Is the Number Line Both Continuous and Porous?
Date: 05/13/2005 at 16:01:44 From: Holly Subject: Continuity of the Number Line vs. Infinite Porosity I am having trouble reconciling the idea that the number line is continuous and yet between every number is an infinite number of other numbers, and between each of those another infinite number, and yet, it is said the number line is DENSELY POPULATED. I must be the dense one here, because it seems to me it is INFINITELY POROUS as well as DENSELY POPULATED. The notion of "elbow room" in the number line called "dense" is making me nuts. Why is the line called dense instead of gappy? There are just as many gaps as numbers, aren't there? I only think about it. I don't have the formal math knowledge to write out equations about it, sorry. Thanks!
Date: 05/16/2005 at 11:29:01 From: Doctor Schwa Subject: Re: Continuity of the Number Line vs. Infinite Porosity Hi Holly, This is an interesting question. Where exactly are the gaps and the porosity that you see? Where's the elbow room? The "densely populated" is the word exactly for the reasons you say: no matter how close together two numbers might be, there's still TONS more numbers packed into the tiny space between them! So the line is truly full of numbers. You say there are just as many gaps as numbers, but I don't see the gaps. I think it's really hard to point at one. No matter where you point, there are still numbers in there! So that's why it's called "densely populated". If you want gaps, look at just the integers--there's a big gap between 0 and 1 with no numbers in it, for example. But when you have the rational numbers, those gaps start to evaporate, and when you go to the reals, there's even more numbers packed in to those spaces. Both the rationals and reals would be described as "dense". I don't see the gaps you refer to, so please help me out--where are they? I think I'm just not understanding the way you see the number line. I'd love to talk about this more, so please explain it to me! I don't think this is a case where you need lots of equations. Talking about what the words mean, and explaining what you see when you look at a number line should be just fine. Enjoy, - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/
Date: 05/16/2005 at 14:04:49 From: Holly Subject: Continuity of the Number Line vs. Infinite Porosity Dear Dr. Schwa: I am both elated and humbled that you answered me and are willing to discuss the idea of a gappy number line further. I have said that the idea of DENSE and GAPPY are in conflict. However, I do not see INFINITE GAPS - INFINITE NUMBERS as in conflict. I am conflicted about referring to the number line as DENSELY POPULATED and CONTINUOUS and having no mention of GAPPY. Referring to the number line as a "densely populated, continuous line with a infinite number of numbers and infinite number of gaps" seems more reasonable to me. As to seeing the gaps, even with an infinite number of numbers between, for example, the numbers 1 and 2, we must believe that each of these numbers (all infinity of them) is DISCRETE, yes? Each is separate, and can be no other, no matter how tightly packed against its infinite brethren. Didn't some math person (Dedekind or some such) make some sort of "cut" where he cut it cleanly, *between* two of the brethren, not taking a piece of either? He cut them AT THE GAP that exists between each number, is my thought. ............... <---- If I could push these closer, they would appear to be a solid line. Yet, there is a gap between each. If each point in a line could be named with its number, each would have its accompanying gap on either side. There must be a gap or they would "blend" into some monstrous single number. Well, that is the limit of my thoughts. Thank you for listening. Sincerely, Holly
Date: 05/16/2005 at 16:26:13 From: Doctor Schwa Subject: Re: Continuity of the Number Line vs. Infinite Porosity Hey, I think we all here find your type of question interesting. Perhaps it's a bit more of a philosophy question than a math question but a lot of the greatest mathematicians did a lot of philosophy, too. Your description is reasonable--it's just not what I see when I look at the line! I see numbers, and any "gaps" are filled in by more numbers, and any "gaps" between those are filled in by yet more numbers. It's just numbers everywhere! This is what a lot of 18th century mathematics was all about. Each number is itself, yes, and can be no other. I agree. But the "separateness" is the part I'd argue about. There's a technical definition, from topology, of what it would mean for the number line to be "discrete", and indeed you can IMPOSE a discrete view on the number line if you wanted to. So I'm not saying you're wrong! In the discrete topology, a number's "neighborhood" is by definition just the number itself. So it lives alone, separate, no matter how tightly packed it may be. At the same time, the NATURAL way to look at it is in terms of distance. A number's "neighborhood" consists of all the other numbers within some small distance of it, in the natural topology. And then no matter how small a neighborhood you look at, there are still infinitely many other numbers in the neighborhood! So this is a way of looking at the number line in which there are no gaps, just densely packed numbers all the way down. Indeed, it's called the "natural topology", which shows quite a big philosophical