How to write tedious algebraic manipulations

Question

In some of my papers, some of the results require tedious but straightforward algebraic manipulations. I am not sure what is the best way to present these manipulations:

• I can write them at the paper body, then they might disturb the continuous reading of the text and distract the reader from the main results.
• I can put them in an appendix, but then they will appear out-of-context and will be difficult to verify by the few readers who will want to.

One solution I thought of is to present them in their place in the paper body, but surround them in a gray box and in a smaller font. Then, most readers will be able to skip them easily, but the few interested readers will be able to read and verify them in context. Is this a good practice?

Answer 1

For this sort of thing, I always fall back on the advice of one of my professors from my PhD coursework. He said you first write out all the algebra in the paper, then you move it to the appendix because it's too long, then you delete it from the appendix because that's too long, and stick it on your website. Finally, you delete it from the website because no one looks at it.

Of course this depends entirely on how crucial the computation is to your paper, and it might be good to look at what's done in other papers in your specific field. But generally speaking, I think you can frequently trust your reader's ability to do tedious algebra if they so desire.

One bit of middle ground that I see a lot is to describe what you do without showing it. For example, if you present three equations and two of them are to be substituted into the first at some point, you can write something like "We first solve equation 1 for lambda, then substitute in equations 2 and 3 for alpha and beta. This gives us equation 4..." That way in one simple sentence you've told the reader exactly how to get to the end result without showing anything more messy than your starting equations and ending equation.

May be unrelated to your field, but in economics I was always fond of Alfred Marshall's advice from 1906:

(1) Use mathematics as a shorthand language, rather than an engine of inquiry. (2) Keep to them till you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life. (5) Burn the mathematics. (6) If you can't succeed in (4), burn (3). This last I did often.

Not really indicative of modern economics, but I like the principle regardless.

Answer 2

I feel that the key is to think carefully about your intended audience: what they want, what they need and what skills they have.

First, your field matters. In some academic fields, mathematical derivations are not necessary in papers; in others, they are not expected; in others still, they are not common but bring some cachet when they are present, and in yet others they are really not wanted and will be viewed negatively. So first figure out how essential a mathematically complete argument is to your paper.

From your profile and site presence, I gather you are working in a branch of CS that is rather mathematical: fair division. If your goal is to prove theorems, then mathematical completeness is obviously quite important. If your goal is to present a new algorithm, then a proof of correctness of the algorithm and/or a rigorous run-time analysis will be very important to some audiences, but others will be more interested in its practical applicability. So I think you should ask yourself How important is it to my readers to be able to understand and verify the mathematical soundness of my work? If it is not that important, then by including a lot of tedious algebra you are putting things in the paper that they will not value that much. I wouldn't recommend doing that.

The next question to ask is To what extent will my readers be able and willing to supply tedious but straightforward algebraic calculations if I omit them? I hope you know that most math papers do not spell out every single detail. On the contrary, when mathematicians are writing for an audience of peers in their subfield, they often omit lots of routine things because they expect that their peers know or can easily figure out the omitted things (and sometimes for less good reasons, honestly, but this is a good reason). It is not rare at all to encounter in a math paper "A routine calculation shows X." (And nowadays, when a calculation is of the sort that a standard software package can do, it is rather common to mention the software package and omit the calculation entirely.) If I had to rule on mathematics as a whole, I would say that our culture is probably a bit too willing to say and do things like this: it is super easy to say "see math.uga.edu/~pete/Gory_Details.pdf if you want the gory details," and this is done sometimes but arguably not often enough. (And indeed, though that's my homepage, there is no such pdf file!) And perhaps the calculation is omitted more because of the length and difficulty of typing it up than because it's so straightforward...unfortunately.

Anyway, I hope these are useful questions, but the answers are not easy. In the end you have to exercise your best judgment as to what will make your paper most readable and most valuable to the audience. Try something sensible and see what happens. For instance, if you think the paper is being cluttered up by too many, too-routine calculations, why not try omitting them from the paper but actually making a Gory_Details file that contains them? The referees and editors may suggest that you do something else -- an appendix, a journal-hosted supplementary file, etc. -- but they will see that you are working hard to do the right thing.

One solution I thought of is to present them in their place in the paper body, but surround them in a gray box and in a smaller font. Then, most readers will be able to skip them easily, but the few interested readers will be able to read and verify them in context. Is this a good practice?

Unless this is commonly done in your field, I don't think so. To me it mostly telegraphs your uncertainty and/or lack of confidence in the value of the calculations. As long as your paper is well written, readers don't need a gray box to be able to skip your calculations, and a small font literally makes it harder to read them (please remember that many people, especially older people, have trouble reading small print), which could be annoying.