Is a line/line segment not a composition of points? At exactly 3:00 in this video, William Lane Craig states that a line is not a composition of points: a line is logically prior to any points that you specify on it. To assume that a line is a composition of points is to already beg the question in favor of an actually infinite number of things.
My understanding:
To create a line segment of length 10cm, I would need 10 line segments of length 1cm each. I would need 20 if the line segments were 0.5cm each. I would need 40 if the line segments were 0.25cm each. I would need 80 if the line segments were 0.125cm each, and so on.
Now, how many line segments would I need if their lengths were 0cm each? Side note: these "line segments" have ceased to become line segments: they are now points. Interesting question. As the segment length is getting shorter, the number of segments is getting bigger. So, if the segment length is very small, then the number of segments will be very big. If we drive this point to its logical conclusion, if the segment length is 0, then the segment number will be infinite: we would need an infinite number of those "line segments" to create a line segment of length 10cm.
Therefore, any line is composed of an infinite number of points; to make a line, you always need an infinite number of points.
Is Prof Craig correct or am I correct?
 A: The question is: how do you know that if you put these "0cm line segments" together, you actually get the entire line back? And: what does it even mean to put together infinitely many things?
You are operating in a modern context where our understanding of the relationship between lines and points is driven by the following very significant and relatively recent conceptual revolutions:

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*Cartesian geometry, the reduction of geometry to arithmetic by the use of coordinates.

*Atomism, the idea that matter is made of atoms.

*The reduction of all of mathematics to set theory.

These concepts are so comfortable and familiar that it's easy to use them without noticing it, but Euclid was doing geometry literally over a thousand years before they became mainstream. So they are not necessary for doing geometry or for thinking about lines and points.
In modern mathematics it is true that the standard construction of, say, the line segment $[0, 1]$ as a subset of the real number line causes it to consist of uncountably infinitely many points. This is a specific technical foundational choice, and other foundational choices are possible. In any case, doing things this way requires significant conceptual technology (set theory, together with either Dedekind cuts or Cauchy sequences) to construct the real numbers, technology that is less than 200 years old. Euclid was doing geometry over 1500 years before this.
This is not the only kind of line in mathematics either. There is an object called the affine line in algebraic geometry, and it is genuinely a mistake to think of it as solely being composed of a collection of points; it is actually a much more interesting object than that. For example, two curves which are tangent in algebraic geometry have a scheme-theoretic intersection which is not a point, nor a collection of points: it is a new kind of object called a "non-reduced scheme," and it carries infinitesimal information (loosely speaking, it is "two infinitely close points") which the set-theoretic intersection is not capable of capturing.
So, there is no reason to limit one's conception of lines to the idea that they are made of points. Euclid didn't need this 2000 years ago and algebraic geometers don't need it now.
