As I understand it, it is assumed that the discovery of non-Euclidean geometry undermines Kant’s understanding of a priori truth. However, Kelley Ross disagrees, on the grounds that the ‘synthetic a priori actually accommodates the possibility of non-Euclidean geometry. There’s an essay on it here.

Many thanks for this, ill take a read.

Non-Euclidean geometries are just as intuitive (synthetic a priori) within their contexts as Euclidean geometry is within the context of everyday experience, so I don't see the problem. Kant considered all of mathematics and geometry to be synthetic a priori knowledge, unless I am mistaken.

Since Kant was tremendously intelligent, and not only understood physics but even published important papers in it, I have no doubt he would have very rapidly understood non-Euclidean Geometry and its usefulness if it had been around when he was alive.

I see no reason to suppose that it would undermine Kant's notion of the Transcendental Aesthetic, which is that we process raw inputs within a framework of three space dimensions and one time dimension. Kant would not have said that the framework could not be non-Euclidean, since the concept was not known in his day. He may well have mentioned Euclid, but that would have been because at the time Kant was writing, the name of Euclid represented all known geometry. Non-Euclidean geometry only became well-known after Gauss wrote about it in the 1813. So Kantian references to Euclid can be interpreted simply as unqualified references to geometry.

Being a great scholar and physicist, Kant, upon being introduced to non-Euclidean geometry, would have rapidly noticed that an essential feature of the non-Euclidean geometry that is used in physics is that Euclidean geometry is an extremely accurate approximation to the non-Euclidean geometry at non-cosmic scales. Since humans live, work and think in non-cosmic scales, I expect Kant would have been entirely comfortable with the notion that our in-built mechanism for arranging information is an approximation to a paradigm whose differences are only visible at scales that are beyond ordinary human experience.

I don't think non-Euclidean geometries would be considered intuitive or come under the transcendental aesthetic because they are concepts of the understanding and have to be described mathematically or modelled within Euclidean space. So they would be in the transcendental analytic.

A quote from the essay that summarises the problem:

"Impressed by the beauty and success of Euclidean geometry, philosophers -- most notably Immanuel Kant -- tried to elevate its assumptions to the status of metaphysical Truths. Geometry that fails to follow Euclid's assumptions is, according to Kant, literally inconceivable."

Frank Wilczek

The problem is that for Kant space is not merely a three dimensional framework but the very basis of any possible intuition whatever.

A quote from the essay that summarises the problem:

"....Geometry that fails to follow Euclid's assumptions is, according to Kant, literally inconceivable."

Frank Wilczek — Perplexed

Non-Euclidean geometry does follow Euclid's assumptions at the scales that are meaningful to humans. So there is no conflict.

A geometry that is not locally Euclidean would have to be fractal - infinitely wiggly - and most of us (certainly me!) find that inconceivable. I can do formal calculations about infinitely wiggly things (the Ito Process that is used as the basis of pricing most financial instruments is infinitely wiggly), but I cannot visualise them, and I have no intuitions about them.

I expect Kant would have been entirely comfortable with the notion that our in-built mechanism for arranging information is an approximation to a paradigm whose differences are only visible at scales that are beyond ordinary human experience. — andrewk

I highlighted the interesting part, because what does it mean for Kant for something to be beyond ordinary human experience, particularly in context of:

we process raw inputs within a framework of three space dimensions and one time dimension. — andrewk

Also, modern physics has entertained higher dimensional space and colliding branes, along with multiple universes. I wonder if those concepts could fit into the transcendental aesthetic since they're so far beyond the normal framework of space/time.

Isn't the issue that Kant elevated the postulates of Euclidean geometry to the level of a metaphysical certitude, and the Non-Euclidean Geometry shows that such a position is not the case?

I wonder if those concepts could fit into the transcendental aesthetic since they're so far beyond the normal framework of space/time.

For me they seem to. When I visualise many-dimensional branes colliding in even higher dimensional space, I visualise hanging, wobbly two-dimensional sheets banging into one another in 3D space. The calculations will be different from the 3D case but for me the visualisation has to remain 3D (or at most 4D - I sometimes use time as proxy for a fourth spatial dimension) as I am not capable of visualising anything higher.

