclidean objects of the senses, in any case. And perceptual space is definitely not intuited as being curved; in fact we cannot even visualize curvature of a three dimensional space, and non-Euclidean geometries deal only with curvatures of two-dimensional planes.

I think it poses a problem, but I don't think it's devastating to his project. While Euclidean geometry and Newtonian physics are the backgrounds upon which he's clearly thinking from, neither are necessary.

One way of arguing is that our intuition is still Euclidean. So in spite of non-Euclidean geometry, our form is Euclidean.

Another way of arguing: you could say that our intuition of space is actually non-Euclidean (or whatever happens to be the correct geometry of space, supposing non-Euclidean geometry is superseded), and Euclidean geometry was merely an empirical concept of that form.

Unlike spacetime curvature, which is coordinate-independent, the curvature of space (strictly, of constant-time hypersurfaces) varies according to the coordinate system being used to make measurements.

I have just been informed by an impeccable source that there are coordinate systems in which space near a large mass like the Earth is locally perfectly flat. Here's the wiki page that describes those systems.

Problem solved! Immanuel Kant has been vindicated. X-)

As an aside, my source pointed out that, even in the Swarzschild coordinate system that is more typically used near a planet, the spatial curvature near Earth would be about one part in a billion, and probably not possible to detect with current equipment.

Another way of arguing: you could say that our intuition of space is actually non-Euclidean (or whatever happens to be the correct geometry of space, supposing non-Euclidean geometry is superseded), and Euclidean geometry was merely an empirical concept of that form. — Moliere

Nice!

You've got this twisted. The "criticism" was simply that if indeed he believed the axioms of Euclidean Geometry were metaphysically necessary, then Non-Euclidean geometries seem to falsify this notion. So I don't understand why you brought up the historical matter of when such maths were developed. I wasn't criticizing him, I just said that (unless I'm missing something) his view on this matter was incorrect. So bringing up that Kant didn't specifically mention the parallel postulate is entirely beside the point since he was referencing the geometry of the day.

The "criticism" was simply that if indeed he believed the axioms of Euclidean Geometry were metaphysically necessary — MindForged

Kant didn't believe that anything was metaphysically necessary. His whole project involved refuting rationalist metaphysics such as those of Leibniz, Spinoza, the Scholastics and the Ancients. Although he didn't specifically state it this way, his project aimed to show what was phenomenologically necessary for human experience.

and Euclidean geometry was merely an empirical concept of that form. — Moliere

I'm having trouble making sense of the idea of Euclidean geometry as an "empirical concept", other than it being obviously a conceptual scheme derived directly from everyday experience (taken in its broadest sense of both "inner" and "outer" experience). If you mean it in some other sense, perhaps you could explain how it would qualify as such?

The "criticism" was simply that if indeed he believed the axioms of Euclidean Geometry were metaphysically necessary, then Non-Euclidean geometries seem to falsify this notion. — MindForged

Fair enough. I had assumed - wrongly, it now seems - that you were aligning with the group that @Wayfarer identified in this post ( ) that assume the discovery of non-Euclidean geometry undermines Kant’s understanding of a priori truth. If all you are suggesting is that Kant may have had a wrong idea about the necessity of the parallel postulate, then you are not adopting the assumptions of that group. The suggestion seems not to damage Kant's thesis at all, and I do not argue against it.

Since most people in Kant's time believed the parallel postulate was necessary in order to be able to do geometry at all, it is no adverse reflection on Kant, or on the Critique of Pure Reason or the usefulness of the Transcendental Aesthetic, if he believed that as well.

I'm having trouble making sense of the idea of Euclidean geometry as an "empirical concept", other than it being obviously a conceptual scheme derived directly from everyday experience (taken in its broadest sense of both "inner" and "outer" experience) — Janus

That's what I mean. Surely it's sensible that we could be wrong about the form of inuition. So, supposing non-Euclidean geometry is the true geometry of the space we experience it doesn't seem like a large step to say that we were simply wrong before about the form of intuition. At least not to me. If that were the case, then it would just be an empirical concept, though -- since a priori concepts of space are apodeictic.

