Very short indeed, at 114 pages of text. Develops some important concepts, such as variance, standard deviation, regression, Bayes’s theorem, and sampVery short indeed, at 114 pages of text. Develops some important concepts, such as variance, standard deviation, regression, Bayes’s theorem, and sampling, and gives a brief discussion of probability. Also touches on many ways in which statistics can go wrong if misapplied, misunderstood, or based on faulty data. Left me wanting more, which isn’t a bad thing....more

From a commentary on my blog assessing Turing's position and this book's role in it:

Last year’s film, an affecting drama on its own terms, is both ina

From a commentary on my blog assessing Turing's position and this book's role in it:

Last year’s film, an affecting drama on its own terms, is both inadequate and inaccurate in terms of its subject; anyone longing to know more about Turing’s life and work can do no better than to consult this account. In its 2014 edition, Hodges’s book contains 32 pages of prefatory material and 679 pages of text, including a final author’s note, plus a further 57 pages of notes and an index. It may appear daunting, but the book’s majestic—Tolstoyan? palatial?—scale is entirely justified. One finds here many figures in the carpet, many patterns in the numbers, many leitmotifs at play in the music. Like Tolstoy, Hodges captures a soul on its adventures within society; like a late-Romantic symphony, the book is vast, dense, highly organized, profoundly moving.

Albert Einstein published his theory of general relativity in 1916, so it’s nearly 100 years old. In this clean and clear account for the interested lAlbert Einstein published his theory of general relativity in 1916, so it’s nearly 100 years old. In this clean and clear account for the interested layman, Pedro G. Ferreira presents what could be called a biography of the theory: its family, so to speak, and the environment into which it was born; the sensation surrounding its early years; a relatively quiet spell, in which it was to some degree ignored; and then a long period of tumult, in which unresolved conflicts that had been present all along began coming to the fore. Ferreira doesn’t use this metaphor of a life story—the book is really just a straightforward history—but it’s appropriate in that general relativity has been and still is a living thing, sometimes kicking at its father (Einstein was troubled by some of its implications), often giving enormous help to many others in their work, and showing little sign of fading away quietly despite its advanced age.

As with many genuine biographies, the story here is much more complicated and intriguing than you might expect, with some fascinating characters, lots of factoids, and even a little local color. Astronomer Arthur Eddington had to travel to the tropical island of Principe and confront cloudy weather in order to make observations that would test a prediction about gravity’s effect on light. Einstein, during his years at Princeton’s Institute for Advanced Study, used to walk and talk with logician Kurt Gödel, whose analysis of general relativity allows for time travel. A cosmologist named Alfred Schild, based at the University of Texas at Austin, set up a symposium whose first meeting was planned for December 1963 in Dallas, which was almost canceled as a result of President Kennedy’s assassination late in November. After some maneuvering, the symposium came off as scheduled; the attendees included Robert Oppenheimer, whose name I assume is pretty well known, along with such luminaries as John Wheeler and Roger Penrose, and the discussions led to the coining of the term “quasar.” (The book doesn’t mention it, but Penrose contributed to the decorative arts as well. Though they're challenging to work with in practice, one can use Penrose tiles to cover such things as a bathroom floor,a shower floor, or a large outdoor surface.) One more example: in the early 90s, a research group working on numerical modeling of relativity developed the Mosaic graphical browser for sites on the World Wide Web. Surely it was the most influential of early Web browsers, if not truly the first—contrary to common belief and to what Ferreira says. Many of Mosaic's features are still a fundamental part of Internet Explorer, Google Chrome, Safari, and Firefox. And so, in reading this, you are almost certainly using an offshoot of research into black holes. ...more

Logic is one thing, war another: that seems clear enough. So you wouldn’t expect to see the complex, abstract issues that are involved in logic, matheLogic is one thing, war another: that seems clear enough. So you wouldn’t expect to see the complex, abstract issues that are involved in logic, mathematics, and philosophy discussed in a book that also deals with the similarly complex but much more concrete and consequential issue of whether one should support a war. Yet that’s just what this extraordinary book does—although that’s not all it does.

