So I submit, your honour, that the parallel postulate is not intuitive. — andrewk
So I submit, your honour, that the parallel postulate is not intuitive. — andrewk
The parallel postulate does not say what, based on your post, you appear to think it says. — andrewk
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, 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 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.
They always look parallel, and that's what matters to our intuitions. — andrewk
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. — Agustino
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. — Metaphysician Undercover
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is actually non-Euclidean (or whatever happens to be the correct geometry of space, supposing non-Euclidean geometry is superseded), and Euclidean geometry was merely an empirical concept of that form. — Moliere
He'd have said it was obvious if you decide that triangles can exist across 3D space, or on the surface of spheres. — charleton
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. — charleton
I am not saying he did.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. — Perplexed
The parallel postulate does not define parallel lines. They are defined in Book I Definition 23, as being two lines that never 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 — Metaphysician Undercover
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.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. — Janus
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.Are you saying that Kant would have denied that any physics experiment could reveal that spacial geometry was not Euclidean? — Perplexed
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. — andrewk
So the postulate neither defines parallel lines, nor asserts that there are any. — andrewk
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.if two parallel lines cross a line segment, they have equal angles — Metaphysician Undercover
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. — Perplexed
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. — Agustino
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.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. — Perplexed
how is it that it can be applied to the world? — Perplexed
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