Excellent mathematical explanations

Question

In the Stanford Encyclopedia of Philosophy there is an entry on mathematical explanation.

The basic philosophical question is: What makes a proof explanatory?

Two main "models" of mathematical explanation are mentioned:

1) Steiner's model, which asserts that an explanatory proof can be distinguished from a non-explanatory proof if only the explanatory proof contains a so-called "characterizing property", which is (roughly) a property unique to a particular entity or structure in a family of structures. Here `family' is taken as primitive. (For a more complete description, follow the above link.)

2) Kitcher's model, which asserts that a proof is explanatory if it provides a unification of disparate methods. (Again, follow the link for a more complete discussion.)

Philosophers of mathematics find neither of these models adequate. The above philosophical accounts of mathematical explanation proceed from the top down, but in each case it has been possible to find good examples of mathematical explanations which don't fit. Quoting the linked entry above:

Recent work has shown that it may be more fruitful to proceed bottom-up, by first providing a good sample of case studies before proposing a single encompassing model of mathematical explanation.

I confess, I've been unable lately to figure out how to ask a good `philosophy of mathematical practice' question here on MO and so this is my attempt to do so. I think we can provide the philosophers (and each other) with a storehouse of proofs that are also excellent explanations, and reason why we think so. If this works, I may send the link to Paolo Mancosu as a gesture of good will toward the philosophers who are studying contemporary mathematical practice.

Please volunteer an example of an excellent explanatory proof, and the reason you think the proof also provides a good explanation of the phenomenon it deals with.

Since the philosophers are looking for case studies, I think that a link to the proof in question will be enough (if the proof is not short). I'm sure that you can be contacted later to explain.

(I'm still not sure if philosophical language is appropriate for MO, but the above question has clear value and will admit precise mathematical answers!)

Answer 1

For your purposes, it may be better to exhibit pairs of proofs of the same result, one of which is considered "more explanatory" than the other.

• The first example that comes to my mind is the alternating sign matrix conjecture, for which Kuperberg's proof is widely regarded as being "more explanatory" than Zeilberger's original, highly computational proof.

• There are surely examples from Lie theory or finite group theory where some result is first proved by invoking a classification theorem and checking each case separately, and then later someone finds a uniform proof that does not rely on the classification. Unfortunately I can't think of any specific examples off the top of my head.

In general my sense is that proofs that involve long, formal calculations with no guiding idea, or that involve exhaustively checking a large number of disparate cases, are considered non-explanatory, unenlightening, and ugly.

Finally, I think that certain proofs without words would be regarded by many as being excellent explanations.

Answer 2

Bill Thurston's Three-Dimensional Geometry and Topology, Volume 1 (Ed. Silvio Levy) contains many examples. Here is one. On p.74, he starts with "Some computations in hyperbolic space," and with this overarching view:

Ultimately, what we seek when we study mathematics is a qualitative understanding. But precise, quantitative manipulations—the nitty-gritty of mathematics—are also important as a way to reach this end, and as a test that our qualitative understanding is correct.

He then proceeds with ten pages of examples and calculations before reaching

Proposition 2.4.12 (ideal triangles). All ideal triangles are congruent and have area $\pi$.

along the way proving the spherical law of cosines:
          

Answer 3

For the outstanding examples of explanatory proofs I would refer to Archimedes, Euler (complete works), and the book of Kepler, Stereometry of wine barrels. (I mention only one book of Kepler, because this is the only one for which I could find a good translation from German to a language that I can read and enjoy).

Of the more recent examples, I would mention the work of Pierre Fatou, Sur les equations fonctionnelles. More examples from the XX century can be given, but I do not want to include a long list.

The common feature of these writings is that their authors explain HOW they arrived to their results, not only the final results themselves. On my opinion, it is THIS feature that makes a proof really explanatory. Unfortunately, this style of writing becomes more and more rare.

Answer 4

I think explanatory is a group concept: a proof is explanatory when it affects a group of readers in such a way that they can explain the proof to others after they read it. There may be a more philosophical way to say it, but leaving the concept to an individual is a practical mistake in my opinion.

As an example, I proffer the compactness theorem for first order logic: a set of sentences has a model if and only if every finite subset of that set has a model. One can check several sources to get the full meaning and impact of the theorem and underlying concepts, and then go through several technical details in many of the proofs, but I will offer an explanatory hint: proofs are finite. This is used as a mnemonic for the challenging part: if the set does not have a model, there is a (finite) proof of contradiction, which suggests the appropriate finite subset to choose as not having a model. I invite the mathematical logic part of the MathOverflow community to enhance this into an explanatory proof example.

Gerhard "Doing The Writeup Costs More" Paseman, 2012.12.18

Answer 5

There was an interesting thesis about analyzing the natural language used in writing mathematics, that seems a bit relevant to this discussion. The thesis is no longer on line but some info about it is here: http://people.ds.cam.ac.uk/mg262/

I forgot where I got that link from, but it was another MO thread a while back.

Answer 6

My analogy for an explanation relates it to explaining the way to the station: you describe the general landscape on the way, not every crack in the pavement. You are assuming a certain known structure, or type of structure. (Such as: turn left at the traffic lights, then right at the Post office.) You do draw attention to any major holes in the path.

In mathematics we often have to invent the structures which make the ideas clear. I heard Raoul Bott in 1958 say of Grothendieck that he was prepared to work very hard to make a proof tautological. Grothendieck was very keen on understanding the underlying processes and forms.

One importance of this is for computational help: we would like computers to deal with the various hierarchical levels in mathematical practice, and not to have to search for a proof using a low level of structure.

Answer 7

In my experience one can study a huge amount of mathematics (group theory and number theory) before encountering the idea that the law of quadratic reciprocity has anything to do with the signature of a permutation. Quadratic reciprocity is so easily a black hole - it works, but why?

This is such a beautiful connection, which so naturally lends itself to thinking "how would I generalise that" as life goes on - I wish I had known it earlier than I did.

Answer 8

One idea of what makes a proof explanatory follows. I suppose it is similar to Steiner's idea, but not identical. I think, however, that the quality of being explanatory is not something neatly definable -- it is instead a function of several disparate things. What makes people satisfied with an explanation (of anything)? It's context dependent, varies person-to-person, and isn't even dependent on correctness (people can be satisfied with incorrect explanations, and dissatisfied with correct ones).

You (I suppose) have some mental model of similar situations to the one you are proving something about. If a proof gives you results that you can immediately realize about the similar situations, then it is explanatory (for example, that X, which is true for your initial situation A, can be / cannot be true for similar situation B -- a long computation will not give this immediate realization until you carry it out). Thus, the property of being explanatory is dependent on your own power of formal manipulation and reasoning -- what you can immediately realize -- and your model of what similar situations are.

For an example, there are image filters that involve sequences of matrix computations that highlight the top edges of objects in the image. A proof of this is trivial, given an image -- just compute. However, it's not explanatory because you don't immediately know from the computational proof that it would work on another image (other images make up our model of similar situations).

On the other hand, a proof that a filter inverts the colours in a particular image might be easily made explanatory, because one can give as a lemma that the filter inverts the colour of any pixel. Then you can see that any other image will be inverted by the same filter.