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If one assumes that the geometry of pure intuition is something other than Euclidean, how does that damage anything in the Critique?

I mean can we still have a grasp of space-time both as an intuition and as an objective thing, to continue his way (by the notion of intuition) and to damage his project (by mentioning its objectivity)?

Farhad Rouhbakhsh
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    It would go against the facts; we immediately *see* space as Euclidean ... an obvious point rather overlooked by his critics. – Mozibur Ullah Sep 11 '17 at 14:10
  • What do you mean by facts? – Farhad Rouhbakhsh Sep 11 '17 at 14:14
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    Just try looking at the world around you...does it look hyperbolic, elliptic or Euclidean? – Mozibur Ullah Sep 11 '17 at 14:23
  • So can we say he was right about our intuition, although modern physics talks about space and time objectively? – Farhad Rouhbakhsh Sep 11 '17 at 14:34
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    I would think so; modern notions of space and time aren't really applicable to the situation he was talking about, they're theoretical notions that apply to situations far from our direct experience. If space and time had significant curvature in our immediate environment we wouldn't be alive to notice it. I think it's also fair to say that Kant actually opened up the way to non-Euclidean geometry by stating that space need not necessarily have the properties we generally think it does. – Mozibur Ullah Sep 11 '17 at 14:38
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    @MoziburUllah "Just try looking at the world around you" that statement lacks nuance, are you arguing that everything in the world is flat and euclidean? Go and draw a triangle on the surface of an apple and ask yourself if the geodesics are curved or flat. "Most things we see in the world aren't curved" isn't a good enough response to the question. – Not_Here Sep 11 '17 at 16:29
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    @MoziburUllah The world *looks* Newtonian and classical to us, the world does not *look* quantum mechanical. My point is that saying "well the world *looks* euclidean" is not good enough. – Not_Here Sep 11 '17 at 16:48
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    One of my great pet peeves is the way Socrates is still presented, by philosophy teachers?, as corrupting the youth of Athens when it is pretty clear to historians that this was a pretext charge, and that his true "crime" was to have been against the Peloponnesian War in its later stages, and to keep making this an issue (Essentially "I told you so") after Athens lost the war. "Athens : a portrait of the city in its Golden Age" C. Meier, 1998 (last part of the book). So I am very open to "new" knowledge on Kant. – Gordon Sep 11 '17 at 17:15
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    It took awhile for non-Euclidean geometry to filter down even to the community of mathematicians in the 19th century. Certainly a man like Gauss could make a lot of progress from merely a hint from Kant. The key is Gauss took Kant seriously, and as I've learned, so did Einstein. My understanding is that Einstein had some trepidation before publishing one of his major papers because he realized it was "overturning" Kant. Einstein was the one who really brought all this to a head. – Gordon Sep 11 '17 at 17:23
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    Fitting non-Euclidean geometry into Kant's theory of intuitive space is what Helmholtz, Riemann and Poincare saw themselves as doing in 19-th century, see [Which school of philosophy motivated thinking about spaces of higher dimension?](https://hsm.stackexchange.com/a/673/55) Their solution was roughly that Kantian intuition of space was too imprecise to single out just one geometry, and one needs empirical observations to pick one of the alternatives consistent with it. – Conifold Sep 11 '17 at 20:32
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    Possible duplicate of [What was the impact of the discovery of non-euclidean geometry on Kantian thought?](https://philosophy.stackexchange.com/questions/6096/what-was-the-impact-of-the-discovery-of-non-euclidean-geometry-on-kantian-though) – Dennis Sep 11 '17 at 23:59
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    @Dennis that other question is talking about the impact but im talking about fitting so they are not duplicate – Farhad Rouhbakhsh Sep 12 '17 at 12:08
  • @FarhadRouhbakhsh "how does that damage" vs. "how does that impact" seem to be getting at roughly the same thing. Both this question and the other one go on to ask, given a certain impact, whether Kant's theory be salvaged in a certain way (how can it be made "fit" with non-Euclidean geometry?). Where they do diverge, on closer reading, is that you're focused on building an objective non-_a priori_ component into intuition (which is to chuck the Critique out of the window, imho) whereas the other question asks about changing the categories. So not exact duplicates, but close. – Dennis Sep 13 '17 at 03:21
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    Comments are not for extended discussion; this conversation has been [moved to chat](http://chat.stackexchange.com/rooms/65537/discussion-on-question-by-farhad-rouhbakhsh-can-we-fit-non-euclidean-geometry-in). – Philip Klöcking Sep 13 '17 at 15:17

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The problem is that our model of space is meant to be a 'form of intuition' for Kant. It should not, then, be modified by experience. There should be nothing out there on the basis of which to modify it, if it is itself an aspect of ourselves and not of nature.

