Mathematical Numbers are Theoretical; Not Proven

[[SomeGuy]]

Mathematical Numbers (numbers used in mathematical expressions; ie. 2x+9=8y) are impossible. For example, there are an infinite amount of numbers between 0 (nothing) and 1 (a whole). If 1 is impossible to reach, then it is only theoretical, not proven.

But, I have a problem with my statement: Real-Life Numbers can be used as Mathematical numbers, and mathematical operations can be used to manipulate Real-Life numbers.

Example of Real Life Numbers: There are two eggs on a table. It's pretty obvious that there are two eggs: Therefore, Real-Life numbers are already proven to exist. But mathematical numbers are only theoretical. You can't have 1.333... eggs.

So, most of us in maths class do problems like:
Graph 2x^2+4=y...blah blah blah. Why does that work? Is mathematics only a theory, because real maths numbers are only theory? Or, maybe we use real life numbers instead of mathematical numbers, but use them as a maths medium?

More on the impossibility of maths numbers: 0.3333... is possible. It exists. 1/3 = 0.333... It is a number made from a simple expression. Any rational number can be expressed this way. So, what about irrational numbers? They can be expressed in basic maths as well. So all numbers between 0-0.999... can be proven. But, you will never reach 1, or 2, or 3. There are an infinite amount of numbers between them. That's confusing for me; if the numbers exist, and are proven to exist, then why are they impossible when put together as a whole? 1+ is only theoretical, but it is made up of an infinite amount of numbers, and each one can be proven. I don't get it.


Summary:
1. Maths numbers are theoretical; in maths expressions, we use Real Life numbers. We don't use maths numbers, because the only maths numbers, the only non theoretical numbers, are between 0-0.999...
2. WTF why can we prove each number between 0-0.999... exists, but we can't prove that 1 exists?

My conclusion for 2: 0.999... MUST be equal to 1, in order to prove that maths numbers are not theoretical; and the same onward. Ie. 2.999...3.999... etc.


What are your thoughts?

[Inside This Fantasy]

Your wording is all very confusing to me. First of all, just because there are an infinite number of numbers between 0 and 1 doesn't mean you will never reach 1. There are an infinite number of numbers period. Every number exists. Saying you can't reach it because there are an infinite number of numbers in between just doesn't make sense. Also, you can add an infinite number of numbers and get a whole number.

The number 1 does exist. Its value is 1.0000000000000000000........ repeating forever. It is also equal to .999999... repeating forever. Source (http://en.wikipedia.org/wiki/0.999...)

Second, saying numbers are only theoretical doesn't make much sense. They aren't real, but that doesn't exactly make them theoretical. Within the realm of mathematics a number has a precise and defined value. It isn't theoretically 1, mathematically it is 1. The only reason you can't have 1.33333333333333333........ eggs is because the real world isnt infinitely divisible. In math, things are, so you could have that many eggs. Infinity doesn't exist in the real world, and I think that is what is confusing you.

I'm sorry if my response is vague or doesn't exactly address what you were saying. Your post kind of confused me and i couldn't tell exactly what you were trying to say.

[archdreamer]

I have no idea what the OP is trying to convey. Not only am I unable to decipher the relationships between the various points made, but I also cannot understand what most of the statements mean at all, even in isolation. It's almost as if the post is in another language. I'd be happy to reply, but I need a rephrase here.

[Photolysis]

If 1 is impossible to reach, then it is only theoretical, not proven.

The argument fails here, as 1 is not impossible to reach. Just because there are an infinite amount of numbers between 0 and 1, this does not stop you from reaching it.

In the same way that there are an infinite amount of ways you can orientate a shape. That doesn't stop you from rotating it 90 degrees.

[Kushna Mufeed]

Infinite gradient =/= infinite magnitude

[mysterious dreamer]

What you're saying is the same as the movement paradox. It's a nice theory, but it doesn't work.

[SnakeCharmer]

Example of Real Life Numbers: There are two eggs on a table. It's pretty obvious that there are two eggs: Therefore, Real-Life numbers are already proven to exist. But mathematical numbers are only theoretical. You can't have 1.333... eggs.


There is no such thing as "two eggs" in real life. That is just a mental model. No two eggs are identical. You just classify those two different objects into an artificial category.
Therefore eggs do not exist?

[Xei]

You're basically making no sense at all.

Please try and reiterate a little more succinctly.

[Artelis]

I think it has been determined that this argument is full of holes. Big ones.

[[SomeGuy]]

Ha, sorry if it sounds a bit confusing. I'll try writing it with more form and repost.

[ninja9578]

Math is the one science that is concrete and universal.

Gravity pulls mass together - theory
Genetic mutations and natural selection creates evolution - theory
God created everything - myth
1 * 10 = 10 - fact

Math is 100%, unlike every other science. It's not based on observable reality (which is variable), it's based on absolutes.

[Universal Mind]

Math is the one science that is concrete and universal.

Gravity pulls mass together - theory
Genetic mutations and natural selection creates evolution - theory
God created everything - myth
1 * 10 = 10 - fact

Math is 100%, unlike every other science. It's not based on observable reality (which is variable), it's based on absolutes.

Ninja, I agree with you. :bigteeth: My partner in philosophical agreement! :cheers:

[[SomeGuy]]

I love maths, I just was having some confusion about why whole numbers are possible. I see what my holes are, so maybe we should forget this thread...lol. I kind of answered my own question earlier today. No need to rewrite it.

[Xei]

The problem is that the absolutes are completely arbitrary, so mathematics isn't an entity at all.

I could say that 3 * 4 = 2, and 2 * 9 = 8, and 6 * 6 = 6, and so on (and then 1 * 10 = 0). It wouldn't be wrong, in fact it would form a completely coherent system. The thing is that whether it has any relevance to what I'm doing depends solely on what I'm trying to do.

Maths generalises reality, that's all.

[[SomeGuy]]

Maths is reality. Maths is the purest form of science and study you can find.

[Universal Mind]

The problem is that the absolutes are completely arbitrary, so mathematics isn't an entity at all.

I could say that 3 * 4 = 2, and 2 * 9 = 8, and 6 * 6 = 6, and so on (and then 1 * 10 = 0). It wouldn't be wrong, in fact it would form a completely coherent system.

It would be wrong, unless you changed the meanings of the symbols. Humans invent the symbols, but not the realities they represent.

[Xei]

Maths is reality. Maths is the purest form of science and study you can find.
Actually, there will always be statements that you can't prove with any set of mathematical rules.

Thus, truth is higher than mathematical proof.

Read about Godel. :)
It would be wrong, unless you changed the meanings of the symbols. Humans invent the symbols, but not the realities they represent.
You can change the symbols to mean anything you want, and the rules of manipulating the symbols. Each time you would get a new, different system, only true in itself. You say one system of maths is real; then what about other, contradictory systems?

[[SomeGuy]]

If you make + = subtraction, then 2 + 2 = 0, where the standard system is 2+2=4. There are an infinite amount of systems you can make, is what he's saying.

