Phi

[altonhare]

Or If I wrote down "30 degrees" and "60 degrees" on a sheet of paper and asked to add the angles together you could write "90 degrees" or "a right angle" and everything would be cool. But if I handed you two circle drawing compasses, one stretched to thirty degrees and the other stretched twice as much and asked you to "add the angles together" you'd look at me like I was crazy.

Yes I would!
My first thought would be:
How can this guy think he can add 30 degrees to 30 degrees ("twice as much") to his compass angle, and then - within the same breath - think he poses some kind of unresolvable problem when he asks someone to add another 30 degrees to his compass angle?
My second thought would be:
Someone must have dropped a brick on his head...

Imagine you had to actually put the two compasses together to make a 90 degree angle. You could adjust one of them to be shaped like a right angle and discard the other, but the angles of the compasses cannot be added together directly the way that numbers can.

When Klypp shows me his "line without area" (without width) I'll concede the debate and name him God.

[webolife]

Or If I wrote down "30 degrees" and "60 degrees" on a sheet of paper and asked to add the angles together you could write "90 degrees" or "a right angle" and everything would be cool. But if I handed you two circle drawing compasses, one stretched to thirty degrees and the other stretched twice as much and asked you to "add the angles together" you'd look at me like I was crazy.

Yes I would!
My first thought would be:
How can this guy think he can add 30 degrees to 30 degrees ("twice as much") to his compass angle, and then - within the same breath - think he poses some kind of unresolvable problem when he asks someone to add another 30 degrees to his compass angle?
My second thought would be:
Someone must have dropped a brick on his head...

Imagine you had to actually put the two compasses together to make a 90 degree angle. You could adjust one of them to be shaped like a right angle and discard the other, but the angles of the compasses cannot be added together directly the way that numbers can.

When Klypp shows me his "line without area" (without width) I'll concede the debate and name him God.

So what y'all are saying is that neither the top nor bottom base of the trapezoid has any length, or that the length is imaginary, or that it has no physical significance? Is that true also for the altitude of the trapezoid [the thickness of the line]... if it has no physical significance then why even argue that the line is a trapezoid... Of course my line of questioning assumes that a "line" is a valid physical concept, as well as circle, sphere, or any other shape...

[altonhare]

Or If I wrote down "30 degrees" and "60 degrees" on a sheet of paper and asked to add the angles together you could write "90 degrees" or "a right angle" and everything would be cool. But if I handed you two circle drawing compasses, one stretched to thirty degrees and the other stretched twice as much and asked you to "add the angles together" you'd look at me like I was crazy.

Yes I would!
My first thought would be:
How can this guy think he can add 30 degrees to 30 degrees ("twice as much") to his compass angle, and then - within the same breath - think he poses some kind of unresolvable problem when he asks someone to add another 30 degrees to his compass angle?
My second thought would be:
Someone must have dropped a brick on his head...

Imagine you had to actually put the two compasses together to make a 90 degree angle. You could adjust one of them to be shaped like a right angle and discard the other, but the angles of the compasses cannot be added together directly the way that numbers can.

When Klypp shows me his "line without area" (without width) I'll concede the debate and name him God.

So what y'all are saying is that neither the top nor bottom base of the trapezoid has any length, or that the length is imaginary, or that it has no physical significance? Is that true also for the altitude of the trapezoid [the thickness of the line]... if it has no physical significance then why even argue that the line is a trapezoid... Of course my line of questioning assumes that a "line" is a valid physical concept, as well as circle, sphere, or any other shape...

The "top of a trapezoid" does not have length because it is not an object. Conceptually, we can imagine "chopping off" a top piece of a trapezoid to examine it closer. But this piece will still be a trapezoid, although its length will vary less. You can take this piece and chop off the top of it, and you will again end up with a trapezoid. Trapezoid's have length, their extent in a direction. Their length just happens to vary constantly. This is the definition of a trapezoid and is explicitly incommensurable with the definition of a line (whose length is perfectly constant).

This "imaginary" and "irrational" business comes about because the mathematicians want to pretend that we can get a line by chopping off the top piece of a trapezoid! But of course this is impossible by the very definition of a trapezoid. On no scale will you ever find that a trapezoid is a line. They are two different objects. They are defined differently. We have two objects and two definitions for a reason.

This does not stop the mathematician from trying to quantify everything with lines. Even circles! A circle cannot be formed from a line, a circle can only be formed from a trapezoid. A circle's circumference has nothing to do with the length of a line wrapped around. It has everything to do with the length of a specific trapezoid (I'll go into detail about this later) wrapped around.