So, supposing non-Euclidean geometry is the true geometry of the space we experience it doesn't seem like a large step to say that we were simply wrong before about the form of intuition. If that were the case, then it would just be an empirical concept, though -- since a priori concepts of space are apodeictic. — Moliere

OK, the problem I have now is with the notion that we experience space. Space is the pure form of intuition, according to Kant, which means that intuitions (visual perceptions in this case) must take spatial form, and our a priori apprehensions of that space are intuitively obvious to us. Our visual perceptions do not take the form of curved space, and as I said before we cannot even visualize such a thing. So, even if spacetime is curved by mass (whatever that actually means beyond our mathematical models and predictions, I still don't see why we would say that spacetime could be the pure form of our intuitions.

OK, the problem I have now is with the notion that we experience space. Space is the pure form of intuition, according to Kant, which means that intuitions (visual perceptions in this case) must take spatial form, and our a priori apprehensions of that space are intuitively obvious to us. — Janus

I don't think "intuition" in Kant means the same thing as intuitive. Space isn't intuitively obvious to us. Others have been wrong about space -- like Leibniz and Newton, for instance. So while the examples Kant uses are from Euclidean geometry it seems to me that one could modify the philosophy without losing the core of the aesthetic. It's not that something is obvious, but rather that we are able to have synethic a priori knowledge about space due to our knowledge of geometry. If one geometry is wrong then, just like Newton could be wrong, we could understand such sciences as something which wasn't part of our cognitive faculties but was derived from them, and is therefore empirical in that sense (and not synthetic a priori knowledge, but instead rests upon that)

As far as I understand it "intuition" for Kant means something pretty close to what we would call 'perception'. I think space is intuitively obvious to us, and that's why the axioms of Euclidean geometry are self-evident. But what space is (merely an abstraction from objects for Leibniz versus an empty but absolutely existent container for Newton while Kant disagreed with both) is certainly not intuitively obvious.

You say "we are able to have synethic a priori knowledge about space due to our knowledge of geometry" but if this were true then it would not be "synthetic a priori knowledge" at all but synthetic a posteriori knowledge. I think it is more to the point that we are able to have knowledge of geometry due to our synthetic a priori knowledge of space. I think that is certainly what Kant thought.

I don't think it makes sense to say that Euclidean or non-Euclidean geometries are "wrong"; both are intuitively obvious in their contexts. This is not say that it is, or even can be, intuitively obvious that spacetime is curved, because, to repeat myself, I don't think we have any reason to think that spacetime is the same thing as perceptual space, for the simple reason that we cannot perceive, or even visualize, the curvature of spacetime. Is there any reason you can think of why we must believe they are the same?

I don't really have much to say about Kant in general, I've scarcely read his works directly. :)

I would say there is no "physical space" for Kant; space is not a physical object. — Janus

Yes, I am already aware of that. It's necessary for me to talk of "physical space" because Kant was wrong.

The space studied by physics is this phenomenal, empirically real space (which of course is a contradiction - Kant wouldn't claim physics studies space, that would be the job of mathematics). — Agustino

space is not a physical object. — Janus

Sure.

Can you provide a citation that supports your idea that Kant was specifically concerned with showing "why mathematics is so effective at describing physical space"? — Janus

Well, for Kant, there is only one space and mathematics (geometry) describes it with apodictic certainty.

Of course geometry is effective at describing perceptual space, because it just consists in formulations of our intuitions of the nature of our visual perception. — Janus

Nope. This is wrong on two counts. (1), our perception may not be Euclidean. Parallel lines do meet, in our perception, at the horizon. So if you want to argue for this point (that our natural intuition of space is Euclidean), with which I actually agree, you cannot appeal to the "nature of visual perception". (2), there is no "perceptual" space as differentiated from "physical" space (the space we encounter when we do our physical experiments) in Kant - there is only one space.

