Gödel’s Proof.

My latest Insider post covers eight top 100 prospects who took a step back this year. I’ll also hold a Klawchat here at 1 pm ET.

I read Rebecca Goldstein’s biography of Kurt Gödel, Incompleteness, last summer, and I believe it was within her book that I read about James Newman and Ernest Nagel’s book Gödel’s Proof that attempts to explain the Austrian logician’s groundbreaking findings. The 114-page volume does a great job of building up to the final proof, but I have to concede that the 19-page section near the end that reveals the fatal blow Gödel delivered to Bertrand Russell, David Hilbert, and others who believed in the essential completeness of mathematical systems lost me in its nested language and ornate symbols. (The newest edition includes a foreword by Douglas Hofstadter, who wrote about the proof in Gödel, Escher, Bach, which won the Pulitzer Prize for Non-fiction.)

Gödel was himself a fascinating figure, a philosopher, mathematician, and logician who wrote a paper with two theorems at age 25 that stunned the world of mathematics in their method and conclusions, proving that any axiomatic system of arithmetic that is consistent cannot be complete. Completeness here means that every true formula that can be expressed within the system can be proven within the system. Gödel’s trick was to create an entire system of expressing logical formulas via what is now called Gödel numbering, and then to craft a formula that says itself that it is unprovable within the system. His proof further stated that even if you could add an axiom to this system of mathematics to cover this new exception, the formula could always be rephrased to pose a new exception, and thus the system is essentially incomplete.

Nagel and Newman do a great job of getting the reader – or at least in getting this reader – to the edge of understanding by building up the history of the question, giving a lay explanation of Gödel’s basic method of numbering and delineating what a simple axiomatic system like that of Russell’s Principia Mathematica (the system Gödel targeted in his proof) would look like. Russell and other logicians of the time were convinced that systems of mathematics were complete – that we could define any such system in terms of a finite number of axioms that would cover all possible formulas we could craft within that system. Any formula that could be proven true at all could then be proven true using only the axioms of that system. Gödel’s proof to the contrary was scarcely noticed at first, but when it spread and others in the field realized it might be true, it blew apart a fundamental assumption of number theory and of logic, while also making Gödel’s name as a major figure in the history of mathematics and logic.

All of which is to say that I just couldn’t follow the nested statements that constitute Nagel and Newman’s explanation of Gödel’s proof. I haven’t read Gödel’s original paper, because it is a truth universally acknowledged that you’ve got to have some serious math background to understand it, so I will accept the claim that Nagel and Newman made it much easier to grasp … but I still only get this at a superficial level. When the authors compare this to Richard’s Paradox, an earlier device that Gödel cited in his paper, I could understand it; these are all descendants of the “This statement is false” type of logical trick that causes an inherent contradiction. Gödel appears to have done the same thing for arithmetic. I just couldn’t quite get to the mental finish line on this one. I guess you could say my understanding of the topic remains ….


Next up: I finished and will review Laurent Binet’s HHhH, and have begun Clifford Simak’s Hugo-winning novel Way Station.

Stick to baseball, 7/30/16.

It’s been a busy week already and I assume the next 52 hours will be even more so; here are my three Insider posts on trades from the last seven days:

• The Aroldis Chapman trade
• The Texas/Atlanta trade and the Blue Jays’ two deals
• The Andrew Cashner and Eduardo Nunez trades

I also have a draft blog post up on last week’s Under Armour Game, and I held my regular Klawchat on Thursday.

I’ll be on ESPN’s trade deadline show on Monday from 1 to 4 pm ET, after which I’m taking a few days off to work on my book and on some other personal projects.