Isn't the issue that Kant elevated the postulates of Euclidean geometry to the level of a metaphysical certitude

Did he do that? I don't know. I never read the original, being a secondary-source kind of chap. I think we'd need to dig out a quote, both in the original German and a diversity of English translations, and analyse it to see whether we can reach that conclusion from it. My loose observation about Kant scholarship is similar to that commonly made about economics: If we put n Kant scholars in a room there will be n+1 opinions about what Kant meant by any particular passage he wrote.

Yes thank you for clarifying my point. In the transcendental aesthetic Kant speaks of space not as an objective absolute background like Newton, nor as a system of relations like Leibnitz but as an integral part of our perceptive apparatus. If Euclidean geometry is meant to be a necessary and timeless truth constrained by the nature of our perceptions then any deviation from this in nature would seem to prove Kant wrong. However, later on in the critique he does seem to talk about space as a coordinate measurement system so this is rather confusing and I'm not sure if he was consistent.

I'm not sure we can rely on geometry remaining Euclidean at human scales because if one takes general relativity into account then masses are warping spaces and causing gravitational geodesics that support our entire existence.

No problem.

I believe Kant did as I said, though as with you, I'm relying on a secondary source (and potentially worse, my hazy recollection of that source). Mostly, I find it plausible that my memory is correct here since Kant also made a similar sort of assertion (that turned out to be wrong) about Aristotelian Logic ("we have no need for more logicians", the century before we got Classical Logic, lol) and something similar about biology. Maybe I have a poor recollection and that has colored my view of Kant.

I'm not sure we can rely on geometry remaining Euclidean at human scales because if one takes general relativity into account then masses are warping spaces and causing gravitational geodesics that support our entire existence. — Perplexed

The curvature that gives us our fairly modest Earth gravity is a curvature of spacetime, not a curvature of space. The difference between the two concepts is crucial. IIRC, it is possible to have a spacetime that is curved but for which all spatial slices are flat. The curvature is only in the relation between space and time, not in the space itself.

Kant saw space and time as separate, so curvature in the relations between them, would have meant nothing to him, and could not contradict his 3D + 1D model* that is the Transcendental Aesthetic . Indeed, the maths needed to express that curvature had not even been developed when Kant was around. All that mattered was that spatial slices ('constant-time hypersurfaces') were flat, and that is extremely close to being the case in a site of very small gravity like the surface of the Earth.

* As distinct from Einstein's 4D model.

Also, modern physics has entertained higher dimensional space and colliding branes, along with multiple universes — Marchesk

and they’re in an absolute schemozle for having done so.

I think the people (which is everyone here pretty much) who think that non-Euclidean geometry is not a problem for the Kantian have not carefully read what Kant (and other transcendental idealists, like Schopenhauer) wrote about the axioms of geometry. Some people who have posted in this thread have read a lot of secondary works about Kant, but not the actual Critique itself - they are lazy thinkers.

There is an old thread about this I opened here:

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"Both Schopenhauer and Kant take space to be an a priori form of representation, applied by the cognitive faculties to the senses. They understand this to mean that the propositions of geometry are synthetic a priori judgements, and are therefore apodeictic - certain.

Space is not an empirical concept which has been derived from outer experiences. For in order that certain sensations be referred to something outside me (that is, to something in another region of space from that in which I find myself), and similarly in order that I may be able to represent them as outside and alongside one another, and accordingly as not only different but as in different places, the representation of space must already underlie them. Therefore, the representation of space cannot be obtained through experience from the relations of outer appearance; this outer experience is itself possible at all only through that representation — Kant

Space is not something objective and real, nor a substance, nor an accident, nor a relation; instead, it is subjective and ideal, and originates from the mind’s nature in accord with a stable law as a scheme, as it were, for coordinating everything sensed externally — Kant

Space is a necessary a priori representation that underlies all outer intuitions. One can never forge a representation of the absence of space, though one can quite well think that no things are to be met within it. It must therefore be regarded as the condition of the possibility of appearances, and not as a determination dependent upon them, and it is an a priori representation that necessarily underlies outer appearances. — Kant

