Point and Line
Date: 04/07/2001 at 20:13:55 From: Shiva Kalyanaraman Subject: A philosophical mathematical question I'm in high school, but this problem has really nothing to do with school, it's just been bothering me for a while. A point has no dimension (I'm assuming), and a line, which has dimension, is a bunch of points strung together. How does something without dimension create something with dimension? Going even farther, a point essentially is nothing, because it has no dimension. My question is, How does a bunch of nothings (a point) create a something (a line)? Thanks for taking your time with my question.
Date: 04/08/2001 at 01:35:04 From: Doctor Jordi Subject: Re: A philosophical mathematical question Hi Shiva. Let me try to address your philosophical concern. The thing is, it's not just a bunch of nothings adding to something. It's infinitely many nothings adding to something. Infinity is a strange creature, to be sure, but there's another way of thinking about lines that does not involve your adding anything infinitely many times. A line is a location that only extends in one dimension. It goes infinitely long one way and infinitely long the other way. Between any two points in the line, there are infinitely more points. Alternatively, you can think of a line as a single point that moved in two opposite directions forever. Actually, this whole idea of motion spurred another branch of mathematics called calculus. All these ideas will become a bit clearer once you take calculus. Calculus is all about toying with infinities, or with their counterparts, limits. Meanwhile, you may want to look over our archives to get a little food for thought: http://mathforum.org/dr.math/faq/faq.large.numbers.html Scroll down to the part about infinity and read a bit around there. Poke around the archives, as well; there's plenty of interesting stuff all over to learn. I hope you like it! - Doctor Jordi, The Math Forum http://mathforum.org/dr.math/
Date: 04/08/2001 at 17:57:05 From: shiva kalyanaraman Subject: Re: A philosophical mathematical question I still have a problem with the explanation used in calculus. According to calculus we are supposed to imagine a line as a stretched point. We all know that there are points on a line, and therefore you're saying there are points within points, and going even farther, since a line is a stretched point, there are lines within lines. Please explain this to me. Thanks a lot.
Date: 04/08/2001 at 18:49:03 From: Doctor Jordi Subject: Re: A philosophical mathematical question Hello again, Shiva. Points within points? I am not very sure what you mean. Between any two points, ANY two points, no matter how close they may be, there are always infinitely many more points. Of course, if two points are on the same spot, then they are the same point. Are you comfortable with the notion of a number line? Numbers are useful things for locating points on a straight line. Take any line and put a zero on it somewhere. Then take an arbitrary distance in either direction on the line and mark that point as 1. Then keep taking the same number of distances in the same direction and you will get to 2, 3, etc. Go back to 0 and take the same distance in the opposite direction. You will get -1, -2, -3, etc. That's a number line. Do you want to locate a point on this number line? Just give the number that's assigned to it. Every point on this number line has one and only one real number assigned to it. There are infinitely many numbers, and infinitely many points. Between any two points there is another point. Between any two real numbers there is another real number. Do you accept the notion that between any two points (numbers) you always have another point (number), hence infinitely many points? How about the idea that if you have have a number (point), you can always make a number (point) that is larger (farther away from point zero) than that number? These are two guiding principles for infinity. Two points are the same point if they have the same number assigned to them. No number can be assigned to two different points, nor can two points be assigned to the same number. Are you still comfortable with these ideas? Two lines in Euclidean geometry can have either no points in common, a single point in common, or infinitely many points in common. If they have at least two points in common, then they have infinitely many in common. If that happens, the two lines are indistinguishable. If two lines share two points, then they are the same line. Lines contain line segments. Such segments on a number line can be marked off as an interval between any two points, so we can talk about the line segment that contains all the points between 4 and 6, including 4 and 6. You can unambiguously determine whether any point you give me is in this line segment. Is pi in this segment? No, because pi is approximately 3.1416, and that's outside 4 and 6. How about 5? Yes, 5 is obviously in that interval. Is sqrt(17) in there? Yes, it is. (Why?) I'm not sure whether I am addressing all your concerns. As I said before, infinity is a funny concept; a bit mind-boggling. If you still have doubts, please let me know about them. I'll try to make them clear. - Doctor Jordi, The Math Forum http://mathforum.org/dr.math/
Date: 04/08/2001 at 19:00:00 From: shiva kalyanaraman Subject: Re: A philosophical mathematical question I'm sorry, I guess I did not explain my question very well. In calculus they imagine a line as a stretched point. A line has points in them. Well, since this stretched point is being referred to as a line, it has to inherit the property of lines, one being that it contains points in them. Therefore you are saying this line, which is actually a stretched point, has points in it. Therefore, you are saying that this stretched point has points in it. I hope that cleared up my question.
