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 ….

…incomplete.

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

Infinitesimal.

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 Elegant Universe.

My latest column at ESPN looks at five potential callups for contenders.

Brian Greene’s 1999 bestseller and Pulitzer Prize finalist The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory is more like two books in one. The first half to two-thirds is a highly accessible history of the two main branches of physics, the macro world perspective that culminated in Einstein’s discovery of general relativity, and the micro (I mean, really micro) perspective covered by quantum mechanics. The two theories could not be unified until the advent of string theory, which Greene lays out in still somewhat easy to follow language. The last third of the book, however, delves into deeper topics like the nature of spacetime or the hypothesis of the multiverse, and I found it increasingly hard to follow and, unfortunately, less compelling at the same time.

String theory – more properly called superstring theory, but like the old basketball team in Seattle, the theory has lost its “super” somewhere along the way – is the prevailing theoretical framework in modern physics about the true nature of matter and the four fundamental forces. Rather than particles comprising ever-smaller subparticles that function as zero-dimensional points, string theory holds that what we perceive as particles are differing vibrations and frequencies of one-dimensional “strings.” String theory allows physicists to reconcile Einstein’s theories of general and special relativity with the explanations of three of those four forces (strong, weak, and electromagnetic) provided by quantum mechanics, resulting in a theory of quantum gravity that posits that that fourth force is the result of a massless quantum particle called the ‘graviton.’ Gravitons have not been observed or experimentally confirmed, but other similar particles have been, and all would be the result of those vibrating strings, open or closed loops in one dimension that, under the framework, are the most basic, indivisible unit of all matter and energy (which are the same thing) in the universe.

Strings are far too small to be observed, or to ever even be observed – you can’t observe a string with a particle, like a photon, larger than the string itself – but physicists believe string theory is accurate because math. And that’s one of the biggest challenges for Greene or anyone else writing about the topic: the proof isn’t in experimental results or great discoveries, but in equations that are too complicated to present in any text aimed at the mass audience.

In fact, the equations underlying string theory require a universe of not four dimensions – the ones we see, three of space and one of time, which Einstein treated simply as four dimensions of one thing called spacetime – but ten or eleven. These “missing” dimensions are here, at every point in the universe, but are tightly curled up in six-dimensional forms called Calabi-Yau manifolds, as if they exist but the universe simply chose not to deploy them. They must be there, however, if string theory is true, because the calculations require them. This is near the part where I started to fall off the train, and it only became worse with Greene’s discussions of further alterations to string theory – such as higher-dimensional analogues to strings called 2-branes and 3-branes – or his descriptions of what rips or tears in spacetime might look like and how they might fix themselves so that we never notice such things. (Although I prefer to think that that’s where some of my lost items ended up.)

The great success of this book, however, is in getting the reader from high school physics up to the basics of string theory. If you’re not that familiar with relativity – itself a pretty confusing concept – this is the best concise explanation of the theories I’ve come across, as Greene uses simple phrasing and diagrams to explain general and special relativity in a single chapter. He follows that up with a chapter on quantum mechanics, hitting all the key names and points, and beginning to explain why general relativity, which explains gravity in a classical framework, cannot be directly coupled with quantum mechanics, which explains the other three forces in an entirely different framework. Building on those two chapters, Greene gives the most cogent explanation of superstrings, string theory, and even the idea of these six or seven unseen spatial dimensions that I’ve come across. We’re talking about objects smaller than particles that we’ve never seen, and the incredible idea that everything, matter, energy, light, whatever, is just open and closed one-dimensional entities the size of the Planck length, 1.6 * 10-35 meters long. To explain that in even moderately comprehensible terms is a small miracle, and Greene is up to the task.

This was a better read, for me at least, than George Musser’s book on quantum entanglement, Spooky Action at a Distance, which covers a different topic but ends up treading similar ground with its descriptions of spacetime and the new, awkwardly-named hypothesis “quantum graphity.” Quantum entanglement is the inexplicable but true phenomenon where two particles created together maintain some sort of connection or relationship where if the charge or spin on on of the particles is flipped, the charge or spin on the other will flip as well, even if the two particles are separated in distance. This appears to violate the law of physics that nothing, including information, can be transmitted faster than the speed of light. How do these particles “know” to flip? Musser’s description of the history of entanglement, including Einstein’s objection that provided the title for this book, is fine, but when he delves into new hypotheses of the fabric of spacetime, he just completely lost me. Quantum graphity reimagines spacetime as a random graph, rather than the smooth four-dimensional fabric of previous theories, where points (or “nodes”) in space are connected to each other in ways that defy traditional notions of distance. This would provide a mechanism for entanglement and also solve a question Greene addresses too, the horizon problem, where disparate areas of the universe that have not been in direct physical contact (under the standard model) since a tiny fraction of a second after the Big Bang currently have the same temperature. I didn’t think Musser explained quantum graphity well enough for the lay reader (me!), or gave enough of an understanding that this is all highly speculative, as opposed to the broader acceptance of something like string theory or absolute acceptance of quantum theory.