Space is not a discursive, or as one says, general concept of relations of things in general, but a pure intuition. For, firstly, one can represent only one space, and if one speaks of many spaces, one thereby understands only parts of one and the same unique space. These parts cannot precede the one all-embracing space as being, as it were, constituents out of which it can be composed, but can only be thought as in it. It is essentially one; the manifold in it, and therefore also the general concept of spaces, depends solely on limitations. It follows from this that an a priori intuition (which is not empirical) underlies all concepts of space. Similarly, geometrical 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 apodeictic certainty — Kant

Schopenhauer follows Kant in conceiving of space (and geometry - the study of space) as transcendentally ideal:

Perception, partly pure a priori, as establishing mathematics, partly empirical a posteriori as establishing all the other sciences [...] We demand the reduction of every logical proof to one of perception. Mathematics, on the contrary, is at great pains deliberately to reject the evidence of perception peculiar to it and everywhere at hand, in order to substitute for it logical evidence. We must look upon this as being like a man who cuts off his legs in order to walk on crutches [...] Whoever denies the necessity, intuitively presented, of the space-relations expressed in any proposition, can with equal right deny the axioms, the following of the conclusion from the premises, or even the principle of contradiction itself, for all these relations are equally indemonstrable, immediately evident, and knowable a priori — Schopenhauer WWR Vol I §14-15

Now Schopenhauer's ontological idealism (and I refer here to the phenomenon/noumenon distinction largely) critically requires that the stage on which experience occurs be transcendentally ideal, for this stage being transcendentally ideal is what enables experience to be called the veil of Maya - appearance - and hence necessitates the noumenon, the thing-in-itself. Without space, time and causality (which constitute the stage in/on which experience occurs) being transcendentally ideal, the distinction between noumenon/phenomenon is in danger, as is idealism - for if at least part of the stage on which experience occurs is real, then Schopenhauer's ontological idealism is false.

Schopenhauer laughed at mathematicians trying to prove Euclid's Fifth Postulate, thinking that it is known from pure perception 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

Non-Euclidean geometry came along, and it turns out that we have empirical proof that Euclid's Fifth Postulate is actually false, with regards to space as investigated by physics. Now the curvature of space cannot be perceived - we perceive objects in space - things in space curve - but how can space itself curve - that is anathema to our perception. What does this mean for Schopenhauer? Well for one, Euclid's Fifth Postulate isn't apodeictic, and neither is it a priori - contrary to what Schopenhauer thought. So Schopenhauer was wrong at least about this one truth of mathematics, and if he was wrong about this one, why should the other mathematical propositions that he was certain of be anymore certain than this? Indeed, it is his method that is wrong. Grounding mathematical propositions in a priori perception without appeal to experience is wrong. As Einstein said:

As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality

More importantly, there is one feature of space - its non-Euclideanness which is NOT a synthetic a priori, but rather a synthetic a posteriori, and therefore not transcendentally ideal, but empirically real - for it takes experience (physical experiments) for us to know of it. This means that at least part of the stage on which experiences occur isn't imposed on reality as a structure by our cognitive faculties, but rather is empirically real. If part of the stage is empirically real, then Schopenhauer's ontological idealism falls apart."

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To summarize - read the 10 or so pages of the Transcendental Aesthetic. Kant is very clear about this - space and time are the pure forms of the sensibility and they are not derived from the contents of sensation. They are NOT empirical intuitions.

I mean what makes geometry - if anything - certain? Before, it was taken to be certain since it was merely the reflection, in the understanding, of the a priori spatial form of the sensibility, and as such, there was nothing "external" involved in it, but rather everything external had to conform to it. — Agustino

From axioms and postulates. — Response

And where do the axioms and postulates get their certainty from?

In Euclid's Elements Book I, Euclid gives definitions, postulates, common notions and propositions (the latter of which are derived from the definitions, postulates and common notions or each other).

There are 5 postulates given:
1. To draw a straight line from any point to any point. {this is to be read in the sense of "it is always possible to draw a straight line from any point to any other point"}
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any centre and distance.
4. That all right angles are equal to one another.
5. 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 there are angles less than two right angles.

