# The art of proof summarizing. Are there known rules, or is it a purely common sense matter?

When a proof is long and difficult, it can be really nice vis-à-vis the reader to give a summary or an outline of the deduction before beginning hard work.

Are there known rules to give a good proof summary?

Are there known rules to find which point to emphasize in a proof summary? I mean rules to find the "nervus probandi".

• My congratulation. This is the first time that I see a question tagged as proof-writing which is really about proof-writing. – José Carlos Santos Apr 24 '19 at 14:20
• @JoséCarlosSantos. Thanks :) – user654868 Apr 24 '19 at 14:21
• I think this is a good and difficult question to address. I certainly don't think it is just "common sense". For instance, it may be that a quick outline of the main steps of the argument does not emphasize properly the places where genuinely new ideas come into play. Also, in some cases such an outline may not be too helpful anyway (for instance, there are results where we know that to get to C from A we should prove B and to prove B probably we prove D first; so giving such an outline won't clarify matters, and you will need to delve into specific details to indicate how you succeeded). – Andrés E. Caicedo Apr 24 '19 at 14:26
• On the other hand, there may be cases where the quick outline is really all you need to help guide the reader. – Andrés E. Caicedo Apr 24 '19 at 14:27
• @EleonoreSaintJames, not really difficult; the inversion principle simply states that "whatever follows from the direct grounds for deriving a proposition must follow from that proposition", it's the justification for the elimination rules of natural deduction. Cut elimination is simply a method for pushing up and up an instance of CUT in sequent calculus until it's eliminated; it's what Gentzen used to form his Hauptsatz. – Simone Apr 24 '19 at 14:38

I agree with the commentors that this is a difficult question to answer, and what I write here may be only a partial answer. I've not encountered any set of guidelines for summarising a mathematical proof, but in general the same as are used for summarising any large or complex topic can be applied. (If you are interested in mathematical proof in particular you might want to look at https://www.math.wustl.edu/~sk/eolss.pdf which is a history of mathematical proof writing by Steven Krantz.)

A good summary tells the reader what to expect from the story(proof) that follows, should highlight any requisite knowledge, and should motivate the reader into making the effort to follow the argument. If you look at the Bourbaki style of proof, for example, there is often none of this -- a theorem is given and is proved and it is up to the reader to contextualise it, link it to previous work and knowledge and find a reason for remembering it. However if you look at some of Steven Krantz's books you will find that he actually spends the majority of a chapter motivating and explaining the ideas, and relegates the actual mathematical proof to the very end of the chapter -- the complete antithesis of a Bourbaki proof.

To write a good summary the author must understand the material thoroughly: in fact, being able to summarise a proof well is a good indication that the author has properly understood it. If at any point the author finds themselves waving their hands, or glossing over a detail, the chances are that there's something there they themselves don't have clear in their own mind.

As an example then: consider the stalwart of calculus lectures, the Intermediate Value Theorem. This says that if we have a continuous function defined on a contiguous set (a 'closed interval') of points, and one endpoint is smaller than zero while the other is greater than zero, then there is a point inside that interval where the function value is zero. This is nicely summarised by saying "the graph of a continuous function has no breaks in it". This immediately suggests a way to start thinking about it (draw a graph), it connects it to other things the reader already knows about (how to graph a function), and highlights that the reader should know what a continuous function is before proceeding.

EDIT: ruakh points out in the comments that I've summarised the theorem and not the proof, for which I apologise. To summarise the proof then:

Since $$f$$ is negative at some points and positive at others, and is continuous, we can show that the supremum of the set of negative points is both $$0$$ and is achieved by $$f$$. $$0$$ plays a key role here, so we can expect that we can find points where $$f(x)=c$$ by considering $$f(x)-c$$, and that we can find roots of polynomials by finding points $$a$$ and $$b$$ with $$f(a)<0$$ and $$f(b)>0$$ (which might lead us to the bisection method).

This achieves our goals in summarising: we know what to expect coming up (the study of the set $$\{x: f(x)<0 \}$$); we know we need to know what a supremum is, and we can see how we might use this in future.

• @postmortes.Thanks for your answer and for the link to Krantz's paper on the History and Concept of Proof. – user654868 Apr 24 '19 at 14:57
• The example in your last paragraph seems strange to me; surely "the graph of a continuous function has no breaks in it" is an informal description of the Intermediate Value Theorem itself, not a summary of its proof? – ruakh Apr 24 '19 at 19:44
• @ruakh thank-you, you are of course right. I've edited the answer; please feel free to downvote though – postmortes Apr 24 '19 at 20:24
• @postmortes: Good heavens, I certainly wouldn't downvote for that! Thanks for the edit. :-) – ruakh Apr 24 '19 at 20:26
• But such summary could substitute the actual proof in a less formal paper or even suggest to the reader a lower level of formality of the paper. – ZonkoProducts May 2 '19 at 8:41