→ March 19, 2016
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Stick to baseball, 3/19/16.

I had a big scouting blog post from Arizona for Insiders this week, leading with Dodgers outfielder Yusniel Diaz, plus a draft blog post on UVA’s Connor Jones and Matt Thaiss, including thoughts on why the Cavaliers have never churned out a big league starter. My weekly Klawchat transcript is up as well.

I reviewed the simple abstract strategy game Circular Reasoning for Paste.

And now, the links…

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→ November 9, 2015
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Ada’s Algorithm.

My top 50 free agent rankings went up Friday for Insiders, following by the “deleted scenes” post with capsules on four guys whom I wrote about before their employers picked up their club options. I’ve also got buyers’ guides to catchers and to corner infielders up, with middle infielders due on Tuesday.

Everything seems to be coming up Ada Lovelace lately; largely overlooked in her own time because she was a woman in the early Victorian era and was better known as the one legitimate offspring of the rake Lord Byron, she’s now widely recognized as the creator of the first machine algorithm, the primary ancestor of the modern computer program. The Department of Defense named a programming language (Ada) after her in the early 1980s, and she’s appeared in numerous works of fiction (such as William Gibson’s The Difference Engine) and non-fiction (including a brand-new short work aimed at schoolchildren called Ada Byron Lovelace and the Thinking Machine) over the last 25 years. Since my daughter was working on a short presentation on Lovelace – all the kids were asked to pick a scientist, and she was pissed off because there was only one woman (Marie Curie, of course) on the original list of assignments – I picked up James Essinger’s 2014 biography, Ada’s Algorithm: How Lord Byron’s Daughter Ada Lovelace Launched the Digital Age, which had most of the key details but is padded with a lot of less critical material.

Ada Lovelace’s place in history comes from her friendship with Charles Babbage, who designed (but never built) the first computers, one called the Difference Engine, of which he built one-seventh, and another called the Analytical Engine, which he never built at all due to the prohibitive cost and lack of manufacturing facilities capable of building all of the cogswheels the device required. Babbage was a bit of a mad scientist, prone to emotional outbursts and self-destructive arguments that cost him any shot to gain the funds necessary to build even part of either Engine beyond what he built. He also lacked Ada’s communications skills, and when the Italian mathematican (and later Prime Minister of Italy) Luigi Federico Menabrea wrote a paper describing Babbage’s Analytical Engine, Lovelace translated it into English and supplemented it with her own Notes, the latter of which ran more than twice as long as Menabrea’s original article, and included the algorithm that earned her posthumous fame. She saw the potential of Babbage’s machine that even Babbage did not – that programmers could use it to solve all kinds of mathematical problems beyond mere arithmetic, as long as the programmer could conceive the necessary series of steps for the calculations.

Lovelace died of uterine cancer at 36, and much of the detail of her life is lost both to time and, it’s believed, to her mother’s decision to destroy much of Ada’s correspondence after the latter’s death. Even many of the letters she exchanged with Babbage are gone, leaving any biographer with relatively meager material from which to construct a story of her life. Essinger barely makes it past 200 pages, and even to get to that point he has to fill with material that’s not all that relevant to the reader primarily interested in Ada’s Notes and the algorithm of the book’s title. For example, we don’t need two chapters on Lord Byron, and I was certainly glad I got the book away from my daughter (who found it boring anyway) before she got to the mentions of his incestuous relationship with his half-sister Augusta or the story of how his nanny would take him into her bed, masturbate him, and then later turn around and beat him, often doing both things in the presence of her friends. (That material would seem essential in any biography of Byron himself, though, since it probably explains his later promiscuity and other “immoral” behavior relative to the mores of the era.) Byron was out of Ada’s life for good while she was still an infant, and including such details on his life seems more than just out of place but almost pandering.

Essinger gives us too much of the text of some of her less relevant letters, and inserts his own speculation on things like whether she might have met certain personages of the era, like Charles Darwin, or whether Babbage was in love with Ada, for which there’s no tangible evidence. The first hardcover edition also has numerous typos and minor errors in the text – for example, using “inconceivable” when he meant “conceivable,” which is kind of a weak word anyway – that further added to my impression that I was reading Essinger’s thoughts and opinions rather than a narrative rendering of her life. It seems that we don’t know enough about Ada Lovelace for a full biography, but that doesn’t quite justify surrounding what we do know with speculation or tangential details.

Next up: Speaking of Gibson, I’m reading Mona Lisa Overdrive, the third book in his Sprawl trilogy, which began with the Hugo-winning Neuromancer.

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→ August 1, 2015
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Saturday five, 8/1/15.

So I was kind of busy this week, writing these pieces for Insiders on the major trades leading up to Friday’s trade deadline.

