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In a philosophical context, I’m currently thinking about how best to explicate mathematicians’ judgements that some correct proofs are ‘explanatory’ while others are not. In this vein, I’m trying to collect examples of theorems that have two salient proofs, one of which is judged to be explanatory whilst the other is not (even better if the examples exhibit strong disagreement regarding which proof is more explanatory). Other things being equal, simpler examples are preferred, and I’m especially interested in examples from abstract algebra, order theory and topology. Pointers towards relevant debates in the history of math would also be appreciated.

(Disclaimer: this question is related to but distinct from the question below, which concerns the relationship between explanation and beauty in mathematical proof: An example of a proof that is explanatory but not beautiful? (or vice versa).)

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    $\begingroup$ By asking for "strong disagreement" you make this question opinion-based; I don't think it suits this site. $\endgroup$
    – Chris Wuthrich
    Commented Nov 5, 2019 at 12:45
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    $\begingroup$ @ChrisWuthrich If there's other sources they can point to which show people strongly disagreeing then that will still be an objective statement. $\endgroup$
    – JoshuaZ
    Commented Nov 5, 2019 at 14:14
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    $\begingroup$ In my opinion the most important criterion for deciding how "explanatory" a proof is, is the extent to which it provides techniques that can be used to prove other things. (And this criterion is what mathematicians really care about, I think.) $\endgroup$
    – Nik Weaver
    Commented Nov 5, 2019 at 17:13
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    $\begingroup$ There is a literature on this — have you consulted plato.stanford.edu/entries/mathematics-explanation? $\endgroup$
    – user44143
    Commented Nov 5, 2019 at 18:10
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    $\begingroup$ @NikWeaver : I'm not sure I agree. Consider algorithmic methods such as the WZ method or Groebner bases, which allow one to crank out proofs of certain types of statements mechanically. These certainly provide "techniques that can be used to prove other things" but I hesitate to call the resulting proofs "explanatory." $\endgroup$
    – Timothy Chow
    Commented Nov 5, 2019 at 23:55

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Paul Halmos once gave the following example in a talk for a general audience. Suppose there is a tennis tournament with 128 players. In the first round 64 of them are paired off with the other 64, they play their games, and all the losers are ejected from the tournament. In the next round the remaining 64 players are paired off and this time the 32 losers are ejected. Eventually one player is left, who wins the tournament. How many games were played in total?

The most obvious way to solve this is to add up 64 + 32 + ... If you remember a formula about geometric series, you can find this sum quickly.

But the explanatory proof is different. Every player aside from the eventual winner loses exactly one game, the game in which they are ejected. So the total number of games = the total number of losers = 127.

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    $\begingroup$ This reminds me of the anecdote about John von Neumann and the fly and two trains problem: http://mathforum.org/dr.math/faq/faq.fly.trains.html. $\endgroup$
    – daneel
    Commented Nov 5, 2019 at 20:57
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    $\begingroup$ I saw a variation of this problem in a Martin Gardner article, where the number of players was not a power of 2, and there were rules for byes. The contrast between the two proofs is even more striking here because the pedestrian proof involves keeping track of the byes, which could get complicated and confusing. $\endgroup$
    – Timothy Chow
    Commented Nov 5, 2019 at 23:41
  • $\begingroup$ I was not familiar with this ... simply beautiful and explanatory! $\endgroup$
    – EGME
    Commented Nov 6, 2019 at 19:21
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This is a well-known and well-documented example: the first proof of the Alternating Sign Matrix theorem was a complicated, inductive "manipulation of generating function"-style argument by Zeilberger; shortly thereafter Kuperberg gave a shorter proof based on a connection to the six-vertex model of statistical mechanics and the Yang-Baxter equation. The history of the Alternating Sign Matrix conjecture is beautifully told in the book Proofs and Confirmations by Bressoud.

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  • $\begingroup$ Good example. This example, and many others that might interest the OP, can be found in another MO question. Usually, though not always, proofs that require a lot of calculation are considered to be less explanatory, and of course Zeilberger's proof was extremely computational. $\endgroup$
    – Timothy Chow
    Commented Nov 5, 2019 at 23:47
  • $\begingroup$ @sam hopkins, thanks for nice example, that's very helpful. $\endgroup$
    – King Kong
    Commented Nov 6, 2019 at 8:12
  • $\begingroup$ @TimothyChow, thanks for pointing this out -- that does look helpful. Importantly though, i'm interested specifically in judgements regarding whether proofs are 'explanatory' rather than whether they are `nice', i.e. short, efficient etc. While there is undoubtedly a strong correlation between e.g. proof length/complexity and explanatoriness, I don't want to conflate any of these properties from the get go. $\endgroup$
    – King Kong
    Commented Nov 6, 2019 at 8:13
  • $\begingroup$ @KingKong : If you're trying to collect data for a formal research project in sociology then you probably need to go the standard route of sending out surveys rather than posting a question on MO. $\endgroup$
    – Timothy Chow
    Commented Nov 6, 2019 at 16:44
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Here is a related situation which may guide later posts: plant ten trees in five rows with four trees per row.

After pushing dots around on paper and realizing that every tree has to belong to two rows, one ends up with ten dots in a pentagram configuration, usually regular. This is a solution, but does not yield much in the way of understanding.

If one instead pushes rows around, one finds many more solutions, since any arrangement of five lines in which no two are parallel (edit: and no three concurrent, oops) leads to another configuration. Then one sees that the problem is about incidence structures, and understands how to generalize the problem and solution set.

Gerhard "Really, It's In The Telling" Paseman, 2019.11.05.

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  • $\begingroup$ I remember hearing this puzzle and being utterly confused by it. It has not occured to me that the rows needn't be parallel (like rows of a matrix, say) $\endgroup$
    – Wojowu
    Commented Nov 5, 2019 at 17:10
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    $\begingroup$ I had to look up pictures to understand this one, too...I think this could be improved by making the "five rows with four trees per row" part clearer. $\endgroup$
    – user3067860
    Commented Nov 5, 2019 at 19:35
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    $\begingroup$ Part of the point of the puzzle is to have the reader imagine a model where the statement makes sense. This is not quite what the original poster asks for, but it underscores my point that the difference between explanatory and non is a matter of perception and forming a good model. Gerhard "Hoping For More Model Answers" Paseman, 2019.11.05. $\endgroup$
    – Gerhard Paseman
    Commented Nov 5, 2019 at 19:45

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