Phi

[altonhare]

If you make yourself a measure stick and mark 1, 2, 3 and so on, you may just as well mark phi on the same stick - with exactly the same precision as your 1, 2 and 3 marks. The length phi meters, is a precisely described length.

This is kids stuff.

Klypp, I'm afraid you have been duped by the establishment. Nobody learns physics in school anymore, nobody learns what an object is or what numbers mean. As rcglinsk so cogently pointed out, you will never be able to mark phi or pi on your stick. Whatever you mark will be rational. Eventually you have to terminate the series. Irrationals do not belong on slide rules or meter-sticks. Ignorant mathematicians of the establishment put them there and hope you won't think too hard on this issue.

[Drethon]

Ratios are rational numbers, the problem is where most people try to handle math without precision. The square root of 2 is not a real number because 2 is not a real number. However the square root of 2.0000000000 is 1.4142135623 because we have a precision of 10^-10.

The question is not one of the irrationality of ratios but rather the fact that no one knows the precision with which they are calculating ratios. Is true precision limited to 2*32 such as in 32 bit architecture computers? Is true precision based on the size of the atom a material is made up of? If so do we know the actual size of a hydrogen atom or does it change, do we average the size of all atoms of the material? Is precision infinite?

No one actually knows this answer because it is impossible to measure a unit that is equal to the smallest possible precision because that measurement would have an inaccuracy of +/-100% exclusive...

[webolife]

Fun debate... I concede, because I agree with the argument, just not the vehemence of the thesis of physical insignificance.
Altonhare, some day you and I need to get together and draw pictures of ropes and rays. Feign, phi, foe, friend.

[altonhare]

Fun debate... I concede, because I agree with the argument, just not the vehemence of the thesis of physical insignificance.
Altonhare, some day you and I need to get together and draw pictures of ropes and rays. Feign, phi, foe, friend.

You agree with the argument, but you'll disagree anyway so you don't appear too extreme?

[Plasmatic]

This debate as with most is resolved when we ask "what in perception gives rise to the concept one "?The number one corresponds to the concept entity. We percieve a world of particular individual entities. The number one as with ALL numbers begins with this perception. There are NO 1/2 entities nor are there any infinite entities. to be is to be something specific apart from other individual particulars. To assert the opposite is to misintegrate the percept that gives rise to the concepts involved. In the conceptual heirarchy of "number" 1/2 is derivative of 1 precisly because that which gives rise to the concept [number/i] is percieved individul entities.

[klypp]

altonhare:

"How can the length from here to there be pi or 2.5? Think about it! Just because we can write an equation or get a result doesn't mean it is physically significant."
"Instead of actually measuring this distance you have assumed it follows some standardized mathematical relation."
"In fact, if you measured the hypotenuse you would always find that it's rational. Even down to the scale which humans cannot see, the distance is rational. sqrt(2) is an abstract mathematical concept with no physical significance."
"If I am measuring the distance from A to be with my brick I may have to break my brick in half to get it right. I may have to do it again etc. An irrational number implies that I can never break my brick in half enough times to measure the distance from A to B. Ever. I will be breaking my brick in half incessantly but never getting the final answer. I concede that I may need to break my brick in half many gazillions of times, but surely I will eventually get to an answer. The distance from A to B is what it is, it's not forever expanding as I add more decimal points to my calculation is it? Surely that's a ludicrous proposition."

Well, this goes on and on... It's the old pythagorean view.
Ironically it was a pythagorean who first contradicted this, Hipassus. Now, how do you think he came up with the weird idea that you cannot measure the hypotenuse with units derived from the sides? By just sitting still until some equations popped up in his head? Maybe...
But a more likely way to discover this would've been to actually try measuring the hypotenuse of this triangle, or likewise the diagonal of a square, just like you are recommending. After all, that's where most of us run into problems. It's experience that first shows us that we can not measure both the side and the diagonal of the square with the same unit...
At some stage in his long line of measuring Hipassus must have asked. How come? Can this really be? How?
Unfortunately he came up with an answer. Very unfortunate, because the story has it that he was on board a ship with other pythogoreans - and he told them. He was immediately throwed overboard. Heresy has its prize.

You are the one assuming things here, altonhare. You assume you'll find a unit that can measure both these lengths, but you cannot prove it. Breaking bricks won't help you. It has been tried before...

Plasmatic:

This debate as with most is resolved when we ask "what in perception gives rise to the concept one "?The number one corresponds to the concept entity. We percieve a world of particular individual entities. The number one as with ALL numbers begins with this perception.

I don't see how this "resolves" the debate. Isn't this where it all starts?