• Yoenis Cespedes to the Mets
• Mike Leake to San Francisco
• Latos/Olivera/Wood three-team trade
• David Price to Toronto
• Joakim Soria to Pittsburgh
• Carlos Gomez/Mike Fiers to Houston
• Brandon Moss to St. Louis
• Cole Hamels to Texas
• Jonathan Papelbon to Washington
• Ben Zobrist to Kansas City
• Troy Tulowitzki to Toronto
• Tyler Clippard to the Mets
• Johnny Cueto to Kansas City
• Several smaller trades
• The Mets/Carlos Gomez trade that didn’t happen

I also have a scouting post up on some Mets and Yankees AA prospects.

And now, the links… saturdayfive

  • Earlier this month, a fan at a Brewers game was hit in the face by a line drive, severely injuring her and missing killing her by centimeters. There’s a fundraising page for her medical bills if you’d like to donate.
  • Twitter is now hiding plagiarized jokes and other tweets if the original authors file complaints. It’s a minor issue compared to some of the abuse hurled at women and minorities on Twitter, but I’ll take any step toward greater editorial control on Twitter as a positive.
  • Molly Knight talked to Lasorda’s Lair about her book on the Dodgers and her history of anxiety disorder. If you haven’t yet, you should buy her book.
  • The Shreveport Times has a sharp opinion piece on how the Lafayette massacre won’t change anything. The piece specifically singles out Louisiana’s “weak and non-existent gun control.” It’s on us, though; you vote for candidates who take money from the NRA, this is what you get. If you don’t like it, get out there and campaign for the other side.
  • Is the song “Happy Birthday” still protected by copyright? It appears it may not be, although we’ll need the judge’s ruling to be sure. There’s a big fight coming in 2018 over expiring copyrights, one that puts me (in favor of putting many older works in the public domain) on the opposite side from my employer (Disney, which has a fair concern about Mickey Mouse falling into p.d.).
  • The Fibonacci shelf takes the mathematical sequence and turns it into stackable furniture. I want this.
  • Three “next-level” recipes for rum punch. That first one, a planter’s punch with homemade grenadine, sounds right up my alley; planter’s punch is the first strong (may I say “grown-up?”) cocktail I liked.
  • Go ahead, be sarcastic, at least with people you know well: it can boost creative thinking, according to a new study by three business school professors.
  • A fantastic profile of prodigy turned mathematician Terry Tao, considered (per the piece) “the finest mathematician of his generation,” and more broadly a piece on number theory. I share Tao’s love of the original computer game Civilization and the difficulty in putting it aside; it occupied a huge portion of the fall semester of my junior year of college, unfortunately. That said, it kills me that the article’s author felt that “prime number” required a definition. You shouldn’t be able to get to high school without knowing what that means.
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→ June 15, 2015
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The Golden Ticket.

Lance Fortnow wrote a piece for Communications of the Association for Computing Machinery in 2009 on the state of the P vs NP problem, one of the most important unsolved problems in both mathematics and computer science. That article led to the short (~175 page) book The Golden Ticket: P, NP, and the Search for the Impossible, which I recently read in its Kindle edition (it’s also on iBooks); Fortnow does a solid job of making an abstruse problem accessible to a wider audience, even engaging in some flights of fancy describing a world in which P equals NP … which is almost certainly not true (but we haven’t proven that yet either!).

P vs NP, which was first posed by Kurt Gödel in 1956, is one of the seven Millennium Problems posed by the Clay Mathematics Institute in 2000; solve one and you get a million bucks. One of them, proving the Poincaré Conjecture (which relates to the shape of the universe), was solved in 2010. But if you solve P vs NP affirmatively, you can probably solve the remaining five and collect a cool $6 million for your problems. You’ll find a box of materials under your desk.

Of course, this is far from an easy question to solve. P and NP are two classes of problems in computer science, and while it seems probable that they are not equivalent, no one’s been able to prove that yet. P is the set of all problems that can be quickly (in deterministic polynomial time – so, like, before the heat death of the universe) solved by an efficient algorithm; NP is the set of all problems whose solutions, once found, can be quickly verified by an efficient algorithm. For example, factoring a huge composite number is in NP: There is no known efficient algorithm to factor a large number, but once we’ve found two factors, a computer can quickly verify that the solution is correct. The “traveling salesman problem” is also in NP; it’s considered NP-complete, meaning that it is in NP and in NP-hard, the set of all problems which are at least as hard as the hardest problems in NP. We can find good solutions to many NP-hard problems using heuristics, but we do not have efficient algorithms to find the optimal solution to such a problem.

If P does in fact equal NP, then we can find efficient algorithms for all problems in NP, even those problems that are NP-complete, and Fortnow details all of these consequences, both positive and negative. One major negative consequence, and one in which Fortnow spends a significant amount of time, would be the effective death of most current systems of cryptography, including public-key cryptography and one-way hashing functions. (In fact, the existence of one-way functions as a mathematical truth is still an unsolved problem; if they exist, then P does not equal NP.) But the positive consequences are rather enormous; Fortnow gives numerous examples, the most striking one is the potential for quickly developing individualized medicines to treat cancer and other diseases where protein structure prediction is an obstacle in quickly crafting effective treatments. He also works in a baseball story, where the game has been dramatically changed across the board by the discovery that P=NP – from better scheduling to accurate ball/strike calls (but only in the minors) to the 2022 prohibition of the use of computers in the dugout. It’s Shangri-La territory, but serves to underscore the value of an affirmative proof: If we can solve NP problems in deterministic polynomial time (as opposed to nondeterministic polynomial time, where NP gets its name), our ability to tease relationships out of huge databases and find solutions to seemingly intractable logical and mathematical problems is far greater than we realized.

