UnderGround Forums
 

PhilosophyGround >> Math/philosophy ?


11/25/03 11:19 PM
Ignore | Quote | Vote Down | Vote Up
marck
Send Private Message Add Comment To Profile

Edited: 25-Nov-03 11:35 PM
Member Since: 01-Jan-01
Posts: 2124
 
Also posted on the OG. I have always wondered how it is that a line has the property of length when a line is defined as a set of points that extend indefinately that has no height or width, but DOES have length. But points, which make lines, have no length, width or height. So how can something made up only of things without length have length?
11/26/03 2:16 AM
Ignore | Quote | Vote Down | Vote Up
Dogbert
Send Private Message Add Comment To Profile

Edited: 26-Nov-03
Member Since: 01-Jan-01
Posts: 11643
Well, the simple answer is: by definition. You basically define a function that associates with two points (thought of as end points) a number and call that number the length of the line between these points. The more fundamental reason is that modern mathematics is built upon set theory, so we have to think of geometric objects as sets of points and try to find definitions using only points that roughly coincide with our intuition of what space looks like. Some properties the preserve, some intuitions (like every line being thick) are not preserved.
11/26/03 2:25 PM
Ignore | Quote | Vote Down | Vote Up
Socrates
Send Private Message Add Comment To Profile

Edited: 26-Nov-03
Member Since: 02-Aug-01
Posts: 353
Simply put, lines are not "made of points" and cannot be concieved of as such. I believe that a huge problem with modern math, which would say otherwise, is that it is not concerned with being intelligible.
11/26/03 2:28 PM
Ignore | Quote | Vote Down | Vote Up
Dogbert
Send Private Message Add Comment To Profile

Edited: 26-Nov-03
Member Since: 01-Jan-01
Posts: 11652
"Simply put, lines are not "made of points" and cannot be concieved of as such." Please tell what lines "really" are. "I believe that a huge problem with modern math, which would say otherwise, is that it is not concerned with being intelligible." Read the American Mathematics Monthly, presentation does matter in Math
11/26/03 7:54 PM
Ignore | Quote | Vote Down | Vote Up
marck
Send Private Message Add Comment To Profile

Edited: 26-Nov-03 09:57 PM
Member Since: 01-Jan-01
Posts: 2127
"Simply put, lines are not "made of points" and cannot be concieved of as such." I have looked up several definitions on the internet and defining a line as "a set of points that extend indefinately..." is not uncommon.
11/28/03 5:01 PM
Ignore | Quote | Vote Down | Vote Up
ChemicalSage
2 The total sum of your votes up and votes down Send Private Message Add Comment To Profile

Edited: 28-Nov-03
Member Since: 22-May-02
Posts: 1326
Dude, a line segment is what you get when you connect two points. continue that onward, and you get a line. Think: point - 1 dimensional. line - 2 dimentional. plane - 3 dimentional. I can't believe you've been thinking about this for any length of time.
11/29/03 3:30 PM
Ignore | Quote | Vote Down | Vote Up
Socrates
Send Private Message Add Comment To Profile

Edited: 29-Nov-03
Member Since: 02-Aug-01
Posts: 354
Dogbert and marck- I believe that the orginal definition of a line (as defined by Euclid in the first book of his Elements) is "a breadthless length". Doesn't the claim that a line is "made up of points" lead to Zeno's paradox? I don't think that the presentation of modern math is the problem. I believe that its fundamental concepts, like the existance of "actual infinities", are simply unintelligible.
11/30/03 1:59 AM
Ignore | Quote | Vote Down | Vote Up
marck
Send Private Message Add Comment To Profile

Edited: 30-Nov-03 02:39 AM
Member Since: 01-Jan-01
Posts: 2142
"Dude, a line segment is what you get when you connect two points. continue that onward, and you get a line. Think: point - 1 dimensional. line - 2 dimentional. plane - 3 dimentional. I can't believe you've been thinking about this for any length of time." I honestly can't tell if this is an attempt at some kind of self-parody or if you're being serious. "Doesn't the claim that a line is "made up of points" lead to Zeno's paradox?" I think it probably would, since there are an infinite amount of points between the ends of any line segment. But points don't have any extension whatsoever, so they wouldn't move you forward at all. Whereas with numbers, they do at least move you further than the last number. This problem is similar though in that it 'apparently' sheds light on a problem when dealing with abstract things(numbers and points)and their relationship to the concrete. Many definitions do start with "A set of points that extend...". My problem is only with that definition.
11/30/03 3:34 AM
Ignore | Quote | Vote Down | Vote Up
RGoodfellow
Send Private Message Add Comment To Profile

