Autolykos,
I am not saying you should go away. Indeed I am the one who left the field, returning only as a personal favor to you.
OK, in detail:
Right - a point is an abstract mathematical construct. It has zero extension in any dimension under consideration. Therefore, any extension can be said to consist of an infinite number of points. Yet the extension is (presumably) finite. How does this make sense?
The extension is not finite in the sense of of containing finitely many points. What is finite is a numeric value we assign to a particular set of points. For example, to the the infinite set of points between the point we call 4 and the point we call 5 we assign the number 1. To make sure we do not confuse this 1 with the point we call one on the number line, we call it Length 1. Length is thus a function from a set of points to the set of numbers. Note that many functions can be assigned to the same set of points. For example, the set of points on the number line between 4 and 5 has Length 1, but Area zero [and Volume zero].
I think it makes sense if one stops thinking of a line, area, volume, etc. as being composed of points at all.
I find it hard to see how you can do this. Consider the line connecting zero to one. Is not the point 1/2 on the line? Of course it is, as are all the other points between zero and one. This is the point of view Euclid and everyone else has adopted that I was referring to.
One can instead think of these things as separations or relations between/among points. Indeed, the very notion of location implies a relation to something else.
This sounds like it contains the germ of the idea of function I mentioned above, plus extraneous stuff [e.g that a line is not composed of points] that has been rejected by Euclid and everyone else, as above.
Geometry isn't even necessary to think about this. How many rational numbers are there between 0 and 1? Infinitely many. Yet the extension between 0 and 1 is said to be finite. Is there a paradox here? I don't see how.
Indeed, given the explanation above.
But Zeno's Paradox is about something else. He is not talking about points at all, but rather about intervals of finite length. All his paradoxes notice that in some way a finite number, like 1, can be "sliced" into an infinity of other finite numbers. Thus an interval of length 1 can be sliced into the following infinite number of finite intervals: 1/2, 1/4, 1/8, 1/16 etc. And yet 1 is not the sum of these numbers, because summation can only be performed on a finite amount of numbers.
He parlays this observation into the various paradoxes he lays out.
The reason I was short is because all this, as well as the refutation of some ridiculous notions tossed around in this thread as if they were brilliant new pearls of wisdom, is standard material in a second or third year math course in a university. Which means there is a lot to explain with a lot of prerequisites on the one hand, and on the other hand nothing original or new to someone who studied math. Which is why I referred you to invest a year or so of your time studying the stuff. The topic is usually called something like "construction of the real numbers from the integers". Anyone to whom that phrase doesn't ring a bell [=Oh, yes, I remember we did that in some course] is missing the background needed to discuss this topic intelligently, [as evidenced by the contributions of some people to this thread].
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It's easy to refute an argument if you first misrepresent it. William Keizer
Another equivocation: between "on the line" and "is part of the line."
Disclosure: Pure mathematics was my major, I did graduate research in number theory as well. I failed to question the sneaky stuff at the time, but apparently so did just about everyone because the holes are not even well-hidden.
Why anarchy fails
Pure mathematics was my major
Impossible. Unless you got an F in every course.
Absurd.
Smiling Dave:Autolykos, I am not saying you should go away. Indeed I am the one who left the field, returning only as a personal favor to you.
With all due respect, your last paragraph seems to belie your benign intentions.
Smiling Dave:OK, in detail: Right - a point is an abstract mathematical construct. It has zero extension in any dimension under consideration. Therefore, any extension can be said to consist of an infinite number of points. Yet the extension is (presumably) finite. How does this make sense? The extension is not finite in the sense of of containing finitely many points. What is finite is a numeric value we assign to a particular set of points. For example, to the the infinite set of points between the point we call 4 and the point we call 5 we assign the number 1. To make sure we do not confuse this 1 with the point we call one on the number line, we call it Length 1. Length is thus a function from a set of points to the set of numbers. Note that many functions can be assigned to the same set of points. For example, the set of points on the number line between 4 and 5 has Length 1, but Area zero [and Volume zero].