Next up: Back to fiction with Eowyn Ivey’s Pulitzer Prize finalist The Snow Child.

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.

Beyond Einstein.

Some great boardgame apps still on sale, including Splendor for $0.99 (iOS or android) and Ticket to Ride for $2.99 (iOS or android).

I enjoyed physicist Michio Kaku’s book Einstein’s Cosmos, a biography of the founder of relativity theory that didn’t skimp on details of Einstein’s work, so when I spotted another of Kaku’s books, the 1995 work Beyond Einstein: The Cosmic Quest for the Theory of the Universe (co-authored by Jennifer Thompson) for half price at Changing Hands in Tempe during my annual AFL trip, I picked it up without a second thought. The book covers a little of the same ground as the Einstein bio, but is primarily a history of superstring theory and the search for a “grand unified theory” (up to 1995, of course) that would bring together the four fundamental forces of physics, building the reader up from the mid-19th century forward through various stops and starts that included the proposal, discarding, and resurrection of string theory from the 1950s to the 1980s.

Strings, in particle physics, are theoretical subparticles that would constitute all types of matter and energy in the universe: the hundreds (or more) types of subatomic particles known to physics may all be manifestations of strings, with different vibrations of the strings showing up to our devices as different subatomic particles. String theory would solve a large number of problems with our current understanding of the nature of matter and energy, from the existence of the aforementioned four forces (gravity, the strong nuclear force, the weak nuclear force, and electromagnetism, although the last two have been shown to be the same thing) to the origins of the universe itself. Most theoretical physics has rested on the assumption that the universe is orderly; the complexity involved in having hundreds of fundamental particles, or even in having four independent forces, has in and of itself led physicists to try to unite these under a single umbrella, with string theory the leading candidate and quite possibly the only game in town.

Where Kaku and Thompson succeed is in guiding the reader to a basic understanding of string theory by gradually working their way through the various milestones in physics research over the 120 or so years before string theory became widely accepted as a serious candidate for the “theory of everything.” That means we get our fill of Maxwell and Einstein, but we also get Feynman diagrams (which apparently are rather a big deal, but were new to me as a lay reader) and the best concise explanation of Schrodinger’s cat paradox I’ve come across. Kaku also explains symmetry and supersymmetry, the suspected nature of dark matter, and the connection between Lie groups (from group theory) and quantum field theory, without ever drowning readers in math unless you go to the footnotes. I wouldn’t say that the book taught me enough about string theory – I think I’ll have to get Brian Greene’s best-selling The Elegant Universe for that – but it gave me more than just a superficial explanation along with plenty of the mind-bending stuff that makes theoretical physics seem fun to someone like me.

There are some sections at the end of the book that seemed to me to go beyond science and into the highly speculative, although some of you may be able to tell me that my impression is wrong. Some of it is just strange, like the argument that the universe was originally in ten dimensions but collapsed into two separate universes, ours with four dimensions and another, minuscule universe that held the other six (are dimensions really additive?). Some seemed borderline metaphysical, like the argument that the universe came from nothing in a sort of quantum leap, even though sudden state shifts like that don’t occur … well, ever, or wouldn’t we stand in constant risk of winking out of existence (or perhaps into another, parallel universe)? Kaku’s book leaves lots of questions unanswered, but I suppose it fits, since theoretical physics has yet to answer many of those same questions.

Next up: Lois McMaster Bujold’s Paladin of Souls, another Hugo Award winner.

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.

The Sixth Extinction.

My annual column on players I got wrong is up for Insiders.

I was feeling okay until I read Elizabeth Kolbert’s The Sixth Extinction: An Unnatural History, winner of the most recent Pulitzer Prize for Non-fiction, an unbelievably well-written and thorough accounting of the history of mass extinctions with a particular emphasis on the current one that is the first to be caused by another species – us.