What’s the connection that Logicomix finds between logic and war? It’s Bertrand Russell, who did important work in mathematical logic, became known as a dedicated pacifist, and over the course of decades was often called upon to discuss the prospects of yet another war. Leaving aside a framing story, the entire book dramatizes a single lecture, titled “The Role of Logic in Human Affairs,” that Russell gave on September 4, 1939, at an American university. As Great Britain had decided on that very day to enter the war in Europe, the stance that Americans should take was of the greatest possible moment. To settle the question, Russell proposed looking to logic, and regarding logic, he said, “My way of telling you the story of logic will be through the tale of one of its most ardent fans, myself.” So begins a vivid, down-to-earth, personal account of Russell’s life and work. It’s a canny piece of biography, an immensely well-informed but easy-to-grasp recap of major episodes in early-20th-century thought, and on top of everything else it’s a story in graphical form.

Russell had been born in 1872, so he had nearly seven decades to account for at the time of this lecture, as well as a number of adventures in his professional work. The easiest way into his work is through his youthful introduction to Euclid’s geometry, which set him on a long path and, as it happens, also brought forth the emotional poles that governed much of his thinking. He was tremendously excited to discover, thanks to one of his private tutors, that a proposition could be proven; in geometry, he says, “I encountered for the first time the delicious experience of knowing something with total certainty.” Not many pages later, though, when Russell’s tutor explains that Euclidean geometry takes some things for granted (its so-called axioms), the moment “marked a terrible disappointment…but ignited the rest of my life.” Why the disappointment? It’s because you haven’t really demonstrated something if your proof depends upon unproven assumptions. Russell wasn’t satisfied with that situation, and he set out to fix it.

The most important direct result of this experience took a while to emerge: the attempt that Russell made, with the help of Alfred North Whitehead, to set all of mathematics upon a reliable foundation in logic. An aside: In saying this, I’ve made a bit of a jump. How did Russell get from the simple matter of Euclid’s assumptions to the “foundational quest,” as it’s called? Such things are hard to explain simply, clearly, and briefly. Readers will find that the book suggests a lot about the difficulty and importance of Russell’s work while avoiding arcane details. This is one of the marvels of Logicomix—you’ll come away with a good sense of how and why it all matters. Now, to return to Russell and Whitehead: Their work unexpectedly took them the entire first decade of the 20th century, and when it was published, under the title Principia Mathematica, it ran to a thousand-odd pages in three volumes. Logicomix takes a moment to delight in something that’s often noted about the Principia: it requires 362 pages to establish that 1+1=2.

Now, recall Russell’s emotional experience of Euclid, in which hopefulness was followed by doubt. That’s the way this project went. Russell was optimistic to the point of overconfidence when he launched into it with Whitehead. By the time they decided to end their work and publish, Russell no longer believed that the Principia succeeded in what it tried to do, though he thought that better men could’ve pulled it off. Nor was that the end of the matter. As a remark in the framing story puts it, Russell was “shaken up,” which is putting it mildly, by Ludwig Wittgenstein’s Tractatus Logico-Philosophicus, published shortly after the end of the Great War. Not many years later, Kurt Gödel put the seal on Russell’s sense of failure when he delivered his Incompleteness Theorem, which proved the impossibility of doing what Russell and Whitehead had attempted.

The pattern repeated itself in Russell’s personal life as well. His desire to find a companion kept coming to naught: he had a total of four wives, of whom two figure into this story, and in addition he spent some time in a hopeless love for Whitehead’s wife. Meanwhile, having concluded from World War I that the existing method of raising children was deeply flawed—because it led to an entire way of life virtually destroying itself—he set out to find a new way, but again his efforts didn’t pay off.

Logicomix is a good deal broader than what I’ve outlined so far. The characters of its main story also include Gottlob Frege, Georg Cantor, G. E. Moore, David Hilbert, Ludwig Wittgenstein, John von Neumann, and Alan Turing. To rattle off the names like that, which I did for the sake of brevity, creates a false impression; in the book, it’s clear who each of them is. Some are supporting players; Moore, for instance, is not even given a first name (probably a mistake in the review copy I read) and is involved only in Russell’s early collegiate encounters with philosophy. Others are fully realized characters who get a good deal of attention; this is especially true of Wittgenstein, who bedevils Russell as one of his students, rather crazily enlists in the Austro-Hungarian army in 1914 (Russell, meanwhile, serves time in prison for his activities against the war), and even achieves some fundamental philosophical insights in the midst of his military service. In fact, by the time the war is over, Wittgenstein has constructed the entirety of his Tractatus.

