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This is mainly a historical question. In Gary Hatfields introduction to Kants Prologomena, he says:

After the discovery of non-Euclidean geometry, Kant’s claims for the synthetic a priori status of Euclid’s geometry as a description of physical space came into question.

He doesn't explicitly say, but is it implied that this had an impact on Kantian thought outside of his conception of mathematics.

Neo-Kantians such as Cassirer questioned whether the categories of human understanding are truly fixed, as Kant had suggested, or change throughout the history of human thought.

If geometry can change, perhaps categories can?

My own thinking on this is that mathematicians from antiquity had already recognised the lack of neccessity in the parallel postulate, and that this shows that they already understood Euclidean geometry wasn't a priori as then constituted. That it took millenia for this insight to be incorporated in the main body of mathematics as refutation alongside the discovery of non-Euclidean geometry is a mere side-issue from this essential insight.


I don't see how either the mathematical discovery of non-euclidean geometries or the physical discovery of non-euclidean geometry of spacetime invalidates Kants reasoning. Physically, in general relativity it is the large-scale geometry that is non-euclidean; and in the small-scale, that is locallY - the scale appropriate to direct human perception (that is not magnified by extra-sensory instruments) - it is euclidean. But this is besides the point; even were we to park ourselves close to somewhere where gravitational forces appreciably altered the curvature of spacetime - I think our direct understanding of space and time would remain euclidean. That is we would see for example a ball following a curved geodesic in spacetime as curved in space and through time and not a straightline.

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What do you base your suspicion about antique mathematicians on? –  artm Feb 18 '13 at 6:49
That the parallel postulate was postulated (i.e. "requested" or "begged for") indicates that Euclid, at least, knew of its lack of self-evidence. But, after Euclid, it seems mathematicians (Proclus especially) were deeply uncomfortable with relying on the postulate--esp. given its formulation. –  Jon Feb 18 '13 at 23:15

5 Answers 5

up vote 6 down vote accepted

Under the understanding of a prioricity at issue pre-Two Dogmas of Empiricism, a priori truths were largely conflated with necessary truths. So, if you could recognize the possibility of the failure of the parallel postulate, that would constitute a falsification of its necessity and thus (given the conflation) a falsification of the claim that it was a priori.

Where Kant went wrong, if this was indeed what he held, was in thinking that our intuition of space and time represented the world as it actually is. Frege famously made the same mistake in one of his later articles, "Foundations of Geometry".

This article, if you can get around the pay-wall, discusses Frege's Kantian views on geometry and provides ways to charitably interpret them.

Now, regarding your second question, I don't think you need to see this as showing that geometry or the categories of understanding had changed. I could see someone holding that it isn't the categories that have changed, but merely the classification of certain truths as falling under one or another of the categories.

So, a Neo-Kantian could consistently hold that the categories of understanding remain fixed and what non-euclidean geometry shows us is that geometry doesn't fall under the category Kant thought it did.

A quick look at the SEP article on categories confirms that there are many philosophers, notably P.F. Strawson, who took up the Kantian project under the heading of "descriptive metaphysics". These philosophers were certainly aware of the developments of non-euclidean geometry.

Additionally, the article suggests (rightly) that this sort of empirical falsification wouldn't undermine a Kantian conception of the categories. See, for example:

Nonetheless, it is clear that for Kant the categories find their original source in principles of human understanding, not in intrinsic divisions in mind-independent reality, and are discoverable by paying attention to possible forms of human judgment, not by study of the world itself, nor by study of our contingent manners of speaking.

Thus, even if we have discovered that the mind-independent world doesn't answer to our euclidean geometric conception of it, it does not follow that there is some fault in the division made between categories.

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I am conflating necessary truths with a priori ones, how are you distinguishing them? –  Mozibur Ullah Feb 22 '13 at 5:34
@MoziburUllah Check out section 3 here –  Dennis Feb 22 '13 at 5:48
I don't think Kant was wrong about the geometry we use (see addendum above). "Where Kant went wrong...was in thinking that our intuition of space and time represented the world as it actually is": Representation of the noumena (the world-as-it-is) surely need not be a complete isomorphism, after all we are not directly aware of either small-scale phenomena (quantum tunnelling) nor of large-scale phenomena (spacetime curvature). –  Mozibur Ullah Mar 3 '13 at 15:10
@MoziburUllah Ok, so I talked to our resident Kant scholar about this view of Kant's. Your addendum doesn't help Kant, because he thought that our intuition of the structure of spacetime was prior to our experience of individual regions. He thought that you couldn't experience individual regions as Euclidean and generalize from that. Rather, the local experience presupposes a global form of the experience. Top-down rather than bottom-up, in a sense. –  Dennis Mar 4 '13 at 18:06
I'm in agreement with your Scholar that Kant is saying our intuition of space & time is prior to our experience of space & time. My addendum was supposed to be consistent with this (but I see now I didn't mention this) and I'm saying it is this intuition that Kant is saying is Euclidean. Presumably your top-down and global form refers to this intuition prior experience that Kant supposes? I don't understand what you mean by 'he thought that you couldn't experience individual regions as Euclidean and generalize from that'. –  Mozibur Ullah Mar 8 '13 at 4:39

