What are lines exactly?

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If points are zero-dimensional, then how are lines a bunch of zero-dimensional points connected together?

Lines are sets of points. The sets are compact (hence dense). The reason which lines are one dimensional is they can be represented in a co-ordinate space by a function with just one free variable. Surfaces can be represented in a co-ordinate space by two free variables and hence they are two dimensions. Likewise volumes are three dimensional etc. etc.

ruveyn

Have you ever read Flatland? The author suggests that when you do something like turn a point into a line (or a line into a plane, or a plane into a cube, or a cube into a tesseract, etc.) it's like if you made the point move forward and leave a trail of itself behind it, creating a line. You could move a line in a similar way, outside of the dimensionality of itself, to make a plane. I don't think having them "connected together" makes as much sense, but anyway this is kind of delving into the realms of calculus, which I'm not too familiar with.

This is similar to a problem the Greeks had. They thought that movement was impossible. Let's say you have an arrow. You fire the arrow. In order for the arrow to fly, it must traverse the distance between you and its target. But the simple trajectory the arrow must traverse is made up of an infinite number of tiny points. The Greeks reasoned that if the arrow had to cross all of these points, than it would take an infinite amount of time, so therefore the arrow never moves in the first place. Motion was an illusion, they said.

This was later solved by calculus, which again I am not very familiar with, but I think the idea was that if you have an infinite number of infinitely small points then it adds up to a finite distance. Which kind of makes sense, in a way.

Welcome to Euclid's world. The definition of a line is made with any two points. The space between the two points is undefined, so for the purposes of geometry, it doesn't matter.

http://mathworld.wolfram.com/EuclidsPostulates.html

That is Zeno's Paradox. No Greek really thought movement was impossible because they, themselves moved. What Zeno was showing was that there is something wrong with our way of describing movement.

ruveyn

The really quick version: Euclidian geometry is static, so when you try to use it to model motion, you get paradoxes and impossibilities. You can't move! Calculus introduces motion. When you're using a Euclidian model to represent motion, you have a lot of static points, and no way to get from one point to another. Any time you look at a given point, the Euclidian model assumes you're already there (because it is a static model), so you never actually move. Calculus doesn't care what happens when you already get to a point; it cares what happens as you approach a point and are still in motion. After all, once you get to a point, you're no longer moving towards it. Calculus changes the model to look at what happens when you get really close to a point, but aren't there yet, so you're still in motion. Over time, the calculus model has proved to be a better model of our motion-based universe than the Euclidian one.

Berlinski's "A Tour of the Calculus" explains the conceptual difference at a high level without getting bogged down in delta and epsilon notation.

The Greek motion paradoxes are found in Aristotle's "Physics" Book 6.

the universe i think has 11 dimensions that are simple to conceive.

dimension 0= a point. points have no length or breadth or height. therefore they do not exist except when multiplied by "infinity".
dimension 1= a line. a line is composed of an infinite amount of "points" that are aligned in a straight path. all lines are of infinite length.

dimension 2= a plane. a plane is an infinite amount of parallel infinitely long lines (with no width) laid side by side.

dimension 3 = a volume is composed of an infinite amount of planes (with no height) stacked on top of each other.

dimension 4= the infinite volume in one "point" of time (an instant) that has no duration but contains infinite spacial volume (infinite points).
dimension 5 = is a "line" of time where every instant (in which is contained the actuality of universal reality on this line of time) is aligned in a straight path. this is where i live. everything in this universe on level 5 happens as "one thing after another".

dimension 6 = a plane of time which is composed of all lines of time laid (in parallel) out in the extended "now instant" plane. all things possible to happen at all times in the universe exist on this plane.

dimension 7 = a volume of time which is "all possibilities on all planes of time in every instant in every alternative universal manifestation".

dimension 8 is the "point" of consciousness (yes it becomes "religious" after dimension 7) that can conceive the singularly of each "point" in each infinite plane in the 7th dimension

dimension 9 is a line of simultaneous consciousness of all occurrences that can happen in a linear time frame of one universal aspect of reality that extends from the infinite past to the infinite future.

dimension 10 is the simultaneous consciousness of all alternative universal realities on a single plane of reality.

dimension 11 is the "voluminous" consciousness of everything in all planes of time and it is enlightenment because every "point" in the universe whether it be in time or space is fully understood.


so i say the first 3 dimensions build space, and the next 3 dimensions build time, and the next 3 dimensions build consciousness, but there are 2 more dimensions which are simply dimension "0" of the next dimension.

dimension 0 is a point of space
dimension 4 is a point of time.
dimension 8 is a point of consciousness.

even though it seems like there are 12 dimensions, the last dimension gives rise to nothing else so there are 11 dimensions.

that is my idea anyway. i did not learn it. it is a way i see the universe. and it may be peculiar to me so whatever.

This question is again related to the question "Is there a number right next to 1, or infinitely close to 1?" It's related because your confusion arises by considering that the points that form a line are next to each other.

Consider this. Imagine that there was a point, r, infinitely close to 1, say to the right. Then 1<r. But if 1<r, then 1<(1+r)/2 <r. Therefore, there is no point right next to 1.

Another source of confusion is whether the number line we're talking about is discrete (whole number integers) or continuous (real numbers). If the number line is discrete, there is nothing between two points on the line, so there's no issue. If the line is continuous, you're still okay if you start on a whole number and move by whole numbers. You'll always go to the next whole number. The only time you get in trouble is if you move a fraction of a number. If the fraction is a rational number, you're still okay. If it isn't, then you get in trouble. The only way you'd get almost close to a whole number but not get to one is if you move a non-rational decimal number. To get anywhere, you'll have to use floor/ceiling functions so you don't fall into an infinity near a whole number. For all practical purposes, we can ignore infinities like that and round everything to a few decimal places. The only time infinities become important would be, say, in calculus when you're approaching a whole number but not there yet. Interesting to think about stuff like this as a gedanken experiment, but it doesn't make any real practical difference in real-world math. Even the finest precision machine tools (since the late 19th century) only measure comparatively coarse distances compared to Euclidian infinities.