Thread: Mathematical Numbers are Theoretical; Not Proven

Originally Posted by Xei

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If it has been covered extensively in "layman's terms", let's see you cover it a bit.

You have dodged a ton of my questions.

Essentially gravity is not a force in the traditional sense; what really happens is that space is bent by mass, and then objects travel through this bent space. The most popular analogy is of placing a heavy ball like a bowling ball on a rubber sheet so that it bends, and then pushing a marble across the rubber sheet so that it orbits the bowling ball. In the traditional Euclidian model space would just be some kind uniform straight grid which the ball would intersect without resistance and then it would pull the marble by a 'force'.

There is no known 'reason' for this, it is just how nature works; in a non-Euclidian fashion.

All the questions I can see are repetitions of things I've already explained a couple of times or just nonsenses (like 'I was just taking a guess at what you could possibly be talking about since you are not telling me' - when I'd already explicitly told you that it was general relativity, not special relativity. You clearly just didn't know the difference.).

Originally Posted by Xei

All the questions I can see are repetitions of things I've already explained a couple of times or just nonsenses (like 'I was just taking a guess at what you could possibly be talking about since you are not telling me' - when I'd already explicitly told you that it was general relativity, not special relativity. You clearly just didn't know the difference.).

Nope. For example...

1. Why is Euclidian geometry taught in practically every high school (where it is required), school of engineering, and school of architecture in the world as FACT (not just approximation methods)? (While we are at it, why are Euclidian geometric proofs called "proofs" instead "approximation evidence"?)

2. Why does Euclidian geometry provide for such microscopically close approxiamations (supposedly, since you say it is not exact)?

3. What is your counterargument to my point about equal change concerning the leaning of sides of a parallelogram and therefore congruence of opposite angles?

4. Specifically how does your point about general relativity disprove the rules of Euclidian geometry? Give specific examples. (Insulting me with comments about how it differs from special relativity does not answer the question.)

5. How does NASA determine vectors?

I've explicitly answered everything there already...

1. It isn't taught as universal fact. When did your teachers ever say, 'this is exactly true everywhere in the universe'? They didn't. And if they did they were lying.

2. Because it works extremely well for weak gravitational fields. Mass is what bends space from the traditional flat geometry.

3. Your intuitive axiom was wrong; there's no reason the two angles should bend by the same amount in non-Euclidian geometry.

4. Because the framework of Euclidian geometry is a uniform grid (in which only parallel lines never meet), wheras the grid of general relativity is warped (so that they needn't) and general relativity has been proved by experiment.



5. I'm not sure what you mean by that, but as I said, they have to use the GR value for keeping satellites in orbit, because the classical value is wrong by a significant enough degree for it to cause the satellite to leave orbit. For pretty much all other physical applications we use Euclidian geometry because it approximates extremely well on Earth and is a million times simpler than using GR. However you would not be able to use it at all in astrophysics, as the warping of space on cosmological scales is very large.

Originally Posted by Xei

I've explicitly answered everything there already...

False.

Originally Posted by Xei

1. It isn't taught as universal fact. When did your teachers ever say, 'this is exactly true everywhere in the universe'? They didn't. And if they did they were lying.

Yes it is taught as universal fact. The words "usually" and "in weak gravitational fields" are not in any of the postulates, theorems, or corollaries. A statement is false if there is just one exception (called a "counterexample") to it. That too is one of the teachings of geometry, and also logic. One single exception to the rule that opposite angles of a parallelogram are parallel would prove the rule false.

Originally Posted by Xei

2. Because it works extremely well for weak gravitational fields. Mass is what bends space from the traditional flat geometry.

If a parallelogram is "bent", it is no longer a parallelogram.

Originally Posted by Xei

3. Your intuitive axiom was wrong; there's no reason the two angles should bend by the same amount in non-Euclidian geometry.

But non-Euclidian geometry is a crock. Like there is science fiction, non-Euclidian geometry is math fiction. In non-reality arithmetic, 2 + 2 = 5. So what?

Originally Posted by Xei

4. Because the framework of Euclidian geometry is a uniform grid (in which only parallel lines never meet), wheras the grid of general relativity is warped (so that they needn't) and general relativity has been proved by experiment.

Those lines in 2 and 3 are not parallel. The opposite sides of a parallelogram are parallel.

