Coming up with an elegant mathematical definition is as much of an art as writing poetry. The analogy works even further, for like a well-written piece of poem, a definition in mathematics packs a lot of meaning in a few words. One has to unpack the meaning to appreciate either, and the process can be frustrating. But once the understanding is complete, it is easy to see why the definition/poem works, and it is infinitely preferable to use it rather than explain what one means.

Here are some examples.

Poetry : Definition of a Free group via the universal property – A subset **X** of a group **F** is said to be a free basis for **F** if for every group **G** and every (set) function , there is a unique homomorphism satisfying . A group **F** is called a free group if it has some subset that is a free basis.

Prose: Homomorphisms carry the algebraic structure of an object. If for example the Cayley group of X (i.e. F) has a loop, any (non-trivial) group homomorphism must carry it into a loop. The only way you can extend *any* set function whatsoever to a group homomorphism is if X has no algebraic structure to be carried over. This is the property that this definition captures. Notice that we do not require to generate $F$. But that is exactly what will happen, because we require the homomorphism to be unique.

The second example needs some set-up.

Quasi-isometric embedding. Let and be two metric spaces. Let with and . A map is called a -quasi isometric embedding if

.

Poetry: A Quasi-isometric embedding is called a quasi-isometry if such that with .

Prose: We are looking for a notion that is not quite a surjective map, but close to being a surjective map. That is why we ask that if an element in is not actually in the image of , it is not too far away from such an element. So any element in is close to some element in .

As they say in Latin, *quod erat demonstrandum*.