I didn't mean that the extension is finite in the sense of containing finitely many points. I meant that the extension has a beginning and an end. An infinite extension would have no beginning and no end.
Yes, of course length can be considered a function from a set of points to the set of numbers. But here you're necessarily talking about infinite sets of points. It makes more sense to me to consider the notion of extension to be more basic than the notion of a point, in which case the latter is derived from the former. Note also that extension covers both length and distance.
Smiling Dave:I find it hard to see how you can do this. Consider the line connecting zero to one. Is not the point 1/2 on the line? Of course it is, as are all the other points between zero and one. This is the point of view Euclid and everyone else has adopted that I was referring to.
I don't see how it's so hard. Instead of having to consider an infinite set of points, you only have to consider two, in this case 0 and 1.
Smiling Dave:This sounds like it contains the germ of the idea of function I mentioned above, plus extraneous stuff [e.g that a line is not composed of points] that has been rejected by Euclid and everyone else, as above.
With all due respect, why should I care whether something has been rejected by Euclid and "everyone else" per se? Either something makes logical sense or it doesn't, right?
Smiling Dave:Indeed, given the explanation above. But Zeno's Paradox is about something else. He is not talking about points at all, but rather about intervals of finite length. All his paradoxes notice that in some way a finite number, like 1, can be "sliced" into an infinity of other finite numbers. Thus an interval of length 1 can be sliced into the following infinite number of finite intervals: 1/2, 1/4, 1/8, 1/16 etc. And yet 1 is not the sum of these numbers, because summation can only be performed on a finite amount of numbers. He parlays this observation into the various paradoxes he lays out.
Yet an infinite geometric series is said to result in either infinity or in a finite number. If the latter case holds, the series is said to be "convergent". I'm sure you already know this though.
One way around Zeno's paradox is to point out that each sub-interval can also be broken down into an infinite series of intervals of finite length. Ultimately, you have an infinite sequence of intervals each with "infinitesimal" (i.e. zero) length. This is essentially the calculus problem of integration.
Smiling Dave:The reason I was short is because all this, as well as the refutation of some ridiculous notions tossed around in this thread as if they were brilliant new pearls of wisdom, is standard material in a second or third year math course in a university. Which means there is a lot to explain with a lot of prerequisites on the one hand, and on the other hand nothing original or new to someone who studied math. Which is why I referred you to invest a year or so of your time studying the stuff. The topic is usually called something like "construction of the real numbers from the integers". Anyone to whom that phrase doesn't ring a bell [=Oh, yes, I remember we did that in some course] is missing the background needed to discuss this topic intelligently, [as evidenced by the contributions of some people to this thread].
As I indicated at the beginning of this post, I find it hard to not interpret this paragraph as basically saying, "Shut up, go away, and spend the next year or so studying the construction of the real numbers from the integers and related math subjects, because right now you're too stupid to discuss this topic."
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I can't believe that vive la insurrection and I are the only people on this thread that resolved the paradox so far.
So you say...
It makes more sense to me to consider the notion of extension to be more basic than the notion of a point, in which case the latter is derived from the former. Note also that extension covers both length and distance.
If you start out with a definition of length to be the difference between the numerical value of the beginning and end points [for example the length of a line from 3 to 5 is 5-3=2], then you get into difficulties. Because the set consisting of the endpoints alone [the set {3,5}] will have the same length as the set of all points from beginning to end [the set of all points from 3 to 5]. Thus, a person buying paint to paint his whole fence will need the same amount of paint as one who intends to paint only the endposts.
Point is, you cannot have a meaningful definition of length without getting into the grubby details of points.
One way around Zeno's paradox is to point out that each sub-interval can also be broken down into an infinite series of intervals of finite length.
That does not resolve the paradox.
Ultimately, you have an infinite sequence of intervals each with "infinitesimal" (i.e. zero) length.
