ancient principle of mathematics: figure = varying element

On page 83 in the book Conceptual mathematics by Lawvere et al. it says:

An ancient principle of mathematics holds that a figure is the locus of a varying element.

What does this quote mean? In particular, what is the "locus of a varying element" and in which sense is it the same as a figure?

• Well, locust is one of the ancient plagues brought upon the Pharaoh for not letting the Hebrews leave. I'm pretty sure that was pretty elemental, as disasters go. Jul 26, 2017 at 13:48

An example:

Let's $P(\alpha)$ be the point $e^{i\alpha}$ of the complex plan.

The locus of $P$ when $\alpha\in\mathbb{R}$ is the circle $C$ of radius $1$ whose center is the origin.

$P$ is the varying element, $C$ is the figure.

I am studying the same book (Perhaps different edition). Here is what I think author intended. Consider category Sets. Suppose $f: A \rightarrow B$. Now consider point $1 \rightarrow A$. A varying element V has following properties:

$$V(1 \rightarrow A)=1 \rightarrow B$$ (It appears that $V$ is a map from set of points of $A$ to set of points of $B$.) This can be shown by composing points with $f$ as follows: $$f∘a=b$$ Where $a: 1 \rightarrow A$ and $b:1 \rightarrow B$.

So evaluating $V$ at different points of $A$ gives a family of points of B. For every point of $A$, $f$ assigns a point of $B$. This is one perspective for a map. Now, we can replace 1 by D, and say that a map $f:A \rightarrow B$ is a family of D-elements of B.

What I do not understand though, is last sentence of that paragraph, which is: " For example, we can take D = A, and the identity $D \rightarrow A$, thus revealing that (c) the varying element, as a single thing, is a single figure or element itself. "

Any ideas on this?

• It could be a different question but let it be. I suppose it means that any Domain (or Set) can itself be considered as a figure. For instance, the Domain $\Delta=(1;\infty)$ is the locus of $P(x)$ when $P$ is the identity and $x\in\Delta$, so $\Delta$ is a figure. Aug 7, 2017 at 13:40
• I think the authors mean that if you put in the identity $D\to A$ into the varying element / the parametrized family, the value you geht is itself a varying element.
– user401895
Aug 19, 2017 at 11:08