Unlike most Calculus textbooks, however, Oullette’s book has an extra helping of sympathy for its audience.  Oullette’s goal is not necessarily to make her readers expert mathematics students; instead, she focuses on unifying seemingly disparate types of problems under the umbrella of Calculus.  Included amongst these examples are applications of Calculus to the equations of motion, thermodynamics, surfing, and the spread of disease.  The wheel is not being reinvented here – most of these examples (with the possible exception of what Calculus tells us about the optimal strategy in the event of a zombie outbreak) should be covered in a Calculus course.  But since Oullette’s goal is to appeal to the people who never took Calculus, this overlap is likely intentional.

Writing about mathematics for a general audience can be a difficult balancing act.  If the language is too technical, the average reader may become confused and frustrated; on the other hand, if the language isn’t technical enough, the reader may not learn much of anything, so what’s the point of writing in the first place?  Unfortunately, I found Oullette’s book too frequently to be weighted towards the latter of these two extremes.  Though the book has the word “Calculus” in the title, there’s nary an equation to be found until the appendix, where all the scary mathematics has been relegated.  While the main body of the text offers well-written explanations for the main ideas, ultimately the writing feels too broad because Oullette refuses to get into the nuts and bolts of Calculus even a little bit.  I’m not saying we need another Calculus book in the market, but ultimately this book is extremely limited in its ability to describe what Calculus is actually like, because the very language of Calculus is barely used.

I realize, of course, that I am probably not the target audience for a book like this, so I asked my wife to read through the prologue and give me her thoughts.  While she said she found the beginning of the book interesting, she didn’t really seem compelled to continue reading.  There isn’t much I can infer from a sample size of one, but it seems to me like reading this book to try and overcome a fear of Calculus is a bit like reading books about French cooking and Louis XIV to try and overcome a fear of learning a foreign language.  Useful applications are presented, but the nature of the beast itself is never really tackled.

Having said that, if you’ve never taken a Calculus course, or have no idea what the word “Calculus” really involves, this book may be a good starting point for you.  It certainly won’t make you an expert, or give you many problems to solve, but if you have a fear of mathematics, you can think of this as dipping your toe in the water.

With a payroll roughly a third the size of the Yankees’, Beane understood that the playing field was not a level one from an economic standpoint.  What’s more, at the end of the 2001 season, three of the A’s star players left Oakland for bigger paychecks.  To fill the void, the film (and book) show how Beane took a more analytic approach, and used statistical analysis to uncover players who were undervalued and could be purchased for much less than they were worth.  Beane, together with Paul DePodesta (Peter Brand in the film, and played by Jonah Hill), used a sabermetric approach to lead the A’s to a league-leading 103 wins for the season. While their first-place ranking for number of wins that year was shared with the Yankees, they spent much less per win than their New York counterparts (the A’s spent the least per win, while the Yankees spent the third most).  Here’s a table comparing the teams; the payroll numbers are taken from here, and differ slightly from the numbers that appear in the book.

Team	Wins	Losses	Payroll	Cost Per Win (millions)
Oakland Athletics	103	59	$40,004,167	$0.388
Minnesota Twins	94	67	$40,225,000	$0.428
Montreal Expos	83	79	$38,670,500	$0.466
Florida Marlins	79	83	$41,979,917	$0.531
Cincinnati Reds	78	84	$45,050,390	$0.578
Pittsburgh Pirates	72	89	$42,323,599	$0.588
Los Angeles Angels	99	63	$61,721,667	$0.624
Tampa Bay Rays	55	106	$34,380,000	$0.625
San Diego Padres	66	96	$41,425,000	$0.628
Chicago White Sox	81	81	$57,052,833	$0.704
Philadelphia Phillies	80	81	$57,954,999	$0.724
Houston Astros	84	78	$63,448,417	$0.755
Kansas City Royals	62	100	$47,257,000	$0.762
St. Louis Cardinals	97	65	$74,660,875	$0.770
Colorado Rockies	73	89	$56,851,043	$0.779
San Francisco Giants	95	66	$78,299,835	$0.824
Seattle Mariners	93	69	$80,282,668	$0.863
Milwaukee Brewers	56	106	$50,287,833	$0.898
Baltimore Orioles	67	95	$60,493,487	$0.903
Atlanta Braves	101	59	$93,470,367	$0.925
Toronto Blue Jays	78	84	$76,864,333	$0.985
Detroit Tigers	55	106	$55,048,000	$1.001
Los Angeles Dodgers	92	70	$94,850,953	$1.031
Arizona Diamondbacks	98	64	$102,819,999	$1.049
Cleveland Indians	74	88	$78,909,449	$1.066
Chicago Cubs	67	95	$75,690,833	$1.130
Boston Red Sox	93	69	$108,366,060	$1.165
New York Yankees	103	58	$125,928,583	$1.223
New York Mets	75	86	$94,633,593	$1.262
Texas Rangers	72	90	$105,726,122	$1.468

Their new approach threw out many pieces of conventional baseball wisdom: stealing bases and bunting were strict no-no’s, for example.  Naturally, these changes brought about some tension, and it’s this tension that makes for the dramatic thrust of the film.  In particular, mathematics takes a backseat, though there are some little cameos for those who are paying attention.