Of course, P probably doesn’t equal NP, because that would just be too easy. That doesn’t mean that NP-complete problems are lost causes, but that those who work in those areas – operations research, medicine, cryptography, and so on – have to use other methods to find solutions that are merely good rather than optimal. Those methods include using heuristics that simplify the problem, approximating solutions, and solving a different but related problem that’s in P. If Fortnow falls short at all in this book, it’s in devoting so much more time to the brigadoon where P=NP and less to the likely real world quandary of solving NP-complete problems in a universe where P≠NP. He also gives over a chapter to the still theoretical promise of quantum computing, including its applications to cryptography (significant) and teleportation (come on), but it seems like a digression from the core question in The Golden Ticket. We don’t know if P equals NP, but as Fortnow reiterates in the conclusion, even thinking about the question and possible approaches to proving it in either direction affect work in various fields that underpin most of our current technological infrastructure. If you’ve ever bought anything online, or even logged into web-based email, you’ve used a bit of technology that only works because, as of right now, we can’t prove that P=NP. For a very fundamental question, the P vs NP problem is scarcely known, and Fortnow does a strong job of presenting it in a way that many more readers can understand it.

If this sounds like it’s up your alley or you’ve already read it, I also suggest John Derbyshire’s Prime Obsession, about the Riemann Hypothesis, another of the Clay Millennium Institute’s six as-yet unsolved problems.

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→ June 11, 2015
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Proofiness.

Whew! I’m glad that’s over. For Insiders, my recaps of the drafts for all 15 NL teams and all 15 AL teams are up, as well as my round one reactions and a post-draft Klawchat.

Charles Seife’s Proofiness: How You’re Being Fooled by the Numbers is a beautiful polemic straight from the headquarters of the Statistical Abuse Department. Seife, whose Zero is an enjoyable, accessible story of the development and controversy of that number and concept, aims both barrels at journalists, politicians, and demagogues who misinterpret or misuse statistics, knowing that if you attach a number to something, people are more inclined to believe it.

Seife opens with Senator Joseph McCarthy’s famous claim about knowing the names of “205 … members of the Communist Party” who were at that moment working in the State Department. It was bullshit; the number kept changing, up and down, every time he gave a version of the speech, but by putting a specific number on it, the audience assumed he had those specific names. It’s a basic logical error: if he has the list of names, he must have the number, but that doesn’t mean the converse is true. He rips through a series of similarly well-known examples of public abuse of statistics, from the miscounting of the Million Man March to stories about blondes becoming extinct to Al Gore cherrypicking data in An Inconvenient Truth, to illustrate some of the different ways people with agendas can and will manipulate you with stats.

One of the best passages, and probably most relevant to us as the Presidential election cycle is beginning, is on polls – particularly on how they’re reported. Seife argues, with some evidence, that many reporters don’t understand what the margin of error means. (This subject also got some time in Ian Ayers’ Super Crunchers, a somewhat dated look at the rise of Big Data in decision-making that has since been lapped by the very topic it attempted to cover.) If done correctly, the margin of error should equal two standard deviations, but many journalists and pundits treat it as some ambiguous measure of the confidence in the reported means. When Smith is leading Jones 51% to 49% with a margin of error of ±3%, that’s not a “statistical dead heat;” that’s telling you that the poll, if run properly, says there’s a 95% chance that Smith’s actual support is between 48% and 54% and a 95% that Jones’ support is between 46% and 52%, with each distribution centered on the means (51% and 49%) that were the actual results of the poll. That’s far from a dead heat, as long as the poll itself didn’t suffer from any systemic bias, as in the famous Literary Digest poll for the 1936 Presidential election.

Seife shifts gears in the second half of the book from journalists to politicians and jurists who either misuse stats for propaganda purposes or who misuse them when crafting bad laws or making bad rulings. He explains gerrymandering, pointing out that this is an easy problem to solve with modern technology if politicians had any actual interest in solving it, and breaks down the 2000 Presidential vote in Florida and the 2008 Minnesota Senate race to show that the inevitable lack of precision even in popular votes and census-taking mean both races were, in fact, dead heats. (Specifically, he says that it is impossible to say with any confidence that either candidate was the winner.) Seife shows how bad data have skewed major court decisions, and how McCleskey v. Kemp ignored compelling data on the skewed implementation of capital punishment. (Antonin Scalia voted with the majority, part of a long pattern of ignoring data that don’t support his views, according to Seife.) This statistical abuse cuts both ways, as he gives examples of both prosecutors and defense attorneys playing dirty with numbers to claim that a defendant is guilty or innocent.

