How to decide the appropriate level of explanation for math in academic papers?

In any academic paper dealing at least in part with math, the authors have to make a choice how much explanation to actually give the math.

It can range from pretty much no explanation:

Given a foobar of 12, a ackbar of 57, and general frobnosticating parameter of 9, the resulting nyan level will be 8.

To belaboring the obvious

The first three people went into the city. Later, four more people went into the city. This meant that seven people had gone into the city which is derived by adding the first three people to the next four people. We add because people are individual units and moving them to a new location neither combines or divides those units.

How does one decide the appropriate level of explanation? Is there any sort of standard that can be appealed to decide when something is clear enough?

The actual background for this question is that I've looked at a paper for a colleague. It is in a non-mathematical subject but does engage in some mathematical reasoning. I think the paper doesn't explain enough, but my colleague disagrees. I have to work at figuring out what calculations were done in the paper and the justification for many of the inferences he draws. That indicates to me that the explanation is insufficient, but is there a less subjective standard I can appeal to?

• Lately, it seems, I have been reading plenty of papers that belabor the obvious when unnecessary, but completely omit any detail regarding the actual hard parts of the calculation. I've found that this occurs partly as an attempt to keep the paper to a certain length, and partly (and independently) to attempt to include some level of detailed explanation. It's frustrating. Feb 1, 2015 at 0:04
• @Arkamis, I've also noticed that trend, I've always taken it as people adding explanations where it is easy to add them rather then where its needed. Feb 1, 2015 at 0:14
• It depends on the field. I recently saw a famous paper in Bioinformatics whose main contribution was combining very different datasets in a meaningful way. They just said "we applied a bayesian framework", and one had to go to the supplementary materials to see the formulas, right after 70 pages of a table enumerating the data sources. Feb 1, 2015 at 4:12

I find that the choice of how much explanation to give math is generally a three-way negotiation between three factors:

1. Your estimate of the audience: different communities will need radically different levels of explanation for the same mathematics.
2. Adjustment based on the opinions of the reviewers about what needs more or less details
3. Any length constraints on the paper.

Of this #1 is really the important thing: you really need to understand your audience in order to decide how in-depth to go with your mathematics.

For example, I recently published a paper which spent several pages explaining a mathematical formulation in depth for its target cross-disciplinary audience. The reviewers requested further expansion of the mathematical explanation (which I was happy to provide). Were I writing for the community from which the mathematics came, however, I would instead spend several pages explaining the context of the problem, but then the math itself would be covered in just a few sentences.

How does one decide the appropriate level of explanation?

There are fields, or subfields, of several applied and experimental sciences where equations, long derivations and mathematical explanations are definitely unwelcome.

In fields like mathematics or theoretical physics, equations, theorems or proofs can be the main topic of a paper: the proof of a theorem can be the most interesting part. In other fields equations might be interesting, but their applications much more; and the proofs are frequently considered a nuisance. Tell me how to use your mathematical ideas, what are the benefits and the weaknesses, but please, really please, put all the proofs, derivations and detailed explanations under the rug.

The rug, in this case, can be either an appendix -- but even there avoid too many details -- or, if you think that your proof is worth per se, write a paper in a journal whose readers might be interested in, then cite it. And sometimes, in a few cases, you can even get away without a proof...

• "Beware of bugs in the above code; I have only proved it correct, not tried it." — Donald Knuth Feb 1, 2015 at 6:19

I'm guessing the author thinks that putting too much math into the body of the paper might disrupt the paper's flow. On the other hand, as a reader, you don't want to have to reconstruct the calculations to satisfy your curiosity that everything is correct.

If both of those assumptions are true, perhaps some of the math can be included in an appendix at the end of the article, with a brief reference in the main body.

• I think that's good advice in general. In my case, I think the author just thinks his paper is clear enough without additional explanation and he shouldn't have to put any in. Feb 1, 2015 at 0:18
• @Winston - Perhaps that was his initial assumption, but, after you looked over his paper, you seem to disagree. Now the ball is in his court; he choose to heed your feedback, or ignore it. I've offered a solution that would let him address your concern without disrupting the flow of the paper.
– J.R.
Feb 1, 2015 at 4:03