Kan't position is that space and time are not real, but are imposed on reality by our perception. If space itself teaches us something about our imagination, like the fact it is off a bit at high speeds, then he is just wrong on that count.

This is not very central to the notion. The rest of the underlying mathematics may still be a form of our intuition. Notions like the continuity of space, the basic properties of metrics, etc. may be part of the form that proceeds from us, while its 'flatness' is synthetic, and would be different if we lived at a different scale or speed.

So it is not deeply damaging to the theory as a whole. But since geometry is the most compelling example, it robs the theory of one main 'hook' that makes us pay attention to it.

  • Right. And once that hook was gone, and certainly once Einstein came along, this WAS modernity, or the "crisis" of modernity. This was a sort of parallel track to Nietzsche's overturning of metaphysics. Both tracks resulted in the overturning of traditional metaphysics. Remember, Kant was only offering less than 1/4 loaf anyway, since he had carved off religion (to "save" it) and deemed his "thing in itself" unknowable. – Gordon Sep 11 '17 at 18:24
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    @Gordon This is an incredible oversimplification, and I totally disagree. Modernism surely was not based on Kant, and it continued intensifying long after Gauss and Nietzsche. Most people feel that Logical Positivism could only have killed itself, as it did, and could not have been taken down from the outside. Even after that Lyotard's "failure of metanarrative" required a crisis of cultural boundaries, not any single input from philosophy. –  Sep 12 '17 at 16:02
  • Kany's view survives any kind of space. You say "Kan't position is that space and time are not real, but are imposed on reality by our perception. If space itself teaches us something about our imagination, like the fact it is off a bit at high speeds, then he is just wrong on that count. –  Sep 22 '17 at 10:35
  • Kant was not wrong just because we learn new things about space. I don't know why anyone would think this. He did not argue that space is flat or pear-shaped, just that is is a product of Mind. It is a condition for perception, not an outcome of perception. Poor old Kant gets such a raw deal. . –  Sep 22 '17 at 10:46
  • @PeterJ If something is a product of the mind, then what the mind assumes about it should be correct. If you can't follow that logic, don't put words in my mouth by guessing. I said nothing about flatness. I said that our intuition of space should be correct, and we find out that it is not -- that in order for it to even be internally consistent, at high speeds, it needs to be adjusted. Adaptation to outside reality is something that pure ideas should not have to do. –  Sep 23 '17 at 23:02
  • @jobermark - The idea is not that your human mind creates space. A human mind can see a rope and think it's a snake. The point would be that space is not real, not that it is like this or like that. –  Sep 24 '17 at 10:28
  • @PeterJ If space has no independent reality how can it be like anything other than what we bring to it? The rope and the snake have external realities to us, so we can be wrong about them. So must space, if we are wrong about it -- and it is, therefore, something more complex than a 'form of intuition'. There is no place in Kant between a priori and a posteriori -- and to make sense of this requires there to be one. For me, that place is 'phenotypical' -- I have not learned 'space', as an individual, but we as a species definitely learned it. Kant lacks that intermediate notion. –  Sep 24 '17 at 18:51
  • @jobermark - Not sure I quite see your point. I'd agree that space is not a 'form of intuition'. We cannot intuit what is not there. Kant was close, I'd say. but not quite correct. Still, I see no argument against space being a conceptual phenomenon. As Weyl points out, we do not experience space or time. They are a theory overlaid on experience. Whatever the truth I see no reason why the geometry of space makes any difference to Kant. –  Sep 26 '17 at 12:45
  • It occurs to me that some of the problem here may be the idea that space is real AND created by Mind. This is not the idea. The idea would be that created things are not real. This would go for space and all it contains. –  Sep 26 '17 at 12:52
  • So, he is not wrong, but you agree with me that he is wrong... On exactly the point you object to... Space is not a 'form of intuition'. This is the *last* time I am going to repeat myself against a *complete* lack of argument on your part. What you don't see is still there, and the argument from ignorance remains a fallacy. If outside phenomena, like observations of the perihelion of Mercury are needed to convince us about what the right model of space is, and extensive arguments beyond that are needed, it cannot be totally conceptual. –  Sep 26 '17 at 16:47
  • The answer itself allows for the possibility that there are two parts to the nature of space, one that is totally conceptual, and then something else. If that is your point. then yes -- that is what I said. –  Sep 26 '17 at 16:52
  • @PeterJ Sorry -- forgot to include your ID –  Sep 26 '17 at 19:21
