Phi

[rcglinsk]

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.

We've moved from real things, cylinders, to mathematical concepts, lines, without any coherent path. We don't know beforehand that any ends are connected unless you can point to a set of three objects, two which are equal in length, one sqrt(2) in length, that are connected together. Please point to what you are talking about. Really we're talking concepts, nothing more. If we are talking about a set of three real objects, please point to and describe them.

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.

There exists a mathematical concept, "right triangle," with the characteristics you describe. Please point to and describe an object in the real world with the characteristics of your triangle.

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!

I would! I swear to you I would. Please tell me about this object, how it is isolated in studied in a laboratory, and what the experimental results tell me about the object.

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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 [i]know [/i]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" [i]your own [/i]"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...

I dunno, point to an object with an infinitely complex shape and I'll believe one exists.

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.

It's not a law of math, but a law of reality, that distances have to be real and not irrational.

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!

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I measure the lenght of objects with bricks of some sort. There is no such thing as a straight line in the physical world. If there is, please point to and describe one.

[klypp]

Altonhare, you obviously have a problem with concepts of any kind. And as for abstract thinking, you seem to be completely lost. But I'll give it one last try.

Let's go back to our square. We learned that if we use the side as the unit, then the diagonals would have to be expressed as an irrational number, sgrt(2). Now let's change unit. Let's use the diagonal as our unit. What happens? Now it is the sides that has to be expressed as irrational numbers, 1/sgrt(2).

Nothing changed in our square. No length was altered. The only thing that changed was our choice of unit.

Now, most of us understand that whether or not any specific length found in the real world or anywhere else should be expressed with an irrational number or a rational number, solely depends on our choice of unit.
And thus, saying that rational numbers has "physical significance" while irrational numbers has not, is just a demonstration of ignorance.

If you still don't get this, go buy a math book and try finding an extremely patient person willing to explain this for you!

[altonhare]

Klypp, the premises of the mathematical argument are incorrect. When we assign a single value to the length of the base/height/hypotenuse we are assuming these have uniform length i.e. that they are lines. If they do not have uniform length obviously we cannot express their length with a single value! We are using something with uniform length:

line.JPG (6.28 KiB) Viewed 12959 times

which is identical to saying we are assuming the length of the base/height/hypot can be expressed with a single value. The object under consideration looks like this:

Isosceles-right-triangle.jpg (8.91 KiB) Viewed 17814 times

Can we construct this object out of anything with uniform length? We cannot. A triangle is not composed of lines, it is composed of trapezoids:

trapezoid.JPG (5.33 KiB) Viewed 12919 times

The mathematical arguments presented saying that the hypotenuse is "immeasurable" have been misinterpreted. The premise of the argument is that the triangle is either composed of objects of uniform length, or we can precisely quantify objects without uniform length as if they had uniform length. When we arrive at a contradiction at the end of the argument, it tells us that our original premise was incorrect.

The physical interpretation of the right triangle mathematical argument is that a triangle is not composed of lines, but of trapezoids. We get a result we are unable to calculate (an irrational) because we are trying to assign a single value of length to an object whose length varies over its width and/or height.

Your move Klypp.

[klypp]

Try measuring the inside or the outside of your triangle.

[altonhare]

Try measuring the inside or the outside of your triangle.

I measured it. I got a rational result.

But seriously, the triangle is composed of trapezoids no matter how closely you look at it. You cannot assign a trapezoid a single value for its length. This is the reason for the anomalous result that it is "indeterminate". You are trying to use a line as a trapezoid.

Mathematicians are assuming trapezoids are lines and, when they get a contradiction, declaring that some lengths are "indeterminate". A trapezoid just isn't a line, no matter how closely you look at it.

Telling me to measure the inside or outside is telling me to treat the triangle as if it were like this:

math_trapezoid.JPG (8.34 KiB) Viewed 12819 times

Is this the triangle we're dealing with Klypp? Made of lines? Or is it the one made of trapezoids?

If you are going to claim that your math has physical significance you must ascribe to one or the other!

[klypp]

altonhare:

But seriously, the triangle is composed of trapezoids no matter how closely you look at it. You cannot assign a trapezoid a single value for its length.

Seriously?
If this isn't a joke, you have a serious problem.

Does the sides of a trapezoid have lengths?

[rcglinsk]

altonhare:

But seriously, the triangle is composed of trapezoids no matter how closely you look at it. You cannot assign a trapezoid a single value for its length.

Seriously?
If this isn't a joke, you have a serious problem.

Does the sides of a trapezoid have lengths?

The sides of the trapezoid are trapezoids as well. Two lines can't form a corner, they overlap, or you have to cut the edges and make them into trapezoids.

[klypp]

This gets better and better!

[Grey Cloud]

Hi folks,
Dumb question: why does the triangle have to be made from three pieces? What if it were say, stamped out from a single sheet of metal? Would that not get rid of the trapezoids?

[klypp]

Tricky one, Grey Cloud!
Of course you'd have to use the trapezoid brick for the stamping. But still...

Nah, we'll have to wait for the experts...

[altonhare]

altonhare:

But seriously, the triangle is composed of trapezoids no matter how closely you look at it. You cannot assign a trapezoid a single value for its length.

Seriously?
If this isn't a joke, you have a serious problem.

Does the sides of a trapezoid have lengths?

Length: The extent of an *object* in a direction

The side of an object, is not an object.

Are you dealing with objects (lines, trapezoids, etc.) or not?

Hi folks,
Dumb question: why does the triangle have to be made from three pieces? What if it were say, stamped out from a single sheet of metal? Would that not get rid of the trapezoids?

It's not a dumb question. It does not have to be made from pieces. However, when the mathematician assigns lengths to the sides/hypotenuse s/he is treating it as if it were made from three pieces. Length is the extent of a single object in a single direction. The length of the triangle varies over its width/height. That's more complicated to deal with, so we assign a single value of length to the hypotenuse with the caveat that we are treating the triangle as if this part of it were a separate line.

However you cannot fit a line like this, it is physically impossible. The interpretation of the irrational result sqrt(2) is that a line can never perfectly approximate a trapezoid. You can never specify a trapezoid's length accurately with a single value.

What the mathematician is doing is saying that, if we shrink the triangle down reeaaaally small, the trapezoid looks like a line from far away, so we can approximate it as a line (with a single value of length). So the mathematician either arrives at a contradiction (in the case of the general "proof" of irrationals via the triangle) or at an irrational relation like sqrt(2). This contradiction simply tells the mathematician that the hypotenuse is not a line and that no amount of shrinking will ever make it so.

[Grey Cloud]

Tricky one, Grey Cloud!
Of course you'd have to use the trapezoid brick for the stamping. But still...

Nah, we'll have to wait for the experts...

I could cut one out of plywood if that helps.

[klypp]

It might help, Grey Cloud. But that is only if plywood can be an "object".

[klypp]

I just looked up the definition for plywood. It says:

a board made of thin layers of wood glued together under pressure, with the grain of one layer at right angles to the grain of the next

There are no such thing as "right angles". So I guess that rules out plywood. It's just some kind of mathematical construct...

[Grey Cloud]

I just looked up the definition for plywood. It says:

a board made of thin layers of wood glued together under pressure, with the grain of one layer at right angles to the grain of the next

There are no such thing as "right angles". So I guess that rules out plywood. It's just some kind of mathematical construct...

Try walking out of your local timber merchant without paying for the ply. And then try using your argument with the judge. [See you in six months ]