# The Perfect Math Library Doesn't Exi-

## Introduction

Many mathematicians have a rough mental list of “desert island” books - a small set of books they would choose if stranded on a desert island, forced to spend the winter in a log cabin, or some similiar scenario. Some have published lists of selected books; for example John Baez’s “How to Learn Math and Physics”, “The Mathematics Autodidact’s Aid” by Kristine Fowler, and the “Chicago undergraduate mathematics bibliography” by Christopher Jeris and other contributers. At least one math book is actually set on a desert island: Knuth’s Surreal Numbers.

The MAA (Mathematical Association of America) regularly publishes reviews of mathematical books. They also release the Basic Library List (BLL): a list of books “recommended by the Association for purchase by college and university libraries.” The BLL rates books on the following scale:

rating definition
BLL The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.
BLL* The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.
BLL** The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.
BLL*** The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.

In short, no star means “suggested,” 1 star means “recommended,” 2 stars means “strongly recommended,” and 3 stars means “essential.” The list is being actively updated and revised (though I am not sure how often), and a spreadsheet of the latest version can be downloaded here. For the rest of this post I will be using the version of the spreadsheet that was last updated on August 26, 2019.

While the scale of the BLL is certainly above “desert island” or “log cabin” scale, it is interesting to take a broad look at the books and topics that are on the list. The rest of this post consists of a data-driven exploration of the BLL list. I used R to do the analysis (this post is written in R Markdown), but since this post is less tutorial-focused than others, I’m not going to expose the R code.

## Data Issues

There are several data issues with the BLL list. Skip this section if you’re not interested - the fun stuff comes later.

First, when reading in the file, I found that that the edition for one book was accidently coded as ‘2nd’ instead of ‘2’:

title author edition publication_year
Golden Years of Moscow Mathematics Smilka Zdravkovska and Peter L. Duren, editors 2nd 2007

Fixing this problem and counting the number of books by edition gives:

edition n
1 4
2 375
3 137
4 61
5 40
6 30
7 14
8 11
9 5
10 5
11 3
12 1
13 1
15 2
18 1
27 1
32 1
NA 2296

I’m guessing that “NA” implicitly codes for the first edition. Here are the four books with an explicit label of first edition:

title author publication_year bll_rating
Hex: The Full Story Ryan B. Hayward and Bjarne Toft 2019 BLL
Introduction to Probability with Texas H o l d ’em Examples Frederic Paik Schoenberg 2012 BLL
Math Art: Truth, Beauty, and Equations Stephen Ornes 2019 BLL
The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions Martin Davis 2004 BLL**

There are no obvious problems here (besides the odd spaces in the title of Schoenberg’s book). The books with really high edition numbers also deserve a sanity check. Here are all the books listed with edition greater than 10:

title author publication_year edition
Readings in the History of Mathematics Education James K. Bidell and Robert G. Clason 1970 32
Machinery’s Handbook 27, Toolbox Edition Erik Oberg et al. 2004 27
Family Math Jean Kerr Stenmark, Virginia Thompson, and Ruth Cossey 1986 18
Contemporary Business Mathematics for Colleges James E. Deitz and James L. Southam 2008 15
Mathematical Crystallography M. Boisen and G. V. Gibbs 1990 15
Mathematical Recreations and Essays W.W. Rouse Ball and H.S.M. Coxeter 1987 13
Introductory Mathematical Analysis for Business, Economics and the Life and Social Sciences Ernest F. Haeussler, Richard S. Paul, and R.J. J. Wood 2007 12
Calculus for Business, Economics, Life Sciences & Social Sciences Raymond A. Barnett, Michael R. Ziegler, and Karl E. Byleen 2007 11
College Mathematics for Business, Economics, Life Sciences & Social Sciences Raymond A. Barnett, Michael R. Ziegler, and, Karl E. Byleen 2007 11
Finite Mathematics for Business, Economics, Life Sciences and Social Sciences Raymond A. Barnett, Michael R. Ziegler, and Karl E. Byleen 2007 11

The first book, Readings in the History of Mathematics Education, was published in conjunction with the 32nd Yearbook of the National Council of Teachers of Mathematics, so it is not actually in its 32nd edition. The second entry looks correct, and the book is already up to its 30th edition. Mathematical Crystallography was book 15 in the series “Reviews in Mineralogy.” I didn’t bother checking more or correcting the mistakes, since I’m not going to use the “edition” variable in what follows.