  • @jobermark I'm sorry, but I can't see how you arrive at you conclusion. I said he is wrong about intuition, not about the conceptual nature of space and time. I cannot see why the model we use for space would make a jot of difference to Kant's basic point. It is not plausible that your argument here could finally bring an end to the perennial philosophy and three thousand years of investigation. Even you cannot believe this. Or I hope not. –  Sep 27 '17 at 08:59
  • @PeterJ Hope what you want, nothing three thousand years old enters into this question, you are now making no sense at all. Kant does not espouse the perennial philosophy, he says exactly what he says, that space is only a form of intuition and nothing more. You can agree or not, but you can't change what he said. –  Sep 28 '17 at 02:57
  • @Jobermark Kant argues that space is conceptual and you cannot show that it is not. We can argue this all day about this but regardless of anything else Kant about space it remains the case that you cannot show it not conceptual. So Kant is safe on that score. The unfalsifiability of solipsism is enough to prove that you'll never show that space is not conceptual. You don't have to agree that it is conceptual but you do have to agree that you don't know it isn't. –  Oct 02 '17 at 10:51
  • @PeterJ I am not responding to your continued non-argument. –  Oct 03 '17 at 00:37
  • Just one nitpick regarding the discussion: In Kant, there is the form of pure intuition that is space and there is a *concept* of what space is and how it works. The form of pure intuition *is not to be confused* with the concept bearing the same name. The latter may very well be altered because of perception and in fact relies on experience. It will not change much about the necessary conditions of perceiving something as a particular object. – Philip Klöcking Oct 04 '17 at 22:34
  • @PhilipKlöcking I avoided using the concept of concept in the post itself, but I guess I got pulled back into my improper usage in the comments. –  Oct 04 '17 at 22:41
  • Well, am pretty sure that the form of pure intuition itself is no *concept* for Kant, it really is just that: The form of pure intuition. Although of course there is a concept of it as well; otherwise we weren't able to speak about it intelligibly - but are we, really, anyways? – Philip Klöcking Oct 05 '17 at 07:35
  • A problem here seems to be the idea that human minds create space. This is clearly a non-starter. Humans create a theory of space and time and if it is wrong then nothing follows. It would be Kant's 'Mind-in-general' that creates space and its contents, not you and me. It's lack of flatness seems irrelevant to anything. –  Oct 05 '17 at 11:02
  • @PeterJ There is no such idea here. I am tired of criticism not based on what has actually been said. Something that is a mere aspect of a process is not created at all, it does not exist, and can only be conceptualized after the fact. What 'creates' the volume my desk occupies? –  Oct 05 '17 at 17:29
  • Therefore space, if it is a mere attribute of a process cannot have 'push-back' against the process that it is an aspect of. Nothing should ever happen that would make us refine our idea of space in general, any more than I could really have a different idea that my desk occupies a different shape from the shape of the desk without the desk itself changing. –  Oct 05 '17 at 17:34
  • @PhilipKlöcking I do understand the vocabulary gap here, but it is just that, and it does not intrude into the actual answer. Space is an attribute of intuition which is a process that is part of our awareness. –  Oct 05 '17 at 17:50
  • @Jobermark I'm not sure what you're proposing here. I;m suggesting that space-time and all it contains are products of Mind. If space is flat or not- flat this would change nothing. You cannot just waive this idea away as if it has been falsified. If it has then the world needs to know. . –  Oct 06 '17 at 12:14
  • @PeterJ And that would have nothing to do with Kant. So we are done. I don't need to address philosophies totally unrelated to the question just because you misunderstand the original position. –  Oct 06 '17 at 16:22
  • @PeterJ I hope to be proposing what Kant actually said, and as clarified by Philip above, for Kant the idea of space is a product of the mind. But space is not, it is a part of the process of intuition. It exists as part of the mind, not its product. It is not an artifact that can be separated from intuition itself. As such it is an epiphenomenon and is not independently created by the mind or by anything else. –  Oct 06 '17 at 16:31
  • @jobmark - I still do not quite understand your point but if you're suggesting that space exists as part of Mind and is epiphenomenal then we're on the same wavelength. A reduction of Mind would then be a reduction of space. –  Oct 07 '17 at 12:25
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Kant wrote in his first critique:

Space is not a discursive, or as one says, general concept of relations of things in general, but a pure intuition.

This is simply saying we shouldn't confuse the immediate experience of space with the concepts that we use to talk about it; this actually has been important in both physics and geometry, especially because of the popularity of the Cartesian notion of describing space, where one imposes a system of axes and then gives the coordinates of space; instead, when we look at space we see no cartesian grid, taking this cue leads to the notion of general covariance in physics, and describing geometry intrinsically.

it follows from this an a priori intuition (which is not empirical) underlies all concepts of space.