For my purposes, it’s a good reminder that numbers can be illustrative but also misleading, especially since the line between giving stats for descriptive reasons can bleed into the appearance of a predictive argument. I pointed out the other day on Twitter that both Michael Conforto and Kyle Schwarber were on short but impressive power streaks; neither run meant anything given how short they were, but I thought they were fun to see and spoke to how both players are elite offensive prospects. (By the way, Dominic Smith is hitting .353/.390/.569 in his last 29 games, and has reached base in 21 straight games!) But I’d recommend this book to anyone working in the media, especially in the political arena, as a manual for how not to use statistics or to believe the ones that are handed to you. It’s also a great guide for how to be a more educated voter, consumer, and reader, so when climate change deniers claim the earth hasn’t warmed for sixteen years, you’ll be ready to spot and ignore it.

Next up: I’m way behind on reviews, but right now I’m halfway through Adam Rogers’ Proof: The Science of Booze.

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→ June 3, 2015
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Incompleteness.

My final top 100 draft prospect ranking for 2015 is up for Insiders, and my latest review for Paste covers the Temple of Elemental Evil boardgame.

Rebecca Goldstein’s Incompleteness: The Proof and Paradox of Kurt Gödel, part of the same “Great Discoveries” series that includes David Foster Wallace’s Everything and More: A Compact History of Infinity, takes the abstruse topic of Gödel’s two incompleteness theorems and folds it into a readable, compelling biography of both the man and his ideas. Using the logician’s friendship with Albert Einstein as a hook, Goldstein gives us about as intimate a portrait of the intensely private Gödel as we can get, while also laying the groundwork so the non-metamathematicians among us can understand the why and how of Gödel’s theorems.

There’s very little actual math involved in Incompleteness, because the two theorems in question revolve around the nature of mathematics, notably arithmetic, as an axiomatic system that mathematicians and philosophers of the early 1900s were trying to use as a basis to describe arithmetic as a formal system – that is, one that can be fully described via a system of symbols and syntax, used to construct well-formed formulas (wffs, not to be confused with wtfs), some of which are the axioms that define the whole system. If such axioms can describe the entire system in a way that an algorithm (which is probably easiest to conceive as a computer program) can use those axioms and only those axioms to prove all truths or assertions about the natural numbers, then arithmetic would be considered a complete system, something Gödel proved was impossible in his first theorem. His second theorem went even further, showing that such a system of arithmetic also could not be consistent within itself by demonstrating that a statement of the system’s consistency would generate an internal contradiction.

Gödel imself was an incomplete figure, a hypochondriac who degenerated into outright paranoia later in life, and a socially awkward man who would likely have been diagnosed in today’s world of psychiatry and medicine as a depressive or even somewhere on the autism spectrum. He formed few lasting friendships, feared that no one could understand him (and given the meandering paths of his mind, I don’t doubt this was true), and was often shockingly aloof to what was happening around him. He fled his native Austria before World War II, even though he wasn’t Jewish, because his association with the secular Jewish scholars of the Vienna Circle (most of whom were logical Positivists, arguing that anything that could not be empirically proven could not be considered true) cost him his university position. Yet he remained unaware of the state of affairs in his native country, once (according to Goldstein) asking a Jewish emigré who had fled Nazi persecution for the United States, “What brings you to America?”

Goldstein does a superb job setting the scene for Gödel’s emergence as a logician/metamathematician and as a figure of adulation and controversy. The Vienna Circle was, in Goldstein’s depiction, somewhat insular in its unwavering acceptance of Positivism, and as such certain that Hilbert’s second problem, asking for a proof that arithmetic is internally consistent, would be proven true. Gödel’s second theorem showed this was not the case, leading to his own intellectual isolation from thinkers who were heavily invested in the Positivist view of mathematics, a bad outcome for a man who was already prone to introversion and diffidence. G&omul;del found it difficult to express himself without the use of logic, and while his station at Princeton’s illustrious Institute for Advanced Study – a sort of magnet think tank that became a home for great scholars in math and the sciences – put him in contact with Albert Einstein, it also deepened his solitude by limiting his orbit … although that may have been inevitable given his late-life delusions of persecution. Goldstein did encounter Gödel once at a garden party at Princeton, only to see him at his most gregarious, holding court with a small group of awestruck graduate students, only to disappear without a trace when his spell of socializing was over. He was, in her description, a phantom presence around campus, especially once his daily walks with Einstein ended with the latter’s death in 1955.

While giving a basic description of Gödel’s theorems and proofs, focusing more on their implications than on the underlying math, Goldstein does send readers to the 1958 Gödel’s Proof, a 160-page book that attempts to give readers a more thorough understanding of the two theorems even if those readers lack any background in higher math. Incompleteness focuses instead on the man as much as it does on his work, producing a true narrative in a story that wouldn’t otherwise have had one, making it a book that I could recommend to anyone who can stand Goldstein’s occasional use of a $2 word (“veridical”) when a ten-cent one (“truthful”) would have done.