The most significant piece of mathematics making an appearance in the film is the Pythagorean Expectation, a formula discovered by Bill James that estimates a team’s win percentage in terms of its runs scored and runs allowed.  More specifically, the formula asserts that a team’s win percentage is approximately equal to

For example, the 2002 A’s scored a total of 800 runs, and allowed a total of 654 runs, for a Pythagorean Expectation of

(relevant stats can be found here). This compares to the team’s actual win percentage of 103/162, which is around 0.636.

In the film, Peter Brand applies this formula in order to estimate the number of runs the team needs to score, along with the maximum number of runs it can allow, in order to secure a playoff spot.  In one scene, he tells Billy Beane that he thinks the A’s will need to win at least 99 games to guarantee a playoff spot.  In a 162 game season, this equates to a win percentage of around 0.611.  In order to ensure that the Pythagorean Expectation is at least this large, we set

With a little algebra, this is the same as

Brand then informs Beane that in order for this to happen, the team needs to score at least 814 runs, and can allow no more than 645 runs.  This gives a runs allowed to runs scored ratio of 645/814, or around 0.793 < 0.798 (though, if I were being anal, I would point out that with 814 runs scored, the team could allow as many as 649 runs and still have a runs scored to runs allowed to runs scored ratio that is less than 0.798).

While the math formulas on display in the film are accurate, I would be remiss if I did not briefly discuss Hill’s portrayal of Peter Brand.  Overall, Hill does a good job; though Brand is clearly a nerd, Hill’s portrayal usually avoids caricature.

Like every other film featuring characters who are good at math, though, Moneyball can’t help itself from including a scene where we see how good Brand is at math because he can do mental calculations quickly.  This particular scene takes place when Brand is sitting in his first meeting with Beane and the rest of the baseball scouts, and though it serves to highlight the tension that exists between Brand’s new school of thought and more traditional baseball thinking, I think the scene could have been just as effective without the clichéd math exercise.

Also, in the interest of full disclosure, I should point out that there are some who feel the story told in Moneyball (both the film and the book) is an exaggeration.  More specifically, as this Slate article discusses, many people believe that the reason for the A’s success during the early aughts had less to do with sabermetrics, and more to do with the fact that they had awesome pitchers in Tim Hudson, Mark Mulder, and Barry Zito, none of whom feature prominently in the book or film.  While I don’t feel knowledgeable enough to weigh in decisively on this issue, the role of the defense certainly appears to be underrepresented here.

To try and convince you of this, recall that the A’s made it to the playoffs in four consecutive years, from 2000-2003.  Here is some data on how many runs they scored and how many runs they allowed during each of those years, and in 2004, when they did not make the playoffs:

Observe that especially from 2001-2003, while the A’s offense declined, their defense remained consistent in allowing relatively few runs.  Of course, this should not be viewed in a vacuum, but rather in relation to how baseball as a whole performed.  Therefore, it is better to consider not runs scored and runs allowed, but runs scored and runs allowed as a proportion of runs scored and runs allowed in the American League.  With this slight adjustment, we get the following picture:

Note in the above that a proportion of 1 means that the A’s were performing at an average rate, while a proportion greater than 1 indicates above-average performance, and a proportion less than 1 indicates below-average performance.  As we can see from the data, in 2001-2003, the A’s defense was allowing runs at a rate well below the average; in other words, the defense was relatively strong.  On the other hand, during the same period, the offense consistently weakened year-over-year, so that the number of runs the A’s scored was below the league average in 2003-2004.  In particular, during the 2002 season profiled in Moneyball, the number of runs scored took a sharp downturn relative to the league average, while the number of runs allowed still remained well below average.  This indicates, to me at least, that the role of the defense was certainly an important factor in the A’s playoff runs during the 2002 and 2003 seasons.  Note also that in the 2004 season the number of runs allowed rose sharply relative to the league average; without a corresponding uptick in runs scored, the A’s didn’t make it to the playoffs.