One book is mistakenly listed with a publication year of 2986:

title author edition publication_year
Probability Distributions V. Rothschild and N. Logothetis NA 2986

A quick google search brings up an Amazon listing with a publication date of 1986, which is seconded by a citation in another book. I changed this value to the correct one in what follows.

There also appear to be several duplicate entries in the list. Looking for books with the same title and author gives:

title author count
A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory Miklós Bóna 3
A Primer on Scientific Programming with Python Hans Petter Langtangen 2
A Survey of Modern Algebra Garrett Birkhoff and Saunders Mac Lane 2
Additive Combinatorics Terence Tao and Van H. Vu 2
Advanced Engineering Mathematics Erwin Kreyszig 2
Advanced Modern Algebra Joseph J. Rotman 2
Algebraic Number Theory Richard A. Mollin 2
An Introduction to Manifolds Loring W. Tu 2
An Introduction to Mathematical Cryptography Jeffrey Hoffstein, Jill Pipher, and Joseph H. Silverman 2
An Introduction to Tensors and Group Theory for Physicists Nadir Jeevanjee 2
An Invitation to Morse Theory Liviu Nicolaescu 2
Category Theory Steve Awodey 2
Combinatorics of Permutations Miklós Bóna 2
Elementary Analysis: The Theory of Calculus Kenneth A. Ross 2
Elementary Linear Algebra: Applications Version Howard Anton and Chris Rorres 2
Elements of Algebra and Algebraic Computing John D. Lipson 2
Episodes in the Mathematics of Medieval Islam J. L. Berggren 2
Finite-Dimensional Vector Spaces Paul R. Halmos 2
Galois Theory Steven H. Weintraub 2
Geometry: Our Cultural Heritage Audun Holme 2
Graph Theory Reinhard Diestel 2
Handbook of Parametric and Nonparametric Statistical Procedures David J. Sheskin 2
Introduction to Analytic and Probabilistic Number Theory Gérald Tenenbaum 2
Introduction to Commutative Algebra M. F. Atiyah and I. G. MacDonald 2
Introductory Combinatorics Richard A. Brualdi 2
Math Through the Ages: A Gentle History for Teachers and Others William P. Berlinghoff and Fernando Q. Gouvêa 2
Mathematics and Its History John Stillwell 2
Modern Algebra: An Introduction John R. Durbin 2
Partial Differential Equations: Analytical and Numerical Methods Mark S. Gockenbach 2
Potential Theory Lester L. Helms 2
Proofs and Refutations: The Logic of Mathematical Discovery Imre Lakatos 2
Proofs from THE BOOK Martin Aigner and Günter M. Ziegler 2
Ramsey Theory on the Integers Bruce M. Landman and Aaron Robertson 2
The Fourth Dimension and Non-Euclidean Geometry in Modern Art Linda Dalrymple Henderson 2
The Principia : Mathematical Principles of Natural Philosophy Isaac Newton, translated by I. Bernard Cohen and Anne Whitman 2
The Theory of Island Biogeography Robert H. MacArthur and Edward O. Wilson 2
The Variational Principles of Mechanics Cornelius Lanczos 2
Thinking Mathematically J. Mason, L. Burton, and K. Stacey 2
Using R for Introductory Statistics John Verzani 2

Taking a closer look at some of these cases reveals that they are different editions of the same book. It is debatable whether multiple editions should be included or not. In general, I think only the latest edition should be included, with the exception being long-used classic works where material gets removed in later editions.

One example of this in physics is Goldstein’s Classical Mechanics - the older second edition contains useful annotated “Suggested References” at the end of each chapter that were removed in the third edition. (The third edition, but not the second, is included in the BLL list, with an “essential” rating.)