Here is a link to a site discussing these problems: http://www.learner.org/courses/mathillu ... &pid=2285#
or better, from the start: http://www.learner.org/courses/mathillu ... ook/01.php

I found it very well written and worth reading both by mathematicians and philosophers.
Brickbusters absolutely should read it...

[altonhare]

You are the one assuming things here, altonhare. You assume you'll find a unit that can measure both these lengths, but you cannot prove it. Breaking bricks won't help you. It has been tried before..

This is a simple matter. We can assume one of two things:

1) The distance from A to B is a certain number of bricks of whatever size.

or

2) The distance from A to B is NOT a certain number of bricks of whatever size.

Is the distance from A to B a number of something or not? Pi, Phi, etc. are not numbers, they're relationships. This is a simple question and a simple matter to resolve. If you assume 2, then how do you define distance?

[klypp]

OK, I can see that you didn't read anything in the links I provided. You should have.
We are talking about a problem that has puzzled philosophy and mathematics for 2500 years. You call it "a simple question and a simple matter to resolve".
Simplicity is sometimes a synonym for foolishness. I'm beginning to see why, unless you can explain to me what is wrong with the following argument:

Let's imagine a square with a side of length a and diagonal of length b.

If these lengths are commensurable, as Pythagoras and his followers believed (without proof), then there is a common unit u such that a = mu and b = nu for some whole numbers m and n. We can assume that m and n are not both even (for if they were, it would indicate that the common unit could instead be u, and we would simply make that adjustment). So, we can safely assume that at least one of these numbers is odd.

Applying Pythagoras' theorem to the triangle formed in the square, we have:

a² + a² = b²

That is,

2a² = b²

or, substituting our common unit expressions for the two lengths,

2m²u² = n²u²

We know that our common unit, u can't be zero, so we can cancel the u² term from both sides of the equation, leaving:

2m² = n²

Obviously, n² is even, because it is equal to some number, m², multiplied by two. If n² is even, then n must be even also (if n were an odd number, then n² would be odd). We can express the even number n as two times some number.

n = 2w

Substituting this expression for n into the preceding equation gives us:

(2w)² = 2m²

4w² = 2m²

m² = 2w²

This reveals that m² is a multiple of two, that is, an even number. Consequently, as we reasoned before, m must also be even, and we can write:

m = 2h

Now we have found a contradiction! Remember, we assumed at the beginning that either m or n was odd, yet we have just shown that both have to be even. This logical contradiction proves that there is no common unit, u, that fits a whole number of times into both a and b—therefore, a and b, the lengths of the side and diagonal of a square, are incommensurable.

http://www.learner.org/courses/mathillu ... &pid=2285#

[rcglinsk]

altonhare:

"How can the length from here to there be pi or 2.5? Think about it! Just because we can write an equation or get a result doesn't mean it is physically significant."
"Instead of actually measuring this distance you have assumed it follows some standardized mathematical relation."
"In fact, if you measured the hypotenuse you would always find that it's rational. Even down to the scale which humans cannot see, the distance is rational. sqrt(2) is an abstract mathematical concept with no physical significance."
"If I am measuring the distance from A to be with my brick I may have to break my brick in half to get it right. I may have to do it again etc. An irrational number implies that I can never break my brick in half enough times to measure the distance from A to B. Ever. I will be breaking my brick in half incessantly but never getting the final answer. I concede that I may need to break my brick in half many gazillions of times, but surely I will eventually get to an answer. The distance from A to B is what it is, it's not forever expanding as I add more decimal points to my calculation is it? Surely that's a ludicrous proposition."

Well, this goes on and on... It's the old pythagorean view.
Ironically it was a pythagorean who first contradicted this, Hipassus. Now, how do you think he came up with the weird idea that you cannot measure the hypotenuse with units derived from the sides? By just sitting still until some equations popped up in his head? Maybe...
But a more likely way to discover this would've been to actually try measuring the hypotenuse of this triangle, or likewise the diagonal of a square, just like you are recommending. After all, that's where most of us run into problems. It's experience that first shows us that we can not measure both the side and the diagonal of the square with the same unit...
At some stage in his long line of measuring Hipassus must have asked. How come? Can this really be? How?
Unfortunately he came up with an answer. Very unfortunate, because the story has it that he was on board a ship with other pythogoreans - and he told them. He was immediately throwed overboard. Heresy has its prize.

You are the one assuming things here, altonhare. You assume you'll find a unit that can measure both these lengths, but you cannot prove it. Breaking bricks won't help you. It has been tried before...

Plasmatic:

This debate as with most is resolved when we ask "what in perception gives rise to the concept one "?The number one corresponds to the concept entity. We percieve a world of particular individual entities. The number one as with ALL numbers begins with this perception.

I don't see how this "resolves" the debate. Isn't this where it all starts?