And now, the links…

  • Dr. Mike Sonne, an injury prevention researcher and a baseball fan, argues that pitch clocks may increase pitcher injury risk by reducing recovery time for fatiguing muscles. So maybe pace of game isn’t such a huge problem.
  • If you missed this on Twitter you really should read Eireann Dolan’s story about her autistic brother, from how he was bullied as a kid to the nightmare they all just went through with him.
  • Iowa Republican Steve King says racist stuff on a regular basis and keeps winning re-election. The Iowa Starting line blog looks at why.
  • As always, I’m nobody’s expert on these matters, but I feel like the rejection of state “vote fraud” laws, including this week’s invalidation of North Carolina’s law as racist, is the biggest story of this election cycle. One, with African-American voters favoring Clinton in historic proportions, it seems like striking down these laws could help her in several critical states, including the swing state of North Carolina. Two, killing these laws – based on the entirely fraudulent fear of fraudulent voting – will have an effect on many elections to come, and, one might hope, will slow efforts to disenfranchise entire demographic groups.
  • BuzzFeed political editor (and longtime reader of mine) Katherine Miller wrote a great longread on how Trump “broke” the conservative movement.
  • Trump has faced multiple allegations of sexual assault from women over the last several decades, including one from his ex-wife Ivana. Everyone dismissed such claims against Bill Clinton in 1991-92, but a quarter-century later, the climate around rape and sexual assault is, or seemed to be, much changed. Perhaps Hannibal Burress needs to joke about it before it’ll go anywhere.
  • A large Swedish study on the environmental impacts of organic agriculture versus conventional found differences in each direction, with neither side clearly favored. This is especially important for consumers, in that food labeled “organic” isn’t going to be more nutritious or necessarily better for the environment. But there’s a problem within the problem here – the term “organic” has itself been watered down (pun intended) from what the term meant when Lord Northbourne coined it in 1940. So-called “natural” pesticides aren’t going to automatically better for the environment, for example, and dumping organic fertilizers into the soil won’t have the same effect as using compost and working in crops (like clover or legumes) that increase nitrogen content in the soil.
  • Those “recyclable” disposable coffee cups aren’t recyclable at all, not unless you have access to one of the very few facilities capable of doing so. This means tons of cups end up in landfills every year, so why don’t we demand better?
  • Scientific American explains a card trick that relies on a simple cipher and the cooperation of a partner.
  • A tough longread on a 20-year-old unsolved missing persons case on the Isle of Wight. The police seem to have botched the earliest stages of the investigation, which may render the case unsolvable.
  • German scientists found a bacterium living inside human noses that produces a chemical toxic to Staphylococcus aureus, the bacterium that causes MRSA. Now if only it worked against gonorrhea, the bacterium behind which has evolved resistance to all known antibiotics.
  • Joe Biden has to acknowledge the LIQUID SWORDS tweet at some point, right? If I see him around here I’m going to ask him.
  • Why are police officers enforcing Trump’s ban on Washington Post reporters? They’re claiming it’s a security issue, but that’s clearly not the case.
  • I wrote about a year ago about an essay I read on the unsolved abc problem in mathematics and the abstruse proof offered by a Japanese mathematician, Shinichi Mochizuki, who created a whole new branch of math to solve it – which meant no one was sure if he actually had solved it at all. Scientific American offers an update and some new commentary, including criticism of Mochizuki’s unwillingness to travel or work with others on the proof.
  • In a new book, Innovation and its Enemies, Calestous Juma explains why people often hate new stuff, and talks about what variables affect adoption rates or drive opposition.
  • The National Post gave the fraudumentary Vaxxed zero stars and an admonition not to see it.
  • Speaking of fraud, anything that claims it can “boost your immune system” is lying and even they worked, it’s a terrible idea. If you pay for these “enhanced” water products, or for useless supplements like Airborne, you might as well flush your money down the toilet.
  • The elusive DC-area chef Peter Chang is opening what he calls the restaurant of his dreams in Bethesda. I’ve been to his place in Charlottesville, and I thought it was excellent but have very little history or knowledge of Sichuan cuisine to compare it to.
  • Congrats to Pizzeria Vetri, our favorite pizzeria in Philly and just one of our favorite restaurants there period, for winning Philly magazine’s Best Soft-Serve Ice Cream nod for 2016.
  • Seth Meyers on “Bernie or Bust” twits:


Amir Alexander’s Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World is less a history of math (although there is quite a bit) than a history of the people and institutions who fought a protracted philosophical battle over something we now consider a trivial bit of precalculus. The idea of infinitesimals, at the time of their development called “indivisibles,” sparked vociferous opposition from the supposedly progressive Jesuits in the 1600s, becoming part of their vendetta against Galileo, leading to banishments and other sentences against Italian mathematicians, and eventually pushing the progress of math itself from Italy out to Germany, England, and the Netherlands.

If you’ve taken calculus at any point, then you’ve encountered infinitesimals, which first appeared in the work of the Greek mathematician Archimedes (the “eureka!” guy). These mathematical quantities are so small that they can’t be measured, but their size is still not quite zero, because you can add up a quantity (or an infinity) of infinitesimals and get a concrete nonzero result. Alexander’s book tells the history of infinitesimals from the ancient Greeks through the philosophical war in Italy between the Jesuits, who opposed the concept of indivisibles as heretical, and the Jesuats, a rival religious order founded in Siena that included several mathematicians of the era who published on the theory of indivisibles, including Bonaventura Cavalieri. When the Jesuits won this battle via politicking within the Catholic hierarchy, the Jesuats were forced to disband, and the work involved in infinitesimals shifted to England, where Alexander describes a second battle, between Thomas Hobbes (yep, the Leviathan guy) and John Wallis, the latter of whom used infinitesimals and some novel work with infinite series in pushing an inductive approach to mathematics and to disprove Hobbes’ assertion that he had solved the problem of squaring the circle.