[grasshoppa]

Numbers are the ghosts and spirits of the rational world.

[Xei]

Think about this: you have some mathematical system, S, which you claim can prove whether any statement is true or not.

If this is so, it should be able to determine the truth of the following statment, G:

G: G cannot be proved by S.

If S says yes, I can prove G is true, then it has contradicted itself, and so S is actually fundamentally wrong.

The other option is that S cannot determine whether or not G is true, in which case it is limited in its 'monopoly' on truth. However, we can clearly see that G is true.

Therefore, there is no system of mathematics which can determine everything which is true.

[[SomeGuy]]

Numbers are the ghosts and spirits of the rational world.

I did your mom.

[grasshoppa]

;1059240']I did your mom.

Perhaps, perhaps not.

[[SomeGuy]]

Perhaps, perhaps not.

No, I'm sure I did.


Anyway, Maths:
2+2=4, correct? We all agree.


Moderators, please delete this topic, it was just a test.

[Artelis]

Numbers are the ghosts and spirits of the rational world.

lol, this bullshit in the middle of actual discussion. I agree with Universal. The way we communicate math is made by mankind, but the truths represented are universal constants.

[Inside This Fantasy]

The problem is that the absolutes are completely arbitrary, so mathematics isn't an entity at all.

I could say that 3 * 4 = 2, and 2 * 9 = 8, and 6 * 6 = 6, and so on (and then 1 * 10 = 0). It wouldn't be wrong, in fact it would form a completely coherent system. The thing is that whether it has any relevance to what I'm doing depends solely on what I'm trying to do.

Maths generalises reality, that's all.

You can change the symbols to mean anything you want, and the rules of manipulating the symbols. Each time you would get a new, different system, only true in itself. You say one system of maths is real; then what about other, contradictory systems?

I think you are misunderstanding the difference between the language of mathematics and the world of mathematics. It is similar to the observable universe and human language. There is (so far) only one observable universe. It exists without language. As humans, we create language to describe it. So far, we have created hundreds of coherent sets of languages (english, japanese, hungarian, etc...) that all make sense describing the world around us. Even though they use completely different sets of rules and sometimes symbols, they all describe one unchanging world. Calling a dog different words doesn't change the fact that its a dog.

The world of math is exactly the same. There is only one world of mathematics, and it exists without language. You can use different languages to describe it (decimal, hex, binary, etc...) but they only make sense if they are coherent throughout the whole world of mathematics. You say you can throw random symbols together and they would make a coherent system...it doesn't work like that. I can say "asjdh jkahsldkah jvhkjlhl" and tell you it makes sense in another language, but it doesn't mean it does. Just like you can spout gibberish in language, you can spout gibberish in math. Just like a dog is a dog no matter what the name, adding the idea of one to the idea of one will create the idea of two. Changing the symbols used to represent that just changes how it looks on paper.

[SnakeCharmer]

I can say "asjdh jkahsldkah jvhkjlhl" and tell you it makes sense in another language, but it doesn't mean it does. Just like you can spout gibberish in language, you can spout gibberish in math. Just like a dog is a dog no matter what the name, adding the idea of one to the idea of one will create the idea of two. Changing the symbols used to represent that just changes how it looks on paper.

He's actually saying that he can construct a language such that "asjdh jkahsldkah jvhkjlhl" has a meaning. That doesn't mean just substituting symbols so they represent different letters/sounds , but creating an entire system (grammar, syntax, etc)


Changing the symbols used to represent that just changes how it looks on paper.

That's the point, you don't just change the symbols, you can also change the relationships between them. You can do that and still get a consistent system.

[Inside This Fantasy]

He's actually saying that he can construct a language such that "asjdh jkahsldkah jvhkjlhl" has a meaning. That doesn't mean just substituting symbols so they represent different letters/sounds , but creating an entire system (grammar, syntax, etc)


That's the point, you don't just change the symbols, you can also change the relationships between them. You can do that and still get a consistent system.

If he does create an entire system from those words, then he has succesfully made a new language to describe the world of math. He hasn't changed the world of math. You can change the symbols and the relations to get a consistent system, but it just makes a new language to describe the same world. I was just trying to say you don't change the world itself when you use a different set of symbols to describe it.

[Universal Mind]

You can change the symbols to mean anything you want, and the rules of manipulating the symbols. Each time you would get a new, different system, only true in itself. You say one system of maths is real; then what about other, contradictory systems?

This many ** plus this many *** is this many *****. There is no way around that. (for example)

[Xei]

Indeed, from the axioms of addition, which are really just based upon common experience.

There are other modes of addition, for example addition modulo 4, in which that wouldn't necessarily be true.

And it all becomes a lot less clear when you ask still very simple questions like,

Does x^n + y^n ever = z^n, for n>2?

Obviously the answer is either yes or no, but whether or not you can prove it with mathematics is a different matter (in this case you actually can determine the answer is no, but there are other problems which you can't).

The world of math is exactly the same. There is only one world of mathematics, and it exists without language. You can use different languages to describe it (decimal, hex, binary, etc...) but they only make sense if they are coherent throughout the whole world of mathematics. You say you can throw random symbols together and they would make a coherent system...it doesn't work like that. I can say "asjdh jkahsldkah jvhkjlhl" and tell you it makes sense in another language, but it doesn't mean it does. Just like you can spout gibberish in language, you can spout gibberish in math. Just like a dog is a dog no matter what the name, adding the idea of one to the idea of one will create the idea of two. Changing the symbols used to represent that just changes how it looks on paper.
I'm afraid this isn't true.

Mathematics is essentially the 'language'. To speak of langauge describing the world of mathematics is not true; mathematics is the language which describes various aspects of our reality. Whatever rules you come up with for what you are allowed to do in mathematics, there will always be some statements which are true, but unprovable. Mathematics is not only a language; it is also a limited language.

I didn't say anything about new systems of maths being 'random'. There are still precisely defined rules; it's just that the rules are different.

For example, one consistent mathematical system is that of Euclidean geometry, which has six (I think) axioms, and then builds upon them. This system, as with all mathematical systems, will be limited. There will be some isolated geometric facts which are true but you can't prove with Euclid's rules. Another consistent system is hyperbolic geometry, which uses different, and mutually exclusive axioms, and will establish new facts within that system which are completely wrong in the other.

[Inside This Fantasy]

For example, one consistent mathematical system is that of Euclidean geometry, which has six (I think) axioms, and then builds upon them. This system, as with all mathematical systems, will be limited. There will be some isolated geometric facts which are true but you can't prove with Euclid's rules. Another consistent system is hyperbolic geometry, which uses different, and mutually exclusive axioms, and will establish new facts within that system which are completely wrong in the other.

Don't these systems of geometry fit what I said? Different languages describing the same world?

[Xei]

But the systems describe mutually contradictory facts..?

How can any world exist if it negates its own existence? It simply makes no sense.

[Inside This Fantasy]

But the systems describe mutually contradictory facts..?

How can any world exist if it negates its own existence? It simply makes no sense.

