Should authors or readers fill the gaps in mathematical arguments?

Starting Point: • We want to pass on understanding of some material. • All people involved should have to spend as little time as possible overall. • Everyone can be a reader at one time and an author at another. • Nowadays in many mathematical texts, it is common for readers to fill some gaps. Dinstiction between gap sizes: • By "small" I mean: roughly one line of intermediate steps is missing. • By "large" I mean: more than that is missing. Small Gaps: Small gaps are, I think, okay. Most readers can easily fill them in mentally. This makes the text shorter and easier to navigate. Large Gaps: I would argue, however, that large gaps increase the total time effort: Qualitative argument

1.: Repeated reading effort for one person: Even if you are an expert, you are still a human being who forgets. If you filled a gap once, but years later want to look something up, it may happen that you have to fill the gap again - almost as if it were the first time. 2.: Many readers, few authors A mathematical text can be written once by a few authors, but read by many people, now and in the future.

Quantitative argument

Assumptions:

• Consider one argument whose written-out version would take about half a page. • Two cases:• Case 1: The argument is not written out. • Case 2: The argument is written out.

• one author; • writing effort in Case 1: 5 minutes; • writing effort in Case 2: 60 minutes; • n readers; • each reader reads the argument twice over their lifetime; • in Case 1, each reading costs 40 minutes; • in Case 2, each reading costs 20 minutes;

General formulas

The total time costs are: • Case 1: T_1 = 5 min + 2n* 40 min and • Case 2: T_2 = 60 min + 2n * 20 min. Case 1 takes more time than case 2 exactly when T_1 > T_2. That is equivalent to T_1 - T_2 > 0. The larger this difference, the more time Case 2 saves relative to Case 1. The difference is T_1 - T_2 = (5 - 60) min + 2n(40 - 20) min = (-55 + 40n) min. For n=2 readers this would already be positive meaning that additional writing effort would save time overall. The larger n is, the more time is saved.

Example - A widely read text

Suppose n = 100 000 readers over 100 years. Then: (-55 + 40 * 100 000) min = 3 999 945 min which is about 66 666 hours = 8 333 eight-hour workdays. If one counts about 260 workdays per year, this is roughly 32 years of work.

Conclusion: Large gaps should usually be filled by authors rather than by readers. In mathematical education, I think there is a lot of room for improvement if we shift some work from reading to writing. More generally: I suspect that similar opportunities exist in non-mathematical education as well. I am interested in connecting with people who want to explore and use this potential.

Discuss