The C.P. Snow introduction, which was nearly as long as the Apology itself, was a fantastic inclusion, as it provided context which I felt really brouThe C.P. Snow introduction, which was nearly as long as the Apology itself, was a fantastic inclusion, as it provided context which I felt really brought me closer to Hardy. Without it, I think, I would have found Hardy to have been a bit too full of himself, as he feels dismissive of perspectives other than his own. Having said that, Hardy is very sensible in explaining the limits of his ability to explain his thoughts, and admits that he is unqualified to approach topics like the aesthetics of mathematics philosophically.

I've come away really wishing that I had this book as part of a dialog, rather than a soliloquy which. I think that there is a lot of fertile ground here, especially in comparing his Platonism with other schools of thought at the time, which he must have been aware of, like Wittgenstein.

Hardy's main point (I think) is that there are two types of mathematics: Real (academic, pure) and trivial (basic, simple, everyday). Trivial mathematics is applied to our lives and serves its purpose (and so is commonly known as applied mathematics). But real mathematics has almost no connection to our lives (outside of philosophy and some of the high sciences), even if it is describing a reality that Hardy, as a Platonist, believes is greater than the one we observe. Real mathematics, to Hardy, can only "be justified as art if it can be justified at all."

This may be seen as a sad book, certainly. But I see Hardy's opinion of his own mortality (he committed suicide) as starkly inconsistent with his opinion of pure mathematics. Near the close of the book, Hardy admits "It is plain now that my life, for what it is worth, is finished, and that nothing I can do can perceptibly increase or diminish its value." What a shame that he considers his life finished because he can't increase its value. Hasn't he spent an entire book justifying (or at least enjoying) something which he admits to be useless? If Hardy saw his life the way he saw his mathematics, he would have spent the remainder of it enjoying himself, simply for his own sake....more

The best book I've read in 2010. This was my one big tome I took on vacation w/ me (5 weeks backpacking through China), because I couldn't see myselfThe best book I've read in 2010. This was my one big tome I took on vacation w/ me (5 weeks backpacking through China), because I couldn't see myself finishing it quickly in any other setting.

It's 750-something pages, and dense. But boy is it worth it. Past explaining Godel's incompleteness theorems, and connecting them conceptually through Escher's drawings and Quine's philosophy, GEB discusses topics that relate to fugues (Bach) like recursion and loops. The book also gets into computational theory, networks, Zen Buddhism, artificial intelligence.

I loved this book, because I still can't understand how one person could connect so many disparate topics out of the blue, let alone write about in a way that's understandable, engaging, and oftentimes very funny.

This is a charming way to become introduced to formalism in mathematics, as the authors have very cleverly inserted a lot of math theory into a lightThis is a charming way to become introduced to formalism in mathematics, as the authors have very cleverly inserted a lot of math theory into a light narrative around a Stanford University student's life.

Past some simple and pleasing proofs (pythagoras, no-greatest-prime-number, etc.) The book dives into Georg Cantor's work in set theory and his continuum hypothesis, the discovery of non-Euclidean geometry, and how these discoveries challenged our understanding of mathematical proofs based on axiomatic theory.

Things never get too complicated, so if you're interested in getting a taste for these mathematical ideas, I think this is probably the best place to start.

I was disappointed that Kurt Godel wasn't mentioned until the final 2 pages of the book. He probably made the most significant contributions in this field, and he was certainly relevant, and perhaps entirely conclusive, this narrative. and I think the authors probably had to cut him out because they couldn't find enough time to fit him into the story....more

A very good book, but with some limitations. It's pretty clear that this was written by a philosopher, rather than a mathematician. Yourgrau shows itA very good book, but with some limitations. It's pretty clear that this was written by a philosopher, rather than a mathematician. Yourgrau shows it in his explanation of Godel's incompleteness theorems: rather than work through Godel numbering, recursive furmulae, etc., we get a more philosophical illustration of the Incompleteness theorem, which comes out less robust. In fact, I'd have an awful hard time understanding what Yourgrau was saying if I hadn't already studied Godel's work elsewhere.

