Question about the existence of points and lines.

Question

Say we draw a point on a graph. If the point should not take up any area than how come we could see it. Say we graph $y=x^2$, we obviously could see it. However, because $y=x^2$ is a function made up of a collection of points with no area, how come we see it? How come we see lines if they have no area?

Answer 1

We don't draw a point, we draw a representation of a point. If we drew a point, we couldn't see it. The ideal graph of $y = x^2$ would also be invisible, which wouldn't give us a whole lot of information. The representation of the graph is not the graph, because we can see it, it is inaccurate; but it does give us more information than the real graph would.

Answer 2

The key idea here is that a point and a line are abstract mathematical concepts. When we "draw a point", we are just creating something concrete to represent that point. When we "draw the curve" $y = x^2$, we are creating a physical thing to represent the set of points $(x,y)$ that satisfy that function.

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Also, since you tagged your post with (euclidean-geometry), some of Euclid's definitions are definitely relevant here. From the list definitions in Euclid's Elements:

• A point is that which has no part.

• A line is breadthless length.