Kant as a philosopher taught strategically*. It means he was looking for interesting problems and the clue was top contemporary debates.

At his time the debate between the Leibnizians and the Newtonians concerning the status of space and time lead him to find out there should be a higher abstract view which can support both ideas. At the time Leibnizians had not enough physical evidence and the math to support relatedness of space-time as like as what Newtonians did with their precise equations.

Einstein later made it done. His relativity theory* is based on the fact that space and time are not absolute as Newtonians taught and with enough astronomical data and mathematical support, formulated this relatedness. Without non-euclidean geometry relativity would never born.

Kantian thought helped non-euclidean geometry development. After non-euclidean geometry developed and subsequently relativity theory bounded it to reality science paid back it's debt to philosophy as follow*:

  • Ontology:

    Kant was wrong: space and time really exist beyond human experience, but only relative to masses in motion (there is no absolute, Euclidean metric to which all physical events conform: space curves locally and times are desynnchronized for objects moving in non-uniform inertial frames).

  • Epistemology:

    Kant was wrong: non-Euclidean space can not only be visualized, but measured (the sun, for example, warps local spacetime by approximately four seconds of arc per century)--suggesting that Kant had the relation between what can be conceived and what can be visualized backwards.

  • Cosmology:

    Kant was wrong: although the First Antinomy purports to show the impossibility of conceiving the universe as either finite or infinite in-itself (because both contradictory metaphysical absolutes can be argued and justified with equal force, it follows that neither can actually be proven), Einstein answered Kant by proposing a consistent non-Euclidean (Riemannian) universe that is finite but unbounded (i.e. without an edge).

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On Ontology, Einstein was interested in Machian philosophy but abandoned it. Non-Euclidean spacetime is not directly apprehensible by us, this is the epistemology that Kant is interested in. Einsteins cosmology says nothing about Kants first Antonomy: it does appear to show that spacetime is bounded in the past - the Big Bang - but since the equations breakdown as we approach that purported time zero - we can in fact say precisely nothing at that time. All we can know is that it was an important point in time –  Mozibur Ullah Mar 4 '13 at 0:23
For all we know it may be a kind of phase transition of time rather than a beginning. His geometry is pseudo-Riemannian - the metric is not positive-definite. See addendum to the question above. –  Mozibur Ullah Mar 4 '13 at 0:26
There is a round trip (iteration) between philosophy and science. Philosophy generalizes (up) and science precises (down). There are more conflicts than you already mentioned but I described why Kant get interested to the subject, what he did was somehow the basis for relativity and relativity revealed new connections between some data which was chaotic to us with precise equations. For know, the same debate is where relativity (at macro scale) cannot explain micro scale (quantum world) and this is a clue for contemporary philosophers. I'm working on a hypothesis on this subject. –  Xaqron Mar 4 '13 at 2:41

Just a little sidenote...

I thought that since Kant believed Euclidean geometry to be synthetic a priori and true, and since space is infact bent and non-euclidean (same with Newtonian mechanics, thought to be synthetic a priori, but along came Einstein), that may refute synthetic a priori as an impossibility.

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see addendum above –  Mozibur Ullah Mar 4 '13 at 0:08

In response to . . .

But this is besides the point; even were we to park ourselves close to somewhere where gravitational forces appreciably altered the curvature of spacetime - I think our direct understanding of space and time would remain euclidean. That is we would see for example a ball following a curved geodesic in spacetime as curved in space and through time and not a straightline.

. . . But if the ball were rolling directly away from us, the sightline from the ball to our eye would be following the same curved path as the ball itself. It would be like looking through the end of a curved fiber optic cable.

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I believe that none of the answers understand Kant or Idealism in general.

Please understand: Kant is NOT trying to disprove objective reality. A naive approach to idealism is to think that, and then to conclude that it tries to disprove all our dear science.

For example when you say:

"Kant was wrong: space and time really exist beyond human experience ..."