Irrational relations are a result of using incommensurable units. Lines can only quantify lines and those objects which can be formed from lines. I am not saying you can't take your trapezoidal block of wood, chop off the top, and end up with a piece of wood whose length varies relatively much less than before (giving a good approximation to a line). I'm saying a trapezoid is not a line by definition. We can use trapezoids to quantify other trapezoids or objects formed from trapezoids. The additional criteria is that all the trapezoids involved must be commensurate, in that their length and width vary identically to each other. The "1 1 sqrt(2)" triangle can *only* be formed from two lines and a trapezoid. This is by definition. Therefore the length and width can be quantified perfectly as 1 using a standard line, but the hypotenuse must be quantified by a commensurable trapezoid.

[webolife]

Are you guys sure that the area of the "line"/trapezoid changes when stretched from a circle to a line?
My formula, "average base times height equals area" works for either shape.

Are ratios involving comparisons of shapes of no physical significance because they can not be expressed as counting numbers? Is that the whole point? If the answer to this is yes, then I simply disagree philosophically. You would have to throw out the periodic table, as well, based as it is, not on actual counts, but on averages of empirical data. Analogous to snowflakes, no atom, even Hydrogen atoms, can be shown to be an exact clone of any other. It's not that I can't identify with the intent of the argument, as it seems similar to my intent in arguing the disavowal of complex numbers in the "real" universe.

Regardless of the answer, the premise we're trying to deal with here is that "irrational numbers have no physical significance." What if we concede that only approximations of the values of pi or phi or root2 or root3 have physical significance (or even discrete numbers like 3,6, and 9) then go on to appreciate that significance by calling it "sacred geometry" or whatever? Would that concession be enough to end the debate? Not that the debate needs to end... but perhaps it is an irrational debate.................................................

[altonhare]

Are you guys sure that the area of the "line"/trapezoid changes when stretched from a circle to a line?
My formula, "average base times height equals area" works for either shape.

Are ratios involving comparisons of shapes of no physical significance because they can not be expressed as counting numbers? Is that the whole point? If the answer to this is yes, then I simply disagree philosophically. You would have to throw out the periodic table, as well, based as it is, not on actual counts, but on averages of empirical data. Analogous to snowflakes, no atom, even Hydrogen atoms, can be shown to be an exact clone of any other. It's not that I can't identify with the intent of the argument, as it seems similar to my intent in arguing the disavowal of complex numbers in the "real" universe.

Regardless of the answer, the premise we're trying to deal with here is that "irrational numbers have no physical significance." What if we concede that only approximations of the values of pi or phi or root2 or root3 have physical significance (or even discrete numbers like 3,6, and 9) then go on to appreciate that significance by calling it "sacred geometry" or whatever? Would that concession be enough to end the debate? Not that the debate needs to end... but perhaps it is an irrational debate.................................................

In order to stretch a line into a circle you have to convert it into a trapezoid, period. You cannot build a circle from a line, but you can from a trapezoid.

Ratios involving comparisons of commensurate shapes are physically significant i.e. correct. We can measure lines with lines and trapezoids with trapezoids and circles with trapezoids. Of course we need a specific trapezoid to form a circle, which I will get into later.

So no, the two objects do not have to be identical. You can use something with uniform length/height to measure ANYTHING else with uniform length/height/ You can use a trapezoid whose length and height vary in a specific way to measure ANY other trapezoid whose length/height/width vary in the same way. You can use a specific trapezoid to quantify the circumference of a specific circle.

Again, irrational relations are a result of measuring incommensurable objects. If some object in nature indeed is a trapezoid, a circle, etc. its length, width, and height are exactly what they are. They may be impossible to measure perfectly but they are still something specific. Irrational relations are only significant in the sense that they tell us what something is not. That is, the hypotenuse of a right triangle is not a line. An irrational result is a rejection of a hypothesis where the hypothesis is a line.

I think the debate is important because people don't realize the reason for irrationals is not something mystical, but simply a result of using incommensurable units. Additionally, some get the impression that Nature is somehow "imprecise" or that something is not exactly what it is because of irrationals. Nature is exactly what it is, irrationals are our laughable attempt to make Nature conform to lines.

[rcglinsk]

In order to stretch a line into a circle you have to convert it into a trapezoid, period. You cannot build a circle from a line, but you can from a trapezoid.

Really? Any chance you could explain how?

[klypp]

Are you guys sure that the area of the "line"/trapezoid changes when stretched from a circle to a line?
My formula, "average base times height equals area" works for either shape.

Your formula is fine. But the red line in the image/video is of course just that - a line. If you choose to conceive it as an area, as altonhare does in his desperate search for "arguments", then it becomes meaningless. You can only get a trapezoid or a rectangle out of it if you stretch the shorter side, compress the longer side or combine these two actions.
I called rcglinsk "brilliant" because he discovered that the video became nonsense if you conceived the line as an area. His calculations though, are not very brilliant. These calculations will show that the area of the rectangle is bigger than the "same" area when it's still on the circle. This is because he assumes that both long sides are pi. But that can only be if he stretches the "inner" side.

