Sharing technical content: thoughts about openness and reproducibility
By: Andre Ferreira Da Silva
Tags: Writing, Communication, Mathematics, Technical Writing, Jupyter Notebook

How many times did we struggle to reproduce or simply understand a public technical work or even a private one, in our workplaces, for example? Despite of all the powerful media we have nowadays, it is sometimes difficult to get over the drag forces of the legacy, as usual.

In this article, I try to do a very simplified analysis of the communication mechanism of a technical subject. Choosing a topic as simple as quadratic equations, I show the first known text about this, followed by the one showing the equation solution the way we know it today. Finally, I highlight some points in the act of communication of this article, itself.

Before going further, two important definitions applied here:

Open: means that the information is available and accessible by the potentially interested audience.

Reproducible: means that the reader has all the required information to repeat the described work, being able to obtain the same results.

DISCLAIMER: I am far from being a communication expert.

Once upon a time…

At the British Museum, there is an Old Babylonian clay tablet (BM 13901) from the century XVIII B.C. (and therefore from almost 4,000 years ago), which is believed to be the oldest ever known mathematical text. The text, using cuneiform symbols, describes a set of geometrical problems and the respective solutions, in which the unknown side length of a square is supposed to be discovered, given the sum of the area and the side itself. Putting in our modern and beautifully comfortable algebraic notation, the problem is simply to find the side L of a square, given the value of L²+L. In other words, the problem is to find the solution of a quadratic equation, a problem that has been solved throughout the human history by Egyptians, Greeks, Chineses, Indians, and maybe many others that did not have their work discovered.

Let us focus on the first problem of the tablet BM 13901, already studied and translated to English by Jens Høyrup and summarized by Duncan Melville. (In the original text, the numbers are in sexagesimal numeral system; also there are no enumeration (a) to (f), I did it here so that the reader can link it to the Figure 1 below)

The area and side of my square I added: 3/4 (a) You, put down 1, the projection. (b) Break 1 in half. (c) Multiply 1/2 and 1/2. (d) Join 1/4 to 3/4: 1. (e) 1 is the square root (of 1). (f) Subtract the 1/2 which you multiplied in the 1: 1/2. The side (is) 1/2.

The geometric interpretation of the described solution is shown in the Figure 1. Notice that for Høyrup, the idea behind adding length and area is actually equivalent to adding just areas, since an area of a rectangle of sides lengths measuring L and the unit (1), is numerically equal to L.

Figure 1. Solution of the first problem of the Babylonian clay tablet BM 13901. The letters (a) to (f) represent the corresponding line in the tablet text reproduced above.

Fast forwarding to at least 3496 years later, in a paper by Henry Heaton (1896), we can find the solution the way we probably learned at school. The picture at Figure 2 partly reproduce the very short paper in the American Mathematical Monthly journal.

Figure 2. Quadratic equation solution by prof. Henry Heaton at the American Mathematical Monthly journal, 1896.

Solving the first problem of BM 13901, using Henry Heaton equations, we would have:

Notice that now we even have a negative solution, which is an abstract idea not yet predicted by the Old Babylonians in their side and area problems.

To enhance the illustration of the quadratic equation solution, we implemented a very simple script to solve it using the same results from Henry Heaton’s paper. The Jupyter notebook is available at Google Colab.

Act of communication

The Old Babylonians, the professor Henry Heaton and even the author of this article all had, in some extent, the same goal: to convey a piece of knowledge. How effective were each of them in this objective? How effectively we improved in this assignment throughout these 4000 years?

Inspired by the Mathematics, let us borrow the systematic methods from the Science to quickly analyze the act of communication of each of them, using a model built by Harold Dwight Lasswell, also known as 5W model, and used for mass communication analysis. According to Lasswell, the act of communication can be described by answering the following questions:

Who

Says what

In which channel

To whom

With what effect?

In the following sections, I try to briefly answer each of the Lasswell’s W for the problems of the Babylonian tablet, the Heaton’s paper and also for this article. Additionally, some further questions are posed to guide the final remarks.

Who: an Old Babylonian scribe

Says what : a set of problems corresponding to an algorithmic solution for discovering the side of a square, given the value of a sum of areas. Translating to modern math: the solution of a quadratic equation.

: a set of problems corresponding to an algorithmic solution for discovering the side of a square, given the value of a sum of areas. Translating to modern math: the solution of a quadratic equation. In which channel : a clay tablet with cuneiform symbols.

: a clay tablet with cuneiform symbols. To whom : probably some other Babylonians “scholars”.

: probably some other Babylonians “scholars”. With what effect: probably to teach how to calculate the taxes on lands, to talk to their Gods through the position of the stars or any other instance of quadratic problem in the Old Babylonian civilization.

Considering the context of those times, here are some further questions:

How many other civilizations in the world could understand cuneiform writing?

How many other civilizations, being unaware of Babylonians studies, “re-invented” the quadratic equation solution?

How much of extra knowledge is required to understand what is written? Remind that the graphical explanation in Figure 1 was not available in the stone and it is not that straightforward to infer from the cuneiforms symbols.

How long can that piece of information last?

Who: Henry Heaton

Says what : deduced a formula to calculate quadratic equations solutions.

: deduced a formula to calculate quadratic equations solutions. In which channel : paper written in English and published in a mathematical journal.

: paper written in English and published in a mathematical journal. To whom : audience of the journal, i.e., mathematicians in general, from undergraduate students to research professionals.

: audience of the journal, i.e., mathematicians in general, from undergraduate students to research professionals. With what effect: share with the mathematical community an alternative method for the general solution of a quadratic equation. Also, Heaton wanted to validate if the method was new (the very last phrase of his paper: “is this new?”)

Considering the context of those times, here are some further questions:

How long would it take to a translated version of this paper to reach an undergraduate student in a different continent far away?

What if some other non-English mathematician had already deduced that formula, how long would he take to be aware of this paper and answer the author?

Who: Myself

Says what : compared briefly the dissemination of a piece of technical information in three different times.

: compared briefly the dissemination of a piece of technical information in three different times. In which channel : social media (Medium, LinkedIn, etc.).

: social media (Medium, LinkedIn, etc.). To whom : anyone producing technical content.

: anyone producing technical content. With what effect: promote reflections on improvements in the technical content dissemination.

Further questions:

How long will it take to reach a person in my social media contacts list who lives thousands of kilometers away from me?

How hard is to find the used references or to reproduce the calculations done in the text?

How easy can the audience discuss the text with the author or among themselves?

How long will this text last? How long is it intended to last?

We can improve it

The scientific community agrees that open and reproducible research strongly enables the collaborative work, so that others can give the further steps on top of the published research, pointing out possible weaknesses or simply re-using it. We can take this in a broader sense, assuming openness and reproducibility as values to be pursued whenever we are broadcasting a technical work.

If we look far behind in the history, we can realize the very powerful communication framework we have nowadays, but also this may lead us to think if we are taking the full advantage of this arsenal. As mentioned in the article The scientific paper is obsolete, we communicate scientific results, for example, mostly the same way throughout at least 400 years: text and pictures in a page. The PDF format, even online, mimics a piece of physical paper. Time to step forward.