He's elaborating here what he means by a pure intuition - it's an 'a priori intuition'.

Similarly, geometric propositions, that, for instance in a triangle two sides together are greater than the third, can never be derived from the general concepts of line and triangle, but only from intuition, and indeed a priori with apodictic certainty (A24-5/B39-40)

This is where Kant opens up the possibility for non-Euclidean geometry; if we exchange the axiom he mentions with a similar one (that is easier to work with, and changes nothing in what Kant wrote): that the angles of a triangle need not add upto 180 degrees; then, if they add up to less, we get hyperbolic geometry, and if they add upto more, we get elliptic geometry.

Gauss was known to have read Kants first critique where this extract is taken from (at least five times, according to one source) then one could conjecture that this - which is talking about geometry, his speciality - opened up for him the possibility of making a definite mathematical model of non-Euclidean geometry. Sometimes in mathematics all one needs is a hint or a cue, and Kant may, and more than likely to, have provided this for him.

Mozibur Ullah
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    +1 Fascinating info about Gauss. Would you add a citation to this claim? – David C. Norris Sep 11 '17 at 21:56
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    @DavidC.Norris:I've added a citation, it's from an extract from a book called *The Fifth Postulate* which has been posted on a site belonging to the APS (The American Physical Society). – Mozibur Ullah Sep 11 '17 at 23:15
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    It is known how Gauss arrived at the ideas of non-Euclidean geometry, and Kant, unfortunately, was counterproductive to that. Rightly or wrongly, he was taken at the time not as "opening up the possibility for non-Euclidean geometry" but as foreclosing it by providing philosophical "justification" for it. Investigating the consequences of denying the parallel postulate was done by Saccheri and Lambert (with whose work Gauss was familiar), and Lambert even speculated about "geometry on imaginary sphere". Kant corresponded with Lambert but he apparently overlooked his geometrical work. – Conifold Sep 11 '17 at 23:24
  • Thanks to both. That's really eye-opening. I have been reading Kant, I won't say "in a vacuum", but without appreciating this wider contemporary context. – David C. Norris Sep 11 '17 at 23:28
  • @Muzibur Ullah So it seems that modern physics doesnt damage Kant's theory AT ALL. As it was common before Einstein to talk about space & time without mentioning intuition. Am I right? – Farhad Rouhbakhsh Sep 12 '17 at 12:33
  • @Mozibur Ullah So it seems that modern physics does not tell anything new about Kant to us. It doesnt necessarily results in that we should revise his philosophy, as other-than-intuition models of space & time were common before Einstein came up with Relativity. – Farhad Rouhbakhsh Sep 12 '17 at 12:40
  • @conifold: how was 'it known how Gauss arrived at the ideas of Non-Euclidean geometry'; did Gauss publish something on this himself? – Mozibur Ullah Sep 12 '17 at 20:03
  • @Farhad rouhbakhsh: yes, I don't think modern physics is at variance with Kants philosophy; personally speaking, I think some intellectuals got rather too excited about the curvature of spacetime without really thinking through what Kant had wrote - perhaps they were rather more interested in building reputations by overturning the thought of a great philosopher like Kant. – Mozibur Ullah Sep 12 '17 at 20:09
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    From his diaries and correspondence with Bessel, Bolyai (senior), Taurinus, etc., see [Gauss and the Non-Euclidean Geometry](https://www.jstor.org/stable/2968396) – Conifold Sep 12 '17 at 20:17
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    @FarhadRouhbakhsh Kant would reject the interpretation now used to defend him (the way you'd like) as psychologistic, it reduces space and time to some accidental mental wiring specific to us, which in the end is irrelevant to what physics studies. The problem with Kant's doctrine is not space and time specifically but the fixity of a priori (he even thought Newton's laws were a priori), and those most sympathetic to him, like neo-Kantians, made them fallible and revisable rather than making them psychologistic. So fallibility of a priori is what modern physics taught us regarding Kant. – Conifold Sep 12 '17 at 20:43
