Sunday, February 26, 2017

Should we learn from the Masters or the Pupils (Sequel)

A while back I had a blog entry Should we learn from the Masters of the Pupils? The Masters may have more insights but he Pupils may have a better view aided by a modern viewpoint.

Sometimes the Masters are in a different language or not in the modern style but you still want to know what they did and why. As I blogged about earlier (See  here) Villarino/Gasarch/Regan have a paper which explains Hilbert's Proof of Hilbert's Irreducibility Theorem (see) Tao has a paper on Szemeredi's Proof of Szemeredi's Theorem (on Tao's webpage: here). Villarino has a paper on Merten's Proof of Merten's Theorem (here).

Mark Villarino read that blog entry (good to know someone did!) and then presented me with MANY examples where the MASTER is worth reading, which I present to you.  For all of them reading a well written exposition of what the Master did would also be good (as good? better?) if such exists.
Here is his letter with a few of my comments.

I would suggest the following examples where the original teaches and illuminates more than the modern slick version:

1.  Euclid's proof of the Pythagorean Theorem (and its converse).  Indeed, once you understand the diagram, the proof is immediate and beautiful. See here.

2.  Gauss' first proof (by induction) of quadratic reciprocity.  If you REALLY read it, you see how Gauss was led to the proof by numerous specific examples and it is quite natural.  It is a marvelous example of how numerical examples inspired the structure of the induction proof. (BILL COMMENT:  Here is a Masters Thesis in Math that has the proof and lots of context and other proofs of QR: here)

3.  Gauss' first proof of the fundamental theorem of algebra.  The real and imaginary parts of the polynomial must vanish simultaneously.  However the graph of each is a curve in the plane, and so the two curves must intersect at some point.  Gauss explicitly finds a circle which contains the parts of the two curves which intersect in the roots of the polynomial.  The proof of the existence of a point of intersection is quite clever and natural, although moderns might quibble.  In an appendix he gives a numerical example (BILL COMMENT- Sketch of the first proof of FTOA that I ever saw: First show that the complex numbers C and the punctured plane C- {(0,0)} have different fundamental groups (The fund group of C is trivial, the fund group of C-{(0,0)} is Z,the integers.) Hence there can't be an X-morphism from C to C-{(0,0)} (I forget which X it is). If there is a poly p in C[x] with no roots in C then the map x --> 1/p(x) is an X-morphism. Contradiction. Slick but not clear what it has to to with polynomials. A far cry from the motivated proof by Gauss.)

4.  Abel's proof, in Crelle's Journal, of the impossibility of solving a quintic  equation by radicals.  Abel explores the properties that a "formula" for the root any algebraic equation must have, for example that if you replace any of its radicals by a conjugate radical, the new formula must also identically satisfy the equation, in order to deduce that the formula cannot exist  Yes, it has a few correctable errors, but the idea is quite natural. (BILL's COMMENT- proof- sounds easier than what I learned, and more natural. There is an exposition in English here. I have to read this since I became a math major just to find out why there is no quintic equation.)

5. Jordan's proof of the Jordan curve theorem.  His idea is to go from the theorem for polygons to then approximate the curve by a polygon and carry the proof over to the curve by a suitable limiting process. See here for a paper on Jordan's proof of the Jordan Curve theorem.

6. Godel's 1948 paper on his rotating universe solution to the Einstein Field Equations.  Although his universe doesn't allow the red-shift, it DOES allow time travel!  The paper is elegant, easy to read, and should be read (in my opinion) by any mathematics student. (Added later- for the paper see here)

7. Einstein's two papers on special/general relativity. There are english translations.  They are both elegantly written and are much better than the later "simplifications" by text-book writers.  I was amazed at how natural his ideas are and how clearly and simply they are presented in the papers. English Translation here

8.  Lagrange's Analytical Mechanics.  There is now an english translation.  What can I say?  It is beautiful.  Available in English here.

9. I add "Merten's proof of Merten's theorem" to the list of natural instructive original proofs.  His strategy is quite natural and the details are analytical fireworks. (BILL COMMENT- as mentioned above there is an exposition in English of Merten's proof.)

I could go on, but these are some standouts.

BILL COMMENT: So, readers, feel free to ad to this list!


  1. Well worth reading as masterful models of clarity are Claude Shannon's two seminal papers on the theory of communication: "A mathematical theory of communication" (Bell Systems Technical Journal, 1948) and "Communication in the presence of noise" (Proceedings of the Institute of Radio Engineers, 1949).

    Thought-provoking too is Joseph Doob's critical review of Shannon's work (Mathematical Reviews, 1949, available as MathSciNet MR0026286):

    --- --- --- ---
    "The discussion is suggestive throughout, rather than mathematical, and it is not always clear that the author’s [Shannon's] mathematical intentions are honorable."
    --- --- --- ---

    So, perhaps we should all of us aspire to mathematical intentions as dishonorable as Claude Shannon's? :)

  2. PS Google Scholar lists the above two Claude Shannon articles receiving, respectively, 91880 citations and 7162 citations. The first number of citations, in particular, greatly exceeds any other 20th century "Master" articles (known to me at least).

    E.g., Stephen Cook's "The complexity of theorem-proving procedures" has received 6991 citations. This is outstandingly many, yet not in Claude Shannon's class!

  3. Why learn from the Masters if we can learn from the PhDs?

  4. Though Gödel's proof of his completeness theorem is rather messy.

  5. Given both the publicity bias (people know about the master's result) and the strong human tendency toward reverence to the discoverer/creator I would be shocked if these were really the best version to read. Many of these famous results have a large number of expositions but the original is read/consulted an inordinately large number of times creating the false impression it's the best presentation.

    For instance, there are a great many presentations of special relativity the good ones pluck the best parts of Einstein's papers and combine them with accumulated wisdom about what parts are most confusing and what explanations work.


    People have an absolutely insane bias toward the original author of a work. In fiction fans will unexplainably regard even works generally acknowledged as horrible messes as having some special status that mere fan fiction doesn't. For instance star wars episode 1. There have even been psychology studies showing that we regard objects that have merely been touched by famous individuals as being special and desirable. For instance, people will express a strong preference for seeing the real Mona Lisa as opposed to an atom for atom replica.

    Given these strong cognitive biases we should be very very skeptical whenever someone claims the master is the best exposition. We know we will be inclined to believe that even if it's not true...especially as reading the original has more social cachet.

  6. (Peter- THANKS for intelligent comments both on this entry and the more recent one on easy exams.)

    My MOTIVATION for this post was indeed along your lines--- that Ramsey's original paper is awful to read. I've seen it quoted and revered at times which I found.... very strange, though I didn't think about this phenomena the way you have above.

    I remember a while back when the `original manuscript' of Huckleberry finn was discovered:

    and there was all this excitement- about a FIRST DRAFT of a book that I am sure got BETTER with the rewrites that Twain did- so this is obsession with THE ORIGINAL even if its not very good.

    And I've heard this joke (it probably won't work as well in print):
    They found UNRELEASED BEATLE RECORDINGS!!!!- and its George Harrison's Answering Machine just saying `please leave a message'

    I've seen some VERY GOOD Fan Fiction and such should be appreciated more than it is.