Even worse, Yourgrau spends very little time discussing the work Godel did with Einstein's General Relativity, which is bizarre, because it seems like that's what the book promises. He doesn't even mention closed timelike curves by name, although he spends about a paragrpah describing them. Again, I suspect this is beause Yourgrau is a philosopher by training, not a mathematician or a physicist.

However, Yourgrau does an AWESOME job of explaining formalism, and how formalism as a school of thought developed amongst philosophers, scientists, and mathematicians.

If you want to learn Godel's incompleteness theorems, pick up "Godel, Escher, Bach" or a more technical book. If you want a good biography of Godel, I'd be split between this and Goldstein's "Incompleteness" which I liked, but I read quite a while ago. If you want a good understanding of the philosophical environment Godel...more

I bought this book despite the strong criticism it got from mathematicians who found pretty egregious mistakes in some of the math. But I'd never readI bought this book despite the strong criticism it got from mathematicians who found pretty egregious mistakes in some of the math. But I'd never read David Foster Wallace before (aside from some of his journalism) and I wanted to try him out.

I suspect the criticism is largely unwarranted - DFW provides enough forewarning that he has "dumbed down" much of the math in order to bridge the gap to the difficult and abstract math he is describing. Doing so comes with the sacrifice of some accuracy. Richard Feynman once explained that there is no real substitute to getting down and dirty in the math - no amount of summarizing and translation into layman's terms will ever do. So for those who want a complete and 100% correct understanding of these ideas, well, caveat emptor.

Then again, for those who want a complete understanding, none of these kind of books will really do.

DFW keeps a very conversational tone throughout the book - peppering words like "stuff" around concepts like Fourier series and uniform convergence, which helps keep your attention without blunting the fidelity (can you blunt fidelity???) of the explanation. I also really enjoyed his extensive foot-noting, which I understand turns a lot of people off. DFW defended his footnotes in Infinite Jest on Charlie Rose, and I think the defense works very well for this book also:

"There is a way, it seems to me, that reality is fractured right now (at least the reality that I live in) and the difficulty of writing about that reality is that text is very linear, and I am constantly on the lookout for ways to fracture the text that aren't totally disorienting." http://www.charlierose.com/view/inter...

The history of our grasp of infinity from a mathematical perspective isn't linear, so why should the telling of it be? DFW's constant use of IYI (his invented acronym for "if you're interested") allows for the reader to delve deeper into the history, if they're interested.

In this way, DFW writes Everything And More mimics the way that I amble wikipedia. Or perhaps I go too far, the asides are always brief and are seldom interconnected.

I enjoyed it, and I suspect that I'll pick it up again....more

Prime numbers are powerful things. If you multiply one or more primes together, you can create any other positive integer that's bigger than one. AndPrime numbers are powerful things. If you multiply one or more primes together, you can create any other positive integer that's bigger than one. And we suspect that every even positive integer greater than 2 is the sum of two primes.

But primes are strange as well - there doesn't appear to be any order to their appearance. The higher you count, the less often you run into them and you'll never stop seeing them. But can we tell when the next one will occur? In other words, is there some sort of pattern to their appearance? And could that pattern tell us something fundamental about numbers in general?

Bernhard Riemann made a suggestion about how prime numbers were distributed along the number line in 1859. This suggestion became known as the Riemann Hypothesis, and stated that the pattern of distribution of primes is connected to non-trivial zeros of the zeta function. What the hell does that mean? The majority of the book is spent answering that question by unpacking the zeta function.

We have a proof that describes how prime numbers are distributed - it's called the prime number theorem, and Riemann's work was crucial to its development. However, we haven't yet proven the Riemann Hypothesis, which is a much stronger statement than PNT.