  • @conifold: I've read through the article and it merely establishes that Bolyai, Lobachevsky & Schweikart independently discovered non-Euclidean geometry from Gauss. Nothing in there demonstrates why they were inspired to think a different geometry was possible; and in fact there is very little about Gauss himself there. – Mozibur Ullah Sep 12 '17 at 21:02
  • @conifold: nothing about his correspondance to Bessel either; one interesting point, is that Sweikart mentions is considering the plane as an surface of an 'infinite sphere'. – Mozibur Ullah Sep 12 '17 at 21:03
  • @Farhad Rouhbaksh: I'm a bit nonplussed about it; it's the first time I've heard of Kants ideas as psychology; it strikes me that psychology is oriented towards something very different from what Kant was talking about, which was more about the rational structure of the mind; that's why he's usually described as an idealist philosopher; psychology is something that I associate more with Freud, Jung and so on. – Mozibur Ullah Sep 12 '17 at 22:23
  • @FarhadRouhbakhsh Our "intuitive sense" is Riemann-Poincare's psychologization of Kant, they reduced his a priori to our mental instincts. To Kant those are universal forms of pure intuition that schematize all empirical knowledge, not intuitive or some other vague cognition. Kant's quote in the post says unequivocally that things like "in a triangle two sides together are greater than the third" are "a priori with apodictic certainty", which one can not get from vague feels about space and time. Kant would likely rather have schemata evolve than surrender them to psychology. – Conifold Sep 13 '17 at 00:02
  • Bessel's letter to Gauss is quoted on p. 252, it mentions Lambert. For a more complete picture look at Gauss's scientific biographies and historical works on reception of Kant in the early 19-th century. – Conifold Sep 13 '17 at 00:39
  • @conifold: ok, I had overlooked it; it's a very short extract and written in 1829:"through that which Lambert said, and which Schweikart has disclosed orally it has become clear to me that our geometry is incomplete and which is hypothetical and, *if the sum of the angles in a triangle adds up to 180 degrees vanishes*. That were the true geometry, the Euclidean the practical, at least for figures on the earth". – Mozibur Ullah Sep 13 '17 at 01:00
  • Doesn't it strike you a little peculiar that four people independently had the idea of non-Euclidean geometry in roughly about the same time, around 1800-1830? It suggests to me that somehow the ground had been already prepared to entertain such an idea. – Mozibur Ullah Sep 13 '17 at 01:04
  • @MoziburUllah certainly the groundwork was laid for such a breakthrough, a necessary condition for the four people independently making the discovery. It strikes me as a leap, however, to assume there was a single figure that laid this groundwork, the figure was the common influence on the 4, and that figure was Kant. That's the kind of claim I'd want to see sourced. Otherwise it's a bit like going from the fact that several different ancient civilizations built pyramids to the assumption that they had a single figure influencing them. Ancient aliens, as the History Channel would tell it! – Dennis Sep 13 '17 at 03:39
  • @FarhadRouhbakhsh I think the problem is that Kant is not using "intuition" in its modern colloquial sense of "intuitive insight", his meaning is completely different and is closer to what we call imagination, it is a medium for generating and observing particulars. As for the "real intuition" (usually called intellectual intuition), Kant flatly denied its existence, along with introspection. Apodicticity comes not from insight but from the structure of cognitive syntheses. Your position is closer to Descartes and Fichte, on Kant's and modern view insight is too flawed for apodicticity. – Conifold Sep 13 '17 at 20:22
  • @dennis: I see what you're saying but I'm not sure the analogy is correct in this case; after all, there you're talking about a wide range of geography, and of time; here we're talking about a small and close knit tradition - geometry in the early 18th C, over a period of thirty years; I'm not making the case that all four were influenced by Kant, but that it's certainly a possibility. – Mozibur Ullah Sep 14 '17 at 03:49