Here is a link to a site discussing these problems: http://www.learner.org/courses/mathillu ... &pid=2285#
or better, from the start: http://www.learner.org/courses/mathillu ... ook/01.php

I found it very well written and worth reading both by mathematicians and philosophers.
Brickbusters absolutely should read it...

The issue is there is a difference between the mathematical concept: "The length of the line lying between (1,0) and (0,1)" and any physical object whose length we might like to measure. So when you talk about measuring the diagonal of a square, you weren't talking about measuring a distance between pencil marks on a page, which has a real and rational measure, but rather the conceptual distance between (1,0) and (0,1). I think altonhare's point is about how in the real world irrational lengths don't exist.

[klypp]

I think altonhare's point is about how in the real world irrational lengths don't exist.

Altonhare says that "rational lengths" exist, "irrational lengths" don't. The argument I quoted above, shows that you cannot have the one without the other. "Irrational lengths" are just as real as "rational lengths". Your argument against "irrational lengths" in the real world could equally well be used against "rational lengths". There are no exact lengths in the real world.

Or put another way:
Altonhare says that there is a smallest unit with which you can measure everything. He cannot prove this. It is an assumption.
There is another view saying that in the real world you'll never find two lengths that will match each other exactly, or where any ratio of one of the lengths can be used to exactly measure the other length. This is also an assumption.

So far in this thread we've proved that altonhare's assumption is wrong.

[rcglinsk]

I think altonhare's point is about how in the real world irrational lengths don't exist.

Altonhare says that "rational lengths" exist, "irrational lengths" don't. The argument I quoted above, shows that you cannot have the one without the other. "Irrational lengths" are just as real as "rational lengths". Your argument against "irrational lengths" in the real world could equally well be used against "rational lengths". There are no exact lengths in the real world.

Or put another way:
Altonhare says that there is a smallest unit with which you can measure everything. He cannot prove this. It is an assumption.
There is another view saying that in the real world you'll never find two lengths that will match each other exactly, or where any ratio of one of the lengths can be used to exactly measure the other length. This is also an ass, orumption.

So far in this thread we've proved that altonhare's assumption is wrong.

There have to be exact lengths in the real world because the word length has to refer to some brick, whether it be a hydrogen atom or that meter bar they have in the box in London, that object has an exact length, one. If you measure other objects with your brick you might have to break the brick into pieces to get things to match up, or find the object has nooks and crannies that require a smaller brick to handle, but there is no object with an infinitely complex dimension or a shape that results from translating a mathematical concept directly to reality. I'll buy there is an infinitely complex shape around if we could isolate and study the object in a laboratory. The other half of the problem is the line of mathematics has no physical counterpart. To measure the length of any object one has to set rules at the outset for how you deal with a rough edge or a curve. Once you've set the rules though, if the object has shape, it has real, not irrational dimensions. The only way to create these ratios without irrational definitions is to use math concepts without physical significance. So the square root of two will result from 2 "straight lines" and one "right angle" where neither straight lines nor right angles have obvious physical counterparts. Math is never more than useful. So we know from the right triangle picture that if slender cylinders of equal length are oriented on a flat plane at a perfect right angle, there does not exist a third similar cylinder that can connect precisely the ends of the first two. That's not particularly useful because I can't imagine that situation comes up often.

[Plasmatic]

To measure the length of any object one has to set rules at the outset

Bingo, All units are conceptual standards.The metaphysical or perceptual basis for this system is/are particular entities, which correspond to the concept "one" .

There are no exact lengths in the real world

Everything measurable is "exactly" what it is . The units we isolate and use as a standard [conceptually] must be exact or the unit economy is ruined. Thus we must choose appropriate unit for the particular attributes of said entities [ weight in pounds in stead of inches]

[altonhare]

I think altonhare's point is about how in the real world irrational lengths don't exist.

Altonhare says that "rational lengths" exist, "irrational lengths" don't. The argument I quoted above, shows that you cannot have the one without the other. "Irrational lengths" are just as real as "rational lengths". Your argument against "irrational lengths" in the real world could equally well be used against "rational lengths". There are no exact lengths in the real world.

Or put another way:
Altonhare says that there is a smallest unit with which you can measure everything. He cannot prove this. It is an assumption.
There is another view saying that in the real world you'll never find two lengths that will match each other exactly, or where any ratio of one of the lengths can be used to exactly measure the other length. This is also an assumption.

So far in this thread we've proved that altonhare's assumption is wrong.

Klypp the distance from A to B is either a certain number of something or it isn't. You state that it isn't. You explicitly state that there are "no exact lengths".

You come to these paradoxical and erroneous conclusions as a result of too much math and too little logic. You follow the mathematical rules without understanding them. It is exactly this habit of simply chugging through the math without proper attention to the premises of the argument that has resulted in the state of "science" today.

