Tuesday, September 04, 2012

Should we learn from the Masters or from the Pupils?

The following is a paraphrase of a comment at the end of the Suggested Readings section of Spivak's calculus book:

Abel remarked that he attributed his profound knowledge of mathematics to the fact that he read the masters, rather than the pupils.

Are you better off reading the Masters or the pupils?  This of course depends on the masters and the pupil and other factors.

  1. I have heard that Godel's original papers (even when translated) are well written and show a profound understanding of the subject and why its important.
  2. However, we now have a better understanding of what Godel did and better ways to express it.
  3. The Masters may include the motivation which may be lost in later papers.
  4. Often the first proof of anything is ugly or odd and later proofs really clean it up.
  5. Often the first proof of anything uses only basic concept- later abstractions may hide the heart of the proof.
  6. As a practical matter sometimes the early papers are not available (thanks to paywalls or obscurity) or in a language you do not read.
  7. If Lance and I ever do a book-of-blog-posts I will clean up some of the spelling, make some of the arguments more clear (perhaps indicate where I am being sarcastic in cases where it was not understood), improve the writing. This will make it better than the blog but less authentic.

Here are examples where the Masters papers may not be worth reading:

  1. Recursion theory in the early 1960's had several infinite injury arguments. I have heard that they were known to work only because the lemmas and proofs worked out.  Only after Bob Soare's excellent article on the topic were they really understood. For 0''' priority arguments it is also true that the early papers are not the ones to read.
  2. Example (and the real motivation for this post). I have tried to read Ramsey's original article. I knew that his goal was a problem in logic, and I wanted to know what that problem was. I had a hard time reading the paper.  (I did  my own writeup.) Why was his version so hard to read?  (1) He never uses the words coloring or graph or hypergraph. He doesn't mention that if you have six people at a party either three of them know each other or three of them don't know each other. Perhaps he didn't go to many parties.  (2) He uses odd terms at time.  (3) His paper is rather abstract. If he had just proven a simple case then it would be obvious how to proceed to his abstract case.  This is true for both his combinatorial theorem--- he only proves (what we would call) the hypergraph version, and also the Logic theorem.
  3. The Cliff notes for Atlas Shrugged are far better than the book. Shorter too.  They are online for free here which makes sense since Ayn Rand was known for her altruism.

SO- what do you think? Examples of cases where the Master is better to read?
Examples of cases where the Pupil (or more generally later summaries, surveys, expositions) is better to read?


  1. The fact that the masters are sometimes hard to read may be an explanation of why it can be useful to read them. Since one has some pain to decipher their work, one is forced to really think to the subject and thus to understand everything. Another point could be that the process of finding the desired proof really appears in a first paper but not in the subsequent surveys.

  2. "Another point could be that the process of finding the desired proof really appears in a first paper but not in the subsequent surveys."


  3. Perhaps no one rule applies:

    "I won't say Onsager was the world's worst lecturer, but he was certainly in contention."

    Onsager's writings are not models of clarity either.

  4. At first sight, from the title the most immediate answer is "From both!" It may sound even naive, but as we all know pupils get mastery in their fields whenever they practise enough, always with all the advantages of being wiser but disadvantage of getting less flexible in their thought. As we all get.
    And I cannot agree with your main example, the second one. Dijsktra (search EWD696 link: http://www.cs.utexas.edu/~EWD/ewd06xx/EWD696.PDF), talks about the issues of learning by example.
    Your statement "If he had just proven a simple case then it would be obvious how to proceed to his abstract case." is what makes uncomfortable. In a single case or example you get an idea which is more an approximation to it than the idea itself. We tend to add our own constraints on this approach.

  5. The first one to get an idea may illuminate what it took to get there and what contravention had to be dealt with. Help one in future to do new work. The student may have a plethora of questions regarding what is "obvious" to the master, and these may help the new student to figure out his own understanding. Would have helped me a lot to know more about why SRT was a good solid explanation rather than a cute and disconnected theory. Every student should understand why the twin paradox is not.

  6. Masters are pupils with an new idea. Pupils are masters who refined and completed the old man's thoughts. This is a silly question because masters presume a timeless point in space where some true came into existence.
    If you are studying the life and thoughts of Goedel, then study his writings. If you are studying his ideas, look to the smart people who picked up his idea, and completed it.
    If you are looking to impress freshmen, then appeal to the fallacy that the oldest source is the most credible. I'd suggest something like, "You know, in my day we only had four elements, and that was all we needed..."

  7. I sense sarcasm. Rand was known for preaching selfishness as an economic motivator, not altruism!

  8. Anon 4:03- my question isn't really a YES/NO which is better
    Pupil/Student- my question is an invitation for my readers to share their experiences reading original sources vs reading later versions.

    Anon 9:47- Yes I was being sarcastic.

  9. You mentioned recursion theory in the 60s because of injury arguments... hell, you don't need to invoke injury arguments! Back when people weren't as confident about liberally using Church's Thesis, every paper was an absolute god-awful mess full of gruesome coding that not even a mother could love!

  10. As a novice your Ramsey writeup was made more confusing by an unfortunate typo in the very first Definition, in which clique and independent set are defined identically. I believe it actually meant to clarify the difference by stating that the latter shares NO edges between pairs of vertices.

    1. BECAUSE you made the comment I fixed it!

  11. My high school notion of Euclidean Geometry was that it was based on a set of axioms. Recently, I read a translation from the original Greek and was somewhat surprised to find it is based on a set of definitions.

  12. Another example comes from the Wikipedia article on Peano's arithmetic:

    "Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number. This choice is arbitrary, as axiom 1 does not endow the constant 0 with any additional properties. However, because 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0."

    Peano's original idea may have been more interesting. Zero has at least one additional property - you can't use it as the denominator of a fraction.