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    @MoziburUllah Agreed. I chose an extreme example to illustrate the issue clearly. Your possibility is plausible, but the best answers don't just toss out plausible possibilities. They argue for them (citing sources as appropriate to back up the claims of the argument), or they cite a credible source that has made such arguments. This is simply a suggestion for how you might improve your answer and make it into something that can aid others in further study. Generally, though, I'm a "source your answer" hawk, so we might differ on standards for ideal answers. – Dennis Sep 14 '17 at 03:57
  • Here's some further [evidence](https://www.math.uwaterloo.ca/~snburris/htdocs/noneucl.pdf) that Gauss was on familiar terms with Kants work; in a letter he wrote to Bolyai Senior im 1832: "precisely the impossibility of deciding a priori between ... gives the clearest proof that Kant was not justified that space is just the form of our perception'; perhaps looking through the corpus of Gauss work might turn up more evidence either way. – Mozibur Ullah Sep 14 '17 at 04:07
  • @conifold: how about his theory of judgement? Does not that count as intellectual intuition? – Mozibur Ullah Sep 14 '17 at 04:08
  • @dennis: sure, I know how evidence works in history; besides, if you look at my answer I don't assert the truth of hypothesis, but merely put it forward as a plausible hypothesis - and you've agreed on its plausibility; I wasn't intending to go any further than that other than to also show that Kant didn't hold ideas about geometry that is commonly thought. – Mozibur Ullah Sep 14 '17 at 04:44
  • Ironically, Critique of Judgement is exactly where Kant spends a whole section denying that we have intellectual intuition, and excoriates Spinoza for holding otherwise. There is no need to comb Gauss's texts, it was hard to live in Germany at the time and not hear about Kant's teachings. In a letter to Bessel Gauss complained about "scream of Beotians" (i.e. of fools) at the suggestion of non-Euclidean geometry, his reason for not publishing on it. It would be unfair to blame its entrenchment on Kant alone, but his declaring it a priori was no help. Your quote indicates the same. – Conifold Sep 14 '17 at 19:31
  • @conifold: Really? The SEP seems to think [otherwise](https://plato.stanford.edu/entries/kant-judgment/):"the power of judgement ... is a cognitive capacity ..., a spontaneous and innate capacity ... which is also the faculty of thinking ... the mind for itself is entirely life"; do you have a reference for your assertion? Kant has a very technical vocabulary, so I think it stands really on how Gauss understood him. – Mozibur Ullah Sep 15 '17 at 03:17
  • @MoziburUllah: §76 of CPoJ is quite explicit about how this is not an ability of finite rational beings, which does not prevent us from considering it as possible. – Philip Klöcking Sep 15 '17 at 09:30
  • @FarhadRouhbakhsh The pre-insight view is not uncommon but it is not Kant's. And AVP's answer explains one reason why: there is nothing for him that it can be insight *of*, there is no "really exists" pre-synthesis, so no sense to it. But saying that space and time exist beyond human experience, and we have insight into that, is not the only way to revise Kant, another way is to say that "alphabet and grammar" of a priori is not set in stone, it can and does evolve. Neo-Kantians chose the latter. This gives "apodictic" certainty, but only for as long as we choose to keep the "grammar". – Conifold Sep 20 '17 at 00:08
  • @conifold Let's stop this chat. You don't understand what I am saying. – Farhad Rouhbakhsh Sep 20 '17 at 21:36
  • The discussion of Kant is necessary to answering the question, but his claim that space-time is a conceptual phenomenon remains even if his argument is faulty. We may believe that non-Euclidean geometry causes problems for Kant, although whether he would think so is a difficult question, but it poses no problems for a conceptual space-time. –  Oct 03 '17 at 12:05
  • @PeterJ: he makes no claims about spacetime as opposed to space and time; the notion wasn't available in his time. – Mozibur Ullah Oct 05 '17 at 08:19
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If you were to plug in different modules for space and time in the Transcendental Aesthetic, or if you were to fiddle around with his categories, what would this to to his project of wanting at least something to be "fixed" in place? True for all, if you will. Remember, the world of phenomena is already contingent, so can't something stay put and be true, permanent and pure? So non-Euclidean geometry would have been a bombshell for Kant, it might have shaken him to the core (at least for a while).

