Question about the claim that concrete infinity does not exist
How can one argue that a concrete infinity does not exist, since there are infinitely many points (as in the “paradox” of Achilles and the tortoise)?
I don’t know what ‘existence’ means. It’s true that the number of points on a continuous line (I don’t know whether to call it ‘existence’) is greater than any number you can think of. That’s our definition of the structure of a continuous line, but it doesn’t necessarily mean that the points are existing entities.
Based on what you said, is it possible to say that this is really a potential infinity, since there is no point that can be pointed to and said that there is an infinity like it, and this is a form of mathematical observation of the structure of a continuous line, and not the real "existence" of an infinite number of concrete points.
I didn't understand the argument. There is an infinity like it in the potential sense.
That's exactly the point, there's no problem with there being an infinite number of such points potentially, but they don't really exist in a concrete way (because there's no point that you can point to and say there's an infinity like it).
I lost you. You can point to any point and say that there is an infinity like it. Only the phrase “infinity” is potentially interpreted.
I didn't understand you completely, so I'll try to write better from the beginning.
I want to argue that there is no concrete infinity here, because it is impossible to point to any point (with any size) and say that it exists infinitely many times (because we don't have infinite space), the infinity of points only exists in our heads by looking at the structure of a line, but these infinity of points do not really exist.
(That is, you can think of infinity of points by a mathematical series, but you can't really point to infinity of points, and therefore they don't really exist either. And just like in a normal series of numbers).
I'm not sure you understand what you wrote. What does it mean that a point exists infinitely many times? Why, if it exists infinitely many times (what is that?), should it occupy infinite space (you probably mean lengthwise, and that's not true either, of course).
We can treat a certain line as an infinite number of points only in the form of a mathematical series, but we can never project this series onto reality.
I meant to say that no matter which point you choose from the series, it does not exist infinitely many times (because any quantity multiplied by infinity will equal infinity, and the length of the line is not infinity).
This is apparently exactly the difference between potential and concrete infinity, we can talk about it mathematically, but infinity has no realization.
Nothing to do with concrete or potential. A collection of infinite points does not give infinite length, and in fact does not give length at all. Beyond a collection of points, one also needs the property of density to create a sequence (and in animalistic, Leibnizian language, the sequence is created from differentials and not from points). A differential is a line whose length is 0 (aims at 0), while a point is a creature without length. It's like the difference between a blind person who cannot see and someone who only sees black (neutral appearance).
Okay, so the terminology “points” was wrong, you could instead replace the word “points” with ”segments”. I didn't see a rejection of the argument itself, why is there no connection between potential and concrete?
But now I understand your point that points are not necessarily existing entities. It's just that now I'm asking the same question only instead of about points (which have no length) about segments (which have some length).
There is no fundamental difference between segments and points. Take N segments each of length 1 divided by N, the total length is 1. Now take N to be infinite. The total length is still 1.
I think we're done.
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