The Math Annotation Project#

Author: Steve Kieffer

Books of annotated chess games like Chernev’s Most Instructive Games of Chess Ever Played [Chernev], and Bronstein’s book on the Zurich 1953 tournament [Bronstein], aim to teach by example. They illuminate otherwise lifeless lists of moves, highlighting brilliancies (and blunders), guessing what the players might have had in mind, distilling rules of thumb, and, above all, helping the reader to understand the game.

Mathematical proofs can be taught in the same way, and yet it is rare to find books of annotated proofs.

The Math Annotation Project is a community-driven effort to annotate great proofs, and thereby promote the accessibility of mathematics, by making its literature more understandable.

We believe the right technology can help the mathematical community to better achieve this goal.

An online browser#

If you want to talk about Najdorf’s move 19 pawn sacrifice against Taimanov, in Round 4 of Zurich 1953, you can call this up in a few clicks of the mouse. You can load the game at chess.com, and then step through to the move of interest.

We need this for mathematics! If we want to discuss the step in Part IV of Hilbert’s Zahlbericht where it is established that the conjugates of \(1 - \zeta\) all generate the same ideal in a cyclotomic field, this too should be immediate. In fact, this much is already possible, and you can load this step in PISE Online in one click!

An annotation system#

The Proofscape Integrated Study Environment (PISE), aims to provide the equivalent for mathematics of an annotated chess game at chess.com, or the detailed and clickable driving directions available in Google maps.

In systems of this kind, the display before you has two sides: a graphical side, where all the steps you need to understand are laid out on a type of board or map; and a textual side, where a discussion guides you through the steps. The discussion usually has embedded links or other clickable “widgets” that cause the graphical side to “advance,” “navigate,” or “light up” so as to show the part the text is currently talking about. It is as if the author of the text is able to “point at” the graphics, and make the graphics update to fit the discussion.

PISE lets you do all this with mathematical proofs. The two sides are called proof charts (graphical), and annotations (textual). You author both by coding modules in the Proofscape language [Kieffer].

Like in any IDE (Integrated Development Environment), the charts and annotations can be opened in tabs, and placed side by side. Also like in an IDE, you can edit the source code (Proofscape modules) that generates the charts and annotations, and rebuild. But our emphasis is on study, so we say “ISE” instead.

PISE is free and open-source. It is available to be downloaded and used on your own computer, or accessed online.

A database of steps#

One of the things that facilitates the writing of chess annotations is the availability of massive databases of the moves of past games. We need a similar database of the steps taken in past proofs. Alpine Mathematics is organizing this effort in the The Historical Proofs Archive. You can find more discussion about the structure of this library in A Framework for an Art History of Proofs.

References#

Chernev

Chernev, Irving. The Most Instructive Games of Chess Ever Played: 62 masterly games of chess strategy. Simon & Schuster, New York, 1965.

Bronstein

Bronstein, David. Zurich International Chess Tournament 1953. Translated from the second Russian edition by Jim Marfia. Dover, 1979.

Kieffer

Kieffer S. (2014) Argument Mapping for Mathematics in Proofscape. In: Dwyer T., Purchase H., Delaney A. (eds) Diagrammatic Representation and Inference. Diagrams 2014. Lecture Notes in Computer Science, vol 8578. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44043-8_10