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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.

addendum

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
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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

4 Answers 4

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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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