You would have to throw out the periodic table, as well, based as it is, not on actual counts, but on averages of empirical data. Analogous to snowflakes, no atom, even Hydrogen atoms, can be shown to be an exact clone of any other.

Exactly! And now I found that we have even more to agree on...

But you're right, this debate is irrational. In stead of having a discussion of the philosophical implications of irrational numbers and incommensurable lengths, what they can tell us about the very foundations of science today...

In stead of that I find this forum spammed with nonsense...

[rcglinsk]

Your formula is fine. But the red line in the image/video is of course just that - a line. If you choose to conceive it as an area, as altonhare does in his desperate search for "arguments", then it becomes meaningless. You can only get a trapezoid or a rectangle out of it if you stretch the shorter side, compress the longer side or combine these two actions.
I called rcglinsk "brilliant" because he discovered that the video became nonsense if you conceived the line as an area. His calculations though, are not very brilliant. These calculations will show that the area of the rectangle is bigger than the "same" area when it's still on the circle. This is because he assumes that both long sides are pi. But that can only be if he stretches the "inner" side.

My equations are right. The area of the red part of your movie changed from start to finish. So when we're talking about shapes or objects a hoop is not a rod, but when we're talking about mathematical concepts, the circumference of a circle is a line. I'm skeptical that the universe is made of mathematical concepts. I understand that most physicists aren't though.

[klypp]

My equations are right. The area of the red part of your movie changed from start to finish. So when we're talking about shapes or objects a hoop is not a rod, but when we're talking about mathematical concepts, the circumference of a circle is a line. I'm skeptical that the universe is made of mathematical concepts. I understand that most physicists aren't though.

"My equations are right."
You didn't even get your first formula right: "When it's wrapped around the circle the area = pi(1+dr)^2 - pi(1)^2 = pi(2dr+dr^2)."
What you missed in the video is that it is the diameter that is 1, not the radius. Also, it is the diameter of the outer circle, not the inner circle, that is 1. Thus the correct equation would be:
pi(1/2)^2 - pi((1/2) - dr)^2
That's two mistakes in this simple equations. Not very impressive.

"when we're talking about shapes or objects a hoop is not a rod".
Correct! But you assume you can "roll out" a hoop in such a way that you get a rod where the sides can be described as a rectangle. You can not. There is no way you can do this without stretching or compressing at least one side. We should stop there.
But you won't stop. You assign pi to both long sides of your rectangle and dr to both short sides. On what grounds? Where is the argument showing that this rectangle should have the same area as the hoop? Nowhere! But that doesn't stop you from calculating both areas and "prove" that "The area of the red part of your movie changed from start to finish". Again, it's nonsense.

Most of us can easily see that the hoop in the video is meant to be a mathematical line, not an area. We can also see that this is an elegant way to demonstrate the actual length of pi.
I bet you see this as well, but you won't admit it. Simply because you started off by the wrong foot by postulating that there are no such numbers as pi, "all numbers has to be rational".
And so you and altonhare try to create all sorts of confusion, hoping to conceal your mistake. Like all this nonsense about concepts: "I'm skeptical that the universe is made of mathematical concepts."
Well, who said it was? I am pretty sure the universe is not made of mathematical concepts, nor any other kind of concept. A concept is something we use to describe the world around us, but that doesn't mean that concepts are building bricks of the universe.

[altonhare]

Most of us can easily see that the hoop in the video is meant to be a mathematical line

Really? Show us your mathematical line so we can understand too.

[rcglinsk]

My equations are right. The area of the red part of your movie changed from start to finish. So when we're talking about shapes or objects a hoop is not a rod, but when we're talking about mathematical concepts, the circumference of a circle is a line. I'm skeptical that the universe is made of mathematical concepts. I understand that most physicists aren't though.

"My equations are right."
You didn't even get your first formula right: "When it's wrapped around the circle the area = pi(1+dr)^2 - pi(1)^2 = pi(2dr+dr^2)."
What you missed in the video is that it is the diameter that is 1, not the radius. Also, it is the diameter of the outer circle, not the inner circle, that is 1. Thus the correct equation would be:
pi(1/2)^2 - pi((1/2) - dr)^2
That's two mistakes in this simple equations. Not very impressive.