So it would have damaged Kant's project, but it would not have damaged his way. See Cassirer's idea here: http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/significance_GR_geometry/Einstein_on_Kant.html

We don't seem to mind such changes today. Kant 1.0, 2.0 etc like software updates, but this kind of thinking does not fit well with certain kinds of metaphysics which seek permanent truth, fixity, etc. And I should mention that Kant was trying to scrape together what knowledge he could. It was still limited in the fact we don't know the thing-in-itself per Kant, and this hanging problem of the thing-in-itself served as the irritant-stimulant to the next great round of German philosophy: Fichte, Schelling, Hegel, Schopenhauer.

Gordon
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  • So can we still have a grasp of space-time both as an intuition and as an objective thing, to continue his way (by the notion of intuition) and to damage his project (by mentioning its objectivity)? – Farhad Rouhbakhsh Sep 11 '17 at 14:42
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    I've had a look at Kants *first critique* and he mentions that the angles of a triangle need not necessarily add up to 180 degrees; hence I'm not so sure that non-Euclidean geometry would have come as 'a bombshell'; Gauss by the way, read the critique a number of times, so it might have opened his eyes to thinking about geometry in a different way. – Mozibur Ullah Sep 11 '17 at 14:43
  • I removed my reference to Gauss for various reasons though he is one of my favorites. I am going to modify Kant and bombshell a bit to because Kant was not the type to lament forever, he would have gone on working. – Gordon Sep 11 '17 at 14:50
  • @FarhadRouhbakhsh Well, according to Kant we don't know the thing-in-itself. All we have is contingent phenomena/appearance. So now if you also make the subject (human subject) also completely "subjective" essentially eliminate the Kant's "Copernican revolution", then Hume would certainly have the last laugh on Kant, in a manner of speaking. – Gordon Sep 11 '17 at 15:04
  • @MozibrUllah. I was not aware of that reference, it's interesting. I don't know the context. It doesn't convince me to change my answer. – Gordon Sep 11 '17 at 15:09
  • @Gordan: I once quoted the relevant extract from his first critique in an answer somewhere; I won't dig it up, as my intention isn't to get you to change your answer! – Mozibur Ullah Sep 11 '17 at 16:15
  • @MoziburUllah But if space proceeds from *us* as a form of intuition, our model should not change. In his theory, space is not just synthetic a-priori like arithmetic, it is entirely phenomenal, imposed by the limits of our imagination. In my (not too schooled) understanding, t is the strongest argument for pure phenomena and therefore for the transcendental distinction itself. –  Sep 11 '17 at 17:32
  • @jobermark: I'm pretty sure that he wrote both space & time are synthetic a priori, so not 'entirely phenomenal'; to be fair, he does ask whether such intuitions are even possible. – Mozibur Ullah Sep 11 '17 at 17:38
  • @MoziburUllah No, they are pure intuitions, 'geometry' is synthetic a priori, space is supposed to be entirely a requirement of intuition, arising from within ourselves. This summary seems to reinforce my recollection that this is a real distinction: http://homepages.wmich.edu/~baldner/spacetime.htm Though I am not sure I follow where 'space' stops and 'geometry' commences. –  Sep 11 '17 at 18:09
  • @jobermark: actually the distinction does makes sense... – Mozibur Ullah Sep 11 '17 at 18:13
  • @jobermark: That was my reasoning too, I mean where space and geometry intermixes; but it's made me want to revisit Kant at some point to clarify. – Mozibur Ullah Sep 11 '17 at 18:16
  • A human intuition can be less than adequate and may be modified or enriched by experience. Non-Euclidean space may be just as unreal as the Classical kind. No problem for Kant. Indeed, that we may have to describe space as non-Euclidean could be used as evidence that he is right about it's conceptual nature. Or so it seems to me. –  Sep 14 '17 at 15:53
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I feel that the philosophical consequences of the discovery of non-Euclidean geometry and later its use in Relativity are overstated.

Our imagination is limited to flat space of dimension three. We cannot visualize anything unless embedded in 3-dimensional flat space. Euclid's axioms are a formalization of our intuition of space. This is the result of Greek abstract thinking over centuries and became a pillar of European mathematics. Therefore we tend to identify the formalization by Euclid with the underlying intuition. I think Kant refers to the latter.

The hypothetical case, that another type of geometry were the geometry of our intuition, might have lead to a different attempt of formalization and in the end Kant's arguments would be exactly the same with respect to that geometry (of course we would be also different beings so this is very hypothetical). In other words Kant's arguments do not depend on the specific form of Euclidean geometry but on the fact that it is a formalization of our natural intuition. Of course one can modify any of the Euclidean axioms and obtain other formalisms. However it is questionable that the result still qualifies as a formalization of our intuition the way Kant understood it. Mathematicians have no problems dealing with curved (Riemann) manifolds of any (including infinite) dimension but these are formal constructions far from our basic intuition or imagination. In all these constructions however Euclidean space remains the standard model. Curvature, as example, is described via the curvature tensor as deviation from the flat case, i.e. we describe curved space via comparison with Euclidean space.

The role of space-time is a different question. As far as I know it was not subject of Kant's theory. Space-time is a mathematical concept to describe motion (Galilean or relativistic). We can visualize a an object moving in Euclidean 3-space and one might argue, whether this would qualify as another example of Kant's theory. We still cannot visualize the whole trajectory in 4-dim space.

Space-Time in general relativity is not only (in the presence of mass) curved but there is also no natural separation of space and time: the concept of 3-space is not natural to general relativity. It requires a synchronizable reference system (a bunch of observers who can agree of a common time scale) and this space would only be partially observable. (because of finite speed of light we can only observe objects in our past within the light cone, i.e. close enough that light reaches us). Thus space-time is far from anything intuitive. Taking space-time as objective sounds more like a realist's perspective and moves away from Kant. Curved space-time is a very elegant description gravity, but not the only possibility to describe motion of masses or Einstein equation. One could consider a flat background space theory - less elegant and problematic for a realist interpretation. An example how Euclidean intuition, sometimes subconsciously, guides our thinking: Physicists talk about the effect of light deflection in relativity - deflection from what ? as if there were a notion of straight light rays. In summary, relativistic space-time is far from intuitive, not necessarily "an objective thing" and I cannot see any impact it could have on Kant's philosophy.

