[Math] Techniques for debugging proofs

formal-proofmathematical-writingsoft-question

After writing many proofs, most of which contained errors in their initial form, I have developed some simple techniques for "debugging" my proofs. Of course, a good way to detect errors in proofs is to send them to a colleague for review. But even friendly and helpful colleagues tend to focus on the more interesting proofs and ignore the more technical ones, which are prone to errors. So it is important to have techniques for self-debugging. Some simple techniques that I use are:

  1. Verify that every assumption made in the theorem is used at least once in the proof. Mathematically this is not strictly required, since if "A" implies "C", then "A and B" implies "C" too. But, if an assumption is made and not used, it may indicate an error [there is an analogous technique in programming: many compilers will raise a warning if they detect that a variable is declared but not used.]

  2. Rewrite the proof in the opposite direction. For example, if the proof is by contradiction, rewrite it as a direct proof, and vice-versa. Mathematically it should not matter, but the process of rewrite may help to discover hidden errors.

  3. Read the proof in print. I have no rational explanation for this, but I found out that, when I read proofs in print, I often detect errors that evaded my eyes when I read them on the computer screen.

Question: what other techniques do you find useful for detecting errors in mathematical proofs? I am looking for general techniques, that you would recommend to your research students.

Best Answer

Several basic suggestions.

First, put your manuscript into a drawer, and forget about it for a couple of months. You will discover a whole lot of exciting new things when you take it off from the drawer and re-read anew.

Second, I found that 80% of all mistakes are conveniently marked by the authors with the words like 'evidently", "clearly", and phrases "it is easy to see" etc. Double-check all occurrences of these words/phrases in your manuscript.

The final suggestion: make your manuscript public. Post it to the arXiv, mail it to a colleague, submit it to a journal. You are guaranteed to find a bunch of mistakes in the following ten minutes.

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