Originally Posted by Xei

5. I'm not sure what you mean by that, but as I said, they have to use the GR value for keeping satellites in orbit, because the classical value is wrong by a significant enough degree for it to cause the satellite to leave orbit. For pretty much all other physical applications we use Euclidian geometry because it approximates extremely well on Earth and is a million times simpler than using GR. However you would not be able to use it at all in astrophysics, as the warping of space on cosmological scales is very large.

Vectors are not used for satellites in orbit because they move in curved paths. I am talking about motion in a straight line when an outside force is involved.

How about on a pure math grid. We're not talking about "bent" parallelograms, which wouldn't be parallelograms any more, we are talking on a piece of paper (or computer screen) please explain how opposite angles are not equal (which you still haven't done).

Originally Posted by Xei

Architects don't build things larger than solar systems, remember?

Size doesn't affect mathematics, remember?

If you had one bacteria, and another, you would have *gasp* two bacteria! If you had one galaxy, and another *gasp*, you would have two galaxies!

If you had one straight building and another, there would be two straight buldings, each with 180 degrees. If you had one star and another star, both perfectly round. both would have internal degrees of 360.

Size doesn't affect that. And warped space makes warped shapes, wich are not the same as they were when they started. A parallelogram and a warped parallelogram are !=.

How bizarre... I'm wishing wendylove was here. :l

How about on a pure math grid. We're not talking about "bent" parallelograms, which wouldn't be parallelograms any more, we are talking on a piece of paper (or computer screen) please explain how opposite angles are not equal (which you still haven't done).

But in Euclidian geometry they are equal?



Please keep up.

Size doesn't affect mathematics, remember?

If you had one bacteria, and another, you would have *gasp* two bacteria! If you had one galaxy, and another *gasp*, you would have two galaxies!

If you had one straight building and another, there would be two straight buldings, each with 180 degrees. If you had one star and another star, both perfectly round. both would have internal degrees of 360.

Size doesn't affect that. And warped space makes warped shapes, wich are not the same as they were when they started. A parallelogram and a warped parallelogram are !=.

A parallelogram in Euclidian space is not the same thing as a parallelogram in non-Euclidian space. Correct.

However the universe is non-Euclidian so everything else you said is bunk.

Vectors are not used for satellites in orbit because they move in curved paths. I am talking about motion in a straight line when an outside force is involved.

...yes they are. There is a transverse velocity vector and a radial centripetal acceleration vector. Please stop using words you don't understand. I can't imagine what you're asking for because as far as I know NASA has never built a satellite that was designed to go in a straight line forever. -_-

Those lines in 2 and 3 are not parallel. The opposite sides of a parallelogram are parallel.

In those geometries they are parallel. If you want to restrict the definition to Euclidian geometries then fine, but then in the real world there would be no such thing as a parallelogram.

But non-Euclidian geometry is a crock. Like there is science fiction, non-Euclidian geometry is math fiction. In non-reality arithmetic, 2 + 2 = 5. So what?

I can't work out for the life of me why you're in denial about something so inert...

Here it is again:

http://en.wikipedia.org/wiki/Tests_o...ral_relativity

Yes it is taught as universal fact. The words "usually" and "in weak gravitational fields" are not in any of the postulates, theorems, or corollaries. A statement is false if there is just one exception (called a "counterexample") to it. That too is one of the teachings of geometry, and also logic. One single exception to the rule that opposite angles of a parallelogram are parallel would prove the rule false.

Why on Earth would they be in any of the axioms? These axioms were proposed by the Greeks thousands of years ago, and to the human eye they appear exactly right. And even then, Euclidian geometry is still studied extensively all over the world, firstly because for almost all human intents and purposes the error is so close to zero that it makes no difference, and secondly because the results are used in other branches of mathematics, such as complex number theory.

Originally Posted by universal mind

I don't deny the existence or truth of general relativity. I just don't accept that it negates the rules of geometry.

Hyperbolic geometry is just doing geometry on a surface of which it's not flat but curved. Euclidean geometry is kind of useless if we look at the surface of a sphere as it's not flat, even worse you can't really smooth out a sphere so it's flat.

Originally Posted by universal mind

Why would schools (worldwide) keep teaching them as fact if they are wrong? I have taught out of a lot of geometry books, and I have never once come across even a footnote that says the rules are outdated or just approximations. They are taught as factual, which they are.