The languages are imperfect ways of describing reality. If they are in fact describing the same aspect of reality and make contradictory claims, wouldn't that be a fault of the language? Even if the systems work within themselves, one would have to be wrong when it comes to reality, because the universe doesn't change depending on whether or not you are using euclidean or hyperbolic geometry. They are just different ways of describing what is.

[Xei]

I don't think you're being very consistent. You were saying that there is a mathematical reality. Now you're saying that physical reality is real and mathematics just describes various models which may apply to it, which is what I was saying...

Mathematics is a just series of logical systems in which you start with a small number of facts and rules which you can apply to them, and build up from there. Whether or not they have any bearing upon reality depends on how closely your facts and rules model those of reality. Some will appear quite similar, others will have no relevance at all, but all will only be true in themselves, not describing some single coherent otherworld, and all of them will be limited in the fact that they will be unable to determine various truths in that system.

[grasshoppa]

lol, this bullshit in the middle of actual discussion. I agree with Universal. The way we communicate math is made by mankind, but the truths represented are universal constants.

What else am I supposed to type? This whole forum is Bullshit.

[Universal Mind]

Indeed, from the axioms of addition, which are really just based upon common experience.


They are factual whether they are experienced or not. This many *** ** and this many ***** are the same. A thing is the thing that it is. That is pure logic. It might even be the most fundamental principle of existence.

[[SomeGuy]]

What else am I supposed to type? This whole forum is Bullshit.
Oh ho ho, don't like it? Get the fuck off and quit trolling to screw it up for everyone that does like this forum. What are you supposed to type? Something pertaining to the the topic at hand. That's how forums work.

I agree with ninja: * added to * gives you **.

[Universal Mind]

;1060180']Oh ho ho, don't like it? Get the fuck off and quit trolling to screw it up for everyone that does like this forum. What are you supposed to type? Something pertaining to the the topic at hand. That's how forums work.

I agree with ninja: * added to * gives you **.

Cool. But... you didn't really just call me Ninja, did you? Not Ninja! :makeitstop:

[[SomeGuy]]

Oh crap...wrong guy! I thought it was him whilst I was writing it. I needed to add some maths to it so I wasn't being a troll...while I was yelling at someone for trolling.

[Xei]

They are factual whether they are experienced or not. This many *** ** and this many ***** are the same. A thing is the thing that it is. That is pure logic. It might even be the most fundamental principle of existence.
It is indeed a fact based on reality, but it needn't be true or even have meaning in a mathematical system.

Reality is real; mathematics as an entity is not. There is no such thing as a 'mathematical number' as opposed to a 'real number', there are just numbers. Numbers of real, physical objects or properties of objects.

[Universal Mind]

;1060226']Oh crap...wrong guy! I thought it was him whilst I was writing it. I needed to add some maths to it so I wasn't being a troll...while I was yelling at someone for trolling.

:lol: I was totally joking. I was mainly screwing with Ninja, not you.

It is indeed a fact based on reality, but it needn't be true or even have meaning in a mathematical system.


Mathematical systems are all about reality, except for imaginary numbers and such, which take imaginary concepts and apply reality's rules to them.

Reality is real; mathematics as an entity is not. There is no such thing as a 'mathematical number' as opposed to a 'real number', there are just numbers. Numbers of real, physical objects or properties of objects.

I am a little confused on what you are saying at the beginning with that. Numbers are reality, and mathematics is about the nature of numbers.

Physical objects are not the only things that exist and that numbers apply to. They also apply to thoughts, metaphysical realities, and hypothetical situations.

[Xei]

Mathematical systems are all about reality, except for imaginary numbers and such, which take imaginary concepts and apply reality's rules to them.
Imaginary numbers are rotations of 90 degrees. Rotations of 90 degrees are real. :l

What imaginary concepts are you referring to? I'd say that multiplication of a negative by a negative has no striking physical counterpart.
I am a little confused on what you are saying at the beginning with that. Numbers are reality, and mathematics is about the nature of numbers.
Not really, numbers are idealised concepts. You will never be 100% accurate or certain about any physical number.

Mathematics can be any language with any set of particular rules. What I'm saying is that there is no 'world of mathematics', because different systems of mathematics contradict each other. It isn't really any different from any other language, for example, English; there is no 'world of English' in some other mystical plane. English is just a human invention, and resides only within human minds.

Both attempt to describe a 'world of truths' - that is to say, reality - but both are flawed and limited.

[grasshoppa]

;1060180']Oh ho ho, don't like it? Get the fuck off and quit trolling to screw it up for everyone that does like this forum. What are you supposed to type? Something pertaining to the the topic at hand. That's how forums work.

I agree with ninja: * added to * gives you **.

Ya, I'm the troll...Your the one saying you did my mom. And my initial post was completely serious.

[[SomeGuy]]

The thread was kinda dead when I said that. I wasn,t complaining about how terrible the forums are in the middle of a math discussion. It was quiet until after that.

Anyway, the concept of numbers was created to describe physical things. Math and physical objects are undeniably tied into each other.

[grasshoppa]

no they are not tied to each other, we create the link in our minds. Numbers are a system and product of rational thinking. It helps us systematically categorize objects

[[SomeGuy]]

They are related to each other because of the fact thatwe created a system of numbers to describe physical objects. That seems like an undeniable link. While the link resides in our minds, there is a link.

[grasshoppa]

Don't forget that we drew the lines on the map

[Artelis]

Don't forget that we drew the lines on the map

doesn't mean the world doesn't exist.

[grasshoppa]

doesn't mean the world doesn't exist.

I was referring to the boundaries we put up for ourselves mentally, socially, physically etc..

[Universal Mind]

Imaginary numbers are rotations of 90 degrees. Rotations of 90 degrees are real. :l

What imaginary concepts are you referring to? I'd say that multiplication of a negative by a negative has no striking physical counterpart.

Not really, numbers are idealised concepts. You will never be 100% accurate or certain about any physical number.

Mathematics can be any language with any set of particular rules. What I'm saying is that there is no 'world of mathematics', because different systems of mathematics contradict each other. It isn't really any different from any other language, for example, English; there is no 'world of English' in some other mystical plane. English is just a human invention, and resides only within human minds.

Both attempt to describe a 'world of truths' - that is to say, reality - but both are flawed and limited.

Then why bother making pi = 3.14...? Why not just make it 3, or 1... or 0? Wouldn't that make a ton of computations much easier? Why not change the quadratic formula to a + b + c = 3? Why not make the slope formula m = x1 + x2 + y1 + y2? Why not make the distance formula d = 1?

[Artelis]

Then why bother making pi = 3.14...? Why not just make it 3, or 1... or 0? Wouldn't that make a ton of computations much easier? Why not change the quadratic formula to a + b + c = 3? Why not make the slope formula m = x1 + x2 + y1 + y2? Why not make the distance formula d = 1?

Because they attempt to to describe a 'world of truths.' These formulas don't even attempt to make sense.

But I'm actually on your side. Just pointing out holes in your argument.

