[edby]

Some quotes related to Euclid's parallel line postulate courtesy of the Department of Mathematics at UC Riverside:

http://math.ucr.edu/~res/math133/geometrynotes05a.f13.pdf

We have already mentioned in Section II.5 that the final assumption in Euclid’s  Elements  (the so – called Fifth Postulate) is far more complicated than the others. Furthermore, the proofs of the first  28 results in the Elements do not use the Fifth Postulate. In addition, there are general questions whether this postulate corresponds to physical reality because it involves objects which are too distant to be observed or questions about measurements that cannot necessarily be answered conclusively because there are always limits to the precision of physical measurements.

A nice quote from Immanuel Kant:

"The concept of [Euclidean] space is by no means of empirical origin, but is an inevitable necessity of thought."

Kant, Critique of Pure Reason (1781)

A good description illustrating the parallel line postulate.


As we can see, it is candidly admitted that this idea is not empirical. It is a hypothetical house of cards.

Is your point that parallel lines do meet after all?

[Tom Bishop]

From https://www.encyclopedia.com/people/science-and-technology/mathematics-biographies/euclid --

The fifth postulate concerns parallel straight lines. These are defined in I, definition 23, as “straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction”. The essential characteristic of parallel lines for Euclid is, therefore, that they do not meet. Other Greek writers toyed with the idea, as many moderns have done, that parallel straight lines are equidistant from each other throughout their lengths or have the same direction, 23 and Euclid shows his genius in opting for nonsecancy as the test of parallelism. The fifth postulate runs:

    5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, will meet on that side on which are the angles less than two right angles.

In Figure 1 the postulate asserts that if a straight line (PQ) cuts two other straight lines (AB, CD) in P, Q so that the sum of the angles BPQ, DQP is less than two right angles, AB, CD will meet on the same side of PQ as those two angles, that is, they will meet if produced beyond B and D.

There was a strong feeling in antiquity that this postulate should be capable of proof, and attempts to prove it were made by Ptolemy and Proclus, among others. 24 Many more attempts have been made in modern times. All depend for their apparent success on making, consciously or unconsciously, an assumption which is equivalent to Euclid’s postulate. It was Saccheri in his book Euclides ab omni naevo vindicatus (1733) who first asked himself what would be the consequences of hypotheses other than that of Euclid, and in so doing he stumbled upon the possibility of non-Euclidean geometries. Being convinced, as all mathematicians and philosophers were until the nineteenth century, that there could be no geometry besides that delineated by Euclid, he did not realize what he had done; and although Gauss had the first understanding of modem ideas, it was left to Lobachevski (1826, 1829) and Bolyai (1832), on the one hand, and Riemann (1854), on the other, to develop non-Euclidean geometries. Euclid’s fifth postulate has thus been revealed for what it really is—an unprovable assumption defining the character of one type of space.

[edby]

From https://www.encyclopedia.com/people/science-and-technology/mathematics-biographies/euclid --

The fifth postulate concerns parallel straight lines. These are defined in I, definition 23, as “straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction”. The essential characteristic of parallel lines for Euclid is, therefore, that they do not meet. Other Greek writers toyed with the idea, as many moderns have done, that parallel straight lines are equidistant from each other throughout their lengths or have the same direction, 23 and Euclid shows his genius in opting for nonsecancy as the test of parallelism. The fifth postulate runs:

    5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, will meet on that side on which are the angles less than two right angles.

In Figure 1 the postulate asserts that if a straight line (PQ) cuts two other straight lines (AB, CD) in P, Q so that the sum of the angles BPQ, DQP is less than two right angles, AB, CD will meet on the same side of PQ as those two angles, that is, they will meet if produced beyond B and D.