My question is "which of these 5 postulates are synthetic a prioris, which are synthetic a posteriori, and which are analytic"? — Agustino

The answer will be forced to be either synthetic a priori or synthetic a posteriori, and one must be given a reason why. If all axioms are synthetic a prioris, except for the 5th one, then why is it that the fifth one isn't, and yet Kant and Schopenhauer thought it is? Because clearly, non-Euclidean geometry demonstrates beyond a shadow of a doubt that the 5th postulate is not a synthetic a priori, despite what Kant and Schopenhauer thought.

If the axioms are synthetic a posteriori, then geometry is not apodictic - it is not certain. This means that outer space does not have to obey our geometrical constructs (since clearly, those constructs are not some a priori forms of our sensibility that exist in the mind and through which all phenomenal experience is given). Then the whole of Kantian thought collapses since space is not an A PRIORI form of the sensibility, and therefore (probably) must be determined by the senses itself (or at any rate, by the senses + a (reconceptualized) understanding), being a synthetic a posteriori.

There’s an essay on it here. — Wayfarer

That essay is a pile of manure.

The Euclidean nature of our imagination led Kant to say that although the denial of the axioms of Euclid could be conceived without contradiction, our intuition is limited by the form of space imposed by our own minds on the world. While it is not uncommon to find claims that the very existence of non-Euclidean geometry refutes Kant's theory, such a view fails to take into account the meaning of the term "synthetic," which is that a synthetic proposition can be denied without contradiction.

>:O And Kant wasted his time saying that the propositions of geometry are synthetic in order to tell us they can be denied without contradiction, because that certainly proves the reliability of mathematics when applied to physics (which is what Kant was trying to do - just open a page of the Prolegomena, instead of reading 20,000 secondary sources who don't know what they're talking about). Of course, this isn't what Kant did. Kant aimed to say that the propositions of geometry don't derive their CERTAINTY (because he took their certainty for granted) due to the law of non-contradiction (hence the synthetic part). Rather they derive their certainty from their a priority, rooted as they are in the pure form of sensation, space.

People will go to any heights of absurdity to prove that what they want to believe is true, instead of admitting to the truth.

Non-Euclidean geometries are just as intuitive (synthetic a priori) within their contexts as Euclidean geometry is within the context of everyday experience, so I don't see the problem. — Janus

Nope. This is a mistaken view. 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. So Kant resolved the problem by saying that it is not just a human construct. We have this form of pure intuition, space, from which we derive the axioms of geometry. So they are synthetic, since we can imagine the opposite to be the case, but they are a priori, because they stand as being true prior to experience. His point was that I don't need to run a physical experiment to see that Euclid's Fifth Postulate holds (which of course turned out to be false). It is simply impossible, according to Kant, for Euclid's Fifth Postulate not to hold - not due to its synthetic nature, but 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.

Kant considered all of mathematics and geometry to be synthetic a priori knowledge, unless I am mistaken. — Janus

That is correct.

Did he do that? I don't know. I never read the original, being a secondary-source kind of chap. — andrewk

Exactly... If you had, you'd be amazed by how often Kant mentions the "apodictic" nature of geometrical propositions.

Meh, you miss the point. Kant says nothing of reality (noumena) except that it can't be known. Space is a precondition of understanding, but he says nothing of the true reality of space.. Quoting you quoting him:

Space is not something objective and real, nor a substance, nor an accident, nor a relation; instead, it is subjective and ideal, and originates from the mind’s nature in accord with a stable law as a scheme, as it were, for coordinating everything sensed externally — Kant

Meh, you miss the point. — Hanover

Sorry, but I will disagree with you on this.

Kant says nothing of reality (noumena) — Hanover

Reality is a hazy word. Why is the noumenon reality, and the phenomenon not? Don't forget that it is the phenomenon that is the empirically real according to Kant, not the noumenal. Kant's notion of a noumenon, at any rate, is confused. He talks of the noumenon causing the phenomenon, which is nonsense, since causality is a category of the understanding, and hence can only apply to the phenomenon. It takes Schopenhauer to clarify this aspect of Kant.

Space is a precondition of understanding — Hanover

Precondition of the sensibility, not of the understanding. Kant talks of space (and time) in the Transcendental Aesthetic, and labels them as forms of the sensibility (as opposed to the matter or content of the sensibility, the sensations themselves), which comes before he goes into the categories of the Understanding.

true reality of space — Hanover

Also it is a grave error to think that there is any "true reality" to space according to Kant. Space is transcendentally ideal, given by the forms of our sensibility. There very likely is no space at all out there - external to our phenomenal, empirically real experience. 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).