The various scientists who work on the history of life on earth and of the planet itself agree that we’ve seen five mass extinction events since life first began, including the one we all learned about in school, the massive impact of a foreign body on earth that ended the Cretaceous period, killing all non-avian dinosaurs and three of every four species extant on the planet at that time. That wasn’t even the most damaging to life on earth – the end Permian event wiped out over 90% of extant species – and other extinction events had differing causes, including widespread glaciation or gradual oceanic acidification. But these events did occur, along with numerous smaller extinction events, which is why the current biosphere looks like it does, with our species the dominant one … and causing the current mass extinction event, which could lead to the loss of half of the biodiversity on the planet by the end of the century.

Kolbert has a lot of what could be some very dry paleontological and geological research on the history of mass extinction events, but instead weaves them into numerous narratives around specific species that we’ve lost or are trying to protect. She flashes backward into historical research to discuss long-vanished species like graptolites, which were wiped out in the ice age that ended the Ordovician period roughly 444 million years ago, or to discuss the various natural environmental phenomena that caused previous mass extinction events. In many of these chapters, she traveled to conservation sites, to zoos, or to natural habitats to follow scientists attempting to stave off extinctions or learn the causes of population losses. She travels to Panama to witness the desperate attempts to save various golden frog populations from the chytrid fungus, which eats away at amphibians’ skin, and to a cave in upstate New York where the native bat population was decimated by “white nose syndrome,” caused by another fungus called Pseudogymnoascus (formerly geomyces) destructans that thrives in the cold temperatures the bats favor.

By the end of the book, Kolbert has devoted a chapter to each of the major effects of human development on the biosphere that are now factors in the ongoing mass extinction event, including climate change, ocean acidification, habitat destruction, geographic fragmentation, spreading invasive species, and hunting/poaching. It’s utterly horrifying, not least because there’s so little we can do at this point: Our very existence, and our (temporary) supremacy atop the evolutionary pyramid, has led to numerous extinctions, from our hunting the great auks out of existence to deforestation that has wiped out numerous bird and amphibian species. Global warming is a dire threat to marine life in particular, and ocean acidification, climate change’s “equally evil twin,” is killing the world’s coral reefs. We’re bringing pathogens to ecosystems where the native species haven’t evolved any resistance, and bringing invasive plant, insect, and reptile species to environments where they lack natural predators. We suck.

Of course, we are also the only species in the history of the planet to actually care about stopping extinctions, although our efforts tend to focus on single species and to come very late, rather than trying to stop massive factors like climate change that threaten thousands of species simultaneously. Kolbert can’t even muster a high note on which to end the book, not that she should sugar-coat the truth, and concludes with the open question of what consequences these environmental catastrophes and the consequential loss of biodiversity might have for us.

Kolbert goes into the suspected causes of the mass extinctions, four of which are more or less tied to the current event. The end Permian “impact” event makes for a fascinating story because the hypothesis is so recent (first proposed in 1980) and was so widely derided at the time, including a famous New York Times editorial from 1985 titled “Miscasting the Dinosaur’s Horoscope,” which concluded with the line, “Astronomers should leave to astrologers the task of seeking the cause of earthly events in the stars.” While it’s the one kind of mass extinction event cause we’re not currently putting in play ourselves, it makes for a compelling side story as Kolbert explains both the discovery of the evidence that backs it up and the scientific establishment’s resistance to the idea when it was first proposed.

Einstein’s Cosmos plus seven other books.

I’ve fallen way behind in book reviews, so rather than procrastinate further and get upset with myself for letting this many pile up, here are my thoughts on eight books I’ve read recently.

Theoretical physicist Michio Kaku does a remarkable job of taking a dense scientific topic and making it accessible in Einstein’s Cosmos, part of the same Great Discoveries series that includes Everything and More by David Foster Wallace and Incompleteness by Rebecca Goldstein. Part biography of Einstein, part survey course in theoretical physics, Einstein’s Cosmos takes the reader back to Einstein’s childhood, dispelling some myths about his youth and eventually leading to the best lay explanation of special relativity I’ve come across. Kaku doesn’t stint on some of Einstein’s less flattering moments, such as his early opposition to quantum field theory, but presents him as a man of great principle as well as an uncommon ability to visualize difficult problems in physics, a skill that first allowed him to formulate the theory of special relativity by asking what would happen if he could chase a beam of light while he himself was traveling at the speed of light. Kaku has to give the reader a substantial amount of information to get to the point of special relativity and the equivalence of mass and energy, including a basic discussion of Maxwell’s equations, four partial differential equations that describe the formation and behavior of electromagnetic fields (above the quantum level, which Maxwell’s equations can only approximate). None of this is easy, but Kaku’s explanations are accessible even if you’ve never taken calculus, because his focus is on the meaning of these formulas and theories rather than on their precise functions. He also gives color the portrait of Einstein, who was an eccentric and widely beloved figure, without reducing him to caricature by repeating old tropes about him being a terrible student (he was a superb student when he cared about the subject) or a mere patent clerk (university politics kept him out of academia at first, not a lack of skill or background). I recommend it very highly if you’re at all interested in the man or his discoveries and, like me, are a long way removed from any coursework that might otherwise be necessary to understand it.