I’ll leave the multiple editions alone, and look for cases where author, title, and edition all match:

title author edition publication_year publisher bll_rating topics count
A Survey of Modern Algebra Garrett Birkhoff and Saunders Mac Lane 4 1997 A K Peters BLL*** Abstract Algebra | Algebra | Classic Works 2
A Survey of Modern Algebra Garrett Birkhoff and Saunders Mac Lane 4 2008 A K Peters BLL** Classic Works | Abstract Algebra 2
Additive Combinatorics Terence Tao and Van H. Vu NA 2006 Cambridge University Press BLL Combinatorics | Number Theory 2
Additive Combinatorics Terence Tao and Van H. Vu NA 2010 Cambridge University Press BLL Combinatorics | Number Theory 2
Elements of Algebra and Algebraic Computing John D. Lipson NA 1981 Addison-Wesley BLL Algebra | Applied Algebra 2
Elements of Algebra and Algebraic Computing John D. Lipson NA 1981 Addison-Wesley BLL Applied Algebra | Algebra 2
Finite-Dimensional Vector Spaces Paul R. Halmos NA 1974 Springer Verlag BLL*** Classic Works | Linear Algebra 2
Finite-Dimensional Vector Spaces Paul R. Halmos NA 2017 Dover Publications BLL** Classic Works | Linear Algebra 2
Introduction to Commutative Algebra M. F. Atiyah and I. G. MacDonald NA 1994 Addison Wesley Publishing Company BLL*** Commutative Algebra 2
Introduction to Commutative Algebra M. F. Atiyah and I. G. MacDonald NA 2015 Westview Press BLL** Classic Works | Commutative Algebra 2
Math Through the Ages: A Gentle History for Teachers and Others William P. Berlinghoff and Fernando Q. Gouvêa 2 2014 Oxton House Publishers BLL* History of Mathematics 2
Math Through the Ages: A Gentle History for Teachers and Others William P. Berlinghoff and Fernando Q. Gouvêa 2 2015 MAA Press/Oxton House Publishers BLL** History of Mathematics 2
Proofs and Refutations: The Logic of Mathematical Discovery Imre Lakatos NA 1976 Cambridge University Press BLL*** Classic Works | Philosophy of Mathematics | Polyhedra 2
Proofs and Refutations: The Logic of Mathematical Discovery Imre Lakatos NA 2016 Cambridge University Press BLL*** Philosophy of Mathematics 2
The Fourth Dimension and Non-Euclidean Geometry in Modern Art Linda Dalrymple Henderson NA 1983 Princeton University Press BLL* Non-Euclidean Geometry | Art 2
The Fourth Dimension and Non-Euclidean Geometry in Modern Art Linda Dalrymple Henderson NA 2013 MIT Press BLL* History of Mathematics | Mathematics and Culture | Non-Euclidean Geometry 2
The Principia : Mathematical Principles of Natural Philosophy Isaac Newton, translated by I. Bernard Cohen and Anne Whitman NA 1999 University of California Press BLL* Classic Works | Classical Mechanics | Mathematical Physics 2
The Principia : Mathematical Principles of Natural Philosophy Isaac Newton, translated by I. Bernard Cohen and Anne Whitman NA 1999 University of California Press BLL* Classic Works | History of Mathematics | Mathematical Physics 2
The Theory of Island Biogeography Robert H. MacArthur and Edward O. Wilson NA 2001 Princeton University Press BLL Mathematical Biology | Mathematical Modeling 2
The Theory of Island Biogeography Robert H. MacArthur and Edward O. Wilson NA 2001 Princeton University Press BLL Mathematical Biology 2

There are 10 books here, and the double listings seem to be genuine errors. Perhaps the committee behind BLL should take a closer look (I plan to send them an email when I publish this blog post). Some books seem to be repeated with slightly different topic listings, some with slightly different names for the publisher, and I am not sure why the 1997 printing of A Survey of Modern Algebra gets an “essential” rating while the 2008 printing only gets “strongly recommended.”

Since there are only 10 books out of close to 3000 total that are “double listed,” I decided to just leave them alone for the following analysis.