Thank you for clearing that up andrewk, I think I need to swot up a bit on my general relativity.

The essay that posted is actually very interesting in regard to the different types of curvature.

Kant aimed to say that the propositions of geometry don't derive their CERTAINTY (because he took their certainty for granted) due to the law of non-contradiction (hence the synthetic part). Rather they derive their certainty from their a priority, rooted as they are in the pure form of sensation, space. — Agustino

I was wondering about this. Before Kant, a priori truths were considered analytic and so are true by definition. The a priori synthetic truths may be certain from the perspective of our sensations but does this really make them logically necessary? Could there not conceivably be forms of intuition different from ours that allow for different types of space?

Your intended point of disagreement is not clear. I haven't suggested that space is a "human construct", in case that was it.

No, that wasn't my point of disagreement.

The a priori synthetic truths may be certain from the perspective of our sensations but does this really make them logically necessary? — Perplexed

According to Kant, the a priori synthetic truths must be certain from the perspective of the phenomenon and our experience. One repercussion of this is that you could not do a physics experiment which did not obey the laws of geometry.

We have this form of pure intuition, space, from which we derive the axioms of geometry. — Agustino

Kant talks of space (and time) in the Transcendental Aesthetic, and labels them as forms of the sensibility (as opposed to the matter or content of the sensibility, the sensations themselves), which comes before he goes into the categories of the Understanding. — Agustino

I apprehend some inconsistency here. In the first, you describe space as a "form of pure intuition". In the second you describe space as one of the "forms of sensibility".

As a form of "sensibility", I would assume that space is a condition for the possibility of sensation. But "intuition" I would think only arises from a being which has sensation. So if space is an intuition, then sensation would be prior to space as an "intuition".

Therefore one or the other cannot be correct. Either space is an intuition, in which case it occurs after sensation, or, space is a condition for the possibility of sensation, in which case it is prior to sensation and cannot be an intuition, which only occurs to creatures which already have sensation.

Edit: This is probably why there is such variance in interpretation of Kant on this issue.

I apprehend some inconsistency here. In the first, you describe space as a "form of pure intuition". In the second you describe space as one of the "forms of sensibility".

As a form of "sensibility", I would assume that space is a condition for the possibility of sensation. But "intuition" I would think only arises from a being which has sensation. So if space is an intuition, then sensation would be prior to space as an "intuition".

Therefore one or the other cannot be correct. Either space is an intuition, in which case it occurs after sensation, or, space is a condition for the possibility of sensation, in which case it is prior to sensation and cannot be an intuition, which only occurs to creatures which already have sensation.

Edit: This is probably why there is such variance in interpretation of Kant on this issue. — Metaphysician Undercover

Kant uses intuition in a technical sense. It's not what we mean by intuition in common language. In Kant, intuition is something closer to what we mean in common language by perception.

And so, in Kant's system, there is no inconsistency though my language was a bit convoluted by saying "form of pure intuition" instead of just "pure intuition". Kant says:

Transcendental Aesthetic - Introduction:

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

So pure intuition refers to the perception of space and time. They are pure, because we cannot "not perceive" them. 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.

I've spent quite a lot of time studying Kant, but it's not something I'm proud of, since it is disappointing that he could be so wrong. The first time I encountered Kant's system, I literarily thought he was a genius, who had advanced philosophy far beyond what anyone had before him. But with time, the more carefully I studied Kant and these issues, I came to see that Kant had made some serious mistakes, that despite the initial beauty and order of his system, could not be maintained. Transcendental idealism as developed even more by Schopenhauer was the first philosophical system that I thought was correct, and we fundamentally gained new knowledge from it. The first time I felt that "this must be right" in philosophy, because it was so well closed, and there were few dangling bits, unlike most other philosophical systems which have many dangling bits. Most of that knowledge was in understanding how things fit together, and why things are as they are - for example, why mathematics describes the physical universe so well. But it turns out that those certainties were mere appearance. Ever since then, I have not found a philosophical system with as much internal coherence as that one.