Proofs that inspire and teach

Question

I was just listening to the show "A Splendid Table" (which I'd recommend, if you're interested in food) in which they were discussing how to spot a good recipe: one which you can follow successfully and reinforces your confidence in your ability to cook. It occurred to me that since theorems/proof are like recipes (the best will prepare an intellectually tasty dish), that we could come up with an analogous checklist (with examples) for them.

Some theorems/proofs are like marvelous magic tricks which can excite and thrill you but leave you mystified as to how it's done ("How did he ever think of that?"). Others also inspire you and leave you with the feeling "I could do that". So I'd like examples of the latter. I don't have specific examples right now, but I was thinking that almost anything by Jean-Pierre Serre has that quality for me. Some might object that all of the details that I list below don't always belong in research papers, but should perhaps be relegated to text books or course notes. I'm not so sure. I've found proofs that could have been made much more accessible for me by adding only a few well-chosen remarks.

Here's what Lynne Rossetto Kasper (the host) gave

You can know ahead of time that a recipe will most likely work if you have a checklist of the key things to look for.

What key ideas, other definitions/theorems do you need to understand before launching into the proof?

One bright red flag is the extremely short recipe. It looks so easy and it can betray you in a nanosecond. That brevity often comes from cutting out the specific information you need to know to end up with something worth eating.

Some proofs are so polished and over simplified that they're like an intricate jewel box -- very pretty, but seem to be evanescent -- leave out one little piece and they fall apart.

Here's the rest of the list:

·Does the recipe tell you what you can prepare ahead?

·Does it tell you how to store the food and for how long?

I'm not sure how well this applies, but it might have to do with how to remember the theorem/proof.

·Are the ingredients specific -- not "1 pound beef," but "1 pound well-marbled beef chuck"?

·Do the instructions tell you ...

  ·What kind of pot and utensils to use?

What mathematical techniques are you using?

  ·The level of heat and/or the timing needed for each step?

I'm not so sure about this, but it might be -- into how much detail you need to go.

  ·What the food should look like, sound like, and/or smell like?

I find this quite interesting. This is a guide to tell how to know if your intuition is on the right track as you proceed.

  ·How to know if it's done?

A lot of times this is obvious, but there are a lot of proofs, at whose end, I'm forced to sit and think as to why it's finished!

  ·How to serve?

How to talk/write about this theorem/proof.

Answer 1

I don't think I care to go through a checklist to confirm this, but I have to say that I find virtually anything by Milnor satisfying and dependable. It's not just the smoothness of his exposition; it's the great courtesy he extends by giving full details in proofs (no "it is easy to see that"), and somehow not making them feel too heavy. It's not usually a feeling, "I could have thought of this myself", but more a feeling, "this is something I can really take home with me".

An exemplar of this would be his wonderful monograph, Morse Theory. But you can see it in shorter works too, for example his proof of the Hairy Ball Theorem (American Mathematical Monthly, July 1978, pp. 521-524). For someone like me who is not a born geometer, he is just a tremendous teacher, with arguments you can just hold in your mouth to sift and savor, again and again.

Answer 2

I feel that many of the proofs in Munkres' Topology are like this; some people might say he takes too much time and goes into too much detail. But his exposition of the proof that all subgroups of free groups are free takes an entire chapter, is well organized, and leaves with the feeling that "I could do that". The same is true for many of his lemma/theorems leading up to the Jordan Curve theorem, as well of his proof of Urysohn's lemma and the Tychonoff theorem.

Answer 3

Not really an answer, but it seems the "recipe for a good recipe" you gave applies better to a well-written paper than a good proof. Here's how I think they correspond to each other:

One bright red flag is the extremely short recipe. It looks so easy and it can betray you in a nanosecond. That brevity often comes from cutting out the specific information you need to know to end up with something worth eating.

One bright red flag is the extremely short paper. It looks so easy and it can betray you in a nanosecond. That brevity often comes from cutting out the specific information you need to know to end up with something worth knowing.

·Does the recipe tell you what you can prepare ahead?

·Does it tell you how to store the food and for how long?

·Does the paper tell you what you can prove ahead?

·Does it tell you how to use the result?

·Are the ingredients specific -- not "1 pound beef," but "1 pound well-marbled beef chuck"?

·Do the statements rely less on context -- not "let $a$ be positive," but "let $a$ be a positive integer"?

·Do the instructions tell you ...

$\quad$·What kind of pot and utensils to use?

$\quad$·The level of heat and/or the timing needed for each step?

$\quad$·What the food should look like, sound like, and/or smell like?

$\quad$·How to know if it's done?

$\quad$·How to serve?

·Does the body text tell you ...

$\quad$·What kind of known result and technique to use?

$\quad$·The level of attention needed to pay to each paragraph?

$\quad$·What the result should look like, sound like, and/or smell like?

$\quad$·When the paper concludes?

$\quad$·How to persuade others to read too?