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→ March 21, 2015
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Saturday five, 3/21/15.

My ESPN Insider content from the last week:

* My breakout player picks for 2015.
* A suggested rule change to cover the Kris Byrant situation, plus Jonathan Gray, Tyler Matzek, Yasmany Tomas, and Yoan Lopez
* Javier Baez, Brandon Finnegan, Danny Duffy, Kyle Schwarber
* Carlos Rodon and Tyler Danish
* Taijuan Walker and Rubby de la Rosa
* A draft blog post on Arizona infielders Kevin Newman and Scott Kingery

I’ll be on the ESPN game broadcasts on Tuesday (Phillies at Atlanta) and Friday (Red Sox at Atlanta), as well as some postgame content to be determined.

And now, this week’s links…

  • There’s been a rash of suicides and attempts in Palo Alto, prompting this sound and accessible piece on how parents can try to help decrease the risks in their own children.
  • This week in terrifying food science news: Antibiotic use at pork farms is soaring, and it’s not just in the United States. Of course, we can’t expect other countries to ban the practice if we refuse to do it ourselves.
  • A Virginia middle school suspended a sixth-grader and referred him for substance-abuse counseling because he brought a leaf to school. No, not a marijuana leaf, or any other kind of illicit drug. This is zero-tolerance policy run completely amok, benefiting no one.
  • It’s made the rounds, but just in case you haven’t seen it, Ashley Judd is seriously sick of your misogynistic bullshit. Death threats are illegal, so why aren’t rape threats? More importantly, why does Twitter persistently refuse to do anything about it?
  • Making busy intersections safer. I imagine the initial reluctance to accept these new designs would be huge – never change anything, anywhere – but they’re fascinating to me as someone who used to love road maps and seeing different streets and intersections as a kid, but also to me as someone who drives all the time and worries a lot about getting in or even causing an accident. Although the skateboarder I nearly brained on San Diego Avenue on Thursday shouldn’t have been in the middle of the car lane, even at midnight.
  • “Hands up, don’t shoot” was built on a lie. Or maybe it wasn’t. Hell if I know.
  • Finally, Baltimore Ravens lineman John Urschel co-authored a math paper titled “A Cascadic Multigrid Algorithm for Computing the Fiedler Vector.” Are we praising him for being brilliant, or are we all just relieved that he’s not a wife-beater or a serial rapist? Regardless, graph theory is heady stuff, beyond anything I ever studied in school or on my own; I remember encountering the Königsberg Bridge Problem, a precursor to modern graph theory, but don’t recall learning its (dis)proof.
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→ March 13, 2015
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Two math anthologies.

My latest Insider post covered scouting notes on Danny Salazar, Kendall Graveman, and others from that same game. My weekly Klawchat transcript is up, and I have a new boardgame review over at Paste for the Kennerspiel des Jahres-nominated strategy game Rococo.

My friend Steve is quite familiar with my affinity for just about all things math – we first met in math class in seventh grade – and for Christmas this year bought me a pair of popular math texts, one new and one classic. (I bought him a lot of tea, as he consumes it even faster than I do.) Both were collections of short pieces, with the unevenness that comes with such an anthology, but with high points making both books well worth reading.

The new title was The Best Writing on Mathematics 2014, a book that opens with a sort of dry exhortation on the apparently declining interest in math among students, a theme revisited later in the book, although I think a large part of that is a function of how we teach math in the United States – something that is in and of itself the subject of a separate essay. The separation of abstract math from its practical uses will only sit well with students who are naturally able to deal with math’s abstractions, to hear the music in numbers and formulas, to understand topics like calculus on an intuitive level; modern American instruction tends to make the majority of students, those who don’t grasp this material as quickly, feel less able or competent in the subject. Math anxiety, the subject of so-and-so’s column, isn’t an innate medical condition like anxiety disorder; it is created by teachers and curricula that quickly tell students they’re just not good enough at this stuff.

I enjoy abstract math – one of the best books I’ve ever read was on the highly abstruse Riemann Hypothesis, called Prime Obsession, one of the great unsolved problems in mathematics and one without any apparent practical applications. Yet I also enjoy writing on the pervasive uses of math in other fields, from physics to probability. One of the best essays in the book, and unfortunately one of the shortest, is from game designer and engineer Soren Johnson, who discusses the uses of probability and controlled randomness in creating successful games, specifically citing the random component in Settlers of Catan that has diminished its standing among the most hardcore boardgaming segment that prefers the less random and more complex style of games like Puerto Rico or Agricola. (My issue with Settlers isn’t the randomness but the length of the games. It’s still a classic and one of the best light-strategy games ever created.)