Michael Blanding’s The Map Thief tells the story of rare map dealer turned thief E. Forbes Smiley III, and follows in the footsteps of an earlier book about another crook who cut rare maps from ancient atlases, Miles Harvey’s The Island of Lost Maps. While Blanding’s book is better written and organized, giving a breezy history of cartography and explaining why some of these maps are so rare, the subject of the book, Smiley, is a fairly milquetoast character, even when Blanding tries to give him more dimension by talking about his attempts to remake a small town in rural Maine. This sort of non-fiction book tends to work best when the central narrative involves a literal or figurative chase, but Blanding spends scant time on the portion of Smiley’s story between the discovery that he may have taken some maps (or even that maps were missing) to his arrest. Harvey’s book, on the other hand, tells the story of the appropriately-named Gilbert Bland, an antiques dealer with no apparent personality, by turning into more of an old-fashioned crime book, documenting his crimes and the process of tracking him down in a way that covers up Bland’s lack of character. Both books are solid reads in their own rights, with Blanding’s shorter and more tightly organized, while Harvey’s has more narrative greed.

I’m still gradually working my way through the Pulitzer Prize for Fiction winners, and read two winners from the 1990s that were good-not-great, although in one case I could at least easily understand why it won. Steven Millhauer’s Martin Dressler: The Tale of an American Dreamer reads like a fable, detailing the titular character’s rise from his youth as the son of a cigar-store owner to successful hotelier and entrepreneur, only to find with each new venture that his ambition is unsated, eventually pushing himself to build a hotel so grandiose that it fails. Along the way, Dressler marries the wrong woman, an entirely unconvincing subplot that undermined much of the novel’s narrative force. I could see the Pulitzer committee loving the book for its exploration of the superficiality of the American Dream.

Michael Cunningham’s The Hours, later adapted into a Best Picture-nominated film that starred three of the best actresses of its specific time (Julianne Moore, Meryl Streep, and Nicole Kidman, who won an Oscar for her performance as Virginia Woolf), seemed to fit the Pulitzer Committee’s loose standards less, but was a more literary, well-rounded work. Cunningham crafts three vaguely interconnected novellas and weaves them together with frequent shifts between them, setting them in three different times, with the only overt connection via Mrs. Dalloway: one story follows Woolf as she’s writing it, the other two revolve around women who’ve read the book and felt a deep connection to it. I would probably have enjoyed or appreciated The Hours more if I’d actually liked Mrs. Dalloway or had at least read it more recently, although the way Cunningham eventually connects the two non-Woolf stories, while somewhat predictable, is touching without devolving into mere sentiment, and still left me wanting more of that unified storyline.

I love Evelyn Waugh’s novels, but Helena, a short work of historical fiction, did nothing for me. It’s missing most of his trademark humor, instead telling a fictionalized version of the life of the Empress Helena, mother of Constantine, who made a pilgrimage to Syriana and, according to legend, rediscovered the True Cross on which Jesus Christ was crucified. Waugh converted to Catholicism after writing his first novel, Vile Bodies, and while there are strains of his religious belief through all of his later works, Helena feels maudlin and ends with a passage that you might characterize as magical realism depending on your point of view on Christianity. Waugh apparently considered this one of his best novels, but since his satirical prose and eye were what made him a great novelist, Helena feels inconsequential in comparison.

William Maxwell’s So Long, See You Tomorrow, winner of a National Book Award in 1982, came recommended by my friend Samantha, an avid bibliophile who favors shorter fiction where I go for novels. So Long is a 135-page novella that explores loss and memory through the eyes of an old man remembering his broken connection with a friend when the latter’s father committed a shocking murder. The narrator goes back to the time of the murder and recounts the circumstances that led up to it, although I imagine his account is supposed to be unreliable (as with the imagined recollections of the narrator of James Salter’s A Sport and a Pastime). Maxwell depicts the life of the small town in Southern Illinois in often painful detail, walking through the minds of the three principals in the affair that led to the murder, and actually devotes little page time to his friend, the unfortunately-named Cletus, whom I couldn’t picture as anything but a slack-jawed yokel.

