## What would Kant have made of non-Euclidan geomety?

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"What would Kant have made of non-Euclidan geomety?"

He'd have said it was obvious if you decide that triangles can exist across 3D space, or on the surface of spheres.
He was smart enough to realise that Euclid assumed 2D.

Move along now.... nothing to see here!
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So I submit, your honour, that the parallel postulate is not intuitive.

The parallel postulate is simply true by the law of non-contradiction. Lines which do not meet are parallel lines. Lines which are not parallel meet. Anything else would be contradictory.
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The parallel postulate does not say what, based on your post, you appear to think it says.
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So I submit, your honour, that the parallel postulate is not intuitive.

So, say two lines met a billion parsecs away. In that case do they start out truly parallel? If they do, then at what point do they cease to be parallel? If they don't, then does that rule out the possibility of two lines being parallel tout court? If so, on account of what exactly would that be the case? So, take the case of a Globe like the Earth; we have a great circle at the Equator; is a circle precisely one meter north or south of that not concentric, and if it is would not the two lines qualify as parallel against an imaginary plane 90 degrees to the horizontal?

Of course there are no perfect circles, perfectly straight or parallel lines in nature, but that would seem to be an entirely separate issue.
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The parallel postulate does not say what, based on your post, you appear to think it says.

Yes, it says just what I thought it says, I looked it up before I posted to make sure. It's all a matter of definition, and non-contradiction.
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So, say two lines met a billion parsecs away. In that case do they start out truly parallel? If they do, then at what point do they cease to be parallel?
They always look parallel, and that's what matters to our intuitions. There is no part of the triangle we can look at in which the bit we can visualise doesn't either look like two parallel lines, or one line when they are so close together that we cannot distinguish them.

So if we want an intuitive parallel postulate, I imagine it would have to be something like:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to an amount that is VISUALISABLY less than two right angles, then there is some VISUALISABLE distance such that, if the two lines are extended for that distance in the direction on which the angles sum to less than two right angles, they meet.

and this postulate would be met by any space that is no more than very slightly curved, which would include our real world space.
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They always look parallel, and that's what matters to our intuitions.

A line exists by definition, it is a defined thing. You can't see what it looks like, nor can you even imagine it. I think this throws everyone off, you can only know what it is, by its definition. The principles of geometry are definitions which must be adhered to in creating geometric forms. In the case of the parallel postulate we have the definition of "line", and also some defined relations between lines. There is the defined relation of parallel, and the defined relation between parallel lines which intersect another line. The parallel postulate holds, (is valid) only if one respects the various definitions which support it.

The idea of a "plane", and the relationship between planes is something created by definition, it cannot be sensed nor can it be imagined. In common use, "intuited" means directly apprehended by the mind. This is how these principles are understood. The key thing is to "understand" them, because whether or not they are "true", and how they relate to the physical world is irrelevant to understanding them. They just need to be understood to be used in construction in the physical world. But the fact that they are useful in the physical world justifies the claim that there is some sort of "truth" to them.
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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.

Are you saying that Kant would have denied that any physics experiment could reveal that spacial geometry was not Euclidean?
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The parallel postulate is simply true by the law of non-contradiction. Lines which do not meet are parallel lines. Lines which are not parallel meet. Anything else would be contradictory.

Since we do not have, and cannot have access to infinity then the premise can only work at a mundane level. Since we can never know if the universe is infinite, or tell if it might or might not coalesce into a pinprick then it might be the case that all lines, including apparently parallel ones might at some point meet.
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One way of arguing is that our intuition is still Euclidean. So in spite of non-Euclidean geometry, our form is Euclidean.

Another way of arguing: you could say that our intuition of space is actually non-Euclidean (or whatever happens to be the correct geometry of space, supposing non-Euclidean geometry is superseded), and Euclidean geometry was merely an empirical concept of that form.

But if physical effects external to ourselves can be shown to influence the geometry of space, is this not fatal to the assumption that space is an a priori form of our intuition?
• 70
He'd have said it was obvious if you decide that triangles can exist across 3D space, or on the surface of spheres.

As far as I'm aware Kant did not mention the geometry of the surface of spheres but in any case that is merely a subset within Euclidean space and as such would have no bearing on the form of our intuition.
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Since we do not have, and cannot have access to infinity then the premise can only work at a mundane level. Since we can never know if the universe is infinite, or tell if it might or might not coalesce into a pinprick then it might be the case that all lines, including apparently parallel ones might at some point meet.

