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)?

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    It would go against the facts; we immediately see space as Euclidean ... an obvious point rather overlooked by his critics. Sep 11, 2017 at 14:10
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    Just try looking at the world around you...does it look hyperbolic, elliptic or Euclidean? Sep 11, 2017 at 14:23
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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. Sep 11, 2017 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, 2017 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, 2017 at 16:48

4 Answers 4


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.

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    +1 Fascinating info about Gauss. Would you add a citation to this claim? Sep 11, 2017 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). Sep 11, 2017 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, 2017 at 23:24
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    From his diaries and correspondence with Bessel, Bolyai (senior), Taurinus, etc., see Gauss and the Non-Euclidean Geometry
    – Conifold
    Sep 12, 2017 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, 2017 at 20:43

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, 2017 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.
    – user9166
    Sep 12, 2017 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.
    – user20253
    Sep 22, 2017 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. .
    – user20253
    Sep 22, 2017 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.
    – user9166
    Sep 23, 2017 at 23:02

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.

  • 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)? Sep 11, 2017 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. Sep 11, 2017 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, 2017 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, 2017 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, 2017 at 15:09

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.

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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, 2017 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' Sep 22, 2017 at 9:38
  • @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, 2017 at 2:09
  • @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, 2017 at 2:11
  • 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, 2017 at 2:29

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