Nevertheless, I don’t think the issue is binary; excellent pitching and a sabermetric approach probably combined to help the A’s.  Even though Moneyball only explores one of these issues, it’s still a film well worth seeing.  If you’re no fan of mathematics, don’t worry, there isn’t much on display.  And if you’re no fan of baseball, surprisingly, I think you might enjoy the movie anyway.

The grandson, Ravi Kapoor, delves into this mysterious part of his grandfather’s past while taking a mathematics class analogue of “Physics for Poets” – in other words, a math class aimed at non-mathematics students. The story jumps between classroom discussions and fictionalized historical records in an attempt to make clear the beauty of mathematics and give insight into the quest for truth (including, but not limited to, mathematical truth).

A precarious bike ride.

The cover of the book describes it as “A Mathematical Novel” – to many, this may seem like an oxymoron. In the preliminary author’s note, they elaborate by stating the following:

Our principal purpose in writing A Certain Ambiguity is to show the reader that mathematics is beautiful. Furthermore, we seek to show that mathematics has profound things to say about what it means for humans to truly know something. We believe that both objectives are best achieved in the medium of a novel. After all it is human beings who feel beauty and it is human beings who feel the immediacy of philosophical questions. And the only way to get human beings into the picture is to tell a story.

But does the novel achieve it’s goals? I’m not so sure that it does.

From a mathematical standpoint, the book does a good job of introducing and developing mathematical concepts for a general audience. For the math newbie, this book discusses Zeno’s paradox, the irrationality of , Cantor’s diagonalization argument, and some non-Euclidean geometry, just to name a few.  It also delves into the history of many mathematical discoveries, and tries to explore the personalities of those who have contributed to our collection of mathematical knowledge.

A mention of non-Euclidean geometry gives me an excuse to link to an Escher drawing. Click through to try and discover the connection!

This is all well and good.  The problem is that this frequently gets in the way of the narrative.  It is clear that the authors have a passion for mathematics, but it’s less clear that they have a passion for storytelling.  Ravi’s story sometimes seems to serve only as an inconvenient bridge that needs to be crossed in order to move from one mathematical discussion to another.  Also, there are several fictionalized journal entries from long deceased mathematicians (these are what personalize the history of mathematics), but these frequently serve no purpose in the larger story.

Even Ravi’s story diverges into areas that don’t really seem to matter.  A romantic subplot with one of his classmates is hinted at here and there, and the novel makes it clear that this girl Ravi is crushing on eventually becomes his wife, but their relationship during the events described in the book never really evolves, and because of this, it’s difficult to care about the fact that they eventually married.

I most enjoyed the historical records chronicling Ravi’s grandfather’s arrest, and subsequent discussions with a judge tasked to determine whether or not the blasphemy case should go to trial.  On the whole, though, the book felt like a bunch of mathematical vignettes aimed for those with little mathematical exposure, loosely strung together by a contemporary story.  Ravi’s desire to discover his grandfather’s past is the most compelling part of said story, and deals the most closely with humanity’s quest for truth, but ultimately isn’t enough to salvage the narrative.

Unfortunately, the authors don’t quite seem to strike the balance they are looking for.  In the end, the book is more of a MATHEMATICAL (novel) than a mathematical novel.  Even so, if you have this knowledge going in, there is enough here to make it a worthwhile read.  If you read only one mathematical novel this year, though (and who among us doesn’t read at least one?), I’d recommend Logicomix first.

One part Memento, one part A Beautiful Mind, the book was named a New York Times Book Review Editors’ Choice, and was popular enough in Japan to warrant a film adaptation (the Japanese language trailer for which can be found below).  Clearly, then, the book has resonated with people regardless of language.  But how does the book stack up from a mathematical perspective?

At first glance, it may sound like there’s a lot of fodder for me to complain about here.  After all, how many times have we seen mathematicians in popular culture with some sort of mental handicap?  Granted, memory loss is a new twist – usually insanity is the preferred condition.  Still, though, I don’t think it’s too much to ask for a mathematician who’s just a normal dude (or even, gasp, a normal lady).

Unfortunately, he is also but the latest entry in a long line of mathematicians in popular culture who are socially maladjusted.  He’s also incredibly reclusive – he has a strong aversion to crowds, and when he accompanies the housekeeper and her son to a baseball game midway through the novel, it’s fairly clear that he hasn’t been on an outing in some time.  One could explain these traits as a byproduct of his mental condition, of course: it’s natural for him to be shy around people when he is always meeting them for the first time, and there’s a danger in taking him out for too long lest he should forget what he’s doing out in the first place.  But part of me feels like these are convenient excuses for rehashing familiar tropes about people who study mathematics.

