A grammar in five volumes

Why five volumes?

Five volumes of mathematics. Before opening the first, a user’s guide is in order. And the first question is simple: why five? Because this course defends a thesis, and a thesis of this size doesn’t fit in a single volume. Here it is: mathematics is the grammar by which reality composes itself. Not a tool invented to describe nature: the internal syntax of what is. Now, a grammar isn’t learned all at once. You start with words, move on to rules, then write sentences. Nobody becomes bilingual over a weekend. The five volumes follow that path. Articulated Course in Mathematics. Reader’s guide, episode one: a grammar in five volumes.

The five volumes

Here are the five volumes, from the most elementary to the most advanced. Volume one, Elements: the first words. Number, distance, angle, equation, function, symmetry. It covers secondary-school mathematics, and assumes nothing beyond elementary arithmetic. Twenty-two chapters, one hundred and sixty articles. Volume two, Foundations: the grammar. The structures that govern those words, and the rules by which they combine: groups, vector spaces, metric spaces. Sixty-two chapters, two hundred and ninety-one articles. It assumes the secondary level. Volume three, Graduate Mathematics: the syntax becomes rich enough to form complex sentences. Functional analysis, measure, Galois theory, differential geometry. Eighty-six chapters, four hundred and fifty-five articles. It assumes the foundations. Then research, in two volumes. Volume four, Correspondences: how to pass from one description of an equation to another, without losing the operations that give them meaning. From class field theory to the Langlands program. Thirty-seven chapters, two hundred and thirty-eight articles. Volume five, Mathematical Physics: how mathematical structures become physical descriptions. From symmetries to strings. Forty-four chapters, four hundred and six articles. Volumes four and five both assume graduate mathematics, and can be read in either order.

Three levels

These five volumes form three levels. Secondary school, with volume one. University, with foundations and graduate mathematics. Research, with volumes four and five. Careful: these levels guide your reading, they don’t impose a timetable. No bell, no deadline. The pace depends on what you already know, and on what you want to understand.

Where to start

Where to start? Where your knowledge ends. You know elementary arithmetic: volume one is waiting. You have the secondary level: open volume two. You have the foundations: volume three. And if graduate mathematics holds no more secrets for you: volume four or volume five, your choice. At the start of each volume, bridges explain the change of perspective. And the path needn’t be linear: you can also take cross-volume passages.

The architecture: parts, chapters, articles

Every volume has the same architecture. Parts, which group chapters, which group articles. The article is the basic unit: a single idea. In all: sixty-four parts, two hundred and fifty-one chapters, one thousand five hundred and fifty articles. The next episode opens an article, and looks at how it’s made.

To remember

To remember. One language, five volumes, three levels. You enter at the level of what you know, and move at your own pace. And the unit of all this is the article. Volume one has a free extract: all of chapter one, with its exercises. The link is in the description. Next episode: the article.