A point doesn’t exist PHYSICALLY, it has no PHYSICAL dimensions, but that doesn’t mean we can’t use it as a tool of thought to describe a location, either within another tool of thought like a line, a plane, or a 3-D space, or within the tangible space-time continuum (assuming the space-time continuum is the top level of reality).

Likewise, we can use a line segment as a tool of thought to talk about, for example, the relative positions of other abstract or physical things (like the top of your head and the bottoms of your feet).

Space, time, objects, etc. do of course exist, and obviously we can measure them. But when we describe the LENGTH of an object, for example, there’s no need to say “it’s three feet long, 1 inch thick, cylindrical in shape, and lasts for 25 years.” Most of that information is immaterial to what we’re interested in — its length.

So we use a term, “length”, to describe it. And the “length” that we’re talking about isn’t a trillionth of an inch thick or any other thickness. It just plain doesn’t have any thickness to it.

A line segment equal in length to the object likewise doesn’t have any thickness. The essence of a line segment is all about its length, position, and orientation — not its thickness.

The very fact that we CAN measure time and space might very well indicate that there really is some measure to a point or an instant – we’re just not capable of perceiving them.

Remember that relativity tells us that time slows to a stop in the universe as you approach the speed of light. So the fastest thing in the universe causes everything else to be the slowest. So perhaps as you get larger, the smallest space gets smaller until you occupy all space, essentially becoming the smallest thing in the universe by virtue of being the largest.

If you are the largest(smallest) thing in the universe, you can never get to your destination because your destination lies the farthest possible distance from you – which is also the smallest possible distance from you – zero.

So the farther our goal, the faster we go, the relative distances and times change until we simply become our goal – having never gone anywhere because where we started has become where we are.

I think that Lynds errs (or fails to give the best answer to the riddle) in not recognizing that, for example, a point is a ZERO dimensional abstraction rather than an infinitely small thing.

In other words, a point has no size, not because it’s an object too infinitely small to contain any size, but because it’s not an object at all — just an abstraction — a precise description of a position (and the position isn’t an infinitely small area of space either).

An apple is not a small apple pie. It’s what you’re left with when you remove the sauce, the crust, etc. from an apple pie — it’s an object lacking certain parts of the nature of an apple pie. “Size” is not part of the nature of a point any more than crust is part of the nature of an apple.

Time is not part of the nature of an “instant” (an “instant” is what you’re left with after you remove time from reality).

Once you’ve grasped the notion of completely discarding time when thinking about instants, it’s simply unnecessary to go through all the complexity of slicing time really small and figuring out whether there’s motion in the slice.

Motion is impossible in an instant because instants are abstractions from reality which contain no time and time is a necessary component of motion.

Time is a relative measurement and not a physical constant like length/depth: It consists of the distance between any two points and the time/measuremnt it takes for an object to move between those two points, i.e. the second is 1/60th within a minute and is a specific measurement of space/distance/time and is not an entirely arbitary (Sun/earth rotations) but still completely relative measurement as expressed through the classic instrument of time measurement: the 60 second/60 minute/24 hour western time-piece/watch/clock.

This must, in turn, be opposed/juxtaposed to other equally measured, but relativeistically determined distances, which include miles/kilometers, longitude/latitudes. Time cannot be, but is exluded from the Zeno’s (Un)Paradox as his object/subject only takes ” if it is movement over space then theoretically that space can get smaller, because numbers can always become smaller” which is the length/depth of the movement and NOT the time (fourth dimension) that is the actual transitive productive force that initiates the movement from the two points.

Zeno displaces “time” and thereby creates an infinite (albeit tantilizing) ultimately “absurd” regression:

“For me to act it takes 1 second to… 1 millisecond to… ,” this, also, is an inf. regre. but instead of time being removed, the spatial element of length/depth has been removed.

“I,” Zeno and possibly INTPnerd make the cognitive error of removing one of the 4 dimensions of space/time–which function interdependently and without regard to cognition. They combine mutually to create: space–length (or depth), width, height [which cumulatively create matter, "me" and you] and time [as an intrinsic combinatorial element of space, cannot be removed except cognitively]).

Zeno commutes the fourth element of time into length/depth, when he says, “one must take the half step of a half step of a… ?!?!” Time, which is the distance between two points, becomes depth or its length (THAT IS THE POINT A TO B and verily, as Roundy states, are just “a bunch of points–points [which] are just descriptions of positions on a line.”). That is, the depth or length of the step and is erroneously conflated by Zeno (without malicious intent, I’m sure) as the “time” it takes to make the first step:

“space-time, or the four-dimensional space in which motion occurs, isn’t made up of a bunch of instants–an instant is just a description of a “position” in space-time.”

There are no discrete moments in time [1] such as there are discrete steps or movements relative to two points in time-space because if there were we would then never move as Zeno hypothesized, but wrongly conflated: depth/length of the step as the relative (and fourth dimenstion of) movement it takes for the step, i.e. the time itself it takes to make the movement relative to point A and B.

1. See also Jeff Lynds at: http://www.peterlynds.net.nz/papers.html. Particularly, “Zeno’s Paradoxes: A Timely Solution.”

If, like you say, there is a single smallest unit of movement, such as the binary movement, then it would undoubtably be movement over an amount of space. We also know that space is able to be represented by a number. Whether that number be one inch or one centemeter etc. However, if it is movement over space then theoretically that space can get smaller, because numbers can always become smaller. Therefore Space, time and anything represented by numbers can become smaller and smaller into infinity.

Please respond with any well thought out arguements.

“I’ll start there: a line isn’t made up of a bunch of points–points are just descriptions of positions on a line.”

As was this:

“space-time, or the four-dimensional space in which motion occurs, isn’t made up of a bunch of instants–an instant is just a description of a “position” in space-time.”

In both of those quotes, Antone Roundy tells us what those things are NOT made up of but he does say what they ARE made up of. Here are my thoughts on Zeno’s paradox: It is true that IF you could infinitely divide the amount distance something can move, then you could not move because if there are an infinite number of halfway points, and each of them takes a certain amount of time to get there, then it would take you an infinite amount of time to get there. Therefore, I conclude that there must be a smallest amount of movement possible for anything. At some point, it either moved or it didn’t move. One could call this ‘binary movement’ because at some point something either moved or it didn’t move, with no possible half way point. A bit is the smallest amount of information imaginable because it specifies that something is either true or false, 1 or 0, on or off, yes or no. I suppose you could refer this smallest amount of movement possible as one unit of space. But if there is “binary movement”, then there has to be “binary time” as well because otherwise, there would be a point in time, at which something moved halfway between the 2 points which make up a length of the smallest amount of movement possible, which would mean that it was not the smallest amount of movement possible. Of course if you are moving from point A to point B, and you keep only moving by halfway, you would never get there, because you are moving by a smaller amount each time (not just smaller but half the distance of your previous movement), so there has to be a point at with you or anything else cannot move half a that distance. This makes it seem like time is something more than just a concept because you cannot move the smallest amount of movement possible in less than the smallest amount of time possible, which gives it a real world property of the smallest amount of itself. You could refer to this as one unit of time. So now I have this concept of single units of space and single units of time and the slowest anything can ever move is with a 1:1 ratio between space and time units. I am not completely sure about any of this, but sounds very logical to me. Let me know what you think.

Then a line (segment) being a real or imaginary mark positioned in relation to (non-existent) fixed points of reference…..cannot exist!