> But, there are no circles in the world, empirically
Only if you have an overly strict definition of circle. I don't think it is wrong to call the outline of a ball a circle, or the shape you do if you take an Y shaped object and rotate it along one of those branches, it isn't a perfect circle but it is still a circle.
And crucially, lots of things like the ones you mentioned are often not better approximated by an ellipse than a circle(I realise circles are just a subset of ellipses).
You can't just decide that the circle is contained in the set of all ellipses. Anyway its a philosophical argument, you can't "prove" mathematically that circles are ellipses or vica versa.
Why do circles need to be ellipses anyway, why can't they be absolutely different? If they were absolutely different, then circles would be purely ideal, and yet an organizing principle (or as the say in Greek, an ἀρχιτεκτονική, from when we receive the word architecture). The only way to understand this, ontologically, is if we take the world to be in a constant tension with the "earth," as Heidegger puts it (cf. The Origin of the Work of Art), the thing in which the "rifts," which is the actual discourse of idealism, come about.
You know, I thought about it for a moment, and I don't think the visual circle is even universal. The schema of the circle may be, but the circle itself never appears. See this article below[0].
I'm not who you replied to, but the reason circles are ellipses is because the definition of a circle is equivalent to the case of an ellipse where both foci have the same x, y coordinates. You can read about all the definitions of an ellipse on Wikipedia.
Functionally, they might not be the same if you're programming them. For example, if you have a circle class with members detailing its center and its radius, it might be more efficient to draw an instance of it than an instance of an ellipse class that has two foci members that just happen to have the same values.
Also, that definition of a circle as an ellipse via locus points (which you mentioned) still requires defining ellipses by an imaginary, ideal circle--even if transcendental (and I've mentioned elsewhere, the circle never appears). The circle that appears is always already an ellipse, empirically, so in pure mathematical terms as I've said the ellipse is actually absolutely different from the circle, which always exceeds it. In order for the circle and ellipse to be set in relation one must be forced to be analytically composed under the other.
Edit: Its clear that phenomenology was the most important philosophical influence on the modern theory of computation, but its curious to me that those that study pure math and physics haven't, for the most part, realized that yet, even as they employ computers to do so much of their work.
> Also, that definition of a circle as an ellipse via locus points (which you mentioned) still requires defining ellipses by an imaginary, ideal circle--even if transcendental (and I've mentioned elsewhere, the circle never appears).
How so? I don't understand. It seems like the opposite to me: because a circle can be defined as a special case of an ellipse, then a circle is defined in terms of ellipses, not the other way around. The definition of an ellipse is a generalization of the definition of a circle.
Its quite a difficult problem...but also one I am actually currently working on, so I should probably put some effort into thinking about it. Give me some time and I'll get back to you. If only there was some way to do private messages on here.
If there were two circles, the second would be isomorphic to the first, therefore, in pure geometry there is only one circle. I suppose by some sort of empirical measure, there would be circles of different sizes, but as we already stated, empirically observed circles are actually ellipses, so in fact there is actually only one circle.
Only if you have an overly strict definition of circle. I don't think it is wrong to call the outline of a ball a circle, or the shape you do if you take an Y shaped object and rotate it along one of those branches, it isn't a perfect circle but it is still a circle.