Well, your post would appear obtuse to the layman, and maybe it just is. — Leontiskos
It is obtuse, but I don't think it just is.
A metric is a way of assigning distances to pairs of points. When you consider a space, it has a metric. The usual distance people think of is called the Euclidean distance, and it's the one you're thinking of and measure with a ruler on a piece of paper.
The thing is that the choice of metric is just that, a choice, and you can write down various other spaces with various other metrics. One of those other metrics is called the taxicab metric. Contrasting that to the Euclidean metric:
Imagine you start at a point, and you go 1 step north and 1 step northeast
The taxicab metric says you've travelled 2 total units - you add the steps.
The euclidean metric says you've travelled sqrt(2) total units - you measure the line.
Because a metric defines the concept of an interpoint distance, circles in taxicab geometry are different from circles in euclidean geometry. A circle in taxicab geometry, a set of points defined as equidistant from a single point, looks a lot like a square in euclidean space. 4 corners, 4 right angles, 4 equal sides.
So it is a circle, if a circle is defined by the property of being equidistant from a point. But perhaps it is not a circle, because... well, like you, you could
insist that we're not talking about a circle when we're talking about sets equidistant from a point in the taxicab metric. So for you, you'd have to do something to block what we're talking about as a circle in taxicab geometry being a "real" circle.
That places a burden on you to study the concepts of circularity and square-iness, and to say why the first blocks the latter and vice versa. Which is what I did in the post. I'll go through it for nonmathematicians.
For something to count as a square, it needs to have:
S 1) Four sides of equal length.
S 2) Each side meets exactly two other sides at right angles.
Let's just take that as a given, that is what a square is. Now we need to think about a circle. What's a circle?
C 1 ) A circle is a set of points equidistant from one point.
If ( C 1 ) is the only defining property, the taxicab circle is indeed a circle, it's just a circle in taxicab space. Clearly you don't want it to be a circle, so you need to stipulate a restriction. I could also insist that it is a circle, and how are we to decide between your preference and my preference? Anyway, onwards:
What is round is not pointy — Leontiskos
You specified such a restriction with "what is round is not pointy", which is something similar to what I formalised with the idea of smoothness. The "corners" form the "pointy bits" of the square because the function that defines a square is not smooth at the exact corner point.
There is an ambiguity regarding pointiness, which is similar to the above ambiguity regarding equidistance. In thinking about the corners of the square thing (the taxicab circle) in taxicab space as pointy in the above sense, that requires specifying the roundness concept in terms of the measure of size - smoothness is typically characterised with respect to a measure of size.
Something is differentiable when its derivative exists at every point.
The derivative of a curve exists at a point if and only if at that point the limit of the ratio of the function evaluated at the endpoints of an arbitrarily
small interval divided by the
length of that interval exists (IE it becomes just a number).
A curve is smooth if you can apply the procedure above to it arbitrarily many times.
The concepts of "interval" and "length" there are also doing a lot of work, since they're distance and size flavoured. And should we expect them to work as our prior Euclidean flavour intuitions would in taxicab geometry? What gives us the right to insist that we think of smoothness as we would in a Euclidean space and transfer it onto smoothness in a taxicab space?
Clearly you would want to insist that they do, my intuitions also run that way. But my intuitions can also side with circles not necessarily being smooth since I'm used to dealing with this stuff!
Where we can agree, though, is with lemma incorporation. In which we specify a set of properties that say exactly what counts as a circle (in your sense) and why it can't be a square.
So for you:
A circle is, by definition, a set of points
Euclidean equidistant from one central point.
And thus we've revealed what sneaky hidden presumption you had through lemma incorporation. What we haven't done is decided why that must be accepted as
the definition of a circle.
If you want to join in with this exercise of lemma incorporation, I invite you to stipulate a definition of
pointy! And we will see where it goes.