There was a strong feeling in antiquity that this postulate should be capable of proof, and attempts to prove it were made by Ptolemy and Proclus, among others. 24 Many more attempts have been made in modern times. All depend for their apparent success on making, consciously or unconsciously, an assumption which is equivalent to Euclid’s postulate. It was Saccheri in his book Euclides ab omni naevo vindicatus (1733) who first asked himself what would be the consequences of hypotheses other than that of Euclid, and in so doing he stumbled upon the possibility of non-Euclidean geometries. Being convinced, as all mathematicians and philosophers were until the

nineteenth century, that there could be no geometry besides that delineated by Euclid, he did not realize what he had done; and although Gauss had the first understanding of modem ideas, it was left to Lobachevski (1826, 1829) and Bolyai (1832), on the one hand, and Riemann (1854), on the other, to develop non-Euclidean geometries. Euclid’s fifth postulate has thus been revealed for what it really is—an unprovable assumption defining the character of one type of space.

Yes yes, but are you taking that as proof your claim that parallel lines do meet after all? If not, what is your point in posting this? What does the possibility of non-Euclidean geometry prove, in your view?

Please tell.

Note that all five assumptions underlying Euclid's geometry are just that, namely assumptions. Did you imagine anything different?

[Curious Squirrel]

From https://www.encyclopedia.com/people/science-and-technology/mathematics-biographies/euclid --

The fifth postulate concerns parallel straight lines. These are defined in I, definition 23, as “straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction”. The essential characteristic of parallel lines for Euclid is, therefore, that they do not meet. Other Greek writers toyed with the idea, as many moderns have done, that parallel straight lines are equidistant from each other throughout their lengths or have the same direction, 23 and Euclid shows his genius in opting for nonsecancy as the test of parallelism. The fifth postulate runs:

    5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, will meet on that side on which are the angles less than two right angles.

In Figure 1 the postulate asserts that if a straight line (PQ) cuts two other straight lines (AB, CD) in P, Q so that the sum of the angles BPQ, DQP is less than two right angles, AB, CD will meet on the same side of PQ as those two angles, that is, they will meet if produced beyond B and D.

There was a strong feeling in antiquity that this postulate should be capable of proof, and attempts to prove it were made by Ptolemy and Proclus, among others. 24 Many more attempts have been made in modern times. All depend for their apparent success on making, consciously or unconsciously, an assumption which is equivalent to Euclid’s postulate. It was Saccheri in his book Euclides ab omni naevo vindicatus (1733) who first asked himself what would be the consequences of hypotheses other than that of Euclid, and in so doing he stumbled upon the possibility of non-Euclidean geometries. Being convinced, as all mathematicians and philosophers were until the

nineteenth century, that there could be no geometry besides that delineated by Euclid, he did not realize what he had done; and although Gauss had the first understanding of modem ideas, it was left to Lobachevski (1826, 1829) and Bolyai (1832), on the one hand, and Riemann (1854), on the other, to develop non-Euclidean geometries. Euclid’s fifth postulate has thus been revealed for what it really is—an unprovable assumption defining the character of one type of space.

Yes yes, but are you taking that as proof your claim that parallel lines do meet after all? If not, what is your point in posting this? What does the possibility of non-Euclidean geometry prove, in your view?

Please tell.

Note that all five assumptions underlying Euclid's geometry are just that, namely assumptions. Did you imagine anything different?

He's not coming right out and saying it, indeed he never has, but what I'm getting is that he's essentially saying his and Rowbotham's perspective theories aren't based upon Euclidean geometry, and as such attempting to use Euclidean geometry upon them is a fruitless effort. This is merely a guess I'm coming up with now though, I could be off as he's never actually said this, but it fits with everything he HAS said at least. Although last I heard/saw John Davis was the one more championing a non-Euclidean flat Earth.

[edby]

He's not coming right out and saying it, indeed he never has, but what I'm getting is that he's essentially saying his and Rowbotham's perspective theories aren't based upon Euclidean geometry, and as such attempting to use Euclidean geometry upon them is a fruitless effort. This is merely a guess I'm coming up with now though, I could be off as he's never actually said this, but it fits with everything he HAS said at least. Although last I heard/saw John Davis was the one more championing a non-Euclidean flat Earth.