Wallis’ work with infinitesimals extended beyond the controversy with Hobbes into the immediate precursors of the calculus developed by Isaac Newton and Gottfried Leibniz, including methods of calculating the area under a curve using these infinitesimals (which Wallis described as width-less parallelograms). Alexander stops short of that work, however, choosing instead to spend the book’s 300 pages on the two philosophical battles, first in Italy and then in England, that came before infinitesimals gained acceptance in the mathematical world and well before Newton or Leibniz entered the picture. Hobbes was wrong – the ancient problem of squaring the circle, which means drawing a square using only a straightedge and compass that has the same area as that of a given circle, is insoluble because the mathematical solution requires the square root of pi, and you can’t draw that. The impossibility of this solution wasn’t proven until 1882, two hundred years after Hobbes’ death, but the philosopher was convinced he’d solved it, which allowed Wallis to tear Hobbes apart in their back-and-forth and, along with some of his own politicking, gave Wallis and the infinitesimals the victory in mathematical circles as well.

Alexander tells a good story here, but doesn’t get far enough into the math for my tastes. The best passage in the book is the description of Hobbes’ work, including the summary of the political philosophy of Leviathan, a sort of utopian autocracy where the will of the sovereign is the will of all of the people, and the sovereign thus rules by acclamation of the populace rather than heredity or divine right. (I was supposed to read Leviathan in college but found the prose excruciating and gave up, so this was all rather new to me.) But Alexander skimps on the historical importance of infinitesimals, devoting just a six-page epilogue to what happened after Wallis won the debate. You can’t have integral calculus without infinitesimals, and calculus is kind of important, but none of its early history appears here, even though there’s a direct line from Wallis to Newton. That makes Infinitesimal a truncated read, great for what it covers, but missing the final chapter.

Next up: The Collected Stories of Katherine Anne Porter, winner of the Pulitzer Prize for Fiction in 1966.

The Unfinished Game.

I’m still playing a bit of catchup on stuff I read during March (and just finished Joe Haldeman’s The Forever War over lunch today), but one title I definitely want to bring to everyone’s attention is the delightful, short book by mathematician (and NPR’s “Math Guy”) Keith Devlin called The Unfinished Game, which explains how one specific letter in the correspondence between Blaise Pascal and Pierre de Fermat opened the door to the world of probability and everything that this branch of mathematics makes possible.

The unfinished game of the book’s title was based on a common, popular controversy of the time surrounding games of chance, which were largely seen as incalculable – our modern, simple way of calculating odds of things like throws of the dice just did not exist at the time. Pascal and Fermat discussed the question of how to divide winnings in a game of two or more players where the players choose to abandon the game before any one player has won the requisite number of matches. (So, for example, they’re playing a best-of-five, but the players quit after three rounds, with one player having won two times and the other one.) The controversy in question will seem silly to any modern reader who’s taken even a few weeks of probability theory in high school math, but Devlin is deft enough to explain the problem in 1600s terms, so that the logical confusion of the era is clear on the page.

The confusion stemmed from the misunderstanding about the frequencies of subsequent events, given that the game would not always be played to its conclusion: You may say up front you’re going to play a best of seven, but you do not always need to play seven matches to determine a winner. If you quit after three games, in the situation I outlined above, it is possible that you would have needed just one more match to determine a winner, and it is possible that you would have needed two more matches. Pascal’s letter to Fermat proposed a method of determining how to split the winnings in such an unfinished game; the letter was the start of modern probability theory, and the problem is now known as the problem of points. (You can read the entire surviving correspondence on the University of York’s website; it also includes their conversations on prime numbers, including Fermat’s surprising error in claiming that all numbers of the form 2(2n)+1, which is only true for 0 ≤ n ≤ 4. Those five numbers are now called Fermat primes; Euler later showed Fermat’s hypothesis was wrong, and 2(25)+1 = 4294967297, which is composite.)

Fermat realized you must count all of the potential solutions, even ones that would not occur because they involved playing the fifth game when it was made unnecessary by the first player winning the fourth match and taking the entire set, so to speak. (The problem they discussed was slightly more involved.) Pascal took Fermat’s tabular solution, a brute-force method of counting out all possible outcomes, and made it generalizable to all cases with a formula that works for any number of players and rounds. This also contributed to Pascal’s work on what we now call Pascal’s triangle, and created what statisticians and economists now refer to as “expectation value” – the amount of money you can expect to win on a specific bet given the odds and payout of each outcome.

Devlin goes about as far as you can when your subject is a single letter, with entertaining diversions into the lives of Pascal and Fermat (who corresponded yet never met) and tangents like Pascal’s wager. At heart, the 166-page book is about probability theory, and Devlin makes the subject accessible to any potential reader, even ones who haven’t gone beyond algebra in school. Given how much of our lives – things like insurance, financial markets, and sports betting, to say nothing of the probabilistic foundations of quantum theory – are possible because of probability theory, The Unfinished Game should probably be required reading for any high school student.

Next up: I just started Eimear McBride’s A Girl is a Half-Formed Thing, winner of the 2014 Baileys Women’s Prize for Fiction.

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…

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.

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.

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.


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.


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.