Compared to other math books, Derbyshire does a great job of getting into the meaty details and walking through them in a clear manner. The tone was very conversational, doesn't require much linear algebra or calculus, and stuck to the topic at hand (he would often hint at enormous tangents while declaring that he will be avoiding them for our own sake).

I have one main concern, and is the reason why I didn't give this more stars. While I can appreciate the mystery of prime numbers, and I could follow all of the book's content(after much concentration on my part), I had trouble grasping the importance of the Riemann Hypothesis after Prime Number Theorem was proven. However, I suspect that at some point he did explain it, and I since forgot it. For its density, this is a pretty long book (about 400 pages, including the notes, which I read). I'll need to, at some point, return to the book to re-learn the significance of RH from the detailed explanation of its composition.

At the end of the day, RH doesn't excite me as much as the continuum hypothesis or the incompleteness theorems. I know full well this is because RH is far more obscure than, say, the continuum hypothesis or the incompleteness theorems in relation to my pedestrian ontological/epistemological interests. I guess that's why I'm not a mathematician.

If you're interested in this specific topic matter, pick it up. If you just have a pedestrian interest in mathematics like I do, this book is interesting, fun and clear, but there are probably more engaging topics out there than the Riemann Hypothesis....more

I think this was a pretty good pick for a first book about information science. Despite being rather long (430 pages), it was easy to read (i.e., notI think this was a pretty good pick for a first book about information science. Despite being rather long (430 pages), it was easy to read (i.e., not technical) and had lots of overlap with subjects I was already familiar with.

The majority of this story is told in the early to mid 20th century around Claude Shannon, with supporting roles from Alan Turing and Norbert Weiner. Gleick also diverges into how information theory relates to physics, and even better, microbiology, which I had gotten a taste of last year when I read Godel Escher Bach.

The best part for me was reading about how information as a concept was invented, and how Shannon revolutionized information by divorcing it from meaning. I came away wanting to understand more about how information relates to entropy and randomness.

I'm not sure how to feel about the end of the book, where Gleick attempts to discuss information and the internet. This field is changing very quickly today, and any attempt to discuss it in a book is hopeless. I make a point to avoid any books as a source of information regarding trends in tech (there are loads out there), because they age terribly. In the spectrum between journalism and history, modern books are always going to hold out longest when geared more towards the latter, because anything timely and relevant can just as easily be blogged about, where the ideas will also benefit from evolving through the Web 2.0 ecosystem.

For example, Gleick's focus on Wikipedia makes this feel like a magazine article from ancient history when people were still impressed by what Wikipedia has accomplished. Admittedly, I'm pegging ancient history at around 2005, but that feels right considering the rate at which our treatment of information on the web is evolving....more

This book is an exercise in self reference, and in doing so feels very much like what Godel invented and Turing used for their mathematics. Godel turnThis book is an exercise in self reference, and in doing so feels very much like what Godel invented and Turing used for their mathematics. Godel turned logical statements into numbers, performed his own kind of mathematics on those numbers, and in doing so was able to write mathematical statements that spoke about themselves. Likewise, Turing turned numbers into machines that computed other numbers inside themselves (and solved a deep puzzle posed by Hilbert in doing so). Calvino wrote a book that not only talks about itself, but uses itself to describe why it was made, how it could come into existence, and what good it serves.

Most reviews identify the convoluted nature of this "meta" book/story, so I wont waste bits. Every dozen or so pages, I have earmarked pages and notes in the margins: Calvino wrote something to be marveled over more than, say, a good yarn.

Read it, and pay attention to the sheer number of ways he identifies reading, books, and writing....more

The heart of the book lies in the "Less and Less and Less Wrong" chapter, which details Bayesian probability (and how it compareThoroughly enjoyed it.