That would be amusing if true.

[Bobby Shafto]

He's not coming right out and saying it, indeed he never has, but what I'm getting is that he's essentially saying his and Rowbotham's perspective theories aren't based upon Euclidean geometry, and as such attempting to use Euclidean geometry upon them is a fruitless effort. This is merely a guess I'm coming up with now though, I could be off as he's never actually said this, but it fits with everything he HAS said at least. Although last I heard/saw John Davis was the one more championing a non-Euclidean flat Earth.

There needs to be a way to explain a horizon (H) at a finite distance in plane topography. Rowbotham's explanation connects the horizon with vanishing lines/points of Perspective, but then that invokes the notion that VP is not finite but merely a false matter of perception that it is finite. So there needs to be a way to avoid infinity for H to have actually have a finite value using Perspective as a foundation. Perspective can't merely be perceptual in a FE model employing Perspective as a rationale. It must be physical. Finitely physical. Has to be. Being infinite and merely perceptual kills Perspective as an available answer for the horizon on FE.

[edby]

.. last I heard/saw John Davis was the one more championing a non-Euclidean flat Earth.

But a surface is flat when its geometry obeys all five Euclidean postulates. Drop the fifth, and you have a curved surface. So are Davis, Bishop et al saying that the earth's surface is curved?

'Non-Euclidean flat Earth' is an oxymoron.

[Curious Squirrel]

.. last I heard/saw John Davis was the one more championing a non-Euclidean flat Earth.

But a surface is flat when its geometry obeys all five Euclidean postulates. Drop the fifth, and you have a curved surface. So are Davis, Bishop et al saying that the earth's surface is curved?

'Non-Euclidean flat Earth' is an oxymoron.

John Davis' model was essentially (and this is super simplified, I've been told it's technically wrong but works well enough if you don't/can't dig into the detail) that the Earth was a flat plane, and space around/on it is curved. Which can present the appearance of it being the Earth that's curved. I don't actually know if something like this is what Tom is putting forth, but the more I've thought about what I DO recall of this one, the more it seems to fit what Tom has been saying. John is just the only person I can recall talking specifically in favor of this model. But it would make Euclid's postulates and Euclidean Geometry largely useless from my understanding. (Note, non-Euclidean geometry has always given me a headache. I suggest digging around the other site if you want to look into this more.)

[Tom Bishop]

Mainly what I have been saying over the last ten years is that the perspective lines are finite rather than infinite. There will be a point at the distance where they merge together.

Things which happen where the Ancient Greek model say should happen at infinity will occur a finite distance away.

- The Ancient Greek model says that it would take infinity for the sun to get to the horizon.
- The Ancient Greek model says that a body directly above your head that recedes into the distance would have to increase its altitude by infinity to stop rotating to perspective.
- The Ancient Greek model says that an overhead receding body would have to increase its altitude by infinity to become perfectly constant in its pace across the sky.

All of the above predictions are predicated upon a continuous universe model where parallel perspective lines would only merge at an infinite distance away (read: never).

Merging perspective lines can be backed up empirically -- The horizon is certainly not an infinite distance away. We do witness lines merging in the distance (even if you want to call it limits of optics, they still merge regardless, and perhaps that limit is part of how we experience the world and should not be discounted as an effect separate from perspective).

In contradiction, continuous perspective lines cannot be observed empirically. Rather than something that is seen as in the above examples, they must be imagined to exist.

[Tom Bishop]

None of this is to say that the two rails of a railroad track will physically and literally merge together into one rail in the distance. No one is saying that, just as no one is saying that the sun will literally crash into the horizon. Perspective is how the world presents itself to us, and this is what we are talking about.

As mentioned in another thread, in Earth Not a Globe Rowbotham has made some observations on the horizontal that seem to contradict concepts in Euclidean Geometry.

