# 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:

ratingdefinition
BLLThe 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’:

titleauthoreditionpublication_year
Golden Years of Moscow MathematicsSmilka Zdravkovska and Peter L. Duren, editors2nd2007

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

editionn
14
2375
3137
461
540
630
714
811
95
105
113
121
131
152
181
271
321
NA2296

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

titleauthorpublication_yearbll_rating
Hex: The Full StoryRyan B. Hayward and Bjarne Toft2019BLL
Introduction to Probability with Texas H o l d ’em ExamplesFrederic Paik Schoenberg2012BLL
Math Art: Truth, Beauty, and EquationsStephen Ornes2019BLL
The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable FunctionsMartin Davis2004BLL**

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:

titleauthorpublication_yearedition
Readings in the History of Mathematics EducationJames K. Bidell and Robert G. Clason197032
Machinery’s Handbook 27, Toolbox EditionErik Oberg et al.200427
Family MathJean Kerr Stenmark, Virginia Thompson, and Ruth Cossey198618
Contemporary Business Mathematics for CollegesJames E. Deitz and James L. Southam200815
Mathematical CrystallographyM. Boisen and G. V. Gibbs199015
Mathematical Recreations and EssaysW.W. Rouse Ball and H.S.M. Coxeter198713
Introductory Mathematical Analysis for Business, Economics and the Life and Social SciencesErnest F. Haeussler, Richard S. Paul, and R.J. J. Wood200712
Calculus for Business, Economics, Life Sciences & Social SciencesRaymond A. Barnett, Michael R. Ziegler, and Karl E. Byleen200711
College Mathematics for Business, Economics, Life Sciences & Social SciencesRaymond A. Barnett, Michael R. Ziegler, and, Karl E. Byleen200711
Finite Mathematics for Business, Economics, Life Sciences and Social SciencesRaymond A. Barnett, Michael R. Ziegler, and Karl E. Byleen200711

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:

titleauthoreditionpublication_year
Probability DistributionsV. Rothschild and N. LogothetisNA2986

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:

titleauthorcount
A Walk Through Combinatorics: An Introduction to Enumeration and Graph TheoryMiklós Bóna3
A Primer on Scientific Programming with PythonHans Petter Langtangen2
A Survey of Modern AlgebraGarrett Birkhoff and Saunders Mac Lane2
Additive CombinatoricsTerence Tao and Van H. Vu2
Algebraic Number TheoryRichard A. Mollin2
An Introduction to ManifoldsLoring W. Tu2
An Introduction to Mathematical CryptographyJeffrey Hoffstein, Jill Pipher, and Joseph H. Silverman2
An Introduction to Tensors and Group Theory for PhysicistsNadir Jeevanjee2
An Invitation to Morse TheoryLiviu Nicolaescu2
Category TheorySteve Awodey2
Combinatorics of PermutationsMiklós Bóna2
Elementary Analysis: The Theory of CalculusKenneth A. Ross2
Elementary Linear Algebra: Applications VersionHoward Anton and Chris Rorres2
Elements of Algebra and Algebraic ComputingJohn D. Lipson2
Episodes in the Mathematics of Medieval IslamJ. L. Berggren2
Finite-Dimensional Vector SpacesPaul R. Halmos2
Galois TheorySteven H. Weintraub2
Geometry: Our Cultural HeritageAudun Holme2
Graph TheoryReinhard Diestel2
Handbook of Parametric and Nonparametric Statistical ProceduresDavid J. Sheskin2
Introduction to Analytic and Probabilistic Number TheoryGérald Tenenbaum2
Introduction to Commutative AlgebraM. F. Atiyah and I. G. MacDonald2
Introductory CombinatoricsRichard A. Brualdi2
Math Through the Ages: A Gentle History for Teachers and OthersWilliam P. Berlinghoff and Fernando Q. Gouvêa2
Mathematics and Its HistoryJohn Stillwell2
Modern Algebra: An IntroductionJohn R. Durbin2
Partial Differential Equations: Analytical and Numerical MethodsMark S. Gockenbach2
Potential TheoryLester L. Helms2
Proofs and Refutations: The Logic of Mathematical DiscoveryImre Lakatos2
Proofs from THE BOOKMartin Aigner and Günter M. Ziegler2
Ramsey Theory on the IntegersBruce M. Landman and Aaron Robertson2
The Fourth Dimension and Non-Euclidean Geometry in Modern ArtLinda Dalrymple Henderson2
The Principia : Mathematical Principles of Natural PhilosophyIsaac Newton, translated by I. Bernard Cohen and Anne Whitman2
The Theory of Island BiogeographyRobert H. MacArthur and Edward O. Wilson2
The Variational Principles of MechanicsCornelius Lanczos2
Thinking MathematicallyJ. Mason, L. Burton, and K. Stacey2
Using R for Introductory StatisticsJohn Verzani2