This is a huge confession of ignorance, AJ, I'm sorry. You never wind up with infinitesmal, or zero length.
"Shut up, go away, and spend the next year or so studying the construction of the real numbers from the integers and related math subjects, because right now you're too stupid to discuss this topic."
I do not think you are stupid, nor am I telling you to shut up or go away. But yes, you lack the backgriound to discuss this question.
Smiling Dave:If you start out with a definition of length to be the difference between the numerical value of the beginning and end points [for example the length of a line from 3 to 5 is 5-3=2], then you get into difficulties. Because the set consisting of the endpoints alone [the set {3,5}] will have the same length as the set of all points from beginning to end [the set of all points from 3 to 5]. Thus, a person buying paint to paint his whole fence will need the same amount of paint as one who intends to paint only the endposts.
I'm sorry but I don't see how your analogy necessarily holds. Where did I say anything about creating sets consisting of the endpoints alone vs. the endpoints and the infinite number of points between them?
Smiling Dave:Point is, you cannot have a meaningful definition of length without getting into the grubby details of points.
I'm sorry but I don't see how.
Smiling Dave:That does not resolve the paradox.
Well how not?
Smiling Dave:This is a huge confession of ignorance, AJ, I'm sorry. You never wind up with infinitesmal, or zero length.
You mean "Autolykos", not "AJ". And can you please be so kind as to once again explain yourself?
Smiling Dave:I do not think you are stupid, nor am I telling you to shut up or go away. But yes, you lack the backgriound to discuss this question.
Well guess what? I'm still going to discuss it. Nothing you can say or do will keep from that. Understand? I don't care if you think I lack the background to discuss it.
Sure, knock yourself out.
That's right. But I'd appreciate a response to the rest of my post.
Sorry, you lack the background to grasp my response.
Hey, plenty of other people here know math enough to answer your q's.
But I'll address one point, just cause /i'm a nice guy. You were saying that 1/[2X2X2x2...] , where 2 is multiplied by itself N times in the denominator of that fraction, eventually becomes zero. But it never does. It's an immediate result of the axiom of Archimedes, which itself is the result of the least upper bound axiom, which itself is the result of the construction of the real numbers from the integers. All of which is taught in standard math courses.
Oops. It's an immediate result of arithmetic. say 1/A=zero for some value of A. Multiply both sides by A and get 1=0, a contradiction. Therefore there is no such A.
I was taught that 1 divided by infinity is equivalent to zero. How do you square that with what you posted?
1 divided by infinity is undefined. Whoever told you that either did not know what he was talking about, or [more likely] was telling you enough to pass a test in basic calculus that was not rigorous.
1 divided by infinity equals zero can be a very informal way of making a statement about limits, that the limit of a sequence of numbers, each of which is a fraction with one in the numerator and increasingly large numbers in the denominator, such that the numbers in the denom become arbitrarily large, is zero. But zero is never the value of any of those fractions, and infinity never appears in any of those fractions either.
There is no number 'infinity': such an equation as n=infinity is as it stands meaningless: a number n cannot be equal to infinity, because because 'equal to infinity' means nothing. G.H Hardy, A Course of Pure Mathematics, Chapter Four
I'm a math major myself, and I would like to contribute, but it's hard to see what anyone's really saying. Sort of reminds me of a friend of mine, who rejects the notion of infinite sets citing, "A collection of objects cannot hold infinitely many objects." When I confronted him with
(0, 1) : = { x in R | 0 < x < 1}
Apparently to him, this was a set. He was confusing boundedness with cardinality. I've grown use to laymen who think they have anything useful to say about mathematics. Topics like Godel, 1/0, .999 = 1, occupy a place in their minds that far outstrips their interest to current mathematicians. But it's clear they don't know what they're talking about when they remain silent on ideas more relevant to modern math like computing homotopy groups, the elegance of the Riemann-Roch theorem, or the existence of non-measurable sets as a consequence of the Axiom of Choice.