First detach yourself emotionally from all your research and understanding, then see that Kant can not possible be trying to disprove that, because that assumes that Kant has a concept of "real existence". Kant is not assuming a well defined concept of objective reality, you are. If then you reply: " but his reasoning is not valid vs the reality of Science", then you still do not understand.

Do not try to understand idealism / kant by comparing it to your realism.

Open your mind: Start with OUT the concept of objective reality.

I will make a futile experiment to try to explain what PLATO, DESCARTES or KANT said:

Lets start with your concept of reality as it pertains to that keyboard under your hands. The problem you need to recognize is not in the keyboard, but in what you used to know it is there. Note that the keyboard didn’t pop in your mind as soon as it was built. It appeared in your mind after you saw it right ?


The first problem with all these was noted by PLATO. He realized that things do not pop in our minds by themselves, they appear after a collection of perceptions: Rectangularness, number of keys, color, angles of the sides…. He called these: “IDEAS”. You agree that you cannot describe your keyboard with out using ideas right?

b) IDEAS do not exist in the "real world"

The problem with such “IDEAS” is that they don't exist. There aren't any perfect squares, circles, straight lines, or points in space in the experienced world. What do you mean with a points in empty space in the first place? What is a line between to points in space? A single point moving in space ?? See that those are not "real" things by themselves.

c) IDEAS do not make things REAL

For one thing, none of these “IDEAS” belong specifically to your keyboard, they are general. These “IDEAS” have no connection with the keyboard. You can think about the keyboard whether the keyboard still exists or not. IDEAS have nothing to do with it. Also they apply to other keyboards.

I bet if I replace it behind your back with the same model, you will not notice that your "real" keyboard is not there.

So if you use IDEAS to define your KEYBOARD, and IDEAS are not real, how come you say the KEYBOARD is real ??

d) IDEAS do not come from REALITY

You will say: IDEAS are learned from reality. Thats why I use them to define reality and Einstein and Euclid had discovered the right ones. All observations attest to their veracity and if the are wrong, only Science can correct them.

... so if you think that keep reading...

The problem with that is that it doesn’t take much to notice that, the result of 1 + 1 = 2 is not learned. The concept of parallel lines was not "discovered" by Euclid. These all started 2000 years ago precisely when Socrates noticed that even an illiterate understands that parallel lines do not cross at infinite.

It’s obvious you will say. Well, again, perfect parallel lines do not exist. Where’s the obvious in that? No one has gone to infinity and verified that parallel lines do not cross. So, why do we see that they don’t so clearly? Why do illiterate people can understand that ? If geometry invented that it should be able to change it, but it can’t. We can’t just come and say: From today on, parallel lines cross at 100 meters, deal with that. No matter how many Euclidean geometries you find, or how many times the postulate is discredited.

f) IDEAS make your mind

So, the real problem is: Where did squareness, the number 3, parallel lines, and all those things came from, if they don’t exist in nature, they don’t belong to objects, and, at least some, we didn’t learn from experience or science? Kant is just saying that, as unbelievable as it might sound, our mind has these things "a priori" to be able to think. Those are like the alphabet, but not the story. At first glance it sounds ridiculous, but, at varying degrees, it’s undeniable: We could not have deduced many of these “IDEAS” from nature, simply because they are required to define nature in the first place.


So here lies the problem: If your reason uses all these hard-coded ideas to present you the real world, we need to see the real world with no reason, to get the "real" thing.

Now you see the problem? Now tell me again that :

"Kant was wrong: space and time really exist beyond human experience ..."

So are you saying that our math and geometry is universal?, that that is the only way to understand reality?? Are you saying that space is reduced to Einstein's formulas and all entities in the universe have to abide to them ? That there is no other way?? Theres no better way ?? Will there ever be a right way?? Are you saying that the human mind is the only "objective" witness to reality ??

How do you know that ??

Kant is not presenting idealism to replace your realism, he is just saying that you make no sense when you say:

..really exists..

Writing about Descartes, Schopenhauer claimed,

"… he was the first to bring to our consciousness the problem whereon all philosophy has since mainly turned, namely that of the ideal and the real. This is the question concerning what in our knowledge is objective and what subjective, and hence what eventually is to be ascribed by us to things different from us and what is to be attributed to ourselves." (Parerga and Paralipomena, Vol. I, "Sketch of a History of the Doctrine of the Ideal and the Real")

So, look at your keyboard again. No, there’s no keyboard there. You are “looking” at a representation inside your head, like in a flat screen, where you only see the results of the analysis made by your mind. Please note that the question is not whether the keyboard exists or not, but

what are you expecting as a keyboard outside your mind?

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