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May 29th, 2015, 08:48 PM   #61
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Originally Posted by Tau View Post
That's fuzzy. So zero doesn't have position relative to 1 for example, or indeed in relation to the entire number line being in the dead center?
That's not fuzzy, all numbers have a position arbitrarily fixed in relation to zero: walking in the numerical line is just a cute way to say that you have a mathematical operation (or a function) that takes you from zero to some point. This point can also be zero, no problem here.

It does not matter if the concept of point is strange, if a point exist in reality, i.e., if some geometric objects can have no parts at all, or not. First of all, reality is already contentious. What is reality? Your mind? God's creation? An electrical impulse in the brain? A dream? Who knows, but mathematics does not care.

Given that we speak meaningfully about objects, at that they have some relations between them, the mathematician derives consequences based on definitions and concepts which are accepted by all rational beings. Again, what is a rational being? Contentious, "fuzzy", whatever, but hey, if you think like that you cannot even wake up from bed in the morning by wondering if the coffee you will drink is "real coffee" or snake oil. The best we can do is to rule out bad questions in favor of good questions, and good questions are questions which can have some kind of explanation.

Now, good answers are answers which are economical, simple/clear, plausible and general. This is how mathematicians solve problems. So geometry is a modeling of the real world. You will never encounter a circle with length $\displaystyle \pi$, but all drawn circles are a physical model of this abstract concept, and it works nicely in our medium-sized homo sapiens world, that's all. What are circles? I don't know, but I model the surrounding reality with this idea, because circles drawn fit such structure.

Last edited by skipjack; May 29th, 2015 at 09:23 PM.
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May 30th, 2015, 01:28 AM   #62
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This is hugely sloppy language! If you want to work with the integers, and then come across the equation $2x = 3$ where is $x$? In your scheme it must either be 1 or 2. With gaps between 1 and 2, $x$ falls in the gap.
You conjecture the gaps. There would be gaps if 1 and 2 were points. But they are differentials. 1.5 doesn't fall in any gap, it is contained in 2. Does the 200m race have a gap 1.5-way through the race, or a 200cm tall guy a gap 150cm high on his body? I can call it the race line or the tallness line and still wouldn't matter zilch that the number line is composed of lengths and not points. Give me zero and I return zero, give me one and we can run.

Last edited by Tau; May 30th, 2015 at 01:33 AM.
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May 30th, 2015, 05:40 AM   #63
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You don't have lines in real life that are built on the integers. They are at least rational, perhaps real. They still have "gaps" where lie the solutions to certain equations.

Don't forget that most of matter is empty space because most of an atom is empty space (and there are spaces between atoms too).

Not that this matters hugely, as the number line is a abstract visualisation designed to illustrate the number system. If you want to change the visualisation, it doesn't change how the number line behaves or how numbers themselves behave. If your visualisation doesn't match the behaviour of numbers you must expect to find contradictions. That's when you have to use the scientific method to work out what is wrong. Luckily for you great minds have considered the problem over the last 3000 years, so you can draw on some of their experience in identifying the source of your contradiction.
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May 30th, 2015, 07:18 AM   #64
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Originally Posted by v8archie View Post
You don't have lines in real life that are built on the integers. They are at least rational, perhaps real. They still have "gaps" where lie the solutions to certain equations.

Don't forget that most of matter is empty space because most of an atom is empty space (and there are spaces between atoms too).
Real life is not and can not (uncertainty principle) be as exact as math. So even if the planck length (the integer possibility) is not the unit of space and it's actually continuous, we will never for instance measure π to any degree we like, we will reach a physical limit. Its ongoing computation will forever be the privilige of pure math.

Those gaps are points or lengths, on the number line?
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May 30th, 2015, 07:52 AM   #65
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No. The gaps are other points. The gaps in the number line left by the rationals are the irrationals. The "gaps" left in the number "line" by the reals are the complex numbers.

I only mentioned real life to refute your comparison with real life. The difference between mathematics and physics is clear to me - and had little to do with number lines except by analogy.

Last edited by v8archie; May 30th, 2015 at 08:00 AM.
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