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.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.
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.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.
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.So you make new assumptions:No, you wouldn't!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.
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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.
It's not a law of math, but a law of reality, that distances have to be real and not irrational.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".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.
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.
[/quote]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!
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.