Edited: 30-Nov-03
Member Since: 03-Jul-02
Posts: 5103
"My problem is only with that definition." I think you're right to have a problem with that particular definition. That definition appears to be a poor/inadequate verbal definition of a mathematic function. Sometimes math is best defined using the language of mathematics (a line is a function: f(x) = m * x + c), rather than a standard cultural language such as English, French, Spanish, etc. It seems like the definition that you have a problem with oversimplifies what a line is, and as a result, creates a definition that falls apart under close linguistic scrutiny.
11/30/03 9:00 PM
Ignore | Quote | Vote Down | Vote Up
Socrates
Send Private Message Add Comment To Profile

Edited: 30-Nov-03
Member Since: 02-Aug-01
Posts: 356
marck- I was thinking that if there are an infinite number of points between A and B, then one would have to reach the point half way between A and B, say C, before one got from A to B. But you also have to get to the halfway point between A and C, before you got to C. continue ad infinitum. So, an infinite number of points means that there are an infinite number of line segments. And one cannot traverse an infinite number of line segments. So one can never get from A to B. Motion is impossible. All is rest. All is one. :)
11/30/03 9:19 PM
Ignore | Quote | Vote Down | Vote Up
FudoMyoo
Send Private Message Add Comment To Profile

Edited: 30-Nov-03
Member Since: 01-Jan-01
Posts: 6707
well this is at the top already..
12/1/03 9:46 AM
Ignore | Quote | Vote Down | Vote Up
Dogbert
Send Private Message Add Comment To Profile

Edited: 01-Dec-03
Member Since: 01-Jan-01
Posts: 11661
"I believe that the orginal definition of a line (as defined by Euclid in the first book of his Elements) is "a breadthless length"." His definition plays no role in the text, so we can just forget it. "I don't think that the presentation of modern math is the problem. I believe that its fundamental concepts, like the existance of "actual infinities", are simply unintelligible." Oh, the joys of modern mathematics. "Doesn't the claim that a line is "made up of points" lead to Zeno's paradox?" No, the problem is simply how you define speed then. And Leibniz and Newton have shown how one can do that. "I was thinking that if there are an infinite number of points between A and B, then one would have to reach the point half way between A and B, say C, before one got from A to B. But you also have to get to the halfway point between A and C, before you got to C. continue ad infinitum." And what is the problem? "So, an infinite number of points means that there are an infinite number of line segments. And one cannot traverse an infinite number of line segments." Why not. The smaller the segment, the less time it takes to pass it. So it´s just a problem of an infinite sum, a standard calculus problem. "So one can never get from A to B. Motion is impossible. All is rest. All is one. :)" Diogenes proved the existence of movement. By moving himself...
12/1/03 9:52 AM
Ignore | Quote | Vote Down | Vote Up
Dogbert
Send Private Message Add Comment To Profile

Edited: 01-Dec-03
Member Since: 01-Jan-01
Posts: 11662
On Zeno's paradox: http://www.mathacademy.com/pr/prime/articles/zeno_tort/index.asp
12/1/03 12:27 PM
Ignore | Quote | Vote Down | Vote Up
Socrates
Send Private Message Add Comment To Profile

Edited: 01-Dec-03
Member Since: 02-Aug-01
Posts: 357
Dogbert- I don't understand your comment about Euclid's definition. I am not pursuaded by your appeal to Newton's definition of speed, because it is Newton's use of limits (ie actual infinities) that I am saying is unintelligible. This is the basis of calculus. Perhaps I should have made that critique more explicit. I know that this is a standard problem in calculus, but I am claiming that the basis of calculus is non-sensical. On a conceptual level, Newton relies on "taking things to the limit" (which he also says can never be reached). The point beyond which a process cannot go is treated as the end of the process; and this entails positing actual infinities. This is what I believe is concepually flawed, and will lead to a paradox, even if calculus claims it does not. Calculus, although impressive in many respects, remains unintelligible. I don't think you can refute a critique of calculus by appealing to the results of calculus. I did not look at your link, but I will do so soon. I should say that I don't think Zeno's paradox is sound. Aristotle answers it quite clearly. I believe, however, that calculus' use of actual infinities leads to the paradox.
12/1/03 4:27 PM
Ignore | Quote | Vote Down | Vote Up
Dogbert
Send Private Message Add Comment To Profile