[Xei]

Then why bother making pi = 3.14...? Why not just make it 3, or 1... or 0? Wouldn't that make a ton of computations much easier? Why not change the quadratic formula to a + b + c = 3? Why not make the slope formula m = x1 + x2 + y1 + y2? Why not make the distance formula d = 1?
If we called a dog a fish would it be able to breathe underwater?

pi, for example, has the value it does because it is a consequence of the axioms of Euclidian geometry (you can estimate pi with a series of fractions); it is defined by and only has meaning within that set of axioms. Euclidian geometry is almost perfect when dealing with things in human experience, so we can reliably say that the circumference of any physical circle will be pretty much pi times the diameter.

We don't do any of those things you mentioned because we prove that the real results are true (using the rules of our system). I can't imagine you've never proved anything before... all of those except the pi one are completely trivial.

[Universal Mind]

Because they attempt to to describe a 'world of truths.' These formulas don't even attempt to make sense.

But I'm actually on your side. Just pointing out holes in your argument.

I don't understand the supposed holes. The real formulas do more than attempt to describe a world of truths. They pull it off successfully.

If we called a dog a fish would it be able to breathe underwater?

pi, for example, has the value it does because it is a consequence of the axioms of Euclidian geometry (you can estimate pi with a series of fractions); it is defined by and only has meaning within that set of axioms. Euclidian geometry is almost perfect when dealing with things in human experience, so we can reliably say that the circumference of any physical circle will be pretty much pi times the diameter.

We don't do any of those things you mentioned because we prove that the real results are true (using the rules of our system). I can't imagine you've never proved anything before... all of those except the pi one are completely trivial.

The formulas are what they are because they describe reality accurately, and the phony ones I mentioned are flat out false. The system we have to work in is not our invention; the system is reality. We could not possibly rearrange our system to where pi = this many ***. Of course we can change the language, but we cannot change the realities that are represented by our language.

What do you mean you can't imagine I've never proved anything before? How are the other formulas trivial? Please explain.

[Xei]

What can I say? You're wrong, that's all. I've already clearly explained why. It has even been proved with maths that this whole single true mathematical system idea is wrong, and you don't accept it, so there's really nothing I can say.
What do you mean you can't imagine I've never proved anything before? How are the other formulas trivial? Please explain.
Just that the answer to your question - the 'reason' those things are true in algebra etc. - is that you can prove them to be true using the rules of algebra.

Solve ax2 + bx + c = 0 for example and you get the quadratic formula. It wasn't found heuristically or something.

[Universal Mind]

What can I say? You're wrong, that's all. I've already clearly explained why. It has even been proved with maths that this whole single true mathematical system idea is wrong, and you don't accept it, so there's really nothing I can say.


You are wrong.

Just that the answer to your question - the 'reason' those things are true in algebra etc. - is that you can prove them to be true using the rules of algebra.


The rules of algebra are rules of reality. 2 apples + 2 apples = 4 apples. By the same completely realistic reasoning, 2x + 2x = 4x. In other words, 2 x's + 2 x's = 4 x's. 2 of anything plus 2 of that same thing equals 4 of that thing. That was true before there were humans. It is not some crazy idea somebody had for the purpose of writing a fiction story. There are completely legitimate reasons we use the systems we use. We use them because they are factual and logical. It is not a mere art project.

Solve ax2 + bx + c = 0 for example and you get the quadratic formula. It wasn't found heuristically or something.

The quadratic formula was discovered, not invented. The value of x in your equation is

-b +/- square root of (b squared - 4ac)
2a

That is in fact the value of x, not something some person decided would be neat to merely call the value of x. If the quadratic formula were just some crazy thing somebody decided to invent and do strange things with, it would have been much easier to make it x = a + b + c. There was no choice on the matter. The quadratic formula is what it is, and there is nothing we can do about it even though we can invent a new language.

[Xei]

You are wrong.
Just read about Godel. If you don't understand it, that's your failing.
The rules of algebra are rules of reality. 2 apples + 2 apples = 4 apples. By the same completely realistic reasoning, 2x + 2x = 4x. In other words, 2 x's + 2 x's = 4 x's. 2 of anything plus 2 of that same thing equals 4 of that thing. That was true before there were humans. It is not some crazy idea somebody had for the purpose of writing a fiction story. There are completely legitimate reasons we use the systems we use. We use them because they are factual and logical. It is not a mere art project.
Then what about the true algebraic facts that you can't work out with algabraic rules?

The 'obvious' rules of arithmetic often don't apply on the quantum scale, for example. They're only obvious because the only things we've ever experienced are those in immediate experience.
The quadratic formula was discovered, not invented. The value of x in your equation is

-b +/- square root of (b squared - 4ac)
2a

That is in fact the value of x, not something some person decided would be neat to merely call the value of x. If the quadratic formula were just some crazy thing somebody decided to invent and do strange things with, it would have been much easier to make it x = a + b + c. There was no choice on the matter. The quadratic formula is what it is, and there is nothing we can do about it even though we can invent a new language.
That is pretty much exactly what I said so I don't really know what you're going on about. Like I just said, you just solve ax2 + bx + c. You do understand how to do that, right..?

It was something which was 'discovered' in a system with particular rules - rules which were invented. The rules are those of Peano arithmetic, which include,

1. ∀x, y, z ∈ N. (x + y) + z = x + (y + z), i.e., addition is associative.
2. ∀x, y ∈ N. x + y = y + x, i.e., addition is commutative.
3. ∀x, y, z ∈ N. (x · y) · z = x · (y · z), i.e., multiplication is associative.
4. ∀x, y ∈ N. x · y = y · x, i.e., multiplication is commutative.
5. ∀x, y, z ∈ N. x · (y + z) = (x · y) + (x · z), i.e., the distributive law.
6. ∀x ∈ N. x + 0 = x ∧ x · 0 = 0, i.e., zero is the identity element for addition
7. ∀x ∈ N. x · 1 = x, i.e., one is the identity element for multiplication.
8. ∀x, y, z ∈ N. x < y ∧ y < z ⊃ x < z, i.e., the '<' operator is transitive.
9. ∀x ∈ N. ¬ (x < x), i.e., the '<' operator is not reflexive.
10. ∀x, y ∈ N. x < y ∨ x = y ∨ x > y.
11. ∀x, y, z ∈ N. x < y ⊃ x + z < y + z.
12. ∀x, y, z ∈ N. 0 < z ∧ x < y ⊃ x · z < y · z.
13. ∀x, y ∈ N. x < y ⊃ ∃z ∈ N. x + z = y.
14. 0 < 1 ∧ ∀x ∈ N. x > 0 ⊃ x ≥ 1..
15. ∀x ∈ N. x ≥ 0.

There are plenty of systems which use contradictory or separate axioms.

And as I keep trying to communicate to you, there are many true facts about arithmetic which can't be proved in the above system. So your 'obvious facts' are really completely arbitrary and limited.

[Universal Mind]

Just read about Godel. If you don't understand it, that's your failing.


Appealilng to supposed authority is not going to cut it.