For Kant space is a pure form of intuition, it is not given empirically, rather it gives the empirical. — Janus

Yeah, or rather, the empirical is given by means of space. Space is the form, and the empirical is the content or matter of that form.

Euclidean geometry is the direct intuition of the characteristics of perceptual space. — Janus

I've already tackled this above.

Non-Euclidean geometries are not empirically given either but are intuitively derived models of how geometrical principles would diverge form the Euclidean on curved two-dimensional planes. — Janus

This is incoherent. Can you perceive non-euclidean geometries? If you can't, then they are not intuitive per Kant's understanding. andrewk has still not told us how he "intuitively" perceives that Euclid's parallel postulate is not a priori.

The curvature of space-time is also not empirically given, but is a hypothetical construct, whose predictions have been very precisely confirmed and measured. The point is, though, that spacetime is not the same as space and time understood as pure forms of intuition; it is something else, we know not what, something that we cannot even visualize. — Janus

Space-time is empirically given, that's why it can be empirically validated.

Can you quote a passage from Kant where he clearly claims that all experimental results must be in accord with our synthetic a priori conceptions of the pure forms of intuition? — Janus

No, and you can't give any to the contrary.

But in Kant's system I can tell you for certain that it can be no other way.

"Therefore in one way only can my intuition anticipate the actuality of the object, and be a cognition a prioir, viz., if my intuition contains nothing but the form of sensibility, which in me as subject precedes all the actual impressions which I am affected by the objects"

Kant conceives the intuition as "anticipating" the actuality of objects, since it already has the cognition, a priori, of space, which precedes & organises the sense impressions of the objects. So objects are given in space, and we know space a priori. Thus, objects necessarily must conform to this space. It follows from this that physics - or anything else - cannot invalidate or cause us to revise our conception of space, if it is a priori given. The only way we could revise it is if it's not a priori. At minimum, space must be more than transcendentally ideal.

Unless I'm missing something, the 5th postulate would also be true for an elliptic surface, such as the surface of a sphere. In order to exclude elliptic geometries, the words ',and not on the other side' would have to be added at the end of the sentence. — andrewk

I don't see why "and not on the other side" must be added when "on the same side" and "on that side on which are the angles less than the two right angles" already exists. This is implicit.

locally perfectly flat. — andrewk

And globally?

Problem solved! Immanuel Kant has been vindicated. X-) — andrewk

:-}

As an aside, my source pointed out that, even in the Swarzschild coordinate system that is more typically used near a planet, the spatial curvature near Earth would be about one part in a billion, and probably not possible to detect with current equipment. — andrewk

But it still exists, hence invalidating Kant.

And you still haven't explained:

Why don't you find it intuitive? When you imagine space, isn't this how you necessarily would imagine it? I lean towards saying that my intuition is thoroughly Euclidean, and non-Euclidean geometry wasn't discovered for so long precisely because we don't have an intuition / direct perception of it. Otherwise, why did it take non-Euclidean geometry so long to be discovered? — Agustino

You keep repeating that Euclid's parallel postulate is not intuitive, but you don't explain why.

I think the distinction 'pure and not empirical' is significant, as it refers to any principle which is immediately evident to intuition itself without reference to any empirical or sensory object. This reflects the Platonist distinction between the intellectual intuition which is able to grasp ideas directly, with sensory perception which is of a lower order in only grasping its objects mediately. — Wayfarer

Ummm, no. Plato's intellectual intuition goes more with Kant's Understanding and the categories than with the forms of sensibility. The forms of sensibility ARE sensuous or sensory in nature. So space and time are not like, say, causality, which is a category of the Understanding. And the forms of sensibility are in no way "lower order" or "higher order" - there is a difference in kind between the content of sensibility and the form of sensibility. The latter is a form - it is the organising principle of the matter, the matter is given through it. And the former is the matter or content itself. On the other side, the Understanding provides the organising principles of our judgements.

You keep doing a mishmash of Plato and Kant and this is totally wrong - these two thinkers are not on friendly grounds in most regards. The Eastern / Schopenhaurian Platonic reinterpretation of Kant fails along with the failure of transcendental idealism.