Both Rowbotham and I are talking about geometries that are non-Euclidean, and indeed violate Euclid's fifth postulate/parallel line ideas. However, non-Euclidean geometries are modern geometries that have arisen out of the weaknesses of Euclid's fifth postulate.

See this essay:

http://www.jeremychapman.info/cms/one-too-many-the-role-of-euclid%E2%80%99s-fifth-postulate-in-the-development-of-non-euclidean-geometries

This essay will show that non-Euclidean geometries arose as a direct consequence of a weakness in one of Euclid's own axiomatic postulates, specifically Postulate Five. Further, that should Euclid have omitted this contentious Fifth Postulate, the very possibility of non-Euclidean geometries would be challenged. This paper will trace the history of non-Euclidean geometries by exploring how the generations of post-Euclidean philosophers and scientists struggled with the Fifth Postulate, and how the work that was performed on this task led directly to the formulation of these alternate geometries. In this way it will be proven that without the debate over the Fifth Postulate, non-Euclidean geometries would be unnecessary because, as we will see, the only modifications these geometries make is to the Fifth Postulate, with the rest remaining almost wholly intact. The discussion will encompass the work of many great thinkers, including Euclid, Kant, Gauss, Bolyai, Lobachevsky, Sacchieri, and Poincare.

Some quotes from Poincaré:

- Non-Euclidean geometries have the same logical and mathematical legitimacy as Euclidean geometry.

- All geometric systems are equivalent and thus no system of axioms may claim that it is the true geometry.

- Axioms of geometry are neither synthetic a priori judgments nor analytic ones; they are conventions or 'disguised' definitions.

[AATW]

Merging perspective lines can be backed up empirically -- The horizon is certainly not an infinite distance away.

Sigh...

This is your circular reasoning I was picking you up on the other day
You're right, the horizon is not an infinite distance away. And in the real world we know why that is, we're looking at the edge of the globe earth which curves away from us.
That's why the horizon distance increases with altitude, you can see further over the curve. This is all observable.

You're claiming that the horizon is the "merging of perspective lines" (if it was then why would it be a line, not a point?) and because the horizon is observably a finite distance away you then conclude that this shows the perspective lines merge at a finite distance.

Your "proof" that perspective lines merge at a finite distance is that
1) The horizon is a finite distance - observation
2) The horizon is the merging of perspective lines - untestable hypothesis.

Your proof of 2 is 1, but your explanation of 1 is 2...

[garygreen]

tom, you're conflating two wildly different things.

euclid's fifth postulate is ultimately about the geometric nature of space.  it has virtually nothing to do with any of these arguments about optics because none of the distances/angles you're arguing about are infinite.  they're all finite.  what happens "at infinity" doesn't matter, because all these triangles have sides with finite lengths.

[Bobby Shafto]

I don't understand this at all. Parallel lines do merge at a finite distance, but they don't literally merge?

Are you saying you agree that it is an illusion that they merge, but disagree with the infinite value of that illusion; saying rather that its an illusion upon which a finite value can be placed? In other words, though train tracks don't literally merge, we can put a finite distance on where it is they look like they merge?

[ICanScienceThat]

I don't understand this at all. Parallel lines do merge at a finite distance, but they don't literally merge?

Are you saying you agree that it is an illusion that they merge, but disagree with the infinite value of that illusion; saying rather that its an illusion upon which a finite value can be placed? In other words, though train tracks don't literally merge, we can put a finite distance on where it is they look like they merge?

Actually parallel lines only merge at infinity. All you need to prove this is the fact that light travels in straight lines. Tom is making the claim that light does not travel in straight lines.

To clarify, (and Tom has acknowledged this), parallel lines will reach a point somewhere before infinity where the resolution of the recording device (your eye or a camera) will not be able to see the gap between them. They will have "merged" effectively, and this happens at a point before infinity. Mathematically, you have to go to infinity, but where that appears to happen to you depends upon the resolution and zoom level of your camera.

[ICanScienceThat]

Mainly what I have been saying over the last ten years is that the perspective lines are finite rather than infinite. There will be a point at the distance where they merge together.