Then what about the true algebraic facts that you can't work out with algabraic rules?


What about them? I didn't say all mathematical facts can be proven. It is not an issue.

The 'obvious' rules of arithmetic often don't apply on the quantum scale, for example. They're only obvious because the only things we've ever experienced are those in immediate experience.


They definitely apply in the reality we are now in. If what you have said is true, then maybe you should redesign our reality's math to where it does apply on the quantum scale. How would that work?

That is pretty much exactly what I said so I don't really know what you're going on about. Like I just said, you just solve ax2 + bx + c. You do understand how to do that, right..?


Yes, and the quadratic formula works every time. It is a universal truth. What is your point?

It was something which was 'discovered' in a system with particular rules - rules which were invented. The rules are those of Peano arithmetic, which include,

1. ∀x, y, z ∈ N. (x + y) + z = x + (y + z), i.e., addition is associative.
2. ∀x, y ∈ N. x + y = y + x, i.e., addition is commutative.
3. ∀x, y, z ∈ N. (x · y) · z = x · (y · z), i.e., multiplication is associative.
4. ∀x, y ∈ N. x · y = y · x, i.e., multiplication is commutative.
5. ∀x, y, z ∈ N. x · (y + z) = (x · y) + (x · z), i.e., the distributive law.
6. ∀x ∈ N. x + 0 = x ∧ x · 0 = 0, i.e., zero is the identity element for addition
7. ∀x ∈ N. x · 1 = x, i.e., one is the identity element for multiplication.
8. ∀x, y, z ∈ N. x < y ∧ y < z ⊃ x < z, i.e., the '<' operator is transitive.
9. ∀x ∈ N. ¬ (x < x), i.e., the '<' operator is not reflexive.
10. ∀x, y ∈ N. x < y ∨ x = y ∨ x > y.
11. ∀x, y, z ∈ N. x < y ⊃ x + z < y + z.
12. ∀x, y, z ∈ N. 0 < z ∧ x < y ⊃ x · z < y · z.
13. ∀x, y ∈ N. x < y ⊃ ∃z ∈ N. x + z = y.
14. 0 < 1 ∧ ∀x ∈ N. x > 0 ⊃ x ≥ 1..
15. ∀x ∈ N. x ≥ 0.


All discovered. If you disagree, make 38 the identity element for addition and 714 the identity element for multiplication. I would love to see that.

There are plenty of systems which use contradictory or separate axioms.

And as I keep trying to communicate to you, there are many true facts about arithmetic which can't be proved in the above system. So your 'obvious facts' are really completely arbitrary and limited.

Then create those rules I brought up. Try doing that and telling me you have invented a system of reality and not fiction.

[Xei]

All discovered. If you disagree, make 38 the identity element for addition and 714 the identity element for multiplication. I would love to see that.
Uh.
Then create those rules I brought up. Try doing that and telling me you have invented a system of reality and not fiction.
Uh?
Yes, and the quadratic formula works every time. It is a universal truth. What is your point?
Perhaps the bit I posted below it.

It's a universal truth within Peano arithmetic. Which is a created system.
They definitely apply in the reality we are now in. If what you have said is true, then maybe you should redesign our reality's math to where it does apply on the quantum scale. How would that work?
Okay, put it like this: Euclidian geometry and hyperbolic geometry are two different mathematical systems with different sets of axioms. Which one of these is real?
Appealilng to supposed authority is not going to cut it.
Oh Lord...

Look, why are you here? If you care about this issue, why won't you even read into it? There's absolutely no way you can get into a discussion about mathematical philosophy without first knowing about Goedel for goodness sakes.

You cannot seriously be saying that appealing to a mathematical proof is a logical fallacy...

[Universal Mind]

Oh Lord...

Look, why are you here? If you care about this issue, why won't you even read into it? There's absolutely no way you can get into a discussion about mathematical philosophy without first knowing about Goedel for goodness sakes.

You cannot seriously be saying that appealing to a mathematical proof is a logical fallacy...

My point was that saying, "Well, this other guy says otherwise," does not explain to me how this many ** ** is not this many ****. I can read the book in time, but we will probably be pretty far away from this discussion by then.

Okay, put it like this: Euclidian geometry and hyperbolic geometry are two different mathematical systems with different sets of axioms. Which one of these is real?


I have only studied Euclidean geometry. It says the opposite angles of a parallelogram are congruent and other such undeniable facts. Does hyperbolic geometry say otherwise?

Uh.

Uh?


You are backing down to my challenge? Make it where 714 times any number equals the number. Show me how any number plus 38 equals the number. Demonstrate this for me and tell me how real it is. My eyes are open.

[Xei]

My point was that saying, "Well, this other guy says otherwise," does not explain to me how this many ** ** is not this many ****. I can read the book in time, but we will probably be pretty far away from this discussion by then.
It's not a book, it's a proof... it's certainly one of the more famous proofs in history, and has an extremely important place in the history of mathematics and philosophy in the 20th century. Just Google it, the very informal version is quite easy to grasp and I've posted it in this thread (it's also explained as a part of many famous philosophical books such as Goedel, Escher, Bach and The Emperor's New Mind).
I have only studied Euclidean geometry. It says the opposite angles of a parallelogram are congruent and other such undeniable facts. Does hyperbolic geometry say otherwise?
Yep, in hyperbolic geometry, two lines at angles to each other don't have to intersect. The interior angles of a triangle add up to less than 180.

It's ironic that you refer to those rules as 'undeniable', because actually, in this universe at least, they're wrong. This is just what I mean by basing axioms upon nothing more than common experience.
You are backing down to my challenge? Make it where 714 times any number equals the number. Show me how any number plus 38 equals the number. Demonstrate this for me and tell me how real it is. My eyes are open.
You could do that if you wanted, but the resultant system would evidently be pretty useless.

However, changing other axioms needn't have a degenerative effect. For example, you can do away with Euclid's parallel axiom, and you still get a consistent system.

[Universal Mind]

It's not a book, it's a proof... it's certainly one of the more famous proofs in history, and has an extremely important place in the history of mathematics and philosophy in the 20th century. Just Google it, the very informal version is quite easy to grasp and I've posted it in this thread (it's also explained as a part of many famous philosophical books such as Goedel, Escher, Bach and The Emperor's New Mind).


Like you said, it's in books, like Godel, Escher, Bach. I am familiar with Godel and the fact that he has a "proof". I just don't think merely mentioning its existence suffices for a counterargument.

Yep, in hyperbolic geometry, two lines at angles to each other don't have to intersect. The interior angles of a triangle add up to less than 180.


Is it because of different language or because placing angles congruent to the angles of a triangle next to each other would not form a line? They in fact form a line. Any system that says otherwise is fiction.

It's ironic that you refer to those rules as 'undeniable', because actually, in this universe at least, they're wrong. This is just what I mean by basing axioms upon nothing more than common experience.


Opposite angles in a parallelogram are not congruent?

You could do that if you wanted, but the resultant system would evidently be pretty useless.