There are several pieces built around randomness, including a high-level essay from Charles Seife (author of Zero, which I enjoyed, as well as Proofiness: How You’re Being Fooled by the Numbers, which is on my to-be-read shelf) on the nature of randomness and our inability to understand it, and another essay on the power of the random in musical compositions. The essay by Prakash Gorrochum titled “Errors of Probability in Historical Context” should be required reading for journalists everywhere, covering the gambler’s fallacy, reasoning on the mean, and misunderstanding statistical independence (Bayes’ theorem). One essay tackles the problem of the Jordan Curve – defined a simple closed curve in a plane or planar region, dividing it into exactly two parts, thus never crossing itself – and its applicability to some amazing works of art. I alluded to the closing essay on Twitter the other day; it discusses the proposed solution to the abc problem, for which the alleged solver had to invent a whole new kind of mathematics, which means that only a few dozen people in the world might be able to interpret his proof, let alone test or critique it.

The selection of titles seems idiosyncratic, as some have very little to do with math proper, such as the dreadful essay on various ancient tools and devices used for mathematical calculations, or the too-lengthy chapter disproving the contention that ancient Celts in modern-day Scotland knew and understood the features of the five regular polyhedra a millennium before anyone else seemed to catch on. The collection ends on several high notes, however, including Gorrochum’s essay (which you can read in its entirety online) and that abc problem/solution story, the latter of which is almost creepy because of how bizarre the whole backstory is. I’d never heard of this series before Steve bought me this book but the handful of strong essays in it made it a great read.

—

The other book Steve bought for me was Martin Gardner’s collection Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi, which is the first in the series of books anthologizing Gardner’s many essays on popular math from his long-running column in Scientific American. Gardner’s writing exudes his sheer joy in math itself, yet most of these essays explore tangible questions even when they’re as useless as the hexaflexagons of the book’s title. Those peculiar shapes are formed by folding one or more strips of paper according to prescribed patterns to form regular polygons, in this specific case hexagons, that can be pushed and folded to reveal hidden sides and features, a chance discovery explored by some very famous names from math and science (Richard Feynman was among their earliest practicioners). A similar vein runs through his essays on the games Hex, invented independently by Nobel Prize-winning game theorist John Nash and Danish polymath Piet Hein, a totally nonrandom game of tile placement on a rhomboid board of hexagonal spaces where each player is trying to complete an unbroken chain from one of his sides to the side facing it. The game can’t end in a draw, and on smaller boards there are unbeatable strategies for whichever player goes first, inspiring much mathematical hand-wringing over the search for algorithms to predict perfect plays.

Other essays pose specific logic and math puzzles to the reader, many of which can be worked out in your head (and are much worse if you start putting pencil to paper). He explores the history of the “boss puzzle,” also known as the 15-14 puzzle, where the player is presented with a 4×4 grid with 15 numbered tiles on it, all in ordered rows but with the final row going 13-15-14. The player is told to use the single open space to move tiles around to get all fifteen tiles into the proper order. (The puzzle is unsolvable because it has a parity of one, meaning there’s a single tile displacement.) He also discusses several popular math and logic paradoxes, such as the division of a rectangle into several triangular pieces, then the reassembly that makes it appear that some surface area has disappeared. (It hasn’t.) They’re fun to puzzle over for their own sake, but the sleights of math used here or in the card tricks Gardner describes in another chapter expose holes in our critical thinking processes – ways we can miss obvious fallacies because something looks or sounds “right” on its face.

The chapter that might be most familiar to readers in subject matter discusses the Birthday Paradox. Given a group of 24 people selected at random, what are the odds that at least two members of the group have the same birthday? The answer is better than one half, which seems at first rather hard to believe as there are 365 days in the calendar. The odds that the first two people don’t have the same birthday are 364/365; the odds that the third person added to the group won’t have the same birthday as either of the first two are 363/365; and so on. The probability that n people won’t have the same birthday is thus a product of all of these individual probabilities (the formula is here); the 23rd person added to the group drops the probability that there is no birthday match under 0.5. It seems intuitively incorrect that just 23* people could suffice to raise the odds of a match over 50% when the number of dates is 365, and there are many methods of figuring these odds incorrectly, such as multiplying the apparent odds of a match (2/365 * 3/365 * 4/365…) or adding up the same fractions. Gardner’s explanations of such paradoxes were both clear and a pleasure to read, which is why so much of his work remains in print a half-century after he started writing. The chapter doesn’t discuss the Monty Hall problem, but describes a similar question around hands of cards that might illuminate that more famous question if you’ve struggled to understand its explanation.

* Gardner’s chapter uses 24 as the threshold, but I’m pretty sure it’s 23, using both methods to calculate the odds. If anyone can show the magic number is 24, please post it in the comments, because then I’ve got this wrong too.

