I once thought that mathematical geometry worked by a kind of detail crunching.

If a line is just a systematic set of infinite number of points checking whether two lines intersect would just be a "simple" operation to check whether they contain a point in common. Take points from one line and check whether it is a part of the other line. Doing this with literally infinite number of points would amount to a supertask. So you could only do so to an arbitrary precision but not exactly.

However a very simple math problem like "find the intersection of lines y=2x and y=3x+5" can be done exactly in a finite small number of symbol operations. And actually the description of the infinite number of points on the first line can be done by a very finite expression of "y=2x". There are also an infinite number of such lines but finding each of their intersection doesn't include attending them pair by pair. The procedure of solving the descriptions as a equation pair can be expressed in a expression more meta and "more finite".

So instead of just a big fleet of lowest level comparisons what really happens is a tiny amount of work on different levels. If one would count each symbol manipulation as a single number crunching operation the supertask of point comparisons would seem to be the most demanding. However using multiple levels of symbols means supporting a wider array of symbol manipulation operations.

So while I appears that I compare infinite numbers of points when I am doing simple geometry, it's just that I am bypassing one kind of calculations limits by using another kind of calculation.