In Kant, intuition is something closer to what we mean in common language by perception. — Agustino
So pure intuition refers to the perception of space and time. — Agustino
"The pure form of sensibility I shall call pure intuition" — Agustino
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
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
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.
There is a remark about something similar:did Kant ever specifically insist that Euclid's parallel postulate was part of our a priori processing of intuitions? — andrewk
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
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.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
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?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
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.I don't think one can say he was still correct in insisting EG as metaphysically certain — MindForged
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
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
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.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
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.
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
Nice!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
The "criticism" was simply that if indeed he believed the axioms of Euclidean Geometry were metaphysically necessary — MindForged
and Euclidean geometry was merely an empirical concept of that form. — Moliere
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.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
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
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. — Janus
Yes, I am already aware of that. It's necessary for me to talk of "physical space" because Kant was wrong.I would say there is no "physical space" for Kant; space is not a physical object. — Janus
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
Sure.space is not a physical object. — Janus
Well, for Kant, there is only one space and mathematics (geometry) describes it with apodictic certainty.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
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.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
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.For Kant space is a pure form of intuition, it is not given empirically, rather it gives the empirical. — Janus
I've already tackled this above.Euclidean geometry is the direct intuition of the characteristics of perceptual space. — 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.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
Space-time is empirically given, that's why it can be empirically validated.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
No, and you can't give any to the contrary.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
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.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
And globally?locally perfectly flat. — andrewk
:-}Problem solved! Immanuel Kant has been vindicated. X-) — andrewk
But it still exists, hence invalidating Kant.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
You keep repeating that Euclid's parallel postulate is not intuitive, but you don't explain why.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
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.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
Yeah, this isn't controversial.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
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.hence invalidating Kant. — Agustino
If I could explain it, it wouldn't be an intuition.You keep repeating that Euclid's parallel postulate is not intuitive, but you don't explain why. — Agustino
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?If I could explain it, it wouldn't be an intuition. — 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.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
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.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
Yes, I mean what ordinary people mean by intuition, not what Kant means . He uses words too weirdly for me.For Kant intuition means something closer to perception — 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.is non-Euclidean geometry perceptible in your mind's eye? — 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.So this new reconceptualisation was not a pure intuition as per Kant's definition of the term? — Agustino
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.