So much for the main story. There’s also that framing story I mentioned. It's wrapped around the main story, and Logicomix cuts away to it fairly often, sometimes for only a panel or two. Its characters are the creators of the book: Apostolos Doxiadis and Christos H. Papadimitriou, who wrote it, and Alecos Papadatos and Annie Di Donna, who did the art and coloration. In part, the framing story is about the making of the book, which makes the book self-referential, which plays on the concept of self-including sets, which is an aspect of mathematical logic. But this outer story also comments on Russell’s story, extending its historical and thematic range. Papadimitriou, for instance, emphasizes more than once the importance of the foundational quest, including the Principia, in computer science.

There are a handful of ways to interpret the book: as an intellectual biography of Russell, as a survey of issues in mathematical logic that happens to track Russell’s role, or even as a lesson in how little we can be certain of, which is the ultimate point of Russell’s lecture. None of these views seem to demand a graphical presentation. Still, there’s a certain sense to it. Doxiadis points out that “the form is perfect for stories of heroes in search of great goals,” and he reminds us that a good comic-book hero is passionate, complex, even tortured. I seem to recall that the stories of Achilles and Odysseus, despite lacking pictures, managed well enough with mere words such as “rosy-fingered dawn,” but you can look at it another way instead: Bertrand Russell deserves to be drawn as a hero more than Bruce Banner ever did.

The characters gain a lot from being shown. Gödel’s full lips, nearly expressionless eyes (which are usually invisible behind his perfectly round-lensed glasses), and clean, chiseled features make him the most strikingly drawn character in the book, unless perhaps Frege is, with his wild hair and crazed facial expressions. The dramatization of Wittgenstein’s realization about language is even better. There’s something appropriate and wonderful about seeing this happen.

I have two qualms, one minor and one more important. The book’s authors as well as its publisher insist on calling this a “graphic novel,” a term whose questionable usage has been spreading for years. Just a day ago, I saw a nonfiction, graphical book about Richard Feynmann described as a “graphic novel biography.” If a novel is a work of fiction, it seems to follow that a graphic novel must also be fictional, and yet this book is pretty rigorously factual.

One aspect of its truth concerns me: did this lecture of Russell’s, on which the entire main story is based, really happen? A short section in the back matter discusses the book’s “deviations from fact”: Russell probably never met Frege or Cantor, pretty certainly didn’t attend Gödel’s presentation of his Incompleteness Theorem, and so forth. The lecture isn’t mentioned at all. But the lecture says these things happened, so either the authors tinkered with a talk that Russell did in fact deliver, or—more likely—they invented it wholesale. The lecture is a fine piece of work, a smart thing, tying Russell’s life to his work and turning those to the task of making an urgent decision. Yet it’s only an occasion, after all; if it was fabricated, that would hardly justify labeling the whole enterprise a novel, graphic or not. It’s a pity, and worse, that we’re left uncertain whom to applaud for this....more

A book about infinity and the man who in modern times did most to advance its study, Georg Cantor. Cantor died in a mental asylum, having been drivenA book about infinity and the man who in modern times did most to advance its study, Georg Cantor. Cantor died in a mental asylum, having been driven there, in a sense, by the maddening complexities of his work. Anyone with a mathematical bent or a certain kind of philosophical inclination who enjoyed mind-bending late-night dorm-room discussions will find much to marvel at in this book.

Side note: When I read this, I had already been intrigued by the complexities of Kabbalah as it figured into Umberto Eco's conspiracy novel, Foucault's Pendulum, so I was pretty well prepared to appreciate its role in Cantor's thinking. Such correspondences and linkages are among the pleasures of wide reading....more

I believe this is where I first encountered such oddities of geometry and math, some of which were once thought to be monsters, as Cantor dust, SierpiI believe this is where I first encountered such oddities of geometry and math, some of which were once thought to be monsters, as Cantor dust, Sierpinski triangles, Koch snowflakes, Peano curves, and the like. More important, it's where I first found explicated the idea of fractional dimensionality, from which I believe Mandelbrot derived the term "fractal." (The concept itself comes from a German mathematician named Hausdorff, but Mandelbrot generalized it). A solid understanding of the book is likely to be beyond anyone who lacks some training in higher mathematics, but the intrepid reader can still learn much. And nowadays, the Internet is at hand to assist, which wasn't the case when I read this in the 80s, not very long after its publication....more