Sunday, October 13, 2019

The Sheldon Conjecture (too late for Problems with a Point)

Chapter 5 of Problems with a Point (by Gasarch and Kruskal) is about how mathematical objects get their names. If it was an e-book that I could edit and add to (is this a good idea or not? later on that) then I would have added the following.

The Sheldon Conjecture

Background: Nobel Laureate Sheldon Cooper has said that 73 is the best number because

a) 73 is prime.

b) 73 is the 21st prime and note that 7*3=21.

c) The mirror of 73, namely 37, is prime.

d) 37 is the 12th prime, and 12 is the mirror of 21.

Sheldon never quite said its the only such number; that was conjectured by Jessie Byrnes, Chris Spicer, and Alyssa Turnquist here. They called it Sheldon's Conjecture probably since Sheldon Cooper should have conjectured it

Why didn't Sheldon make Sheldon's conjecture? This kind of question has been asked before:

Could Euler have conjectured the prime number theorem

Why didn't Hilbert (or others) pursue Ramsey Theory?

(readers are encouraged to give other examples)

I doubt we'll be asking this about Sheldon Cooper since he is a fictional character.

I am delighted that

a) There is a Sheldon's Conjecture.

b) It has been solved by Pomerance and Spicer, see here

Actually (b) might not be so delightful--- once a conjecture is proven its stops being called by the name of the conjecturer. If you don't believe me just ask Vazsonyi or Baudet. If you don't know who they are then (1) see here and (2) that proves my point. So perhaps I wish it had not been solved so The Sheldon Conjecture would live on as a name.

Another issue this brings up: Lets say that Problems with a Point was an online book that I was able to edit easily. Then I might add material on The Sheldon Conjecture. And while I am at it, I would add The Monty Hall Paradox to the chapter on how theorems get there names. Plus, I would fix some typos and references. Perhaps update some reference. Now lets say that all books were online and the authors could modify them. Would this be good or bad?

1) Good- The book would get better and better as errors got removed.

2) Good- The book would get to include material that is appropriate but came out after it was published.

3) Good- The book would get to include material that is appropriate but the authors forgot to include the first time around.

4) Bad- For referencing the book or for book reviews of the book, you are looking at different objects. The current system has First Edition, Second Edition, etc, so you can point to which one you are looking at. The easily-edited books would have more of a continuous update process so harder to point to things.

5) Bad- When Clyde and I emailed the final version to the publisher we were almost done. When we got the galleys and commented on them we were DONE DONE! For typos and errors maybe I want to fix them online, but entire new sections--- when we're done we are DONE.

6) Bad- at what point is it (i) a corrected version of the old book, (ii) a new edition of the old book, (iii) an entirely new book? Life is complicated enough.

I would prob like a system where you can fix errors but can't add new material. Not sure if that's really a clean distinction.

1. Isn't the idea of a book which is modified/updated all the time similar to what happens with (open source) software? Bugs get fixed, and features added (and sometimes removed). New releases are published at regular times, and in some cases, any change is immediately available, made easy with services like Github.

Points 4) and 6) are addressed with version numbers. Often of the from X.Y.Z where a change in X indicates a major change, a change in Y a minor change, and a change in Z a tiny patch. Referring to software doesn't always require version numbers ("Java has object"); you'd only need them if referring to something which appears from a certain version onwards ("Perl 5 has objects" (just a major number) or "Bug XYZ got fixed in kernel version 5.3.6" (all components of the version))

1. AH-so e-books which are easily updated but also have a more refined numbering system then Edition 2, Edition 3, etc.
That could work!

2. I differ with Abigail a bit. I offer a few Free books, one of them reasonably popular, and I've exchanged hundreds of emails with students and instructors about them over the years. Books are different than software. With software, if you fix a bug in a subroutine then nobody has to see it. If you add a new flag then the old flags continue to work. But with a book people see everything (apart from makefile-type stuff).

For example, if people are going to use it in a classroom then they want that everyone in the room has the same content, in the same sections on the same pages, matching the same answers (if you offer an answer book). Instructors don't want to hear that half the class got a corrected version of the Jordan Form theorem, in part because that means that half of the class did not.

Also, in my experience they want a paper version, which means periodically freezing at a version since students want to buy it from Amazon or from their school's bookstore.

I don't even intermittently update the git repo, because I've heard from instructors about assigning questions 1-15 odd on page 95 and then having some people say they don't have any questions on that page. The instructor finds out that those folks downloaded an updated version and then is annoyed.

Perhaps it is not logical of the instructor to be annoyed. But they are.