Edited: 01-Dec-03
Member Since: 01-Jan-01
Posts: 11672
"I don't understand your comment about Euclid's definition." Euclids axioms consist of explanations of the terms involved, like the one you gave, and statements about the relationships between these. In the post-Hilbert view, only the latter matters. It doesn´t matter what the relvent objects "are" since that plays no role in any proof. "I know that this is a standard problem in calculus, but I am claiming that the basis of calculus is non-sensical. On a conceptual level, Newton relies on "taking things to the limit" (which he also says can never be reached). The point beyond which a process cannot go is treated as the end of the process; and this entails positing actual infinities." Well, in the early years of calculus people didn´t know what they were doing from a logical point, as smart observers like Berkeley realised. Solid foundations came with the work of Cauchy and Weierstrass, who don´t use any "actual infinities", I will later explain what they did. Another foundation is given by the logican Abraham Robinson who based calculus in a sound way on non-standard numbers, numbers that contain infinitely small and big numbers. I think this approach is very elegant and beautiful, but since you are apposed to actual infinities, let´s look at the work of Cauchy/Weierstrass, the standard approach today. A sequence is a set of numbers, ordered by the natural numbers. In other words there is a first number, a second, a third and so on, a_1, a_2, a_3, a_4... This can be explicitely given by a formula, if you want so (from a technical pov unnecessary). Now the definition of a limit is just that "a" is the limit of the sequence if for every epsilon bigger 0, there is a natural number N, such that from a_n on the distance to a is smaller than epsilon. This is only about usual inequalities, there is no actual infinity involved. Calculus is completely sound from a logical point of view. "This is what I believe is concepually flawed, and will lead to a paradox, even if calculus claims it does not. Calculus, although impressive in many respects, remains unintelligible. I don't think you can refute a critique of calculus by appealing to the results of calculus." What specific part do you think is unintelligible. "I did not look at your link, but I will do so soon. I should say that I don't think Zeno's paradox is sound. Aristotle answers it quite clearly. I believe, however, that calculus' use of actual infinities leads to the paradox." You think there is no movement whatsoever?
12/1/03 8:50 PM
Ignore | Quote | Vote Down | Vote Up
Socrates
Send Private Message Add Comment To Profile

Edited: 01-Dec-03
Member Since: 02-Aug-01
Posts: 358
I don't know who Hilbert is, but I don't think I agree with his position. But I will refrain from further comment now. I'd like first to try to understand your explanation of calculus. "A sequence is a set of numbers, ordered by the natural numbers." Ummm... first, what is a "set of numbers"? How many numbers are in a set? What are "natural numbers"? Are there "un-natural numbers"? I think I need to understand these things before I proceed. "What specific part do you think is unintelligible" Actual infinity. Newton used this when he develpoed calculus, and (from what I've seen), all modern math after him uses it in one way or another. It leads to all sorts of problems. "You think there is no movement whatsoever?" I said that "I don't think Zeno's paradox is sound". That means that I disagree with it, and thus that I do think there is motion. Aristotle points out that motion (and space and time) is continuous, not discrete. Treating motion (or space or time) as discrete leads to the paradoxes.
12/2/03 8:06 AM
Ignore | Quote | Vote Down | Vote Up
Dogbert
Send Private Message Add Comment To Profile

Edited: 02-Dec-03
Member Since: 01-Jan-01
Posts: 11679
"Ummm... first, what is a "set of numbers"? How many numbers are in a set? What are "natural numbers"? Are there "un-natural numbers"?" Well, just think of a collection. NAtural numbers are 1,2,3,4,5... "Aristotle points out that motion (and space and time) is continuous, not discrete." What does that mean?
12/2/03 9:39 PM
Ignore | Quote | Vote Down | Vote Up
Socrates
Send Private Message Add Comment To Profile

Edited: 02-Dec-03
Member Since: 02-Aug-01
Posts: 359
If a set of numbers is ordered by natural numbers (which are infinite), does that mean that a set is "a collection of an infinte number of numbers"? "What does that mean?" In the simpliest terms, by discrete I mean composed of blocks, and by continuous I mean not composed of blocks (the difference between a stone wall and water).
12/3/03 2:43 AM
Ignore | Quote | Vote Down | Vote Up
Dogbert
Send Private Message Add Comment To Profile