"when we're talking about shapes or objects a hoop is not a rod".
Correct! But you assume you can "roll out" a hoop in such a way that you get a rod where the sides can be described as a rectangle. You can not. There is no way you can do this without stretching or compressing at least one side. We should stop there.
But you won't stop. You assign pi to both long sides of your rectangle and dr to both short sides. On what grounds? Where is the argument showing that this rectangle should have the same area as the hoop? Nowhere! But that doesn't stop you from calculating both areas and "prove" that "The area of the red part of your movie changed from start to finish". Again, it's nonsense.

Most of us can easily see that the hoop in the video is meant to be a mathematical line, not an area. We can also see that this is an elegant way to demonstrate the actual length of pi.
I bet you see this as well, but you won't admit it. Simply because you started off by the wrong foot by postulating that there are no such numbers as pi, "all numbers has to be rational".
And so you and altonhare try to create all sorts of confusion, hoping to conceal your mistake. Like all this nonsense about concepts: "I'm skeptical that the universe is made of mathematical concepts."
Well, who said it was? I am pretty sure the universe is not made of mathematical concepts, nor any other kind of concept. A concept is something we use to describe the world around us, but that doesn't mean that concepts are building bricks of the universe.

It seems we are in complete agreement about the meaning of your video. A hoop is not a circumference; a rod is not a line. The video represents the mathematical concepts in a way that teaches about them, but does not prove that lines and circumferences are objects. Whoops on the math error:) I am slightly confused though as I thought the topic of discussion was whether an irrational length could exist, not whether an irrational number could.

[altonhare]

In order to stretch a line into a circle you have to convert it into a trapezoid, period. You cannot build a circle from a line, but you can from a trapezoid.

Really? Any chance you could explain how?

I don't mean we can take a physical trapezoid and form a circle, I'm speaking strictly conceptually. If you run Klypp's video with a trapezoid there's no change in area.

We can also see that this is an elegant way to demonstrate the actual length of pi.

We don't need equations or mathematics to resolve this issue. Pi is not a length. Length is an object's extent in a direction perpendicular to width and height. No object has length, width, or height equal to pi.

Pi is a dynamic concept. 3.1415927 is a static concept. It refers to 31,415,927 standard objects (objects whose length is defined as 1). Pi refers to the act of first laying down 3 standard objects, then breaking them into smaller equal standards and laying down 31 of them. Then breaking them again and laying down 314 of them... ad nauseum. If the standard object we're using were commensurate with what we're measuring we'd always be able to break our standard until we arrive at a precise result. If our standard is incommensurate we will have to keep breaking it smaller in order to get a closer approximation. Pi and other irrational relations are okay as long as they are understood as incommensurate approximations. On the other hand, when people state that an object literally has an "indeterminate length" in the sense that Nature itself is "indeterminate" this indicates they do not understand. When one converts the notion of pi as an incommensurate approximation into the notion of an infinite iteration implying indeterminacy they convert pi into an abstract concept. An abstract concept is one that is impossible, we cannot lay an infinite number of bricks because infinity is not a number.

So if our standard is a line and we are measuring a trapezoid we can lay the line on the trapezoid, but we will find the ends are not flush. We can break the line into pieces, but still the end is not flush. And so on and so forth. We will never actually get to a point where the line matches the trapezoid exactly. They are incommensurate. The trapezoid's length is exactly 1 standard trapezoid length (one of itself). The trapezoid has a different identity than the line. Analogous situation for the circle, we lay down standard lines around in a perimeter in ever-increasing numbers but we never get a circle. Note that this has nothing to do with math but everything to do with the definition of line, trapezoid, and circle.

To humans lines are conceptually the simplest so we try to build everything else out of them. This works to a good approximation. But Nature doesn't care what we find easiest to use. In Nature, if there is a such thing as a trapezoid or a circle, its dimensions are precisely defined independent of our ability to quantify it with lines.

Make sense?

[webolife]

Not really...
All you are actually saying is that:
1. The real universe is made up only of objects describable with counting numbers. A premise.
2. Phi, pi, etc. are not rational numbers because they cannot be expressed with counting numbers. A tautology.
3. Therefore "irrational" numbers such as these have no physical significance, restating the premise.

I actually agree with most of your argument about circles and trapezoids, that's never been the issue for me.

[altonhare]

Not really...
All you are actually saying is that:
1. The real universe is made up only of objects describable with counting numbers. A premise.
2. Phi, pi, etc. are not rational numbers because they cannot be expressed with counting numbers. A tautology.
3. Therefore "irrational" numbers such as these have no physical significance, restating the premise.

I actually agree with most of your argument about circles and trapezoids, that's never been the issue for me.

The "premise" is basically a restatement of Identity. Since you accept Identity I don't see why you wouldn't understand. Unless you don't see why (1) is a restatement of Identity, if you don't let me know and I'll try to illustrate why.

[webolife]

No, I understand... I was just summarizing your logic.