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I need to perform several text reviews, each of them containing lots of equations. I need some methodology for proof-reading the equations. The context is not really academic, but I believe a similar situation happens for peer-reviewers in scientific journals, or when some one proof-reads a book before publication. I'm not allowed to contact writers directly.

In general, I can try to "read paying attention", just like someone would read an english text searching for grammar and spelling mistakes. I believe this would be tiresome and prone to error.

Some equations are "practical", such that I can place numbers on them and test consistency of results. Most are not. A few of them might have quantities associated with physical dimensions, such that I can check for errors in the physical dimensions.

A few equations could be tested by implementing unit test-like software that would simulate them, but this would be a long, time consuming process which itself is also prone to error.

Some equations are referenced in the literature, but the notation is quite different.

I have at least to assume that for any equation:

1-The writer might have come up with something that is just plain wrong.

2-The writer might have misspelled something, and this error may propagate to later equations

3-There might be errors in the derivation of those equations.

My performance guidelines are:

1- I'm expecting around 5% of the equations to have some issue, but If I tell some equation is wrong, I'm allowed to be myself mistaken less than 5% of times. But out of all wrong equations, I need to point out 95% of them. The idea is that after 3 reviews by different people, the chances of any mistake having slipped through are negligible.

2- I need to present my reasoning to tell why an equation is wrong, at least to the point where if I'm wrong myself, my own mistake can be pointed out to me.

Are there rules or best practices used in academia for this purpose? Are there other assumptions one could/should make when performing such kind of review job?

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    As peer reviewer in science I'd expect equations to be either referenced or explained (proof or remark that this follows from xxx easily or along so-and-so lines). I wouldn't need to say that an equation is wrong, it suffices to say "this is unclear, so reference or clarify it" (of course if I know it's wrong I'll say it, but if it's unclear I don't need to make my mind up about whether I believe it or not; science is not about belief). However I suspect that this won't help you because your job seems to be somewhat different, removing more or less all the mistakes. That's very tough indeed.
    – Christian Hennig
    Sep 2, 2019 at 21:48
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    I think it would be very useful if you explained where this assignment comes from. Papers aren't collections of random formulas, but one generally follows from the previous one(s), and so one can usually follow the argument -- and if one can't, then there might be a mistake. The point is that checking formulas is not like checking sentences -- each of which might independently have grammar and/or spelling mistakes unrelated to possible mistakes in the previous sentence.
    – Wolfgang Bangerth
    Sep 2, 2019 at 22:33
  • @WolfgangBangerth : I'm working in the development of scientific software, and I'm doing documentation review myself, and trying to come up with a process such that more people can do this properly.
    – Mefitico
    Sep 3, 2019 at 12:40
  • For consideration: Even math textbooks on edition 10+, one or two decades of time on market, and readership in the tens of thousands still have some errors that need fixing.
    – Daniel R. Collins
    Sep 3, 2019 at 15:36

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If the texts that you have to check are papers written for publication in respected academic journals, then you have a problem that is very difficult to solve unless you are an expert in the subject of each of the papers. The reason for that is that published papers are often highly compressed, with quite important intermediate steps left out. I recently spent several weeks studying a paper, trying to reproduce its results. I eventually realised that there was a mistake in one equation. A little later I found what I thought was another mistake. I contacted the author (who had been very gracious about the first mistake) but he showed me that I was wrong: the second equation was correct. All this in a field that is my speciality.

You might possibly spot obvious typographical errors, but the variety of notations used by authors is so great that things that look obviously wrong turn out to be right - in some special notation.

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