bias, as opposed to the "discrete topology", which is looked at by the mathematicians as being more of a curiosity than a really useful structure to impose. In the natural topology, there's not any real notion of separateness. Essentially, if you take any number x, and say "there's a gap next to it", you then need to answer how BIG that gap is--and of course its size must be 0. That's where thinking of things in terms of distance gives you a view of the number line in which there are no gaps. And the same with separateness: there's a sense, pretty obvious to both of us I think, in which 1 is separate from 2. And indeed, 2 makes the end of the gap that 1 begins. Now, answer a similar question in the real numbers: if there is a gap after 1, what number ends that gap? Or, if 1 is discrete, how far is it to the nearest neighbor? Dedekind cuts are very interesting little structures. They're actually used to CONSTRUCT the real numbers from the rationals. All the stuff I said above, about gaps, and neighborhoods, and in my previous message about having infinitely many numbers in each "gap" between two numbers, applies whether you're talking about fractions (rationals) or decimals (reals). What Dedekind did was to say that by splitting the rationals into two groups, one group being all the numbers less than something, and the other being all the numbers greater than something, you could nonetheless identify gaps in the rationals. For instance, you could look at the split of all the numbers that when squared are less than 2, and those that when squared are more than 2. You'd find that there's a split at some number near 1.4 ... near 1.41 ... near 1.414 ... and so on, but there'd be no exact fraction located AT the split. However, what Dedekind also showed was that the real numbers (decimals) are precisely what you need to fill in those gaps. After adding them to the system, there are no more gaps in the spaces created by the Dedekind cuts. >............... <---- If I could push these closer, they would >appear to be a solid line. Yet, there is a gap between each. If each >point in a line could be named with its number, each would have its >accompanying gap on either side. There must be a gap or they >would "blend" into some monstrous single number. This is the crux of the argument, indeed, and exactly what it took about a century of mathematicians and philosophers to work out. The interesting thing is that there really is a difference between real numbers (decimals) which are a sort of limit as that space gets arbitrarily close to 0, and what actually happens when the space IS 0 and the points are all on top of each other. VERY LONG decimal numbers that eventually stop are still expressible as fractions; INFINITE decimals may not be. Similarly, your dots, when squished together, will turn out to cover only the fractions; there'll still be all the nonrepeating infinite decimals uncovered (at least, as I envision the process). I'm trying to avoid getting mathematical as much as possible, but as I see it, it goes something like this: Take all the integers 0, 1, 2, 3, 4, ... and let's look at 9378 as one example. But the whole list of integers are your infinite list of . . . . . . . spaced out at intervals of 1 unit. Now divide them all by 10 to squish them together. Now you have 0, 0.1, 0.2, 0.3, ..., 937.8, ... And keep dividing by 10 to squish. If you name any terminating decimal, say 0.9378, you can see when it'll get hit. And it'll keep getting hit forever after that! After 4 squishes, 9378 lands on 0.9378. After 5 squishes, 93780 lands on 0.9378. After 6 squishes, 937800 lands on 0.9378. So it's definitely covered! But no INFINITE decimal ever gets covered in your scheme. So as you keep squishing, there really ARE gaps left, in the Dedekind sense. Your . . . . . . ., as it squishes, only covers finite (terminating) decimals. But Dedekind cuts show how to construct the real numbers out of this to fill in the gaps. In summary, then: There are two types of gaps that I see. One is the big finite-sized gaps like the ones in the integers, the gap between 0 and 1. The other is the gap like you see in a Dedekind cut, where you can group the fractions into two sets, one below something and one above something, and then you find that there is no fraction equal to the something! There must be gaps in the fractions, which can be filled in by these somethings (like the square root of 2, or pi, for example). When you fill them in, then you have the real numbers (and in fact Dedekind cuts are often used, around junior year for math majors in college, as an explanation of what the real numbers represent). Then, once the real numbers are filled in, I don't see gaps any more, and I've tried to help you understand why not--but maybe no persuasion will work. It's always allowed to have your own view of things! However, mathematicians find the definition of neighborhood so useful that they are unlikely to adopt the alternative (discrete) definition where every number is a neighborhood by itself. When working with the real numbers, it's almost always more fruitful to say "a neighborhood is any interval centered on your number" and though neighborhoods can be as narrow as you like, they still contain infinitely many numbers. Feel free to keep writing back, this is fun. Enjoy, - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/
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