Date: 04/08/2001 at 19:06:38 From: Doctor Jordi Subject: Re: A philosophical mathematical question Now we're just playing word games. :) When you "stretch" the point, as you envision it, it no longer is a point. It is a line. You have altered it. It is something different. You can't give it the same name, otherwise there would be ambiguity. Stretched points (a.k.a. lines) contain points, yes. A point is not the same as a stretched point. "Stretched point" is another name for a "line." The reason we call it a "stretched point" is just in order to make it clearer how the line was made from the point. You must be careful not to confuse the abstractions with the words we use to describe them. - Doctor Jordi, The Math Forum http://mathforum.org/dr.math/
Date: 04/08/2001 at 19:42:46 From: shiva kalyanaraman Subject: Re: A philosophical mathematical question Okay, sorry to bug you with all these questions, but here's the next one. If you say to imagine a line as a stretched point, that brings me right back to my first question. All it does is redefine a line in a different way. Furthermore, how can you stretch something without dimensions? How can you consider a point anything (it has no dimensions, so it's nothing, I guess). I'm starting to come to the conclusion that this is a question without an answer. My answer would be that a point is simply a made-up thing that helps us deal with graphical math, and really doesn't exist. Anyway, please try to answer me. Loved the discussion, by the way. Thanks again.
Date: 04/09/2001 at 06:37:28 From: Doctor Jordi Subject: Re: A philosophical mathematical question Hello again, Shiva. I am very glad to receive another question from you. Your questions are very stimulating and I am enjoying them very much because they are making me think about certain concepts in ways I don't normally think. Yes, we have redefined line in a different way. Does that bother you? I hope it doesn't, because you will see this happen very often in mathematics: the same concept can be defined in several different ways. It might be important at this point to discuss intuition. Here is a question for you: Does it really matter what you and I have in mind when we say "line," as long as we come to the same conclusions about it? Here are some questions about the essential properties of lines in Euclidean geometry. Tell me if you disagree with any of the answers I give. Can a line be extended in any direction? Yes. Can a line contain points? Yes. How many points does a line contain? At least two. Can it have more than two? Yes. What is the least number of dimensions necessary for a line to exist? One. Can a line be used to graphically represent the real numbers? Yes. Can two completely different lines contain the same two points? No. Can two lines contain only two points in common? No. Can two lines be anything other than parallel, intersecting, or skew? No. Do you disagree with any of those answers? If you don't, then we're in business, because it looks as if you and I are thinking about the same thing; we are having the same intuition about it. We agree in the realm of mathematics. If you do disagree with any of the above answers, then something is amiss: we have not yet arrived at an unambiguous definition of line and we'd better go back to our previous definitions and see where the ambiguity arose. Can you think of any other questions about lines? More importantly, can you think of a question about lines in which you and I might not agree? I can only think about one question you and I might disagree upon, or might give multiple correct answers to: "What is a line?" So, how do we resolve that? In mathematics, in general, we resolve any ambiguity by giving a definition of an abstract object from which the properties can be deduced. One knows lines by how they behave. To partially misquote from a movie: "a line is as a line does." This is a utilitarian approach to the philosophy of mathematics. Stuff is just as good as the abstract use we can give to it. Tell me, bearing that in mind, does it really matter what you and I are thinking about when we say "line" as long as we can both logically arrive at the same conclusions about lines? Or how about this question, not really a mathematical one at all, and something I'm quite sure that must have crossed your mind at some point: How do you know that what I call black I perceive in the same way you do? Maybe I see everything in negative colours, so what I call black you would see as white if you were to see through my eyes. How do you know that I see the same colours you see? Here's the biggie... What difference does it make either way? I won't give you the answer to that question. Just roll it around in your head for a while. As for your other questions... How can you stretch a point into a line? Quite simple. Just imagine that you can. If you can imagine it, it can be done. A point is dimensionless, yes. That means that if you took a ruler and measured it any way you wanted to, you would always obtain 0 inches. Exactly 0 inches. Now, how do you stretch that nothing? It is not nothing. It is a point. It would be nothing if there were no point, but there is a point, hence there is something! Now, grasp that point. C'mon, do it. Grab it with each of your two thumbs and forefingers. Now, streeeeeeetch! Wow, that was a lot easier than it seemed, wasn't it? Does a point exist? Well, apparently it exists in our imaginations, because we have been going on at great length discussing it for the past few e-mails. How can it not exist if you and I are both imagining it? Perhaps you are saying that there is no such thing as a dimensionless thing in our reality? Tell me, what is the difference between the reality around you and the thoughts in your head? Do numbers exist? How about all of mathematics, does that exist or not? What does it mean to exist, anyway? I'll give you one of my favourite mathematical definitions of existence: Existence: n. 1. Freedom from contradiction. There you have it. If something you can imagine does not contradict anything you imagined before it, then it exists. Another quote from a movie, this time properly quoted: "Free your mind." - Doctor Jordi, The Math Forum http://mathforum.org/dr.math/
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