You can't really teach children a third year topic at uni or even get somebody in school to understand it. Alot of geometry is not taught in school for example mobius geometry, wild geometry, algebraic geometry, non communative geometry and communative geometry. I think you should stop reading school books and maybe pick up a uni book on geometry.

Saying, that I heard it's general view of Physicist that the universe at its tiniest scale is euclidean.

Originally Posted by universal mind

If the rules turned out to be wrong, which they did not, they would have been revised.

Actually they did. Look up the work of Hilbert.

Originally Posted by Xei

I'm not; if you define parallel as 'the quality of two lines which can be extended infinitely without intersection' then it can be used quite comfortably in non-Euclidian geometries.

That is not the definition. What you just said applies to skew lines.

Originally Posted by Xei

This brings the issue back to the crux of the whole discussion really, which is that mathematics is not 'real' anyway; every result you learn in Euclidian geometry is only 'factual' within Euclidian geometry; you won't find any disclaimers in a geometry textbook because the textbooks are about Euclidian geometry (which is useful for many other branches of mathematics), not reality (i.e. physics): reality is not Euclidian.

You said the rules are disproven and outdated. Did you change your mind? Are the opposite angles of a parallelogram congruent or not?

If mathematics is not "real", why are you arguing for the truth of hyperbolic geometry?

Originally Posted by Xei

The approximations on Earth are extremely good, I think I read in one of Penrose's books that a line will only deviate from the Euclidian model by something like one atomic radius per the radius of Earth itself, or some ludicrous number.

So one angle of a square is 89 and a whole bunch of decimal places while its opposite angle is 90 and a whole bunch of zeros and then another digit after the decimal?

Originally Posted by Xei

However near strong gravitational fields it is a completely different matter; in the most extreme situations, that is, around black holes, the geometry is so warped that it's basically folded back upon itself.

An angle is a measure of the amount of rotation between two lines. In hyperbolic geometry, those angles would all be > pi/2.

All four angles of a square? Then what would make it a square?

Originally Posted by wendylove

Hyperbolic geometry is just doing geometry on a surface of which it's not flat but curved. Euclidean geometry is kind of useless if we look at the surface of a sphere as it's not flat, even worse you can't really smooth out a sphere so it's flat.

Spheres are covered in Euclidian geometry.

Originally Posted by wendylove

You can't really teach children a third year topic at uni or even get somebody in school to understand it. Alot of geometry is not taught in school for example mobius geometry, wild geometry, algebraic geometry, non communative geometry and communative geometry. I think you should stop reading school books and maybe pick up a uni book on geometry.

That does not answer anything. I asked why Euclidian geometry is taught EVERYWHERE as fact if it is not.

Originally Posted by wendylove

Actually they did. Look up the work of Hilbert.

Look up the postulates and theorems of required geometry.

Originally Posted by Licity

This definition of parallel only applies to Euclidean space within a single plane. http://en.wikipedia.org/wiki/Skew_lines

That is not the definition. What you just said applies to skew lines.

See above.

You said the rules are disproven and outdated. Did you change your mind? Are the opposite angles of a parallelogram congruent or not?

If mathematics is not "real", why are you arguing for the truth of hyperbolic geometry?

Don't make things up. When did I ever say anything along the lines of the rules of Euclidian geometry being outdated or 'disproven'?? It was discovered a century ago that Euclidian geometry doesn't apply to the universe, but Euclidian geometry is still completely valid in itself, and all the proofs are flawless. The only issue is that the axioms - specifically the parallel axiom - aren't true in the universe. So all of the proofs built upon those axioms also don't apply to the universe. They're completely valid within Euclidian geometry though, and have many applications elsewhere, such as when working with complex numbers.

I'm not arguing for the truth of hyperbolic geometry either; again, hyperbolic geometry is only true within the axioms of hyperbolic geometry. The geometry of the universe isn't actually hyperbolic either, it's more complicated than that.

So one angle of a square is 89 and a whole bunch of decimal places while its opposite angle is 90 and a whole bunch of zeros and then another digit after the decimal?

I don't see why not (although they needn't sum to 180 in case that's what you were getting at). It depends on the local amount of curvature. If you're near a very large dense mass such as a black hole the angles could be vastly different.

All four angles of a square? Then what would make it a square?

A square is a polygon with four sides of equal length and four equal angles. Here is what a hyperbolic square looks like:
http://upload.wikimedia.org/wikipedi...olic_plane.png

Originally Posted by Xei

See above.