However, changing other axioms needn't have a degenerative effect. For example, you can do away with Euclid's parallel axiom, and you still get a consistent system.

Are you going to take me up on the challenge or not? It gets to the very heart of our discussion.

Why would the resulting system be useless? Is it because it would be out of line with reality?

[Xei]

It would be useless because every number would equal every other number, so you wouldn't be able to derive any meaningful results. Most of the time we only care about mathematical systems which model reality, like that of arithmetic for macroscopic experiences, although sometimes they are studied for their own sakes.
Opposite angles in a parallelogram are not congruent?
Nope. Space isn't Euclidian. That hypothesis is outdated by about a century now.

[Universal Mind]

It would be useless because every number would equal every other number, so you wouldn't be able to derive any meaningful results. Most of the time we only care about mathematical systems which model reality, like that of arithmetic for macroscopic experiences, although sometimes they are studied for their own sakes.


So it has to jive with reality to at least some extent? It has to jive with reality to the full extent.

Nope. Space isn't Euclidian. That hypothesis is outdated by about a century now.

Outdated? WTF???? It wasn't that long ago that I was teaching it. I still help high school students with it sometimes. It is still taught in practically every high school in the U.S. and probably the world.

Parallelograms are two-dimensional while space is three-dimensional, but the second dimension does exist. Opposite angles of a parallelogram are congruent. I never dreamed I would be debating somebody on that some day.

http://tbn2.google.com/images?q=tbn:1Bx-iL3JiZmfxM:http://giraffian.com/pictionary-files/p/parallelogram.png ([URL]http://images.google.com/imgres?imgurl=http://giraffian.com/pictionary-files/p/parallelogram.png&imgrefurl=http://giraffian.com/dictionary/p/parallelogram&usg=__x_f5Z0h3tlHsdbpw-W6JlFGkH90=&h=250&w=250&sz=5&hl=en&start=2&um=1&tbnid=1Bx-iL3JiZmfxM:&tbnh=111&tbnw=111&prev=/images%3Fq%3Dparallelogram%26hl%3Den%26rlz%3D1T4DK US_enUS206US213%26sa%3DN%26um%3D1)

A square is a parallelogram. I challenge you to do your absolute best and argue that the opposite angles of the square below do not have the same measure. Try to even convince me that all four angles of a square do not have the same measure. I really want to understand this.

http://upload.wikimedia.org/wikipedia/commons/archive/3/32/20080203213602!Square_(geometry).png

[[SomeGuy]]

Lol, I really want to see this argument. It seems like common sense...

[Xei]

General relativity? :l

It has been empirically proven that space is not Euclidian, ie. in the real world, the opposite angles of a parallelogram are almost never equal.

[Universal Mind]

General relativity? :l

It has been empirically proven that space is not Euclidian, ie. in the real world, the opposite angles of a parallelogram are almost never equal.

Please explain. I want you to tell me how the square and the other parallelogram I posted do not have congruent opposite angles. I am not asking for a reference to something else. I am asking you to tell me in your own words how in the world those angles do not have equal measures.

Like I said, parallelograms are two-dimensional. Also, Euclidian geometry is a required subject in practically every high school in my country and probably yours. Engineers and astronauts depend on the truth of it.

[Xei]

Prove that they're equal.

You'll find that you reduce the problem to a few starting facts which you cannot actually prove - you just have to say that they're 'obvious' (these are Euclid's postulates). One of these is the 'obvious' truth that two lines must be parallel if you don't want them to ever meet. This is in fact just wrong. Reality doesn't work like this. It just approximates to it on human scales.

As a result, everything that you prove with Euclid's axioms is also not necessarily true about reality. This includes the idea that the interior angles of a triangle add to 180 (they don't) and that opposite angles of a parallelogram are equal.

You definitely need to read about Einstein's General Theory of Relativity if you want to understand the basics of this. The formal theory is extremely complicated and requires several years of study to understand, but it is so important that it has been covered extensively in layman's terms. Just Google it. As I said, it has been empirically verified many times.

Ironically astronauts depend on Euclidian geometry not being correct. If we tried to use it, satellites would not stay in orbit. It's wrong.

[Universal Mind]

Prove that they're equal.


Take two parallel lines and make them perpendicular to two other parallel lines. All four interior angles created will be right angles. Then lean the first two parallel lines the same amount. The opposite angles will change exactly the same amount. That is because exactly the same thing is being done to them.

Now you... (This is about the fourth request.)

You'll find that you reduce the problem to a few starting facts which you cannot actually prove - you just have to say that they're 'obvious' (these are Euclid's postulates). One of these is the 'obvious' truth that two lines must be parallel if you don't want them to ever meet. This is in fact just wrong. Reality doesn't work like this. It just approximates to it on human scales.


Skew lines never meet and are not parallel. That is because they are not on the same plane. In one plane, only parallel lines never intersect. If you can explain otherwise, it will be the most interesting explanation I have requested.

As a result, everything that you prove with Euclid's axioms is also not necessarily true about reality. This includes the idea that the interior angles of a triangle add to 180 (they don't) and that opposite angles of a parallelogram are equal.


I don't see how in the world they would not. Also, please tell me why pretty much every high school and engineering and architecture school in the world teaches that they do. Can you tell me that?

You definitely need to read about Einstein's General Theory of Relativity if you want to understand the basics of this. The formal theory is extremely complicated and requires several years of study to understand, but it is so important that it has been covered extensively in layman's terms. Just Google it. As I said, it has been empirically verified many times.


I have read about it. When you get the 4th dimension, time, into equations, bizarre things happen, but they do not contradict the postulates and theorems of two-dimensional geometry.

Ironically astronauts depend on Euclidian geometry not being correct. If we tried to use it, satellites would not stay in orbit. It's wrong.

That is not true. The Pythagorean Theorem is used to determine vectors, for example.

Now please explain what I have requested, especially why Euclidian geometry is a requirement at practically every high school in the world and why architects and engineers have to know it and use it.

[[SomeGuy]]

It's simple maths. They wouldn't teach it and make it required if it was false.

|----------------|
| a b |
| |
|c d|
|----------------|
a=90
b=90
c=90
d=x

a+b+c+d=360 (Proof of 360, try it with a protractor.)
90+90+90+x=360
270+x=360
-270 -270
=========
x= 90

[Xei]

Take two parallel lines and make them perpendicular to two other parallel lines. All four interior angles created will be right angles. Then lean the first two parallel lines the same amount. The opposite angles will change exactly the same amount. That is because exactly the same thing is being done to them.

Now you... (This is about the fourth request.)
Sorry, that's not a valid geometric proof. But you've illustrated my point all the same; you had to resort to 'obvious' facts which you could only justify by your experience.
Skew lines never meet and are not parallel. That is because they are not on the same plane. In one plane, only parallel lines never intersect. If you can explain otherwise, it will be the most interesting explanation I have requested.
I'm not talking about skew lines, I'm talking about lines in a plane in hyperbolic geometry, of which there are many non-parallel lines which do not intersect.
I don't see how in the world they would not. Also, please tell me why pretty much every high school and engineering and architecture school in the world teaches that they do. Can you tell me that?
Once again; it's an extremely good approximation on human scales, but on larger scales, it is completely wrong.