Edited: 03-Dec-03
Member Since: 01-Jan-01
Posts: 11705
"If a set of numbers is ordered by natural numbers (which are infinite), does that mean that a set is "a collection of an infinte number of numbers"?" Well, a set ordered by infinitely numbers is infinite, yes. "In the simpliest terms, by discrete I mean composed of blocks, and by continuous I mean not composed of blocks (the difference between a stone wall and water)." Well, points are not blocks, so where do they fit in?
12/3/03 2:52 PM
Ignore | Quote | Vote Down | Vote Up
Socrates
Send Private Message Add Comment To Profile

Edited: 03-Dec-03
Member Since: 02-Aug-01
Posts: 360
Then hasn't an actual infinity slipped in under the phrase "a set of numbers, ordered by the natural numbers"? I think I need to clarify. The term discrete is used in two ways. First, a thing is called discrete when it is an individual unity. In this way, a point is discrete. However, things are also called discrete when they are composed of these individual unities. This is a very different meaning. In this way, a point is not discrete.
12/4/03 8:09 AM
Ignore | Quote | Vote Down | Vote Up
Dogbert
Send Private Message Add Comment To Profile

Edited: 04-Dec-03
Member Since: 01-Jan-01
Posts: 11706
"Then hasn't an actual infinity slipped in under the phrase "a set of numbers, ordered by the natural numbers"?" No that´s just using the language of set theory. If a set contains all elements of a potential infinite, it is actual infinite. It´s just important that there is a first one, a second one and so on. "I think I need to clarify. The term discrete is used in two ways. First, a thing is called discrete when it is an individual unity. In this way, a point is discrete. However, things are also called discrete when they are composed of these individual unities. This is a very different meaning. In this way, a point is not discrete." So lines don´t consist of points. Do squares conmsist of points?
12/4/03 12:23 PM
Ignore | Quote | Vote Down | Vote Up
Socrates
Send Private Message Add Comment To Profile

Edited: 04-Dec-03
Member Since: 02-Aug-01
Posts: 362
"No that´s just using the language of set theory. If a set contains all elements of a potential infinite, it is actual infinite." Forgive me; I'm not trying to sound argumentitive, but your last statement confuses me. Just to be quite clear, according to what you said above, a "set of numbers, ordered by the natural numbers" would be an actual infinite, right? "So lines don´t consist of points. Do squares conmsist of points?" No, I would say a square consists of four equal lines segments, which intersect such that they form four right angles. Technically, the intersection of two lines might form a point (I don't quite remember), so you might be able to say that a square also consists of four points. But this seems superfluous, because the four points are covered by the phrase "intersecting lines".
12/4/03 2:43 PM
Ignore | Quote | Vote Down | Vote Up
Dogbert
Send Private Message Add Comment To Profile

Edited: 04-Dec-03
Member Since: 01-Jan-01
Posts: 11707
"Forgive me; I'm not trying to sound argumentitive, but your last statement confuses me. Just to be quite clear, according to what you said above, a "set of numbers, ordered by the natural numbers" would be an actual infinite, right?" Yes. "No, I would say a square consists of four equal lines segments, which intersect such that they form four right angles. Technically, the intersection of two lines might form a point (I don't quite remember), so you might be able to say that a square also consists of four points. But this seems superfluous, because the four points are covered by the phrase "intersecting lines"." So according to your definition, a square IS a line.
12/4/03 3:18 PM
Ignore | Quote | Vote Down | Vote Up
Socrates
Send Private Message Add Comment To Profile

Edited: 04-Dec-03
Member Since: 02-Aug-01
Posts: 364
Then it seems to me that you mis-spoke when you said "This is only about usual inequalities, there is no actual infinity involved." The sequence is, tacitly, an actual infinity. "So according to your definition, a square IS a line." I do not believe that this follows. You asked what a square consisted of (ie what it was made from). That is not the same as asking what it IS. A whole is not necessarily simply the sum of its parts. I said a square consists of lines. I might even be willing to say that a square consists of one line, bent in a particular way, if you prefer that. BUT because a square consists of a line does not mean that it is a line. I would say that a square is a certain shape, which is formed by the lines I described.
12/5/03 10:09 AM
Ignore | Quote | Vote Down | Vote Up
Dogbert
Send Private Message Add Comment To Profile

Edited: 05-Dec-03
Member Since: 01-Jan-01
Posts: 11709
"The sequence is, tacitly, an actual infinity." DO the numbers 1, 2, 3, 4... exist and with each number n the number n+1? If you accept this, you accept this "actual infinite".

Reply Post

You must log in to post a reply. Click here to login.