a point is defined as a location in space. it has no width, length, or height. no dimensions at all. how come these points can constitute a line which has a dimension? length. how come two points (and the infinitesimal points between them) make up a line segment which has length? anyone?
Because what you’re describing is pretty much the definition of the 1-dimensional. It’s the ***distance ***between the points that adds the new dimension. Imagine how it would be if the two points were in the same location. Without any distance separating them, they’re still 0-dimensional.
Right. The OP’s mistake is in believing that the points “constitute” the line; that 0D plus 0D add up to 1D. They do not, any more than the bottom edges of your living room’s western and southern walls (each 1D) actually comprise the floor space (2D) where you set your furniture.
The response so far has been, correctly, that two points define a line segment, which also consists of all the infinite number of points in between.
Can two points however be a line segment, two points infinitely close together, but yet not the same point?
Yes. If they aren’t the same point, there is distance, and an infinite number of points between them.
At least in standard analysis, there is no such thing as “two points infinitely close together, but yet not the same point.” If two points are distinct (not the same point), then no matter how close together they are, there must be some finite, non-zero distance between them, and there exist infinitely other points in between those two (on the line segment that connects them).
In fact, you could say that the whole idea of differential calculus is to find a way around thsi restriction. You can’t really take one point on a curve and then take a second point infinitely close to the first one, and connect them with a straight line and see which direction the line is going to determine the slope of the curve at that point. But in calculus, you take the limit of such lines as the second point approaches the first one, as the distance between them approaches zero.
(A more modern, less well-known approach, called non-standard analysis, does allow for “infinitessimals.”)
Yes, that differential calculus angle was what I was thinking of. And the clarification made is two points never make up a line segment, no matter how close to the limit one gets. If the two points do not have the same exact identity then the line segment created consists of an infinite number of points. One only goes from 0-D to 1-D by adding up an infinite number of 0-D points, no matter how infinitely small the line segment is.
Take it from pure mathematics into physics now. Two points separated by one Planck length. Given that distance may not have any meaning below that, could a line segment consisting exclusively of two points one Planck length apart be said to be theoretically possible?
Describe in detail the quantum gravity theory you’re using to ascribe significance to the Planck length, and then I’ll answer that question.
Chronos - Could you pick a quantum gravity theory of your choice and give us an example of what such an answer would be like?
Why not? It’s still a “distance,” no matter how small. Since the two points aren’t exactly congruent, it meets the definition of a “line.” Whether or not the line can actually be “drawn” is another matter.
The simplest way to think about the dimension of a space is that it’s the minimum number of coordinates you need to specify to say where you are. If you only have one point, there’s only one place, and so you don’t need any numbers. But as soon as you have a second point, you need one number to say which point you’re at.
Points are abstractions. Every line consists of an infinite number of points. Every plane consists of an infinite number of lines. Space contains an infinite number of planes.
The idea of two points with no space between them that do not define the same point is nonsensical. It’s like asking someone to solve for x where x - x ≠ 0. If there is no distance between two points, they are the same point.
The problem comes from thinking of a point as a really small dot, rather than an infinitesimally small dot. It has no width. The only way for two points to touch is for them to be in the same location.
Quoth TriPolar:
No, I can’t, since no quantum theory of gravity exists yet.
I’m not sure that’s a good way of putting it. A line certainly contains an infinite number of points, but to say it consists of that requires a bit more care. For starters, a countable infinity of points is not enough.
As Thudlow pointed out, there are such things as infinitesimals. Two points may well occupy the same real number location, but differ by an infinitesimal amount.
Infinitesimals are actually the only way to make sense of the way calculus is usually taught to students (since it essentially amounts to asking “what is 0/0 equal to?”).
Not in the real number system.
So if I understand you correctly Chronos, until there is an aceptable quantum gravity theory any thoughts, like say these ones that basically state that based on what we do know already, what does happen at that level likely is not meaningful in terms of distance as we think of it (or as they put it: “In short, it’s weird, but beyond that nobody really knows. To be more precise, the Planck length is the length scale at which quantum mechanics, gravity and relativity all interact very strongly.”) are not worth even reading and thinking about. A speculative comment like “Given that distance may not have any meaning below that, could a line segment consisting exclusively of two points one Planck length apart be said to be theoretically possible?” that includes “may” and “theoretically possible” does not deserve an answer because we do not that it is yet?
Okay then.
But I’ll stand my post. Mathematically there is the infinitely small, but many theoretical physicists do not believe that reality actually holds the infinitely small. According to these speculations there exists a granular level below which distance no longer exists as we know it, and for a variety of reasons some speculate that such a distance is the Planck length - understanding of which would require an adequate quantum gravity theory. If reality is indeed like that then one can consider a line segment that actually consists of just two points. Even if the real number system does not describe that geometry.
If you take it to physics, and want to posite the Planck length as the lower bound of dimensionlity, then there is no such thing as a “point”" (0 dimensional object)
It is always dangerous to confuse mathematical constructs with theoretical physics. That’s how cats die.
I think what you mean is, not infinitely small, but arbitrarily small. Mathematically, there is no limit to how small a (nonzero) distance can be (which is not the same thing as saying that there are “infinitely small” distances), but in physcial reality there may be such a limit.
For a less speculative example, I suppose you could consider lines drawn on a computer screen, where a “line segment” could consist of just two pixels.
More precisely, I’d say something like “many theoretical physicists have a sort of vague guess that reality might not actually hold the infinitely small, but none of them are really sure what it would have instead, if so”. And really, until we get an actual theory of quantum gravity that can be explored mathematically and experimentally, the vague guesses of theoretical physicists aren’t worth all that much more than anyone else’s vague guesses.