Finally, there are several data errors in the “publisher” variable. Counting the most common publishers:

publisher books percent_of_total
Springer 319 10.7
Dover Publications 300 10.0
Springer Verlag 210 7.0
John Wiley 180 6.0
Cambridge University Press 178 6.0
American Mathematical Society 144 4.8
Mathematical Association of America 141 4.7
Princeton University Press 134 4.5
Oxford University Press 101 3.4
Birkhäuser 80 2.7

Springer and Springer Verlag together make up about 18% of the list, which is not surprising for math books. I was a bit surprised by Dover taking second place.

Scrolling through the entire list of publishers reveals several data issues, such as “princeton University Press,” “Princeton Universy Press” and other misspellings and typos. One publisher shows up as “hapman & Hall/CRC,” “Chapman/CRC,” “Chapman Hall/CRC,” “Chapman and Hall/CRC,” “Chapman & Hill/CRC,” “Chapman & Hall,” “CRC/Chapman & Hall,” “CRC Press,” “Chapman&Hall/CRC,” and “Chapman & Hall/CRC.” The final variant is the most common (78 books, or 2.6% of the total), and adding together all of the variants does change the list of the top 10 publishers:
publisher n percent
Springer 319 10.7
Dover Publications 300 10.0
Springer Verlag 210 7.0
John Wiley 180 6.0
Cambridge University Press 178 6.0
American Mathematical Society 144 4.8
Mathematical Association of America 141 4.7
Princeton University Press 134 4.5
Oxford University Press 101 3.4
Chapman & Hall/CRC 93 3.1

As with the “edition” variable, I did not extensively dig into the “publisher” variable or bother fixing the mistakes, since I’m not going to use it. The variables that I am most interested in and that I will explore in the following sections are author, publication date, topics, and ratings.

## Authors

Which authors have the most appearances in the list? Here are the 15 most prolific:

author n
Martin Gardner 15
Ian Stewart 9
Mircea Pitici, editor 9
Serge Lang 8
Irving Kaplansky 7
Miklós Bóna 7
Paul R. Halmos 7
Donald E. Knuth 6
I. M. Gel’Fand and G. E. Shilov 6
Jeremy Gray 6
John Stillwell 6
Leonard E. Dickson 6
Michael Spivak 6
Nathan Jacobson 6
Steven G. Krantz 6
Gardner and Stewart are no surprise in the top 2 spots, and Mircea Pitici edits a series called The Best Writing in Mathematics which comes out with a new book each year. Serge Lang doesn’t receive any top ratings:
title bll_rating
Algebra BLL*
Algebraic Number Theory BLL
Cyclotomic Fields I and II BLL
Elliptic Functions BLL*
Introduction to Linear Algebra BLL*
Linear Algebra BLL**
The Beauty of Doing Mathematics: Three Public Dialogues BLL*
Undergraduate Algebra BLL

Before you protest, remember that this is a list for undergraduate math libraries, and Lang is best-known for his graduate-level books (Algebra in particular).

Miklós Bóna only makes it on the list because of repeated entries, and Leonard Dickson only because an entire series gets duplicated:
title bll_rating publication_year
History of the Theory of Numbers, Vol. 1: Divisibility and Primality BLL* 2005
History of the Theory of Numbers, Vol. 2: Diophantine Analysis BLL* 2005
History of the Theory of Numbers, Vol. 3: Quadratic and Higher Forms BLL* 2005
History of the Theory of Numbers, Volume I : Divisibility and Primality BLL*** 1966
History of the Theory of Numbers, Volume II: Diophantine Analysis BLL* 1966
History of the Theory of Numbers, Volume III: Quadratic and Higher Forms BLL* 1966

## Publication Date

When were the books on the BLL list published? Here is a plot of the number of books published by year:

