Kant and non-Euclidean geometry

Kant and non-Euclidean geometry
andrewk
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#11 - Quote - Permalink
Posted Nov 28, 2012 - 1:41 AM:

The Great Whatever wrote:
Does anyone claim that non-Euclidean geometry is intuitable?
Some of it is. For instance, the geometry on the surface of a large sphere like the Earth seems pretty intuitive. For example the triangle made of the great circles that are the Greenwich Meridian, the Equator and the Meridian of 90 degrees longitude (which looks like it goes through Bangladesh), and that contains Iran and other countries, has three interior angles, each of which is 90 degrees. So its interior angles add to 270 degrees. I think I can understand that in an intuitive enough way.

Curvature of 4D spacetime is entirely another matter though. I think someone would have to be pretty special to have intuition about that.
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Posted Nov 28, 2012 - 1:47 AM:

The Great Whatever wrote:
Kant only claims that space is Euclidean as far as it is intuitable.

Does anyone claim that non-Euclidean geometry is intuitable?

So if non-Euclidiean geometry is not intuitable, then Euclidean geometry must be intuitable.

Excellent argument.

The interesting question might be: what is intuitable
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Posted Nov 28, 2012 - 9:45 AM:

My (very limited) understanding is that Kant is not just talking about our native intuitions* about space, which are surely more Euclidean than anything else. That wouldn't be interesting, especially not today, when we know a lot more about human cognition. What's important about Kant's ideas about space is that for Kant it is not something that inheres in the world, the way it tends to be seen in modern physics. Kant says that space is a "pure (a priori) intuition" - in other words, it's all in the mind. So, regardless of whether space was thought of as Euclidean or not, the physicist's idea of space would be different from Kant's. And since, for Kant, there is no space other than this idea that we hold in our mind, when Kant says that space is Euclidean, he does not mean that we perceive space as Euclidean - he says that space actually is Euclidean.

I could be very wrong about this.

* in the usual sense - Kant uses the word 'intuition' in his own special sense
On Nov 28, 2012 - 3:20 PM, Banno responded: If you are right, then it seems Kant was wrong.
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Posted Nov 29, 2012 - 12:15 PM:

SophistiCat wrote:
My (very limited) understanding is that Kant is not just talking about our native intuitions* about space, which are surely more Euclidean than anything else. That wouldn't be interesting, especially not today, when we know a lot more about human cognition. What's important about Kant's ideas about space is that for Kant it is not something that inheres in the world, the way it tends to be seen in modern physics. Kant says that space is a "pure (a priori) intuition" - in other words, it's all in the mind. So, regardless of whether space was thought of as Euclidean or not, the physicist's idea of space would be different from Kant's. And since, for Kant, there is no space other than this idea that we hold in our mind, when Kant says that space is Euclidean, he does not mean that we perceive space as Euclidean - he says that space actually is Euclidean.

I could be very wrong about this.

* in the usual sense - Kant uses the word 'intuition' in his own special sense

You forgot one thing: space (for Kant) is what makes experience possible. And if I can only intuit Euclidean geometry, why is the space, as I experience it in physics, non-Euclidean? The necessary relation between intuitable space and physical space is gone.

@The Great Whatever

That's not what Kant claims.

Kant: "Without resorting to a priori intuitions, mathematics can’t take a single step. So its judgments are always intuitive. (...) This
fact about mathematics points us to the absolutely basic thing that makes mathematics possible, namely that it is grounded in pure intuitions in which it can construct all its concepts—that is, can represent them in a manner that is concrete rather than abstract, and a priori rather than
empirical. If we can discover this pure intuition and what makes it possible, we will then be able to explain how there can be synthetic a priori propositions in pure mathematics, and thus how mathematics itself is possible."

Elsewhere he says: "Mathematics can’t proceed analytically by dissecting concepts, but only synthetically; so without pure intuition it
can’t take a single step, since only pure intuition provides the material for synthetic a priori judgments. Geometry is based on the pure intuition of space."

For him, space HAD to be Euclidean and had to be intuitable, otherwise geometry wouldn't be possible.

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Posted Nov 29, 2012 - 1:48 PM:

If we had no conception of what Euclidian space was, then how could we have a conception of non-Euclidian space? The particular here entails the conception of the general via an antitheis; while the general does not entail the particular as an antithesis or contrary.
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Posted Nov 29, 2012 - 2:31 PM:

Is there anyone with a PhD. in mathematics on this forum? I want to ask him if non-Euclidean geometry is intuitable (in Kants meaning).

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Posted Dec 1, 2012 - 11:54 AM:

andrewk wrote:
Some of it is. For instance, the geometry on the surface of a large sphere like the Earth seems pretty intuitive. For example the triangle made of the great circles that are the Greenwich Meridian, the Equator and the Meridian of 90 degrees longitude (which looks like it goes through Bangladesh), and that contains Iran and other countries, has three interior angles, each of which is 90 degrees. So its interior angles add to 270 degrees. I think I can understand that in an intuitive enough way.

Curvature of 4D spacetime is entirely another matter though. I think someone would have to be pretty special to have intuition about that.