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:

titleauthoreditionpublication_yearpublisherbll_ratingtopicscount
A Survey of Modern AlgebraGarrett Birkhoff and Saunders Mac Lane41997A K PetersBLL***Abstract Algebra | Algebra | Classic Works2
A Survey of Modern AlgebraGarrett Birkhoff and Saunders Mac Lane42008A K PetersBLL**Classic Works | Abstract Algebra2
Additive CombinatoricsTerence Tao and Van H. VuNA2006Cambridge University PressBLLCombinatorics | Number Theory2
Additive CombinatoricsTerence Tao and Van H. VuNA2010Cambridge University PressBLLCombinatorics | Number Theory2
Elements of Algebra and Algebraic ComputingJohn D. LipsonNA1981Addison-WesleyBLLAlgebra | Applied Algebra2
Elements of Algebra and Algebraic ComputingJohn D. LipsonNA1981Addison-WesleyBLLApplied Algebra | Algebra2
Finite-Dimensional Vector SpacesPaul R. HalmosNA1974Springer VerlagBLL***Classic Works | Linear Algebra2
Finite-Dimensional Vector SpacesPaul R. HalmosNA2017Dover PublicationsBLL**Classic Works | Linear Algebra2
Introduction to Commutative AlgebraM. F. Atiyah and I. G. MacDonaldNA1994Addison Wesley Publishing CompanyBLL***Commutative Algebra2
Introduction to Commutative AlgebraM. F. Atiyah and I. G. MacDonaldNA2015Westview PressBLL**Classic Works | Commutative Algebra2
Math Through the Ages: A Gentle History for Teachers and OthersWilliam P. Berlinghoff and Fernando Q. Gouvêa22014Oxton House PublishersBLL*History of Mathematics2
Math Through the Ages: A Gentle History for Teachers and OthersWilliam P. Berlinghoff and Fernando Q. Gouvêa22015MAA Press/Oxton House PublishersBLL**History of Mathematics2
Proofs and Refutations: The Logic of Mathematical DiscoveryImre LakatosNA1976Cambridge University PressBLL***Classic Works | Philosophy of Mathematics | Polyhedra2
Proofs and Refutations: The Logic of Mathematical DiscoveryImre LakatosNA2016Cambridge University PressBLL***Philosophy of Mathematics2
The Fourth Dimension and Non-Euclidean Geometry in Modern ArtLinda Dalrymple HendersonNA1983Princeton University PressBLL*Non-Euclidean Geometry | Art2
The Fourth Dimension and Non-Euclidean Geometry in Modern ArtLinda Dalrymple HendersonNA2013MIT PressBLL*History of Mathematics | Mathematics and Culture | Non-Euclidean Geometry2
The Principia : Mathematical Principles of Natural PhilosophyIsaac Newton, translated by I. Bernard Cohen and Anne WhitmanNA1999University of California PressBLL*Classic Works | Classical Mechanics | Mathematical Physics2
The Principia : Mathematical Principles of Natural PhilosophyIsaac Newton, translated by I. Bernard Cohen and Anne WhitmanNA1999University of California PressBLL*Classic Works | History of Mathematics | Mathematical Physics2
The Theory of Island BiogeographyRobert H. MacArthur and Edward O. WilsonNA2001Princeton University PressBLLMathematical Biology | Mathematical Modeling2
The Theory of Island BiogeographyRobert H. MacArthur and Edward O. WilsonNA2001Princeton University PressBLLMathematical Biology2

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:

publisherbookspercent_of_total
Springer31910.7
Dover Publications30010.0
Springer Verlag2107.0
John Wiley1806.0
Cambridge University Press1786.0
American Mathematical Society1444.8
Mathematical Association of America1414.7
Princeton University Press1344.5
Oxford University Press1013.4
Birkhäuser802.7

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:
publishernpercent
Springer31910.7
Dover Publications30010.0
Springer Verlag2107.0
John Wiley1806.0
Cambridge University Press1786.0
American Mathematical Society1444.8
Mathematical Association of America1414.7
Princeton University Press1344.5
Oxford University Press1013.4
Chapman & Hall/CRC933.1