Things which happen where the Ancient Greek model say should happen at infinity will occur a finite distance away.

- The Ancient Greek model says that it would take infinity for the sun to get to the horizon.
- The Ancient Greek model says that a body directly above your head that recedes into the distance would have to increase its altitude by infinity to stop rotating to perspective.
- The Ancient Greek model says that an overhead receding body would have to increase its altitude by infinity to become perfectly constant in its pace across the sky.

All of the above predictions are predicated upon a continuous universe model where parallel perspective lines would only merge at an infinite distance away (read: never).

Merging perspective lines can be backed up empirically -- The horizon is certainly not an infinite distance away. We do witness lines merging in the distance (even if you want to call it limits of optics, they still merge regardless, and perhaps that limit is part of how we experience the world and should not be discounted as an effect separate from perspective).

In contradiction, continuous perspective lines cannot be observed empirically. Rather than something that is seen as in the above examples, they must be imagined to exist.

The phenomenon known as perspective arises from the fact that light generally travels in straight lines. Everything else we know about perspective can all be derived straight from that. To make any of Tom's ideas work, we need light to bend. Tom is simply arguing that light bends.

Let's consider these points one at a time:
"- The Ancient Greek model says that it would take infinity for the sun to get to the horizon."
I think we can all agree, to get the FE model to work with a sunset, we need a substantial amount of light bending.
We can also agree, that the RE model acknowledges a much smaller amount of bending via refraction near the horizon.

"- The Ancient Greek model says that a body directly above your head that recedes into the distance would have to increase its altitude by infinity to stop rotating to perspective."
I frankly don't understand what is meant by "rotating to perspective." I googled that and got no hits. Are we trying to explain why we all see the same face of the Sun no matter where we are on Earth? Again, the FE model is going to need some bendy light to pull that off.
Maybe Tom can explain this one.

"- The Ancient Greek model says that an overhead receding body would have to increase its altitude by infinity to become perfectly constant in its pace across the sky."
Agreed. I can only surmise we're talking about the sun again. The Sun can be observed moving at a constant angular rate across the sky. According to the FE model, we're going to need some serious light bending to make it happen.

So if we start from a flat Earth with a close (3000 miles above) Sun, we cannot make observations match without bendy light. Got it. Agreed. This leaves us with 2 possibilities:
a) The Earth is not flat
b) Light bends

Can we prove that light doesn't bend? At close distances we certainly can. If we avoid refraction, light can be shown to be straight as far out as your experiment can reach (100s of feet?). Earlier I showed how crepuscular rays show light to move in straight lines out as far as the distant clouds (10s of miles?). But what do we know about how light moves out farther than that?

Some have suggested that a great dome in the sky causes dramatic refraction as the Sun's light passes through it, and that is beyond the reach of our crepuscular rays.
Others (Tom) have suggested that there is a natural property of light that makes it bend as it travels really long distances.

These things are hard to test empirically. However, we have more information here. We've acknowledged that we need light to bend to make this work, but we can actually say how much we need the light to bend and in what way.
To make the Sun cross below the horizon from anywhere on the FE, what kind of bending does that take? For simplicity, I'll take the AE map, and let's just sit on the equator on the equinox. At noon, the Sun is 3000 miles straight up. At sunset, the sun is 3000 miles up, 6000 miles to the West and 6000 miles North. We need that Sun to appear to be directly West and right on the horizon. This tells us that light has bent 45 degrees counter-clockwise and around 20 degrees upward on its journey to us.

What do you say Tom? Is this basically what we're talking about here? I know you don't like the AE map, should we use the dual-pole map instead?

Let me summarize:
a) We cannot do empirical measurements to prove the straightness of light out to the 1000s of miles we need to reach the FE sun.
b) We need light to bend to make any of the FE models match observations
c) We CAN work backwards from the observations to see what sort of bending the FE model requires.