Yes but you still imagined the sphere (Earth) on a background that is flat. So basically what you did is: you have the Euclidean flat space, within that space you intuited a globe with a figure drawn on that. Thus, we cannot say that you really inuited non-Euclidean space. You only intuited one non-euclidean figure in an euclidean space.

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Posted Dec 1, 2012 - 12:18 PM:

Well Kant is not the only philosopher whose a priori philosophical rational speculations about the nature of intuition,space and time turned out not to correspond to our a posteriori knowledge of space and time based on experiment, empirical evidence and scientific investigation. One of the features of the relationship between philosophy and science is that philosophical speculations must be modified or abandoned in the face of scientific advance. Philosophers can speculate about the nature of ultimate reality but if they are to remain relevant and rational they cannot ignore the facts of science. It is entirely possible that even Einstein’s view of space and time will be modified in the future and is merely a model which approximates but does not represent reality. Kant still has a great deal to tell us about the relationship between intuition, perception and reality.
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#19 - Quote - Permalink
Posted Dec 2, 2012 - 12:42 AM:

Cleantes wrote:
Yes but you still imagined the sphere (Earth) on a background that is flat. So basically what you did is: you have the Euclidean flat space, within that space you intuited a globe with a figure drawn on that. Thus, we cannot say that you really inuited non-Euclidean space. You only intuited one non-euclidean figure in an euclidean space.


I don't think that's right. I have spent a bit of time over the years imagining what it would be like to live in a 2D universe. This is partly my own musings and partly inspired by the excellent 'Flatland' by Edwin Abbott. As part of that I have often imagined what it would be like to live in a non-Euclidean 2D universe, as the surface of a sphere would be, if we were unable to look 'up' away from the sphere or 'down' towards its centre. I don't find any conceptual problem with it. In the case of the triangle around the meridians that I mentioned it would just be a case of:

0. Head in any direction in a straight line until I get back to where I started from. Read the number of steps on my pedometer. Call that x.

1. head in any direction for x/4 steps

2. Turn right by 90 degrees

3. head in the new direction for x/4 steps

4. Turn right by 90 degrees

5. head in the new direction for x/4 steps

6. Voila! I'm back where I started, having circumnavigated a triangle with three internal angles of 90 degrees each.

You could do the same thing on Earth, if it were possible to travel across mountains, plains and oceans without ever losing a sense of what the direction 'straight ahead' was. On land at least, we could use a South-pointing Chariot.
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#20 - Quote - Permalink
Posted Dec 2, 2012 - 5:54 PM:

andrewk wrote:


I don't think that's right. I have spent a bit of time over the years imagining what it would be like to live in a 2D universe. This is partly my own musings and partly inspired by the excellent 'Flatland' by Edwin Abbott. As part of that I have often imagined what it would be like to live in a non-Euclidean 2D universe, as the surface of a sphere would be, if we were unable to look 'up' away from the sphere or 'down' towards its centre. I don't find any conceptual problem with it. In the case of the triangle around the meridians that I mentioned it would just be a case of:

0. Head in any direction in a straight line until I get back to where I started from. Read the number of steps on my pedometer. Call that x.

1. head in any direction for x/4 steps

2. Turn right by 90 degrees

3. head in the new direction for x/4 steps

4. Turn right by 90 degrees

5. head in the new direction for x/4 steps

6. Voila! I'm back where I started, having circumnavigated a triangle with three internal angles of 90 degrees each.

You could do the same thing on Earth, if it were possible to travel across mountains, plains and oceans without ever losing a sense of what the direction 'straight ahead' was. On land at least, we could use a South-pointing Chariot.

Andrew, it is impossible for us to imagine anything with more than 3 dimensions. That for a line, a plane or a space to be "curved" it must occupy a space of higher dimension, i.e. that a curved line requires a plane, a curved plane requires a volume, a curved volume requires some fourth dimension, etc. And this fourth dimension we can never imagine. The surface of the Earth is the classic model of a 2-dimensional, positively curved Riemannian space, and we say the lines on this surface are curved. But we only say they are curved, because the surface itself is curved into the third dimension of Euclidean space.

We cannot visualize any true Lobachevskian spaces or any non-Euclidean spaces at all with more than two dimensions - or any spaces at all with more than three dimensions. We can only visualize a positively curved surface if this is embedded in a Euclidean volume with an explicit extrinsic curvature.

But anyways, I think I found a way to defend Kant. It requires only a reinterpretation of the term 'a priori' when related to geometry. For Kant, a priori meant two things: 1) A priori means it is possible to expand our geometric knowledge independent of experience 2) a priori judgements mean they are universal and are apodictic certain.

Now, my proposal is: we must reject 2) and keep only 1) as a feature of the term 'a priori'. If we do this, it is perfectly possible to maintain Kants system with regarding tot non-euclidean geometry. Gauss and Lobachevsky developed non-euclidean geometry independent of experience, they merely changed 1 axiom of Euclid. It was not before the 20th century that Einstein showed us that the empirical space as a whole is non-euclidean. But this doesn't mean that euclidean geometry is invalid now. We only use the non-euclidean geometry when we deal with black holes, etc. where the distortion of the shape of space is so huge.

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