Architects don't build things larger than solar systems, remember?
I have read about it. When you get the 4th dimension, time, into equations, bizarre things happen, but they do not contradict the postulates and theorems of two-dimensional geometry.
It sounds like you're getting confused with special relativity.

And yes, for the nth time, reality does contradict those postulates (there's no need to restrict them to two dimensions though). Seeing as you've repeatedly refused to look up any of this (it's extremely easy to find out about and is covered in the introduction of both the general relativity and Euclidian geometry pages on Wikipedia), here's independent verification:
For over two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute sense. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. It also is no longer taken for granted that Euclidean geometry describes physical space. An implication of Einstein's theory of general relativity is that Euclidean geometry is a good approximation to the properties of physical space only if the gravitational field is not too strong.
That is not true.
Wrong.

[Universal Mind]

Sorry, that's not a valid geometric proof. But you've illustrated my point all the same; you had to resort to 'obvious' facts which you could only justify by your experience.


I resorted to logic. You reject the geometric proofs, so I gave you another way to look at it. Can you counter the logic I used?

I'm not talking about skew lines, I'm talking about lines in a plane in hyperbolic geometry, of which there are many non-parallel lines which do not intersect.


Explain. (!!!!!!!!!!)

Once again; it's an extremely good approximation on human scales, but on larger scales, it is completely wrong.


How?

It sounds like you're getting confused with special relativity.


I was just taking a guess at what you could possibly be talking about since you are not telling me.

And yes, for the nth time, reality does contradict those postulates (there's no need to restrict them to two dimensions though). Seeing as you've repeatedly refused to look up any of this (it's extremely easy to find out about and is covered in the introduction of both the general relativity and Euclidian geometry pages on Wikipedia), here's independent verification:


That is not an explanation. It is just one more assertion. That is all you are giving me.

Wrong.

Wrong. Are you saying NASA does not use the Pythagorean Theorem to determine vectors? What do you think they use?

Why is Euclidian geometry taught as FACT in practically every high school, school of architecture, and school of engineering in the world????????????????

[Xei]

Once again; it's an extremely good approximation on human scales, but on larger scales, it is completely wrong.
For fuck's sake...

[Universal Mind]

For fuck's sake...

They are not taught as approximations. They are taught as FACTS. Why??????????????????????????????????????????????? ?????????????????????

Why do they work so well as "approximations"?

Explain your assertions when you think you are ready.

[Xei]

An implication of Einstein's theory of general relativity is that Euclidean geometry is a good approximation to the properties of physical space only if the gravitational field is not too strong..

[Universal Mind]

.

That is an appeal to an appeal to authority, not an explanation. You are playing some major dodgeball in this discussion.

[Xei]

The formal theory is extremely complicated and requires several years of study to understand, but it is so important that it has been covered extensively in layman's terms. Just Google it. As I said, it has been empirically verified many times..

[Universal Mind]

.

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.

[Xei]

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.).

[Universal Mind]

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?

[Xei]

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 (http://en.wikipedia.org/wiki/Tests_of_general_relativity).

http://upload.wikimedia.org/wikipedia/commons/5/5e/Euclidian_and_non_euclidian_geometry.png

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.

[Universal Mind]

I've explicitly answered everything there already...


False.

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.

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.

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?

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 (http://en.wikipedia.org/wiki/Tests_of_general_relativity).

http://upload.wikimedia.org/wikipedia/commons/5/5e/Euclidian_and_non_euclidian_geometry.png



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

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.

[[SomeGuy]]

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).


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 !=.

[Xei]

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_of_general_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.

[Universal Mind]

...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. -_-


Again, I was not asking about satellites. You brought them up, and I kept to what I was talking about, which is motion in a straight line with an outside force acting. However, the specific vectors you brought up do concern satellites, and guess what is used to determine those vectors. The Pythagorean Theorem.

http://books.google.com/books?id=U9lkAkTdAosC&pg=PA125&lpg=PA125&dq=pythagorean+theorem+vectors+satellites&source=bl&ots=mh_Ss10up9&sig=rjbmxDhabB6GN2v3nT_W3At1tYs&hl=en&ei=KmHYSb6ZI4TiyAWAnqjpDg&sa=X&oi=book_result&ct=result&resnum=2

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.


Since you are accepting a total deviation from the standard definition of the word "parellel", tell me the specific definition you are going by.

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_of_general_relativity


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

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.

If the rules turned out to be wrong, which they did not, they would have been revised. Were they revised because of general relativity? No. 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.

Since you say the rules are only good for approximations, what kinds of approximations are you talking about? If the opposite angles of a parallelogram are not congruent, how close to congruent are they? For example, if on angle of a parallelogram is 70 degrees, what is the opposite angle's difference? A trillionth of a degree? How much?

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.


http://images.encarta.msn.com/xrefmedia/AEncMed%5CTargets%5CIllus%5CIFG%5C000f26be.gif

That is a square, which is a type of parallelogram. All four of its interior angles are right angles. But you disagree, right? So, what are the measures of the angles, according to you and your mathematical astrology?

[Xei]

Since you are accepting a total deviation from the standard definition of the word "parellel", tell me the specific definition you are going by.
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.
If the rules turned out to be wrong, which they did not, they would have been revised. Were they revised because of general relativity? No. 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.

Since you say the rules are only good for approximations, what kinds of approximations are you talking about? If the opposite angles of a parallelogram are not congruent, how close to congruent are they? For example, if on angle of a parallelogram is 70 degrees, what is the opposite angle's difference? A trillionth of a degree? How much?
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.

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.

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.
That is a square, which is a type of parallelogram. All four of its interior angles are right angles. But you disagree, right? So, what are the measures of the angles, according to you and your mathematical astrology?
An angle is a measure of the amount of rotation between two lines. In hyperbolic geometry, those angles would all be > pi/2.

[wendylove]

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.

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.

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.

[Licity]

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.

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

[Xei]

Indeed; we're talking about 2D Euclidian / hyperbolic geometries at the mo though, in which case the condition is sufficient and necessary.

For higher dimensions I'd imagine it would be either very hard to generalise mathematically, or impossible to put in non-technical language. In 3D Euclid two lines are parallel if their direction vectors are multitples of one another, but already it's pretty hard to put this in English, and god knows what the condition for parallel lines in 3D hyperbolic space is.

[Universal Mind]

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.

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?

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?

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?

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.

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.

Actually they did. Look up the work of Hilbert.

Look up the postulates and theorems of required geometry.

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

:bigteeth:

[Xei]

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/wikipedia/commons/1/16/Square_on_hyperbolic_plane.png

[Universal Mind]

See above.


See specifically what above?

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

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.

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.

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.

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.

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.

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/wikipedia/commons/1/16/Square_on_hyperbolic_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.

[Xei]

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.

[Universal Mind]

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?