Dodie Smith’s name may not be familiar to you, but you know her work: She wrote the children’s book that Disney adapted for 101 Dalmatians. She also wrote a novel, I Capture the Castle, that’s highly regarded in England but seems to have never caught on here, perhaps because its subject is so very British. The 1949 novel starts out like a Jane Austen book: Two sisters move into a remote castle with their author father, who subsequently falls into severe writer’s block and finds himself unable to produce another novel – or any income, with the girls’ stepmother only barely more able to provide. A wealthy family moves into the neighborhood, with two very eligible bachelor sons, one of whom takes a fancy to the narrator’s sister … but Smith avoids the predictable and crafts a compelling narrative by having the younger sister, Cassandra, tell the story through her journal, with scrupulous honesty. I was hoping for a little more humor, but the seventeen-year-old narrator’s voice doesn’t have Austen’s wry comic style. The descriptions of the family’s privations early in the book wore on, but the denouement justified much of the time spent to get there.

The final book in this list gets the shortest writeup. Cesare Pavere’s The Moon and the Bonfires tells of an Italian expatriate’s return to his hometown after the devastation of the Mussolini regime and the second World War, and the tragedies he uncovers while obviously hoping to return to a town unchanged. Without any knowledge of the specific history of Italy under fascism, however, I failed to connect with the story or any of the characters. The isolation of the protagonist and the sparse prose reminded me of Camus, and not in a good way.

Undeniable.

I’ve had three Insider pieces go up in the last 36 hours, on the the Johnny Cueto trade, a few Binghamton Mets prospects, and the Tyler Clippard trade.

Bill Nye’s Undeniable: Evolution and the Science of Creation should be required reading for every American high school student, and I’d hand the book to anyone who indicated s/he plans on voting in our next election. Nye demolishes the many ignorant anti-evolution arguments out there, while eloquently and ardently presenting a case for science in a world of denial and fear-mongering.

The title refers to the persistence of evolution deniers, those folks who refuse to accept the scientific proof of evolution because it interferes with other aspects of their worldview. Nye engaged in a well-known debate with a particularly ardent denier, Ken Ham, who also refuses to accept the actual age of the earth, substituting his own fiction (I believe he says it’s 6000 years old, although some other deniers go with 10,000 years, not that it matters in the least because they’re wrong) for geological fact much as he substitutes his own fiction (that the first book of the Christian Bible is the literal truth of our creation) for biological fact. That debate, in which Nye clowned Ham, who continually referred to the Bible as his “evidence,” was one of the spurs for Nye to write Undeniable, but it serves more broadly as a frontal assault on the anti-science/anti-intellectual movement that hinders or prevents us from facing major societal or global problems, from disease eradication to feeding the planet to slowing anthropomorphic climate change.

The book should convince anyone who still denies evolution yet is willing to listen to some basic facts. We know now that all mammals descended from a common ancestor that lived some 70 million years ago, something demonstrated by patterns in the fossil record and the similarities between our DNA and those of many species seemingly unrelated to us. We’re barely distinguishable at the DNA level from chimpanzees, sharing 99% of our DNA with the related primates called bonobos, while we share about half of our DNA sequences with bananas (themselves the product of cloning; every yellow banana you eat is a Cavendish and is genetically identical to all of the other Cavendishes in the world). NOTE: I edited the common ancestor bit, as I conflated two numbers when writing this review from memory. Thanks to the readers on FB who pointed this out.

He attacks some of the most common (and dumb) creationist arguments against evolution, swatting them down like so many genetically-similar-to-human fruit flies. The argument that evolution violates the second law of thermodynamics fails because that law only applies to closed systems, whereas the Earth – getting energy from that big yellow ball in the sky – is very much an open system. The argument from irreducible complexity, that current organisms are too complex to be explained without an Official Designer™, fails on multiple counts, not the least of which is all of the suboptimal designs we see in nature; Nye even mentions ulnar collateral ligaments for pitchers in an amusing aside on this topic. He points out more significantly that the only reason you’d see the “designs” we see in nature are as the result of a process of incremental changes through genetic mutations, and that the “what good is half a wing?” variation of this argument misstates how features like wings evolved. He takes down the false dichotomy of macroevolution versus microevolution (which creationists claim is only “adaptation”), including how the latter is the inevitable result of the former – and how there’s plenty of tangible proof of the latter, despite what Ken Ham might claim.

