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Sunday, July 24, 2022

100 Best Number Theory books of all Time---except many are not on Number Theory

I was recently emailed this link:


That sounds like a good list to have!  But then I looked at it. 

The issue IS NOT that the books on it are not good. I suspect they all are.

The issue IS that many of the books on the list are not on Number Theory.

DEFINITLY NOT:

A Mathematicians Apology by Hardy

The Universe Speaks in Numbers by Farmelo (looks like Physics)

Category theory in Context by Riehl

A First Course in Mathematical logic and set theory by O'Leary

Astronomical Algorithms by Meeus (Algorithms for Astronomy)

Pocket Book of Integrals and Math Formulas by Tallardia

Entropy and Diversity by Leinster

BORDERLINE:

Too many to name, so I will name categories (Not the type Riehl talks about)

Logic books. Here Number Theory  seems to mean Peano Arithmetic and they are looking at what you can and can't prove in it. 

Combinatorics books:  Indeed, sometimes it is hard to draw the line between Combinatorics and Number Theory, but I still would not put a book on Analytic Combinatorics on a list of top books in Number Theory. 

Discrete Math textbooks: Yes, most discrete math textbooks have some elementary number theory in them, but that does not make them number theory books.

Abstract Algebra, Discrete Harmonic Analysis, other hard math books: These have theorems in Number Theory as an Application.  But they are not books on number theory. 

WHAT OF ALL THIS? 

Lists like this often have several problems

1) The actual object of study is not well defined.

2) The criteria for being good is not well defined.

3) The list is just one person's opinion. If I think the best math-novelty song of all time is William-Rowan-Hamilton (see  here) and the worse one is the Bolzano--Weierstrass rap (see here) that's just my opinion. Even if I was the leading authority on Math Novelty songs and had the largest collection in the world, its still just my opinion. (Another contender for worst math song of all time is here.)

4) Who is the audience for such lists? For the Number Theory Books is the audience ugrad math majors? grad math majors? Number Theorists? This needs to be well defined.

5) The list may tell more about the people making the list then the intrinsic qualify of the objects. This is more common in, say, the ranking of presidents. My favorite is Jimmy Carter since he is the only one with the guts to be sworn in by his nickname Jimmy, unlike  Bill Clinton (sworn in as William Jefferson Clinton- a name only used by his mother when she was mad at him) or Joe Biden (sworn in as Joseph Robinette Biden which I doubt even his mother ever used). My opinion may seem silly, but it reflects my bias towards nicknames, just as the people who rank presidents use their bias. 














1 comment:

  1. David in Tokyo here:
    I'm reviewing (and then hopefully pushing) my (painfully weak (Comp.Sci, MS 1984 level)) mathematics. One thing I've noticed is that the first chapter of a surprising number of math books go over number theory basics (divisibility, gcd, and the like) before getting down to the subject at hand. So your "But they are not books on number theory. " is exactly right.

    But I tend to like lists (especially of books). They often tell more about the person who made them than about the subject matter at hand. But that's often fun. Here's one on math books.

    https://www.youtube.com/watch?v=OU4j5g3iNk4&t=694s&ab_channel=TheMathSorcerer

    My other amusement in my old age is Japanese lit, and there are some great fun YouTube channels by gloriously enthusiastic young Japanese bibliophiles, often with lists of books. Just having the list as a target for arguing with is useful and fun. And they'll come up with off the wall things that I'd never thought of looking twice at, or mainline things that aren't my cup of tea, but I really ought to read. Definately food for thought.

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