The "proofs" you have presented all start from nonphysical premises. A mathematical line has no physical significance because it is defined as having no physical width or height. In reality all objects have length, width, and height. You may call this an assumption, but it follows directly from the definition of "object". You may call this an assumption but you cannot present us with a single "object" that does not have length, width, and height. You also cannot formulate a definition of "object" that can be used consistently that qualifies mathematical points and lines as objects.

Irrational relations are a result of trying to apply nonexistents to existents i.e. trying to use a mathematical line as a physical line. An irrational relation means that we have shrunk our physical objects down to nothing i.e. made them nonphysical. It makes no sense to speak of an "infinitely small" object because "infinitely small" means what we are referring to has vanished. We can talk about shrinking an object down incessantly, but whenever we stop shrinking it we "terminate" our 3.1415927... and end up with a rational number. When we talk about shrinking a physical object down to literally nothing and then claim that the resulting relationship (which we cannot even express because it is an infinite series!) means that something does not have a definite measure this is extremely illogical because we are referring to nothing now, not something!

We either terminated the "shrinking" and got a rational result or we did not terminate the "shrinking" and ended up with nothing, which certainly cannot let us conclude anything about "something"!

[klypp]

Let's have a brief look at what you're saying, rcglinsk. I'll start with your last paragraph.

Math is never more than useful. So we know from the right triangle picture that if slender cylinders of equal length are oriented on a flat plane at a perfect right angle, there does not exist a third similar cylinder that can connect precisely the ends of the first two. That's not particularly useful because I can't imagine that situation comes up often.

No, we don't know that "there does not exist a third similar cylinder that can connect precisely the ends of the first two."
Math doesn't say that these ends cannot be connected. We knew beforehand that these ends are connected. What math tells us is simply what the length of this connection will be, the sgrt(2). There is no mystery here. There is no problem here.
The "problem" arises when someone get the weird idea that this number is some kind of nonsense number and that such number does not have any "physical significance".
You are the one that "know" that these ends cannot be connected . But you cannot deduce that from the triangle. The only way you can "know" this is because you apply some other "mathematics" to this situation, another kind of "logic". But you don't seem to have the faintest idea of what that "math" or "logic" looks like. You talk about "rules", but you don't know what rules your own assumptions are based on.
You assume brickbusting will save you. It is already proven that it can't. No matter what unit you use to draw the sides of the square, the diagonal will be sqrt(2) expressed in this unit.

So you make new assumptions:

there is no object with an infinitely complex dimension or a shape that results from translating a mathematical concept directly to reality. I'll buy there is an infinitely complex shape around if we could isolate and study the object in a laboratory.

No, you wouldn't!
These kind of "shapes" has been on the table for at least 2500 years. The pythagoreans couldn't explain them, but at least they came up with far better arguments than you've done so far. My bet is you don't even know the basis of your own ideas or the historical background they sprung from.
How do you know there is "no object with an infinitely complex dimension"? You give no arguments for this view. You just think or believe that it is like that. What you're doing is "translating" your own "mathematical concept directly to reality"!
But OK, I'll credit you for seeing that these kind of "shapes" is at the issue here. I've met enough jerks with the idea that an infinite number of decimals means an infinite number. At least you didn't fell into this kind of stupidity...

The only way to create these ratios without irrational definitions is to use math concepts without physical significance. So the square root of two will result from 2 "straight lines" and one "right angle" where neither straight lines nor right angles have obvious physical counterparts.

Now again, to repeat myself, an irrational number is a number that cannot be expressed as a ratio. This is the definition of irrational numbers, but it is not an "irrational definiton". You should avoid such expressions. It only creates the impressions that you mix up "not a ratio" with "not reasonable".
There is no law or rule in mathematics that says a number has to be a ratio. That's a rule you just made. Based on what?
Mathematicians first came up with a definition of irrational numbers after they discovered that these numbers had to exist as a consequence of the more fundamental rules.

Arguments like "neither straight lines nor right angles have obvious physical counterparts" is of course nonsense in this context. What do you use to measure lengths in nature? A straight line or what?

Likewise, to exclude irrational numbers because you don't find them "particular useful" is also nonsense. Numbers are what they are.
Some people would never use the number 13. Now some others have decided not to use sgrt(2). There is a word for it...
Ah yes, superstition!

[altonhare]

There is no law or rule in mathematics that says a number has to be a ratio. That's a rule you just made. Based on what?
Mathematicians first came up with a definition of irrational numbers after they discovered that these numbers had to exist as a consequence of the more fundamental rules.

Support this claim. Define "number" so that we may know if sqrt(2) qualifies as one.