No, the parallel lines never meet, it is impossible, because the definition indicates this. If they meet they are not parallel. The point being that we must accept the definitions and adhere to them whether or not there is any such thing as infinite lines, or parallel lines in the physical world.

What we see, sense, and even what we imagine in our minds, is completely different from what we know by the acceptance of definitions. However, we apply the things we know through acceptance of definitions, to the sense things of the physical world, by relating them, and if the definitions are useful this helps us in understanding and using the physical things in the sense world.
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As far as I'm aware Kant did not mention the geometry of the surface of spheres but in any case that is merely a subset within Euclidean space and as such would have no bearing on the form of our intuition.
I am not saying he did.
I am saying that he could. Any child can see the difference between a flat triangle and one which is plastered on the side of a ball.
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While I agree that it would be surprising if Kant did not know about this geometry, he had no reason to apply it to space as a whole and for the reasons mentioned above it is irrelevant to his conception of space.
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No, the parallel lines never meet, it is impossible, because the definition indicates this. If they meet they are not parallel. The point being that we must accept the definitions and adhere to them whether or not there is any such thing as infinite lines, or parallel lines in the physical world
The parallel postulate does not define parallel lines. They are defined in Book I Definition 23, as being two lines that never meet.
What the parallel postulate does is assert that two lines that cross either end of a line segment at non-right angles are not parallel. It doesn't actually say 'not parallel' but rather gives a property that is equivalent to being not parallel.

So the postulate neither defines parallel lines, nor asserts that there are any. I presume the existence of at least one pair of parallel lines must be a theorem that is deduced from that and other postulates. Although, as I said much earlier, I believe that statement of the postulate is incomplete, and needs the words 'and only on that side' to be added. Otherwise the postulate does not exclude elliptic geometries, where all non-coincident pairs of lines meet in two places.
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We don't observe light rays or curvature of space in the way we see cells through a microscope, though. We observe other phenomena about which light rays and curvature of space are explanatory theories.
I disagree. We can set up experiments where we send a beam of light in a straight line passing by the sun and the set up detectors on the other side to see where the beam lands. If it lands not in a straight line, but in a curve, then we have seen the light rays bendings. We don't see the curvature of space, but the curvature of space is that which explains the bending of the light rays, just like atoms (which we don't see) are what explain phenomena such as brownian motion, which we do see. You are a pragmatist, so how did the Peirce go - the whole of the effects is the whole of the conception, or something of that sort, anyways.

Are you saying that Kant would have denied that any physics experiment could reveal that spacial geometry was not Euclidean?
Yes, I'm quite sure he would have. If he found out about non-Euclidean geometry, he would have tried to re-adjust his theories.
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The parallel postulate does not define parallel lines. They are defined in Book I Definition 23, as being two lines that never meet.
What the parallel postulate does is assert that two lines that cross either end of a line segment at non-right angles are not parallel. It doesn't actually say 'not parallel' but rather gives a property that is equivalent to being not parallel.

Well it's not strictly that the angles are "non-right angles", because a parallelogram need not have right angles. What is necessary is that the angles where parallel lines cross the line segment must be equal. If the angles are not equal then the two lines are not parallel, or as you say "equivalent to being not parallel". This must be accepted "as defined", and the definition of "parallel" must be accepted "as defined", in order for the parallel postulate to hold.

So the postulate neither defines parallel lines, nor asserts that there are any.

The parallel postulate follows logically from accepting the definitions, and accepting the law of non-contradiction. It is produced from these definitions: the definition of parallel, that two parallel lines never meet, and the definition that if two parallel lines cross a line segment, they have equal angles. If you accept the definitions, the postulate holds. If you do not accept them, it does not hold.
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if two parallel lines cross a line segment, they have equal angles
That is a claim, not a definition. Observe how it does not say 'we define X to mean' or any of the equivalent forms of words that flag a definition.

The claim is only true if we adopt the parallel postulate or something equivalent to it.
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Any definition can be stated as a claim. That two parallel lines do not meet can be said to be a claim. The so-called claim here, is derived from the relationship between different planes and this relationship is defined by angles, it is not claimed.