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:

authorn
Martin Gardner15
Ian Stewart9
Mircea Pitici, editor9
Serge Lang8
Irving Kaplansky7
Miklós Bóna7
Paul R. Halmos7
Donald E. Knuth6
I. M. Gel’Fand and G. E. Shilov6
Jeremy Gray6
John Stillwell6
Leonard E. Dickson6
Michael Spivak6
Nathan Jacobson6
Steven G. Krantz6
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:
titlebll_rating
AlgebraBLL*
Algebraic Number TheoryBLL
Cyclotomic Fields I and IIBLL
Elliptic FunctionsBLL*
Introduction to Linear AlgebraBLL*
Linear AlgebraBLL**
The Beauty of Doing Mathematics: Three Public DialoguesBLL*

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:
titlebll_ratingpublication_year
History of the Theory of Numbers, Vol. 1: Divisibility and PrimalityBLL*2005
History of the Theory of Numbers, Vol. 2: Diophantine AnalysisBLL*2005
History of the Theory of Numbers, Vol. 3: Quadratic and Higher FormsBLL*2005
History of the Theory of Numbers, Volume I : Divisibility and PrimalityBLL***1966
History of the Theory of Numbers, Volume II: Diophantine AnalysisBLL*1966
History of the Theory of Numbers, Volume III: Quadratic and Higher FormsBLL*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:

titleauthorpublication_year
The Teaching and History of Mathematics in the United StatesFlorian Cajori1890
Projective GeometryOswald Veblen and John Wesley Young1910
Treatise on the Mathematical Theory of ElasticityAugustus E. Love1927
Leçons sur la Théorie Mathématique de la Lutte pour la VieVito Volterra1931
Greek Mathematical Works, Volume I: Thales to EuclidIvor Thomas, translator1939
Rings and IdealsNeal H. McCoy1948
Divergent SeriesG. H. Hardy1949
Elgenfunction Expansions Associated With Second Order Differential EquationsE. C. Titchmarsh1950
Nonlinear Vibrations In Mechanical and Electrical SystemsJ J Stoker1950
Abstract Set TheoryAbraham Fraenkel1953

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.”

topicsnpercent
History of Mathematics42414.2
Statistics1515.1
Mathematical Biology1264.2
Mathematics Education1224.1
Biography1123.7
Combinatorics1113.7
Classic Works1103.7
Number Theory1043.5
Geometry933.1
Real Analysis872.9
Algebra842.8
Mathematical Modeling842.8
Analysis792.6
Mathematical Physics782.6
Partial Differential Equations722.4
Logic702.3
Calculus692.3
Linear Algebra692.3
Abstract Algebra662.2
Surveys of Mathematics652.2

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:

topicsn
History of Mathematics93
Biography | History of Mathematics50
Classic Works | History of Mathematics26
History of Mathematics | Non-Western Cultures14
History of Mathematics | Mathematics Education9
Biography | History of Mathematics | Women in Mathematics8
Calculus | History of Mathematics8
History of Mathematics | Mathematics for the General Reader8
History of Mathematics | Philosophy of Mathematics7
History of Mathematics | Biography6

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:”

topicsn
Classic Works | History of Mathematics26
Analytic Number Theory | Classic Works | History of Mathematics | Number Theory3
Classic Works | Classical Algebra | History of Mathematics3
Classic Works | Geometry | History of Mathematics3
Classic Works | Recreational Mathematics3
Abstract Algebra | Classic Works2
Classic Works | History of Mathematics | Mathematical Physics2
Classic Works | History of Mathematics | Non-Euclidean Geometry2
Classic Works | History of Mathematics | Non-Western Cultures2
Classic Works | Linear Algebra2

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

topicsn
Mathematics Education20
History of Mathematics19
Teaching Mathematics9
Biography6
Statistics6

## 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_ratingnpercent
BLL141947.5
BLL*93131.2
BLL**45115.1
BLL***1876.3

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:

topicbooks
Differential Geometry54
Functional Analysis37
Algebraic Geometry33
Representation Theory25
Technical Mathematics21
Measure Theory19
Proofs and Logic19
Cryptography18
Low-dimensional Topology18
Finance17
Actuarial Science15
Applied Algebra15
Lie Groups15
Engineering Mathematics14
Operator Theory14
Category Theory13
Visualization13
Inequalities11
Liberal Arts Mathematics11
Mathematical Software11
Model Theory11
Multivariate Statistics11
Transformation Geometry11

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:

topicbooks
History of Mathematics19
Classic Works12
Statistics11
Surveys of Mathematics10
Mathematical Modeling8
Mathematical Physics8
Teaching Mathematics8
Algorithms7
Mathematical Biology7
Recreational Mathematics7
Abstract Algebra6
Combinatorics6
Mathematics Education6
Non-Western Cultures6
Number Theory6
Algebra5
Analytic Number Theory5
Biography5
Geometry5

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.