The heart of the book lies in the "Less and Less and Less Wrong" chapter, which details Bayesian probability (and how it compares to Frequentist methods). Silver argues Bayesian thinking is most productive format we have for making predictions, and what explains we have been doing wrong in various fields (by pointing out fallacies in heuristics, non-Bayesian thinking, etc)

The Bayesian focus on asserting a prior probability, testing that against new events to produce a better (posterior) probability and then iterating reminds me very much of another book I had read earlier in the year - Eric Ries's Lean Startup. Ries's manifesto, which is very much in-vogue in the tech community, feels very Bayesian now: Place a small bet with a minimum viable product, test the results, then iterate. The cheaper and faster (i.e., more efficiently and frequently) a company can iterate their product, the more successful they will be.

I don't think Silver or Ries are really talking about anything new, but considering how popular both of these men are today, I wonder if this won't be philosophy that transforms many more disciplines over the next few years.

Two other unrelated thoughts on the book:

Looking at the frequently utilized end-notes, it was clear Nate Silver had a LOT of fun researching this book (he mentions having performed over 100 interviews, and there were a lot of really cool sounding scientific papers that he referenced).

The best explanation for Nate Silver's magic came from someone I heard describe him on Charlie Rose recently. Nate Silver, like Michael Lewis, recognizes the value of of narration and providing context and color for what are traditionally very dry or academic ideas. Silver isn't quite as good of a story teller as Lewis, but he is better at focusing on the concepts....more

"She could at this stage of things, recognize signals like that, as the epileptic is said to - an odor, color, pure piercing grace note announcing his

"She could at this stage of things, recognize signals like that, as the epileptic is said to - an odor, color, pure piercing grace note announcing his seizure. Afterward it is only this signal, really dross, this secular announcement, and never what is revealed during the attack, that he remembers. Oedipa wondered whether, at the end of this (if it were supposed to end), she too might not be left with only compiled memories of clues, announcements, intimations, but never the central truth itself, which must somehow each time be too bright for her memory to hold; which must always blaze out, destroying its own message irreversibly, leaving an overexposed blank when the ordinary would came back. In the space of a sip of dandelion wine it came to her that she would never know how many times such a seizure may already have visited, or how to grasp it should it visit again. Perhaps even in this last second - but there was no way to tell. She glanced down the corridor of Cohen's rooms in the rain and saw, for the very first time, how far it might be possible to get lost in this."

What an incredible little book. At one level, we the reader are "foiled" to never see the conspiracy that may or may not exist which is threaded through the plot exposed. In common culture, this is an annoying and disappointing result - I can remember my anger with how the X-Files ended in high school and my friends' trouble reaching satisfaction with the conclusion of LOST. But the trouble with mysteries is we love the ride so much more than the conclusion. There is neuro-research to support this. Our pleasure is derived from release of dopamine, which occurs during anticipation of a reward and before the reward itself.

Perhaps more importantly, this book is trying to teach us a lesson about the limits of what we can know. The end of the book in particular is rife with allusions to this, long beautiful passages like the one I started with, but my favorite may be a brief one relating to the protagonist's relationship with a former lover whose death triggered the plot and the great mystery:

"Though he had never talked business with her, she had known it to be a faction of him that couldn't come out even, would carry forever beyond any decimal place she might name; her love, such as it had been, remaining incommensurate with his need to possess, to alter the land, to bring new skylines, personal antagonisms, growth rates into being. "Keep it bouncing," he'd told her once, "that's all the secret, keep it bouncing."

The apocryphal story goes that when Hippasus proved the existance of irrational numbers using the hypotenuse of the unit square, his master Pythagoras and his fellow students threw him into the sea to drown. The Pythagorean cult dictated that the all numbers could be known as the ratio of intergers, and this proof would have ended them.

It gets worse, as far as irrational numbers go. Cantor proved that the infinite set of them is larger than even the infinite set of intergers. There are so many irrational numbers, that the universe could never find ways to represent all of them (although I suppose the same could be said about any infinite set).

