It’s strange how mathematics has evolved over the last few centuries, but the way research mathematics is presented hasn’t changed in a meaningful way. Sure, since the advent of internet, it is easier and faster to propagate your work to the mathematical community, but the form in which this happens seems quite archaic, considering the tools available nowadays. What I mean is the following: since the 15th-16th century, people were able to spread their ideas to an audience broader than ever before, due to the invention of printing. This practice of putting your mathematical research into print has carried all the way to today, with the only difference that printed journals are nowadays also available online. Although the opportunities offered by the online medium are enormous, it seems to me that no meaningful changes have been made since the transition from paper to screen. One of the few features is that you are now able to quickly click-through to references and citations using hyperlinks (which is a feature that actually could use improvement in my opinion, see later). Other than that, the linear structure of text on paper (read from top to bottom) has been preserved, as is the lack of interactivity.
Although I realize that many mathematicians like to complain that there is no need to change what works (I also prefer blackboards to smartboards for teaching purposes, I admit), I would like to propose a thought experiment on how research mathematics is presented and how it could be improved. One of the first links you will stumble upon in a quick Google search is Leslie Lamport‘s “How to write a 21st century proof” (pdf). In this note, Leslie proposes a new style of proofs, which admittedly, seems a bit extreme the first time you come across it. However, extreme does not necessarily imply bad or useless. Let me give you some motivation why change is not always bad. As Leslie puts it himself in “How to write a proof”, an earlier version of the aforementioned essay:
Mathematical notation has improved over the past few centuries. In the
seventeenth century, a mathematician might have written
There do not exist four positive integers, the last being greater than two, such that the sum of the first two, each raised to the power of the fourth, equals the third raised to that same power.
How much easier it is to read the modern version
There do not exist positive integers , and , with , such that .
Surely, he has a point. The added value of using notation and symbols to shorten the language in which you want to convey your mathematical idea cannot be understated. It is this same evolution of natural language to a more structured language that Leslie (and I) would like to see in proofs. I will refer to his Section 2 for an explicit example of how he proves things nowadays.
Personally, my frustration with the current limitations of static pdfs arose while reading a paper, and having to scroll all the way down for the nth time to check a reference, only to scroll all the way up. I’m sure this has to be a familiar feeling for other people. It made me reflect on the shortcomings of the current state, and try to think of some improvements, so here’s two improvements which should be feasible in the near future.
References on hover
The first suggestion is related to my own frustration. It is already possible to click through on references in recent articles, however, wouldn’t it be nice to have a little pop up box appear when hovering over an article in text. In the last few years, YouTube introduced a similar concept for their videos, allowing you to hover the mouse over the timeline at the bottom of a video and see a preview of what will be shown at that specific time. It should not be that hard to replicate a similar ‘textbox-on-hover’ when hovering over a reference, no?
I started thinking more about the way a paper is presented in a pdf and came across the idea of hypertext, or what is apparently called accordions. A Google search revealed that Leslie Lamport had had this idea when I was still in diapers. I will show a rough sketch of a similar example to demonstrate the merit of the idea. It gives the opportunity for quick first readings, skip things you already know or understand but also delve into proof steps that might be unclear. You can write for several audiences (e.g. students and more advanced researchers) at the same time while still catering to their respective needs (e.g. students might need more explanations or motivations on why and how certain things are done). However, there is also a certain drawback to this method. It forces authors to more concise and precise, which is not necessarily a bad thing, but surely there’s a lot more work involved. Secondly, opinions on the depth of detail that should be given are inevitably going to differ so clear guidelines on just how much should be explained might not be feasible. It might improve a lot of texts, but new technology will not suddenly solve the problem of badly-written papers as this is a classical PEBCAK problem.
Without further ado, let me demonstrate the simple example which I recently needed in my own research. The implementation is very rough and also doesn’t allow for LaTeX typesetting, so I’ll have to ask to use your imagination a little. Ideally, there would also be some form of hierarchy as described in Leslie’s paper, but again my limitations with HTML prevent me to show this. In short, we want to prove that a certain group acts transitively on a set of points in the affine plane , an odd prime power, which could be an exercise in an undergraduate course.