[QUOTE]Originally posted by addabox
Let's not make the mistake of implying that the instrumental imprecision of defining the smallest possible unit of space time to be used in defining the boundary between objects has anything to do with transcendental numbers, except insofar transcendental numbers might be used to describe some aspect of such a unit. The infinitude of irrationals and such is of a very different order of epistemology than the "fuzziness" of the world at very small scales, which in fact the point of "paradox" in the original post.
I agree to the extent of what I understand. CosmoNut is trying to ask what he is "missing". He has not defined a "proffessional" Paradox like the kinds the "scientific community" is dealing with.
CosmoNut, IMHO, this is what you are "missing"
1. You propose a situation whereby two objects are moving towards each other, each time moving half the distance between them. Yes, at some point in time, the space between them will become sooooo small that essentially Newtonian physics is no longer relevant, we get into the Quantum physics realm and all the "weirdness" associated with it - which we have tried to describe above. This small distance is not as small as you think, we can think in terms of one Angstrom. Ten billion angstroms equal 1 meter. So AFAIK there is no specific definition where Quantum physics take over but certainly once you hit 1 Angstrom and less Quantum mechanics and the weirdness of atomic- and subatomic-particle interactions start to apply. At this stage the "Newtonian-style view" of an electron orbiting the nucleus like the earth around the sun is total rubbish.
2. You made a huge jump
in tying together the maths side of things and the physics side of things. You started with saying, let's assume 0.000000000000....000001 approximates to essentially zero. Then you JUMP
to the conclusion of these numbers relating to a scale of space between two objects
. Then you jump again to the conclusion that "I approximate 0.0000000.....0000001 to zero therefore the real-world space between two objects at that level must also be zero". So like it has been said before, currently in our understanding there is a Physics "real world" situation and we use Applied Maths to tackle the issue. Going the other way round does not make sense in this case because you are taking an abstracted Mathematical idea and then "dumping" it onto the real world. The title of this thread itself is problematic because the way you defined it -- taking an abstracted mathematical situation and then "duct-taping" it onto a physical situation.
3. "Decimals can't be infinite" ... Who says so? If we just look at the Mathematical implications of that, again this is the third issue of what you are "missing". Why is 1+1 = 2? Only because by convention. Mathematicians do all sorts of weird stuff in the "pure maths" area. For example, depending on what they are trying to do and the conventions of the field they are working with, they can say Decimals are infinite or decimals are not, they can say all sorts of things, depending on the problem they are working on. Just take Pi - we think of it as 22/7 and it just goes on forever. But actually 22/7 is GREATER than Pi.http://en.wikipedia.org/wiki/A_simpl...2/7_exceeds_pi
I would assume then from this that those programs that use computers to derive all the values of pie, do not just take 22 and divide it by 7. They use other formulas to work out the millions of digits or whatever. Stupid irrational numbers, so irrational
Disclaimer: Again, as is my understanding at this point in time and I can't believe my brain is still handling at least 10% of this mindf*ck stuff.