The article: one idea, one tension, one principle

What is 2 + 3 × 4?

A mathematics book usually opens with a definition. The first article of this one opens with a question. What is two plus three times four? Nearly everyone, the first time, answers twenty. The answer is fourteen. You may count again, it won’t change. That is how an article begins: with a tension, a surprise. Never with a dry definition. Articulated Course in Mathematics. Reader’s guide, episode two: the article.

One article, one idea

The article is the basic unit of the course. Each article develops a single idea, in two hundred and fifty to eight hundred words, from its motivation through to its interpretation. Articles are grouped into chapters. At the head of each chapter, the list of its articles, numbered, each with the question it asks. Chapter one of volume one has seven.

Anatomy of an article

Let’s open the article from a moment ago. It opens on the question. Then comes the definition, in a box. A figure lights up the idea before it is formalised. Results, propositions or theorems, carry their label: rule, theorem. In volumes two and three, the major theorems are also boxed in red. Next, the example, a concrete calculation. Then the common error, when one lurks, with its counterexample. And the article closes on a sentence that lets you hear a principle deeper than its immediate content.

The status of proofs

A proof, in this course, has a status, and that status is always visible. It is present where it illuminates. Sketched, where full rigour would have obscured the point. Taken as given, when its difficulty exceeds the scope of the article. The reader will be able to tell the three cases apart. A result taken as given is presented as such: the course never passes a fact off as a proof.

The figures

Now, the figures. More than eight hundred in the collection. Drawn with the care a pocket format demands: every stroke has a reason to be there, every label is clear of any curve, and the palette is limited to four colours. They do not replace the proof: they come before it. Whoever looks at the figure before reading the theorem often understands the statement before reading a single word of it.

After the article

And after the article? Practice exercises: a few short ones, to put the idea just presented to work. Then, at the end of the chapter, method studies and expeditions. That is the subject of the next three episodes.

To remember

To remember. One article, one idea. It opens on a tension, unfolds through definitions, results, examples and pitfalls, and closes on a principle. Proofs display their status, and figures come before proofs. Chapter one of volume one is available as a free extract, articles included. The link is in the description. Next episode: the exercises, and the upside-down solution.