Gardner also discusses magic squares, which seem to me to be the logical ancestors of the much simpler sudoku; the Tower of Hanoi problem; and some topological oddities that arise from manipulations of a Mobius strip (or two of them together). He gets a little ahead of himself, perhaps a function of the space limitations of the print world, in the chapter on fallacies by presenting two pure math fallacies without explaining exactly why they fail. Both revolve around attempts to prove that two unequal entities are equal; one fails through a disguised attempt to divide by zero, the other by treating i as a real (rather than imaginary) number, but I wouldn’t assume either fallacy was obvious by the way Gardner presents them.

I first encountered Gardner’s work in junior high school through his now out-of-print Aha! Gotcha! book, which took a similar approach to math tricks and paradoxes but was aimed at a younger audience; Hexaflexagons is the more grown-up version, aimed at math-loving kids like me who just refused to grow up.

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→ November 26, 2014
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Everything and More: A Compact History of Infinity.

I knew David Foster Wallace was brilliant when I read Infinite Jest, a wildly imaginative, sprawling novel that showcased DFW’s prodigious vocabulary as well as his deep knowledge of a variety of seemingly unrelated subjects. Even with that background, I was flabbergasted by Everything and More: A Compact History of Infinity, in which DFW delves into abstract set theory and other similarly abstruse topics from the history of math, explaining much of it lucidly and with humor until he gets too close to the finish to avoid relying on the reader to understand more of set theory than most readers will.

The book is less an explanation of the number infinity – which isn’t a single number, at least not in the sense that 1 or 5 or π or √2 – than the history of mathematicians’ attempts to deal with it. DFW starts with the Greeks, where most math stories begin anyway, even though the Greeks didn’t like or accept infinity or zero or the irrationals. (Zero came from Indian mathematicians, and reached Europe by way of Arab mathematicians quite a bit later.) The Greeks encountered questions around infinity, particularly in the famous paradoxes of Zeno, who liked to play semantic games around what we now refer to as convergent series – a sum of a series of terms that never ends but that approaches a specific limit as the number of terms grows. (In a related note, DFW fails to answer the question of how Zeno never got punched in the face for coming up with these paradoxes.) This discomfort with infinity continued through the writings of Aristotle and the Catholic Church’s influence over all manner of academic research, which included the idea that infinity was the sole province of God rather than of man, meaning we never got anywhere with infinity until the end of the Dark Ages and the separation of mathematics and religion during the Renaissance.

The pace of the narrative picks up at that point thanks to the explosion of advances in math and related areas of science. The empirical foundation that limited mathematical explorations until the 1600s is tossed aside in favor of more abstract thinking, with appearances by Kepler, Newton, and my homeboy Galileo, as trigonometry and eventually calculus displace geometry as the central philosophy guiding mathematical thinking and what we now think of as number theory. DFW presents an extraordinarily clear explanation of calculus, especially the infitesimals that underpin differentiation and integration and, as the name implies, connect it to the main topic of the book. The goal here is to get to Georg Cantor, the brilliant and mentally ill mathematician whose work remains the foundation of modern set theory and who was the first to recognize that there are different degrees of infinity (ℵ0 and ℵ1, at the least) but died unable to prove that those two infinities had no other infinities between them.

DFW’s writing is clear and witty thoughout the book, with many examples drawn from a former professor of his that help elucidate many of the more recondite concepts around infinity. His explanations of one-to-one mapping and Cantor’s diagonalization method of proving that real numbers are nondenumerable are outstanding, especially the latter, which I knew was true but still wanted to disbelieve because it just sounds impossible. Unfortunately, in the last 40-50 pages of the book, DFW gets so far down the set theory rabbit-hole that I found it increasingly hard to follow, such as discussions of ordinality versus cardinality and power sets of power sets. I got off the math train in college after multivariate calculus with vectors, in part because continuing meant pushing into more abstract areas – linear algebra was the next course, which starts the shift from empirical math to abstract – but that left me a little lost as Everything and More slid into Cantor’s work on the various infinities and work on numerability of sets.

Cantor’s transfinite numbers are the real goal of the narrative here, rather than what I would call the lay opinion of ∞ (what Cantor referred to as “absolute infinity”). A transfinite number is infinite in that it is greater than all of the finite numbers, but has some properties in common with the finites. If you’re familiar with the ℵ0 I mentioned above – the first transfinite cardinal number, corresponding to the number of members (cardinality) of the set of natural numbers (non-negative integers). Cantor’s continuum hypothesis, which appeared first on the famous list of unsolved math problems David Hilbert presented in 1900, posited that there was no set with cardinality (number of members) between the natural numbers and the real numbers (the cardinality of which Cantor designated as ℵ1). The hypothesis itself may be unprovable, at least within the confines of Zermelo-Fraenkel set theory … which DFW mentions but doesn’t explain, concluding instead with the explanation that later work by Kurt Gödel (the incompleteness guy) and Paul Cohen (who proved that the hypothesis and the ZFC’s axiom of choice were independent) set the question aside without really solving it. At least, I think that’s what he said, because I was just barely treading water by the final page. Which also made me wonder if all of these reviewers quoted as giving the book raves actually finished and understood the whole thing; I imagine the number of people who have sufficient math background to follow DFW down to the bitter end is pretty small.

