# 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. – Asaf Karagila Jul 26 '17 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. – Evargalo Aug 7 '17 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. – user7280899 Aug 19 '17 at 11:08