Edit: Planes only exist by definition.
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Let's not discuss that any more.
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â†ªcharleton While I agree that it would be surprising if Kant did not know about this geometry, he had no reason to apply it to space as a whole and for the reasons mentioned above it is irrelevant to his conception of space.

I puzzled at the way you seem to view this idea.
All maths is a conceit. It's a means by which humans are able to describe space. There are no points, straight lines, nor perfect shapes in nature.
It's not that he 'did not know about it' because it had yet to be invented by Gauss, by the time Kant was DEAD.
• 70
The difficulty is that if maths is a conceit, how is it that it can be applied to the world? This was one of the fundamental questions Kant was trying to answer.
Is it invented or discovered?
• 11.3k
The difficulty is that if maths is a conceit, how is it that it can be applied to the world?
What do you mean by applied to the world?
• 70
I mean it gives us an applicable framework with which we can make sense and order our experiences. It is a pure form which the principals of physics take up to accurately predict events that we all perceive.
• 4.3k
Yes, I'm quite sure he would have. If he found out about non-Euclidean geometry, he would have tried to re-adjust his theories.

Why would an empirical discovery (an a posteriori truth) impact an a priori one? If a priori truths are determined by a posteriori truths, then they're not known prior, but post experience, and therefore are defeated definitionaly, not empirically. The point being that an a priori truth need not comport to an a posteriori discovery. Neither type of truth is primary or of higher order. And by "higher order," I mean more consistent with reality, which is noumenal anyway.
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I mean it gives us an applicable framework with which we can make sense and order our experiences. It is a pure form which the principals of physics take up to accurately predict events that we all perceive.
No, this gives you the wrong idea of mathematics. If you go back 2000 years ago, the math we had back then was completely different from the math we have today. Math has developed over the centuries - parts of new developments were kept, others were thrown out. For example, some cultures have thought that zero was not a number. It was a huge conceptual breakthrough to introduce negative numbers! Imagine what is a negative quantity? Makes no sense. Imaginary numbers? Give me a break! And complex numbers too?! Trigonometry applied outside of triangles? Limits? Calculus? Etc.

These are all mathematical inventions - and these are only the small set of inventions that were accepted over time. But there are a whole host of other ones that were rejected. Of course, you don't take all those into account. Human beings have created math, and continue to create math - and they keep what is useful, and throw everything else away.

Now, you and I, are affected by a bias - Kant too. Namely, we look at all the math that has proven useful and we have kept, and we say "Ahh what an achievement! How was this at all possible?" But we forget all that we've thrown away along the way - all the math that didn't work. We also forget about the ridiculousness of some mathematical concepts like imaginary numbers, which are still used, to this day, and practically applied in physics and engineering. Imagine for a moment that you did not know about the state of mathematics, and someone started telling you about imaginary numbers! You'd kick him out. What about stuff like 0! = 1? n! = 1*2*3*...*n . So how does it make sense that 0! = 1! = 1? But it does! Again - these are just modes of behaving and thinking about things that have proven useful. We tend to celebrate successful "method" and ignore the productive disclosing that made it possible to begin with. We are blinded by success.
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For example, how math grows. We have no negative factorials. Some people are trying to introduce negative factorials and form a coherent concept of them. If this invention proves useful, it will be kept, otherwise, to the dustbin of history.

https://mathoverflow.net/questions/10124/the-factorial-of-1-2-3
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Why would an empirical discovery (an a posteriori truth) impact an a priori one?
Simple - the latter isn't a priori, it was only mistaken to be a priori.
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how is it that it can be applied to the world?

Kant was good at thinking 'Copernican turns'. You might want to think about your question backwards?
Maths and the world is a dialogue. Whilst we invent maths, we draw our instances from the world.
But if you are yet to be convinced, please show me PI or any other irrational number in the world.
The world seems to be a round hole and maths is a square peg, as it relies on integers, which also do not exist. 1=1 might be true. But an orange is never equal to an orange.
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This doesn't address the question. Why does the a posteriori determine the a priori? It's as if you're saying non-euclidian a posteriori discoveries are true and a priori euclidian intuitions are false. How do you know which is true when truth is noumenal? All we're talking about is phenomenal, and it's perfectly logical to say that euclidian intuitions are necessary for phenomenal experience even if a posteriori knowledge might be discovered that is incomprehensible on an intuitive level.
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