Apropos of nothing else, the biggest laugh I got from the book was when DFW referred to a mathematician as a world-class pleonast, which is the pot writing a three-page letter to the editor about the mote in the kettle’s eye.

Next up: Ned Beauman’s 2012 novel The Teleportation Accident, recommended by a fellow bibliophile I met in New York in August.

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→ July 9, 2014
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The Solitude of Prime Numbers.

The Italian physicist Paolo Giordano became the youngest winner ever of the Premio Strega, Italy’s equivalent to the Pulitzer Prize for Fiction, when his debut novel, The Solitude of Prime Numbers, took the award in 2008. It became a feature film in Italian in 2010 and made its way here in an English translation that same year, earning very positive reviews around the world for its prose and the development of its two central characters. It is a beautiful rendering of those two horribly broken individuals, and one of the saddest novels I have ever read.

Giordano begins the novel by breaking those two characters in harsh, haunting ways. Alice, pushed too hard by a father whose impetus is never quite clear, suffers a horrible accident while skiing that leaves her scarred, disabled, and bitter. Mattia, meanwhile, is saddled with a twin sister who is severely developmentally disabled, and one day, while taking her to a birthday party to which they’ve both been invited, he leaves her in a park and tells her to wait there for him rather than taking her to the party. She vanishes and is never seen again, leaving Mattia a shell who fears the outside world and inflicts compensatory punishment via self-mutilation.

Solitude cover The two end up meeting in secondary school and forge a friendship based on their mutual recognition of each other’s willful isolation. Mattia is a math genius who has a single friend, Denis, who himself is gay and in love with Mattia but, of course, closeted and himself ostracized from the cruel society of his classmates. Those same classmates taunt or ignore Alice, and eventually the school’s mean girls clique targets her, both because of her disability and her late entry into puberty – the result of her anorexia, which worsens as she gets older. Mattia and Alice seem like a perfect couple, the two peas in the pod on the paperback’s cover, but Giordano argues through his prose that people this detached from others cannot be together. They are twin primes: Two prime numbers with a difference of two, like 11 and 13 or 59 and 61, as close as a pair of odd primes can get, but never actually adjacent. (Only 2 and 3 are neighboring primes. Because 2 is the only even prime, all other pairs must be separated by at least two places on the integer scale.)

Eventually, other circumstances drive Mattia and Alice apart, both as a couple and as neighbors, as Mattia takes a job in Germany and Alice ends up in another relationship. But a strange coincidence, one that Giordano wisely never confirms in full, brings them back together for one final attempt at … something, a connection if not an actual romance, because Giordano hasn’t given us any reason to believe these two broken people can heal themselves enough to be with each other. If this were Hollywood, they’d promise to fix each other (with Coldplay softly playing in the background) and that would be the ending. Giordano gives us ambiguous realism rather than pat endings, and while it doesn’t offer the catharsis a book this sad might call for, it keeps the ending in tune with the remainder of the story.

Alice’s character felt more familiar than Mattia’s, probably because we’ve all known someone who had one of her major issues – a physical disability, leading to social isolation; or an eating disorder. Piling both on Alice might have made her more pathetic, yet Giordano gave her more strength of character, more forcefulness than Mattia, to balance the scales. Mattia’s disappearance into math, especially into research on prime numbers (and specifically Riemann’s zeta function, a key component in the search for a proof or disproof of the Riemann Hypothesis, an unsolved problem detailed extremely well in the book Prime Obsession), further underscores his difficulty with communication – as if a man that comfortable with numbers and order could ever be comfortable in the subjective, anarchic world of words and feelings. He comforts himself by counting objects or looking for familiar shapes or structures in the world, but eventually ends up hurting himself or drawing his own blood in almost every disturbing situation. Is it really right to expect two “primes,” two loners whose self-inflicted solitude has become inescapable, to be able to save each other when neither is capable of helping him- or herself?

As gorgeous as Giordano’s rendering of his characters, even secondary ones like Denis or Alice’s housekeeper Soledad, can be, Solitude can also be intensely painful to read because of the damage he inflicts on them, as if he were pushing and prodding them to see how far they can bend without breaking. Where Alice responds with anger, Mattia responds by becoming increasingly insular, as if even the solitude he finds in numbers isn’t alone enough for him. I was also a little surprised that none of Mattia’s or Alice’s parents seemed to take an active role in trying to draw their children out, showing more resignation, perhaps provoked by guilt (on Alice’s father’s side) or shame (for Mattia’s parents). The only hints of this come with the primes’ rejection of their parents as adults, something I found even more painful now as a parent – to love and raise a child, only to find that child has no use for use once she’s grown, would be the dementor’s kiss of parenthood.

If you’ve already read Solitude, Giordano’s second novel, The Human Body, will be released in the U.S. on October 2nd.

Next up: J.K. Rowling’s first non-Potter novel, The Casual Vacancy.

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