It’s not all bad, though.  In fact, I found myself able to forgive much of what I didn’t like about this portrayal of mathematicians, because there are many positive features about the professor as well.  For starters, the professor is able to form a close relationship to the housekeeper’s son (who he nicknames Root, because the child’s flat head of hear reminds him of the square root sign).  Even though the professor can’t remember who Root is from day to day, every time the boy comes to the professor’s house, the professor dotes on him like a father.  He obsesses over the safety of Root more than the housekeeper, and keeps Root in his mind as much as he can, given his circumstances.

Of course, his concern about Root wouldn’t be complete if it didn’t include concern over his mathematics education.  Here’s another thing Ogawa does quite well – she is able to not only show the professor’s love of mathematics, but she is also able to illustrate how that passion can inspire others.  The professor is a number theorist, and he is always looking for meaning behind numbers (something which, in the hands of a poorer story-teller, would no doubt incite my rage).  What makes the professor’s interest significant is that he never discusses numbers for the sake of random numerological connections – instead, he is able to take small observations and use them to hint at larger mathematical ideas.  Here’s one such example, from the day the three of them went to a baseball game (this may be my favorite passage from the book):

And when he noticed that his seat number was 714 and Root’s was 715, he began to lecture again and completely forgot to sit down.

“The home run record Babe Ruth set in 1935 is 714.  On April 8, 1974, Hank Aaron broke that record, hitting his 715th off of Al Downing of the Dodgers.  The product of 714 and 715 is equal to the product of the first seven prime numbers: 714 x 715 = 2 x 3 x 5 x 7 x 11 x 13 x 17 = 510510.  And, the sum of the prime factors of 714 equals the sum of the prime factors of 715: 714 = 2 x 3 x 7 x 17, 715 = 5 x 11 x 13; 2 + 3 + 7 + 17 = 5 + 11 + 13 = 29.  A pair of consecutive whole numbers with these properties is quite rare.  There are only 26 such pairs up to 20,000.  This one is the Ruth-Aaron pair.  Just like prime numbers, they are more rare as the numbers get larger.  And 5 and 6 are the smallest pair.  But the proof to show that those pairs are infinite in number is quite difficult*. . . . The important thing is that I’m sitting in 714 and you’re in 715, instead of the opposite.  It’s the young who have to break the old records.  That’s the way the world works, don’t you think?” (90)

Happy to have broken the home run record, or happy to be part of an interesting number phenomenon?

There are vignettes like this peppered throughout the book, where the professor will link an everyday observation to some kind of mathematics.  In the story, which is told from the housekeeper’s perspective, we see how this inspires the housekeeper to think about simple mathematical problems.  In spite of her lack of formal training, the professor is able to inspire in her a sense of mathematical curiosity (something which all math teachers should aspire to do).  Those of the mathematical persuasion are rarely shown as being able to interest people who are less mathematically inclined.  I’m glad to see that this book bucks that trend.

Moreover, this book does a better than average job of discussing what makes mathematics so appealing to those of us who study it.  Consider the following exchange between the housekeeper and the professor, after she apologizes for sending a proof of his to a journal via regular mail instead of express:

“No, there was no need to send it express.  Of course, it’s important to arrive at the correct answer before anyone else, but it’s just as important that the proof is elegant.”

“I had no idea a proof could be beautiful . . . or ugly.”

“Of course it can,” he said.  Getting up from the table, he came over to the sink where I was washing the dishes and peered at me as he continued.  “The truly correct proof is one that strikes a harmonious balance between strength and flexibility.  There are plenty of proofs that are technically correct but are messy and inelegant or counterintuitive.  But it’s not something you can put into words—explaining why a formula is beautiful is like trying to explain why the stars are beautiful.” (16)

While I don’t necessarily agree with the last part, I think it’s refreshing to find a discussion of what accounts for mathematical beauty in a book like this.  This type of discussion between a mathematical expert and a mathematical amateur is not often present in works that center on mathematics – more frequently, the conversation is between math experts, or is not about mathematics at all.  Providing this type of simple insight to a reader who may not have a mathematical background is certainly a plus.

Despite falling into some tired stereotypes, the professor emerges as a fully realized character.  His memory problems are much more than a gimmick, and while they enable certain stereotypes to persist, Ogawa also uses his disability to showcase a degree of empathy for other people that is not often found in portrayals of those who study mathematics.  Overall I found that I quite enjoyed the book – if you’ve got a lazy Sunday coming up (the book is short, so you could easily finish it in a weekend), I’d certainly recommend giving this story a shot.

*Actually, the question as to whether or not there are infinitely many such pairs is actually still open.  See here for more information.