[Tom Bishop]

The phenomenon known as perspective arises from the fact that light generally travels in straight lines. Everything else we know about perspective can all be derived straight from that. To make any of Tom's ideas work, we need light to bend. Tom is simply arguing that light bends.

If I were to describe the mechanism further I wouldn't call it a bending light theory. I would lean more towards it being caused by a fundamental structure to the universe.

As speculation, I would say that it has something to do with the quantized nature of the universe. In the traditional Continuous Universe of the Ancient Greeks the following is assumed:

        - That perfect circles can exist
        - That one could zoom into a circle forever and see a curve
        - That any length of space can be divided into infinitely smaller parts
        - That the space can be infinitely long
        - Time can likewise be infinitely divided, or infinitely long
        - The Perspective Lines receded infinitely and continuously into the distance

However, from Quantum Mechanics, a Quantized Universe is predicted. Space is made up of a fundamental unit called Plank units, which cannot be divided further. The fundamental unit of time is called Planck time. There is increasing evidence in QM that our universe is quantized.

The angles of perspective lines match the nearest possible path through finite and quantized space, and cannot truly match a hypothetical Continuous Universe.

As the resolution of reality is not infinite, the perspective lines of a distant body may appear to, and perhaps for all intents, merge, much like distant bodies in a computer game or in a computer simulation may merge to a pixel since the resolution of the computer simulation is not infinite.

Light travels along the possible paths of perspective lines, of which are limited by the resolution of the universe. Limits of optics may also play a secondary part in this.

[ICanScienceThat]

The phenomenon known as perspective arises from the fact that light generally travels in straight lines. Everything else we know about perspective can all be derived straight from that. To make any of Tom's ideas work, we need light to bend. Tom is simply arguing that light bends.

If I were to describe the mechanism further I wouldn't call it a bending light theory. It's a more caused by a fundamental structure to the universe.

As speculation, I would say that it has something to do with the quantized nature of the universe. In the traditional Continuous Universe of the Ancient Greeks the following is assumed:

        - That perfect circles can exist
        - That one could zoom into a circle forever and see a curve
        - That any length of space can be divided into infinitely smaller parts
        - That the space can be infinitely long
        - Time can likewise be infinitely divided, or infinitely long
        - The Perspective Lines receded infinitely and continuously into the distance

However, from Quantum Mechanics, a Quantized Universe is predicted. Space is made up of a fundamental unit called Plank units, which cannot be divided further. The fundamental unit of time is called Planck time. There is increasing evidence in QM that our universe is quantized.

As the resolution of reality is not infinite, the light from a distant body may appear to, and perhaps for all intents, merge, much like distant bodies in a computer game or in a computer simulation may merge to a pixel since the resolution of the computer simulation is not infinite.

The angles of perspective lines match the nearest possible path through finite and quantized space, and cannot truly match a hypothetical Continuous Universe.

Cool stuff. QM is mind-bending for sure, and the idea that the universe is quantized according to Planck length and Plank time is interesting. It's cool, and if we really wanted to, we could apply those out to the distances predicted by FE.

I'd prefer to start off with the observations we have. Are you on board with the back-of-the-envelope calculations I've got started here? The light from the Sun bends upward around 20 degrees during sunset while coming straight towards the viewer at lanai noon? The horizontal bending estimates really need some kind of hypothetical FE map to work from. Could you provide one for discussion purposes? Otherwise, I'll stick with this one.

[garygreen]

In the traditional Continuous Universe of the Ancient Greeks the following is assumed:

        - That perfect circles can exist
        - That one could zoom into a circle forever and see a curve
        - That any length of space can be divided into infinitely smaller parts
        - That the space can be infinitely long
        - Time can likewise be infinitely divided, or infinitely long
        - The Perspective Lines receded infinitely and continuously into the distance

flat earth or no, the sun is a finite distance from you, yes?  so what difference does it make what ancient greek dummies thought about infinity?  none of the distances you're arguing about are infinite.

"but what if there are planck units" doesn't have anything to do with optics.