I had written here (“My entire world almost disintegrated during an exchange about infinity and math”):
Does 0.999… = 1 ?
No, it does not. It can never equal 1 because that contradicts the “…” at the end there – that is, the “…” involves infinity, and infinity as we have seen, is something that is often beyond the scope of math. The infinity here means there will always be a 9 at the end, and therefore, that is not sufficient to equal 1. That is the point!
So… this provides us with the beginning of an answer to our dot problem (and how many lines could you trace on a sphere).
A dot exists, but we can never “catch hold of it” because it’s too small, it is truly infinitely small. However it will never be nothing, that is the beauty of it. It is obviously the inverse of the above 0.999… issue. A dot is what exists just before we get to nothing – and we, as humans, can never “catch” that. It is the magic moment when you go from nothing to something, the very beginning, like the conception of light and life, if you will, in mathematics.
And then I had another thought, based on the above:
If we take my definition of a point as the infinitely small dot, what do we have? Take a blank white piece of paper. Now imagine a black thin line on this paper. Now, if you wanted to pinpoint a point in the line, you have a problem. Because as a human, the dot, or your point, just keeps getting smaller and smaller, in such a way that you could almost see gaps appear between the points, because they are so small. They are, as I postulated above, infinitely small.
But, and here is the big “but”, if we had gaps between the points, it wouldn’t be a line, but several line segments, or simply several lines, interrupted by these gaps.
The fact that a line exists therefore means there are no gaps between its points, even though the points composing it are infinitely small. This means the line cannot contain more dots, even though they are the smallest that can be. Now here comes the big conclusion.
Consequently… a line of no matter what length is the embodiment of infinity.
Recall what this math person had posited here https://www.mathsisfun.com/numbers/infinity.html :
Infinity is the idea of something that has no end.
In our world we don’t have anything like it. So we imagine traveling on and on, trying hard to get there, but that is not actually infinity.
So don’t think like that (it just hurts your brain!). Just think “endless”, or “boundless”.
Infinity does not grow
Infinity is not “getting larger”, it is already fully formed.
Sometimes people (including me) say it “goes on and on” which sounds like it is growing somehow. But infinity does not do anything, it just is.
If you have no gaps between infinitely small points in a line, you have infinity.
A line, infinity.
I just thought of something – and it complicates what I just wrote above. What if you just take two infinitely small points, one adjacent to another, which still qualifies as a line?
That doesn’t sound very infinite. Oops.
Anyways, a longer line still does.
And you could say about the two adjacent infinitely small points: here was one point and she was lonely. And next to her was another point, a he-point. Then they got together and said: we’re no longer just infinitely small points, now we’re a line, a little part of infinity.