But we function today in a world with irrational numbers, transfinite and infinite sets and infinitesimal distances, quantum uncertainty, problems that have such computational complexity that they can't ever be solved in the life of the universe. What's more, we harness the unknowable and use it to our advantage. Doing so has taken time, and we have a rich history of mathematicians and physicists starting perhaps with Newton/Lebinitz to thank.

I think Pynchon wanted his protagonist to discover this parallel shadow world of the unknowable and to grow from it. One of Oedipa's first signs of strength came when she consoled a strange old man who held for her a clue but more importantly needed emotional support. Obviously, I'm biased to this quote also because the calculus reference ties well to my thoughts above :-)

"She knew, because she had held him, that he suffered DT's. Behind the initials was a metaphor, a delirium tremens, a trembling unfurrowing of the mind's plowshare. The saint whose water can light lamps, the clairvoyant whose lapse in recall is the breath of God, the true paranoid for whom all is organized in spheres joyful or threatening about the central pulse of himself, the dreamer whose puns probe ancient fetid shafts and tunnels of truth all act in the same special relevance to the word, or whatever it is the world is there, buffering, to protect us from. The act of metaphor then was a thrust at truth and a lie, depending where you were: inside, safe, or outside, lost. Oedipa did not know where she was. Trembling, unfurrowed, she slipped sidewise, screeching back across grooves of years, to hear again the earnest high voice of her second or third collegiate love Ray Glozing bitching among "uhs" and the syncopated tonguing of a cavity, about his freshman calculus; "dt," God help this old tattooed man, meant also a time differential, a vanishingly small instant in which change had to be confronted at last for what it was, where it could no longer disguise itself as something innocuous like an average rate; where velocity dwelled in the projectile though the projectile be frozen in midflight, where death dwelled in the cell though the cell be looked in on at its most quick."

"Just look along the road, and tell me if you can see either of them" "I see nobody on the road," said Alice. "I only wish I had such eyes," the King re

"Just look along the road, and tell me if you can see either of them" "I see nobody on the road," said Alice. "I only wish I had such eyes," the King remarked in a fretful tone. "To be able to see Nobody! And at that distance too! Why it's as much as I can do to see real people, by this light!"

I had never read Alice in Wonderland or Through the Looking Glass, and bought the book almost out of a feeling of obligation. My favorite book, "Godel, Escher, Bach" uses characters from a Lewis Carroll story as its primary characters. Expecting only a simple children's story, I was surprised by how much I enjoyed both books. This book has doubled the length of the original text by annotating it with hundreds of footnotes, which were hit-or-misss in value for me.

Carroll has found so many creative ways to be silly that he really seems to have made a science out of it. In fact, the footnotes I valued most in this annotated edition were the ones that interpreted a passage through the lens of mathematical logic, recursion, lorentz contraction, information theory, etc. Some of these interpretations are a bit of a stretch (Carroll certainly didn't have information theory in mind when he was writing, as he predates it by over 100 years), but seeing that he was on the math faculty at Cambridge, these notes are still in the same spirit of his writing.

There are also some truly emotionally endearing passages - I loved the chapter about the White Night near the end of Looking Glass, as well as the closing pages in Alice in Wonderland....more

There is surprising little math and science for a book that is entirely about mathematicians and scientistI read about 3/4 of it, and just got bored.

There is surprising little math and science for a book that is entirely about mathematicians and scientists working at a research institution. And what little there is tends to mostly be brief references to mathematicians in history and scattered astronomy factoids.

The characters are all very wacky, but seem stilted after my reading the pantheon of wacky characters in Infinite Jest and Oblivion.

Maybe Delilo is doing something I didn't notice to capture the strangeness / exploratory feeling of culture among scientists when their research skirts the boundaries of scientific canon. ...more

I enjoyed Logicomix, but was disappointed in this book. I was hoping for a more creative/interesting way to blend mathematics into a narrative, but AI enjoyed Logicomix, but was disappointed in this book. I was hoping for a more creative/interesting way to blend mathematics into a narrative, but A Certain Ambiguity is a much better book for that....more