Once Nye has explained the theory of evolution by way of the various insubstantial criticisms levied at it by creationists, he takes on multiple issues that are related to or follow naturally from our understanding of evolution, all of which are significant issues in the science policy sphere.

* GMOs. Nye has already walked back some of his criticisms from this chapter after taking fire from the scientific community at large, although the concerns he raises about the introduction of DNA from distant species into food crops – notably that their effects on the crops’ ecosystems are difficult to predict – are valid. Humans are particularly terrible at foreseeing unintended consequences, as explained in The Invisible Gorilla and demonstrated in nearly every public-policy decision or the entire Bud Selig reign in MLB, and such genetic modifications entail lots of unpredictable ramifications. Nye has continued to raise the alarm about the massive reduction in the monarch butterfly population thanks to the widespread use of glyphosphate, the enzyme-inhibiting herbicide marketed as Roundup, which has decimated the natural supply of milkweed plants. You should plant some in your yard if you’re in the right part of the country; we have for the last two summers and were rewarded in 2014 with several visiting monarch caterpillars.

* Abortion. Nye points out that the claim that life begins at conception is untenable, as a successfully fertilized human embryo may fail to implant in the uterine wall or fail to successfully undergo gastrulation; if such eggs are considered to be alive and human, then a woman who miscarries for these reasons has committed murder. Nye broaches the topic when discussing stem cells and the concerns, most of which are baseless, about harvesting such cells from fertilized embryos that would otherwise be headed for the sewer.

* Antibiotic drug resistance. If you’ve read my stuff for a while, you know this is a huge issue for me, particularly as it relates to food safety. The problem exists because evolution is true: bacteria that have beneficial mutations that allow them to survive an antibiotic purge reproduce and eventually spread, leading to resistant strains that defeat our drugs. We can’t ever win this battle, but we can certainly fight it more intelligently than we do now.

* Race. It’s not real – that is, not biologically real. Race is a social construct, and Nye explains why.

* Space exploration. Ah, here’s where Nye and I diverge in our views. Nye discusses the possibility of life on other bodies within our solar system, naming a few likely candidates (Mars, Europa, Enceladus), and argues in favor of fairly expensive missions to try to determine if there is life of the microbial variety on any of these planets or satellites. I won’t try to paraphrase his case for fear of doing it an injustice, but I did not find the case satisfactory. A multi-billion dollar mission like this has to have a significant potential payoff for us, and he doesn’t provide one. Knowing there’s life on other worlds would be interesting, but does it advance our knowledge in any practical or meaningful fashion? How would it? Perhaps we’d find microbes that can produce energy in a novel way, or that can consume chemicals that are pollutants on earth … but he doesn’t even broach those possibilities. And, of course, that $10 billion or $20 billion mission has a very high probability of finding no life at all, so the potential payout has to exceed the cost by a significant factor.

* Another chapter, on the evolutionary explanations for altruism, also fell a bit short of the mark for me, but for different reasons. I’m strictly a lay reader on this, and can’t put my opinions on the matter on par with those of Nye or his sources, but it seems even after reading Nye’s explanation that the evolutionary psychology explanation for human altruism is too post hoc – crafting a narrative to fit the facts, rather than working from the facts forward as evolutionary biologists have done. The comparison of human altruism to altruistic behavior in other species also struck me as facile, an argument by weak analogy that did not address the extent or nuance of human altruistic behaviors.

Nye does not explicitly offer any arguments against religion or theism, although he is arguing heavily against creationism, Intelligent Design, and any sort of Creator force behind life on this planet. He also makes several points that are inherently anti-religious, such as the fact that we are not “special” from a genetic perspective and the fact that we aren’t the end product of evolution because evolution has no end product. Nye points out that some readers may find these points depressing, but says he finds evolution and the march of science inspiring, especially because of the breadth of knowledge out there waiting for us to discover it.

I listened to Nye’s narration of the Audible audio edition of Undeniable, and there is no question in my mind that he made the book more enjoyable for me. He brings tremendous enthusiasm to the subject, and his comic timing and delivery are effortless and natural. It’s hard to hear him exude over these topics and not feel his excitement or his indignation. Nye says he wrote this book because teaching anti-scientific topics like creationism hurts our children and our country, a point with which I agree wholeheartedly. Hearing those words from his mouth made the message seem more potent.

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.