C.Gunther
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  • That Kant refers to the psychological intuition of space was a re-interpretation of Kant by Helmholtz about 50 years after his death. Kant is explicit that he does not refer to our mental predispositions, and his argument for apriority of space is not based on introspection or "formalization of our natural intuition". What you describe is a common way to "justify" Kant today, and it has "intuitive appeal", but no historical basis in his works. This is why non-Euclidean geometries had a knock-out effect on the a priori part of his philosophy, he went too far and claimed too much on this one. – Conifold Sep 21 '17 at 23:49
  • +1: for 'the philosophical consequences of the discovery of non-Euclidean geometry and it's later use in relativity is over-stated' – Mozibur Ullah Sep 22 '17 at 09:38
  • @Conifold Could you elaborate? How does psychology enter here ? Just tried to raise 3 points: 1.Euclidean Geometry is a formalization of our cognitive capacity which Kant calls space. It is the geometry, which is a priori, not the axioms. (the word intuition in this context may be misleading, just used the questions wording). 2.Non-Euclidean geometry is mere a modification of the axioms, a technicality. Kant might have called it poetry. 3. Relativistic Space-time geometry does not describe a property of space or time, so it does not even relate to Kant's theory of space. – C.Gunther Sep 23 '17 at 02:09
  • @Conifold It would be nice to learn more where you see a knock-out effect, and which parts of Kants philosophy would be affected. – C.Gunther Sep 23 '17 at 02:11
  • For Kant synthetic a priori are not merely "formalizations of our cognitive capacity", he claims that all and any empirical knowledge, and not even just human one at that, must be schematized in (Euclidean) space and time to be brought under the categories of understanding at all. In other words, the mere possibility of empirical non-Euclidean geometries and relativistic spacetime flatly contradicts his conception. As I said, he went too far and claimed too much. Fudgy "capacities" and "intuitions" related to our species' constitution are traditionally called "psychologism" in epistemology. – Conifold Sep 23 '17 at 02:29
  • @Conifold - You make a point but don't really back it up. I see no reason why non-Euclidean space would make any difference to Kant and would agree with Gunther. Nobody has ever suggested that non-Euclidean space falsifies the Perennial philosophy, which agrees with Kant on the ontological status of space-time. It simply makes no difference how we describe space. I would disagree with Kant about intuition since one cannot intuit something that isn't real. So I'd modify Kant by reference to Weyl (who knew about Riemann etc). The end result is the same. Space would be conceptual. . . –  Sep 26 '17 at 11:59
  • @PeterJ It may make no difference to Perennial philosophy how we describe space but Kant held very strong opinions about it. They are not however of the kind described in this post. But I think you'll agree with Kant on "intuition", he simply did not have the conception of it found in Poincare, Brouwer, Weyl, etc., it was a revision developed partly in response to the discovery of non-Euclidean geometries. As I explained in comments to another post in this thread, for him we do not have spatial "intuition" in the modern sense of the word. – Conifold Sep 26 '17 at 17:58
  • @Conifold Okay. I'm now a little confused about where we agree and disagree. No problem. Kant does tend to cause arguments. I find that Weyl makes a much simpler and better argument for the conceptual nature of space-time than Kant. –  Sep 27 '17 at 09:04
  • @Conifold I am not questioning you interpretation of Kant although it appears quite strict to me. But I do disagree with this interpretation of Non-Euclidean geometry: Non-Euclidean geometry in dim >2 is not empirical (it is a mere abstraction) and has never been applied to space alone. In dim 2 it describes just subsets of Euclidean space and was implicitly known at Kant's time from ocean navigation.Space-time on the other hand has, as far as I know, never been considered by Kant. Space-time (at least Relativistic) does not even define empirical 3-space. – C.Gunther Sep 28 '17 at 17:55
  • There was an intermediate period before the modern idea of geometry as formal abstraction took hold, so the "implicitly known" seems somewhat anachronistic to me. In 19th century non-Euclidean geometry (including 3D) was first pondered as the (non-implicit) empirical geometry of observed space, Riemann describes the latter as only locally Euclidean a priori, and subject to measurements. But even this empiricization was read out of the popularization of Kant by Herbart, his successor at Königsberg. Geometry of spacetime subsumes geometry of space, which is even more at odds with original Kant. – Conifold Sep 28 '17 at 19:48