This is a very interesting time series. First, here are the 10 earliest books:

title author publication_year
The Teaching and History of Mathematics in the United States Florian Cajori 1890
Projective Geometry Oswald Veblen and John Wesley Young 1910
Treatise on the Mathematical Theory of Elasticity Augustus E. Love 1927
Leçons sur la Théorie Mathématique de la Lutte pour la Vie Vito Volterra 1931
Greek Mathematical Works, Volume I: Thales to Euclid Ivor Thomas, translator 1939
Rings and Ideals Neal H. McCoy 1948
Divergent Series G. H. Hardy 1949
Elgenfunction Expansions Associated With Second Order Differential Equations E. C. Titchmarsh 1950
Nonlinear Vibrations In Mechanical and Electrical Systems J J Stoker 1950
Abstract Set Theory Abraham Fraenkel 1953

The book by Volterra is in French; Google translate gives “Lessons on the Mathematical Theory of the Struggle for Life.” I don’t think that many American undergraduate math majors can read French. Out of curiousity, I checked and the list does contain Whittaker and Watson’s A Course of Modern Analysis, but a 4th edition published in 1996 (the first edition was published in 1902).

After a brief dip in 1972 (9 books), things pick up in 1973 (26 books), and stay fairly consistently high until reaching a peak in 1990 (103 books). There is then a collapse that bottoms out in 1993 (26 books), and then things pick up again and reach a global maximum in 2013 with 120 books. Maybe someone with a closer connection to mathematical publishing can explain this. What was so special about 1990 and the preceding years, and why do things decline so quickly right after that?

The list includes 12 books with a publication year of 2019, and I assume that the number will grow as the list is updated.

## Topics

Each book has one or several associated topics. Here are the top 20 by the number of books tagged with that topic. The topics are not exclusive, so the percentages can add up to more than 100. For example, a book could be tagged with both “Classic Works” and “Algebra.”

topics n percent
History of Mathematics 424 14.2
Mathematics for the General Reader 192 6.4
Statistics 151 5.1
Mathematical Biology 126 4.2
Mathematics Education 122 4.1
Biography 112 3.7
Combinatorics 111 3.7
Classic Works 110 3.7
Number Theory 104 3.5
Geometry 93 3.1
Real Analysis 87 2.9
Algebra 84 2.8
Mathematical Modeling 84 2.8
Analysis 79 2.6
Mathematical Physics 78 2.6
Partial Differential Equations 72 2.4
Logic 70 2.3
Calculus 69 2.3
Linear Algebra 69 2.3
Abstract Algebra 66 2.2
Surveys of Mathematics 65 2.2

The vastly predominant category is “History of Mathematics.” I was surprised that “Mathematics for the General Reader,”, “Mathematical Biology,” “Mathematics Education,” and “Biography” are all more numerous than “classic” categories like “Real Analysis,” “Number Theory,” and “Algebra.”

I thought that perhaps “History of Mathematics” might be so dominant because each subfield has its own history, so there would be history of number theory, history of algebra, etc., each tagged with the subfield as well as “History of Mathematics.” To check this, I looked at the categories most often combined with “History of Mathematics.” Here are the top 10:

topics n
History of Mathematics 93
Biography | History of Mathematics 50
Classic Works | History of Mathematics 26
History of Mathematics | Non-Western Cultures 14
History of Mathematics | Mathematics Education 9
Biography | History of Mathematics | Women in Mathematics 8
Calculus | History of Mathematics 8
History of Mathematics | Mathematics for the General Reader 8
History of Mathematics | Philosophy of Mathematics 7
History of Mathematics | Biography 6

It looks like my hypothesis is wrong - the only grouping of the type I was thinking of is “Calculus | History of Mathematics.” These 10 category groupings contain 229 books, so “History of Mathematics” would still be on top only including these 10. I am not sure what I think about this. History is important, to be sure, but should the ideal math library lean so heavily towards history, especially an undergraduate library? Often tracing the historical development of a subject is not the best way to learn that subject. A good example from physics is quantum mechanics. Learning quantum mechanics for the first time by reading through the history doesn’t really work - the fundamental principles tend to get obscured by unneccessary historical detail. On the other hand, history can catch the interest of students and provide glimpses of mathematical “culture.”

