Originally posted by SwlabrI think it's relevant. It's impossible for any continuum variable to be exactly equal to something, but interger-like variables can be exactly equal to something.
Well yes, but then re-labelling 1/2 to be 1 we have just created an integer out of a fraction.
Come to think of it, it is possible to re-label all the rational numbers as integers (as they have the same 'size'😉, but not so with the real numbers. Is this perhaps significant?
Originally posted by KazetNagorraFractionals are all countables. So yes, we can count them, one by one, and therefore we can also label them as 1, 2, 3, ... (See George Cantor)
I think it's relevant. It's impossible for any continuum variable to be exactly equal to something, but interger-like variables [b]can be exactly equal to something.[/b]
Reals, however, are uncountable, so there we cannot label them as 1, 2, 3, ...
Originally posted by SwlabrIf I invent a new measurement unit (the twit) and define it as 1cm=sqrt(2)twit then your 1cm line is now exactly sqrt(2)twit.
Assuming you can draw two lines of length precisely 1, then you can draw a line of length precisely sqrt(2), by Pythagoras, which is an irrational number. However, surely it is impossible to draw a line of length sqrt(2), as we live in a finite universe?
The point I am making is that distance only has a relation to numbers when dealing with proportions. There is no such thing as absolute distance. Everything is relative (as pointed out by Einstein).
What do you mean by 'finite universe'? The overall size of the universe has nothing to do with its internal structure so I can only assume that you mean that the universe is made up of a finite number of bits of space. If so, I still do not see how your logic would apply. If space is made up of spherical pieces in a sort of crystalline structure then it should still by possible to calculate the distance in spheres between any two spheres and some of those distances will be irrational. The fact that you cannot place an exact number of spheres along the line is irrelevant.
Originally posted by FabianFnasMathematics is not invented. At best you could call it 'discovered'. Pythagoras' Theorem works just as well in the real world as it does in hypothetical ones because it is a fact not an invention.
Mathematics is the best of all worlds. Because it is invented. It is not the real world. More as an approximation, of course.
In the real world, nothing is exactly as in mathematics. Say a circle, as an example. Circles doesn not exist in the real world.
Originally posted by twhiteheadI disagree. Mathematics is basically one big circular argument, which happens to have practical applications. There may be a "real" counterpart to Pythagoras' theorem, but where is the practical application of a 23-dimensional sphere with radius i? (not saying there should be one)
Mathematics is not invented. At best you could call it 'discovered'. Pythagoras' Theorem works just as well in the real world as it does in hypothetical ones because it is a fact not an invention.
Originally posted by KazetNagorraIt matters not whether or not a 23-dimensional sphere with radius i exists. It remains true that if one did exist it would have certain properties that could be determined mathematically. If those properties are a necessary outcome then they can hardly be said to have been 'invented'.
I disagree. Mathematics is basically one big circular argument, which happens to have practical applications. There may be a "real" counterpart to Pythagoras' theorem, but where is the practical application of a 23-dimensional sphere with radius i? (not saying there should be one)
I realize that a lot of what we call mathematics is actually methods of understanding relationships between abstract ideas (theories) and methods of convincing ourselves of the validity of those relationships (proofs) yet one can hardly call them 'inventions'. 'Discoveries' is a far better fit.
Originally posted by twhiteheadHow do you discover anything when you cannot do any observation of it? There are no such things in universe as a 23 dimensinal perfect circle with a radius of i? Show me one, and I will surely admit that I am wrong.
It matters not whether or not a 23-dimensional sphere with radius i exists. It remains true that if one did exist it would have certain properties that could be determined mathematically. If those properties are a necessary outcome then they can hardly be said to have been 'invented'.
I realize that a lot of what we call mathematics is actually methods o ...[text shortened]... ips (proofs) yet one can hardly call them 'inventions'. 'Discoveries' is a far better fit.
Until then we have to invent it.
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Originally posted by FabianFnasInteresting. How 'bout the other sacred shapes? Are there any perfect cubes, pyramids, etc. in nature? I'm just curious--I read somewhere that the Greeks declared 5 of the shapes to be "perfect", but I don't remember which ones. (Or was it 6?)
Circloids, perhaps, but never exact mathematical circles.
Exact mathematic circles don't exist in nature.
Originally posted by SwlabrAn irrational number is a number for which we cannot provide an exact decimal representation.
Assuming you can draw two lines of length precisely 1, then you can draw a line of length precisely sqrt(2), by Pythagoras, which is an irrational number. However, surely it is impossible to draw a line of length sqrt(2), as we live in a finite universe?
A line can very easily have that length -- just like the circumference of a circle can have a given
length -- it's just that we lack the capacity to offer a finite rendering of that given length.
The line itself isn't infinitely long; only its numerical representation is infinitely long.
Nemesio
Originally posted by PinkFloydMarbles are spheroids, not circles (which are two-dimensional).
The marbles I had as a child existed as a glob of lots of circles, very tightly compressed and touching each other.
We're speaking very precisely here, PinkFloyd. For all intents and purposes, marbles are
spheres, but they are imperfect -- if you look at them under a microscope, you can see how bumpy
they really are. This is why, if you're being super-precise, you don't call them spheres, but spheroids.
But, I don't think the imperfectness of real-life mathematical has anything to do with the original
question, for reasons I stated above.
Nemesio
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Originally posted by PinkFloydThink of a cube. What is it made of? Atoms. Are atoms qubes? No. So how can you make a perfect cube of round objects?
Interesting. How 'bout the other sacred shapes? Are there any perfect cubes, pyramids, etc. in nature? I'm just curious--I read somewhere that the Greeks declared 5 of the shapes to be "perfect", but I don't remember which ones. (Or was it 6?)
Hence: There are no perfect mathematatical cubes in nature, only in mathematics.
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Originally posted by FabianFnasAtoms are not round (or spherical) nor do they fill space. Most of space is empty. I suppose that in a dense enough star in which atoms have collapsed down to a sea of protons and neutrons we could have a cube perfect to the width of a proton. I do not know if quarks have volume.
Think of a cube. What is it made of? Atoms. Are atoms qubes? No. So how can you make a perfect cube of round objects?
Hence: There are no perfect mathematatical cubes in nature, only in mathematics.
The real question then is what you really mean by stating that a mathematical shape 'exists in nature'. Must it be filled completely with matter whilst having empty space for a region outside its boundaries? If so, can space be truly filled with matter?
But in the original question I see no reason to believe that there can never be two quarks that are an irrational number of centimetres apart.
Originally posted by twhiteheadIn your first paragraph I can only see that you agree with the idea that there are no perfect cubes in universe. Every one imaginable is distorted in one way or another.
Atoms are not round (or spherical) nor do they fill space. Most of space is empty. I suppose that in a dense enough star in which atoms have collapsed down to a sea of protons and neutrons we could have a cube perfect to the width of a proton. I do not know if quarks have volume.
The real question then is what you really mean by stating that a mathematic ...[text shortened]... believe that there can never be two quarks that are an irrational number of centimetres apart.
In the second paragraph. Perhaps that there are, but you will never know when.
Originally posted by twhiteheadIn the Standard Model, all particles have no volume.
Atoms are not round (or spherical) nor do they fill space. Most of space is empty. I suppose that in a dense enough star in which atoms have collapsed down to a sea of protons and neutrons we could have a cube perfect to the width of a proton. I do not know if quarks have volume.
The real question then is what you really mean by stating that a mathematic ...[text shortened]... believe that there can never be two quarks that are an irrational number of centimetres apart.