See specifically what above?

Originally Posted by Xei

Don't make things up. When did I ever say anything along the lines of the rules of Euclidian geometry being outdated or 'disproven'??

Originally Posted by Xei

That hypothesis is outdated by about a century now.

You said that in response to my question about whether opposite angles of a parallelogram are congruent. It is in post #61.

Originally Posted by Xei

It was discovered a century ago that Euclidian geometry doesn't apply to the universe, but Euclidian geometry is still completely valid in itself, and all the proofs are flawless.

Earlier, you said they are nothing more than good approximations.

Originally Posted by Xei

The only issue is that the axioms - specifically the parallel axiom - aren't true in the universe. So all of the proofs built upon those axioms also don't apply to the universe. They're completely valid within Euclidian geometry though, and have many applications elsewhere, such as when working with complex numbers.

The second dimension is in the universe.

Originally Posted by Xei

I'm not arguing for the truth of hyperbolic geometry either; again, hyperbolic geometry is only true within the axioms of hyperbolic geometry. The geometry of the universe isn't actually hyperbolic either, it's more complicated than that.

How is it true within itself if it is just something humans made up? Does it have any more reality than truth within The Wizard of Oz? What is the difference between math and science fiction, in terms of truth? The Death Star is real within Star Wars.

Originally Posted by Xei

I don't see why not (although they needn't sum to 180 in case that's what you were getting at). It depends on the local amount of curvature. If you're near a very large dense mass such as a black hole the angles could be vastly different.

Then it would not be a square.

Originally Posted by Xei

A square is a polygon with four sides of equal length and four equal angles. Here is what a hyperbolic square looks like:
http://upload.wikimedia.org/wikipedi...olic_plane.png

That makes no sense. It is not even a polygon, much less a square. A polygon is completely enclosed, unlike that figure, and the sides of a polygon are segments, which are straight.

See specifically what above?

My post to Licity who made the same (fair) point.

You said that in response to my question about whether opposite angles of a parallelogram are congruent. It is in post #61.

No, I said, "Space isn't Euclidian. That hypothesis is outdated by about a century now."

Which is exactly what I've been saying all along. Do you still not understand the distinction?

Euclidian geometry is only true within the axioms of Euclidian geometry; the same condition applies to all branches of mathematics. In reality, however, space is not Euclidian; it is warped by mass.

However that does not make Euclidian geometry redundant, for two important reasons:

- Euclidian geometry is an incredibly accurate approximation on Earth so engineers etc. don't even have to worry about it.
- The axioms of Euclidian geometry can be applied perfectly to other areas of mathematics, such as vectors, the Argand plane, etc.; other areas of maths which can then either be applied to the real world as models (with varying degrees of accuracy), or studied for their own sake (pure mathematics).

Earlier, you said they are nothing more than good approximations.

Nope I'm still repeating exactly the same things for the tenth time. Read more carefully I guess.

The second dimension is in the universe.

I can't glean any sense from this at all... if you take any 2D plane through the universe, it will be non-Euclidian, because it's a cross section of a 4D non-Euclidian space.

How is it true within itself if it is just something humans made up? Does it have any more reality than truth within The Wizard of Oz? What is the difference between math and science fiction, in terms of truth? The Death Star is real within Star Wars.

Well exactly. Humans 'made it up'. Maths is not a physical/objective entity. The axioms and resultant models are often extremely good approximations to the real world though so they do have practical uses.

That makes no sense. It is not even a polygon, much less a square. A polygon is completely enclosed, unlike that figure, and the sides of a polygon are segments, which are straight.

It's a representation of hyperbolic space, not Euclidian. Those lines are straight within hyperbolic axioms.

The little bits in the corners aren't part of the square.

I don't feel like hunting down another one of your quotes, but you did say that Euclidian geometry is taught and required everywhere because it provides for great "approximations". You also denied and denied that the opposite angles of a parallelogram are congruent and said the idea that they are is outdated. That was the point of your response I quoted, which was in response to my point that the opposite angles of a parallelogram are congruent. I don't think you were talking to yourself when you said it.

The second dimension is part of the multidimensional system of our universe, but that does not mean it starts acting like some greater dimension. The opposite angles of a parallelogram are congruent, no matter what you say happens in greater dimensional aspects.

If these systems of math are fiction ideas that are only true within themselves, why do they provide for such astoundingly good "approximations" concerning reality?