On a similar note, here are the most common topic groupings containing “Classic Works:”

topics n
Classic Works | History of Mathematics 26
Analytic Number Theory | Classic Works | History of Mathematics | Number Theory 3
Classic Works | Classical Algebra | History of Mathematics 3
Classic Works | Geometry | History of Mathematics 3
Classic Works | Recreational Mathematics 3
Abstract Algebra | Classic Works 2
Classic Works | History of Mathematics | Mathematical Physics 2
Classic Works | History of Mathematics | Non-Euclidean Geometry 2
Classic Works | History of Mathematics | Non-Western Cultures 2
Classic Works | Linear Algebra 2

What were the topics of books published in that prolific year of 1990? Here are the top 5:

topics n
Mathematics Education 20
History of Mathematics 19
Teaching Mathematics 9
Biography 6
Statistics 6

## Ratings

Now we come to the most exciting part of the BLL list - the ratings! First, a breakdown of the number of books given each rating:
bll_rating n percent
BLL 1419 47.5
BLL* 931 31.2
BLL** 451 15.1
BLL*** 187 6.3

So about 1 in 16 books get the coveted “essential” rating (BLL***). At an average price of $50 per book, you could purchase the “essential” library for under $10,000. (I don’t actually know if the average price would come out to \$50 per book. It might be higher.)

How does the rating relate to the publication year? (In this plot and the following one, I only included years in which more than 1 book was published.)

Another way to look at this is to plot the percentage of books published each year that received a given rating:

The proportion of books that are just “suggested” has increased rapidly in recent years: from 36% in 2007 to 92% in 2019. This could be because more low-quality books are getting published, or it could be because it takes longer to evaluate the quality of “essential” books - some of the 2019 “suggested” books might have their ratings upgraded to “essential” as time goes on.

What is the makeup by topic of the “essential” undergraduate math library? There are 122 distinct topics for the 186 books. In the entire list (all ratings included), there are 354 distinct topics. Here are the 20 most common topics that are not included in the essential library:

topic books
Differential Geometry 54
Functional Analysis 37
Algebraic Geometry 33
Transition to Advanced Mathematics 31
Representation Theory 25
Technical Mathematics 21
Measure Theory 19
Proofs and Logic 19
Cryptography 18
Low-dimensional Topology 18
Finance 17
Actuarial Science 15
Applied Algebra 15
Lie Groups 15
Engineering Mathematics 14
Operator Theory 14
Category Theory 13
Visualization 13
Inequalities 11
Liberal Arts Mathematics 11
Mathematical Software 11
Model Theory 11
Multivariate Statistics 11
Transformation Geometry 11

Many of these topics are specialized and lean more towards graduate-level than undergraduate level. One that I find surprising is Inequalities - in my opinion a book like The Cauchy-Schwarz Master Class by Steele deserves an “essential” rating rather than just the “strongly recommended” rating it was given.

Here are the 20 most common topics that are included in the essential library:

topic books
History of Mathematics 19
Mathematics for the General Reader 17
Classic Works 12
Statistics 11
Surveys of Mathematics 10
Mathematical Modeling 8
Mathematical Physics 8
Teaching Mathematics 8
Algorithms 7
Mathematical Biology 7
Recreational Mathematics 7
Abstract Algebra 6
Combinatorics 6
Mathematics Education 6
Non-Western Cultures 6
Number Theory 6
Algebra 5
Analytic Number Theory 5
Biography 5
Geometry 5

I looked specifically for some books and was surprised by their omission - for example Gradshteyn and Ryzhik was only included at the base “BLL” level. From the physics point of view, I was surprised that Goldstein’s Classical Mechanics was included in the top-rated books, while the Feynman Lectures on Physics were not included at all. Goldstein’s book is often used as a graduate level text, and the Feynman Lectures certainly seem more appropriate for undergrads.

But these are small quibbles - any list of books inherently invites criticism from individual opinions of its sins of omission and commission. That is part of the value of the list - it invites you to argue with it! Lists are also valuable for their serendipity. While working with the BLL list, I came across several books that I made note of to peruse in the future.

##### Landon Lehman
###### Data Scientist

My research interests include data science, statistics, physics, and applied math.