Some Platonist ideas are common to all Western philosophy. All the comment was about, was the distinction that Schopenhauer recognises between ‘truths of reason’ and empirical observations. It does indeed reflect a distinction which is basic to philosophy, generally.

Some Platonist ideas are common to all Western philosophy. All the comment was about, was the distinction that Schopenhauer recognises between ‘truths of reason’ and empirical observations. It does indeed reflect a distinction which is basic to philosophy, generally. — Wayfarer

Yeah, this isn't controversial.

hence invalidating Kant. — Agustino

I don't agree that it does, but I was wondering who might be an example that strong view that Wayfarer mentioned some people holding. So now I know.

You keep repeating that Euclid's parallel postulate is not intuitive, but you don't explain why. — Agustino

If I could explain it, it wouldn't be an intuition.

The other postulates seem obvious and undeniable to me. That one doesn't. I suppose it must be just the way my brain's wired.

At least I can tell you why it took so long to discover the other geometries though. It's because it wasn't just a question of removing the parallel postulate. It needed to be replaced by something, otherwise we're taking away too much. In fact, what was needed was a complete re-axiomatisation, starting with a completely new set of axioms that does not resemble the existing ones at all. In fact a completely new language was needed, involving things called manifolds, vector spaces, tensors and metrics.

That was a very difficult task, and needed to wait for some extremely clever people to first realise that's what was needed, then secondly work out how to do it.

If I could explain it, it wouldn't be an intuition. — andrewk

For Kant intuition means something closer to perception. So I assumed you were using that term, otherwise, it has no bearing on what Kant was writing about anyways. So is non-Euclidean geometry perceptible in your mind's eye / imagination?

The other postulates seem obvious and undeniable to me. That one doesn't. I suppose it must be just the way my brain's wired. — andrewk

Why doesn't the parallel postulate also seem undeniable? There must be a reason for it, otherwise, I think we will have to attribute it to habit. Are you a mathematician? If so, perhaps you have trained for long enough in non-Euclidean geometry that this training has become second-nature to you.

For me, Euclid's parallel postulate seems undeniable. I am not a mathematician, and from early on I was taught geometry according to Euclid. Euclid's postulates are second nature to my practice of geometry and thinking about geometric problems. Indeed, this may be exactly why I find it easy to perceive the truth of Euclid's parallel postulate, and not to perceive the opposite. It may just be habit, as Hume said, that has entrenched these unprovable things that we take for granted, that Kant now pulls out of the hat as a magician, and calls them "synthetic a prioris". There have also been other critiques of Kant along Marxist lines which claim that Kant's Categories themselves are conceived in the praxis of economic exchange, and then philosophy (& science) is misled to consider them properly basic.

At least I can tell you why it took so long to discover the other geometries though. It's because it wasn't just a question of removing the parallel postulate. It needed to be replaced by something, otherwise we're taking away too much. In fact, what was needed was a complete re-axiomatisation, starting with a completely new set of axioms that does not resemble the existing ones at all. In fact a completely new language was needed, involving things called manifolds, vector spaces, tensors and metrics.

That was a very difficult task, and needed to wait for some extremely clever people to first realise that's what was needed, then secondly work out how to do it. — andrewk

So this new reconceptualisation was not a pure intuition as per Kant's definition of the term? It arose by means other than intuition, such as conceptualisation, right? It took several minds to adjust the conceptualisation so that it all made sense.

For Kant intuition means something closer to perception — Agustino

Yes, I mean what ordinary people mean by intuition, not what Kant means . He uses words too weirdly for me.

is non-Euclidean geometry perceptible in your mind's eye? — Agustino

Yes. It may be, as you say, cos I'm a mathematician. Or maybe I'm a mathematician cos I look at things that way.

So this new reconceptualisation was not a pure intuition as per Kant's definition of the term? — Agustino

I think the concepts are a lot easier than the axiomatisation. The concepts are intuitive (again, maybe only to me), but the axioms are not.