[RedfishBluefish]

Good "approximations" on small scales. The curvature of space by gravity becomes increasingly irrelevant as the scale reduces.

If I draw a line on a sphere, then draw a line at 90° to it, and another one 90° to that line, are the first and last lines parallel? Do they intersect? The answers are not the same as in Euclidean geometry (on a flat plane). This is because a sphere is curved. But if you take a small enough piece out of a large sphere, it looks completely flat as far as a human eye is able to see, and Euclidean axioms hold as far as we can see by experiment. Now generalise that to 3D (or more dimensions).

[Xei]

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?
In reality, the opposite angles of a parallelogram are not equal, because reality is not Euclidian. I have never denied that they are equal within Euclidian geometry - indeed I asked you to prove it - the point is that the physical universe is not actually Euclidian.

There aren't really any objective 2 dimensions 'out there', but if you were to take a 2D cross section of the universe, it would also not be Euclidian. Standard hyperbolic geometry is 2D. It is really just a human assumption that Euclidian geometry and not some other geometry should apply well to larger scales than the ones we percieve; a completely understandable one, but ultimately a flawed one. It is just like how it was assumed for a few hundred years that Newtonian mechanics was objectively true; turns out that at small velocities it is only an approximation, and at very large velocities it is pretty much useless.

Euclidian geometry is an extremely good approximation on Earth because the axioms are almost perfectly true. It's only the parallel axiom which is incorrect, and even then, only by a tiny amount. The gravitational field of the Earth is really extremely weak, and nowhere near strong enough to cause geometry to diverge from the Euclidian model by a significant amount. Redfishbluefish explained it quite nicely with the paper analogy.

[Universal Mind]

So (both of you), if one angle of a parallelogram is 70 degrees, what is its opposite angle? I am not asking for what it almost is. I am asking you what it is. You might need to answer that with a range, based on what I know about your view.

[Xei]

In which geometry?

In Euclidian, it is 70 degrees.

In hyperbolic, I am not 100% sure about this, but I think it could be any n ∈ ℝ+, depending on the amount of local curvature.

In reality, as in, if you actually made a parallelogram out of bits of wood or something, or if you were just talking about any abstract lines in space (you could define them by the path taken by a photon), I think you get the same result as above. It depends entirely on the amount of local spacetime curvature. It would only be 70 if you were in some volume of space not affected by gravitation, and as far as we know there is no such location.

[Universal Mind]

In which geometry?

In Euclidian, it is 70 degrees.

In hyperbolic, I am not 100% sure about this, but I think it could be any n ∈ ℝ+, depending on the amount of local curvature.

In reality, as in, if you actually made a parallelogram out of bits of wood or something, or if you were just talking about any abstract lines in space (you could define them by the path taken by a photon), I think you get the same result as above. It depends entirely on the amount of local spacetime curvature. It would only be 70 if you were in some volume of space not affected by gravitation, and as far as we know there is no such location.

I think that's crazy, but I think we are going to have to agree to disagree.

[Xei]

I guess so. You just have to bear in mind that our visual cortexes have billions of years of evolutionary history and an entire lifetime of developmental history based upon Euclidian axioms so it's unsurprising that it seems crazy. That's why we have science, really; human intuition is often either limited or wrong, so objective evidence is the only path to truth.

[wendylove]

I guess so. You just have to bear in mind that our visual cortexes have billions of years of evolutionary history and an entire lifetime of developmental history based upon Euclidian axioms so it's unsurprising that it seems crazy. That's why we have science, really; human intuition is often either limited or wrong, so objective evidence is the only path to truth.
To be fair to Euclid the parrallel postulate has been proved independent of the other four postulates. Mathematically speaking mathematical ideas should be allowed to be developed really freely and abstractly with only the person worrying about if it's logically consistent. As a system which can describe arithmetic would be incomplete.

Anyway, Universal Mind is very stubborn for example he won't accept Quantum mechanics. It's sad to find that UM doesn't even agree with general relativity.

To add a further point. If you look at visual illusions like Penrose triangle then it shows that humans can be tricked. That's why you really should take a more mathematical view of life.

P.S. UM look at a experiment where they showed that light is getting bent by the suns gravity.

[Universal Mind]

It's sad to find that UM doesn't even agree with general relativity.


http://mikeely.files.wordpress.com/2008/04/strawman2.jpg

[wendylove]

UM but you don't. Einstein general relativity is based on non-Euclidean geometry which you don't agree with.

[Universal Mind]

UM but you don't. Einstein general relativity is based on non-Euclidean geometry which you don't agree with.

Please quote where Einstein said parallel lines look like the ones in 2 and 3...

http://upload.wikimedia.org/wikipedia/commons/5/5e/Euclidian_and_non_euclidian_geometry.png

Did Einstein ever claim that alternate interior angles created by parallel lines are not congruent? Did he ever say that there are not 360 degrees in a circle? Did he ever say a central angle of a circle does not intercept an arc of the same degree measure? Did he ever say that if two sides of a triangle are congruent, the angles opposite those sides are not congruent? Can you quote Einstein saying that the diagonals of a rhombus do not bisect each other?

[RedfishBluefish]

Actually I think I did read in something about relativity once that a triangle drawn around the sun would not have 180° in it...

But anyway I thought the whole basis of general relativity was that the force of gravity is not a force, but just a result of movement in a straight line through space curved by a mass. So it follows directly from relativity that if gravity can bend light so parallel rays intersect, space is not euclidean.

[Universal Mind]

Actually I think I did read in something about relativity once that a triangle drawn around the sun would not have 180° in it...

But anyway I thought the whole basis of general relativity was that the force of gravity is not a force, but just a result of movement in a straight line through space curved by a mass. So it follows directly from relativity that if gravity can bend light so parallel rays intersect, space is not euclidean.

Wouldn't that mean the triangle ceases to be a triangle and the rays cease to be parallel?

[RedfishBluefish]

The sides of the triangle are straight in the same way lines from the north pole to the equator along the earth's surface are straight. If you could step "outside" the universe and look at space-time itself from the 5th (or whatever high enough) dimension, sure, you would probably see that the lines aren't straight, they are bent by being drawn on a curved surface.

But in our 3 dimensions they are straight line segments connecting three vertices - ie. a triangle.

[Universal Mind]

The sides of the triangle are straight in the same way lines from the north pole to the equator along the earth's surface are straight. If you could step "outside" the universe and look at space-time itself from the 5th (or whatever high enough) dimension, sure, you would probably see that the lines aren't straight, they are bent by being drawn on a curved surface.

But in our 3 dimensions they are straight line segments connecting three vertices - ie. a triangle.

Maybe there is something to what you are saying, but the figure can no longer be classified as a triangle once the sides are bent.

[Xei]

They aren't bent as far as I am aware; they are straight within the geometry of general relativity.

I searched for the triangle thing and found this nice page on Wikibooks:

http://en.wikibooks.org/wiki/General_relativity/Curvature

And this one's good too:

http://members.tripod.com/~noneuclidean/applications.html