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4.15 AVERAGE
4.15 AVERAGE
I read Godel Escher and Bach long before I studied any set theory. This was a great read to revisit both.
Some of the most revolutionary ideas are deceptively simple. Not that this was easy to read or comprehend, but Gödel's starting point is a relatively trivial idea which he takes to startling conclusions. Much of what you will hear about the incompleteness proof is likely false. This book handily explains the context, general proof, and what it shouldn't be interpreted as. It is extremely profound and useful. Anybody interested in math, logic, or philosophy should read this. I'd say that it is essential.
Generally well-written but not sure who the audience is; offers some mathematical depth but not enough
A short refresher of some of my mathematical logic courses, one of which focused specifically on incompleteness. Conversational for the most part. Should be approachable by most (maybe with some work), except for perhaps some of the finer details if it's a first exposure to formal languages or formal logic.
What caught me most, however, and which is certainly a tangent from the topic of the book, was in the forward where Hofstadter very clearly has connections and conversations with world class intellectuals a young man as largely a result of what appears to just be class privilege and familial acquaintance. Vacations and whatnot all described with a carelessness and apparent lack of awareness of the unusualness of such access. The inclusion of some of these autobiographical remarks just set a bad taste in my mouth to start. Doesn't affect the text, it just made me aware of how the production of intellectuals is still largely informed by class dynamics. An appropriate reflection as one reads about a powerful result in abstract logic.
What caught me most, however, and which is certainly a tangent from the topic of the book, was in the forward where Hofstadter very clearly has connections and conversations with world class intellectuals a young man as largely a result of what appears to just be class privilege and familial acquaintance. Vacations and whatnot all described with a carelessness and apparent lack of awareness of the unusualness of such access. The inclusion of some of these autobiographical remarks just set a bad taste in my mouth to start. Doesn't affect the text, it just made me aware of how the production of intellectuals is still largely informed by class dynamics. An appropriate reflection as one reads about a powerful result in abstract logic.
Like most of the folks who pick this up, I'm not a professional mathematician. Nor have I, up until quite recently, begun to appreciate the allure of pure logic. A common recommendation for newcomers is [b:Logicomix: An Epic Search for Truth|6493321|Logicomix An Epic Search for Truth|Apostolos Doxiadis|https://i.gr-assets.com/images/S/compressed.photo.goodreads.com/books/1312031956l/6493321._SX50_.jpg|6684749]. It was brought to my attention by Stanford's logic class on Coursera and, I'm inclined to say that it is probably the best place to start.
Ok, now on to Nagel and Newman...
Even if everything remotely mathematical flies right over your head, which it probably won't until the second section, you will come out of your experience richer. If you reread the second section, assuming you don't let the unfamiliar symbols intimidate you, all of it should stick. Gödel's argument is subtle, and its implications are hard to extrapolate from which may be why, in contrast to the discoveries of twentieth century physics, incompleteness has attracted comparatively few cranks.
I'd describe it as mind candy and as something more nutritious. Some of the statements tickle our brains, those about things being true but potentially unprovable. I'd call this candy. Substance can be added when we start to reflect on what this means, if anything, for our areas of endeavor and expertise.
Ok, now on to Nagel and Newman...
Even if everything remotely mathematical flies right over your head, which it probably won't until the second section, you will come out of your experience richer. If you reread the second section, assuming you don't let the unfamiliar symbols intimidate you, all of it should stick. Gödel's argument is subtle, and its implications are hard to extrapolate from which may be why, in contrast to the discoveries of twentieth century physics, incompleteness has attracted comparatively few cranks.
I'd describe it as mind candy and as something more nutritious. Some of the statements tickle our brains, those about things being true but potentially unprovable. I'd call this candy. Substance can be added when we start to reflect on what this means, if anything, for our areas of endeavor and expertise.
challenging
informative
reflective
slow-paced
I was first introduced to Gödel’s theorem when I was in undergrad and I was always fascinated by their conclusions. However, I had never read the original paper, nor I am in expert in mathematical logic. I cannot say how faithful and correct the exposition is when it comes to the finer details, but the purpose is to only give an sketch of the history of the proof and of the proof itself, emphasizing the ideas rather than the details.
The book is short, but it packs a lot, and both Nagel and Newman does marvelous job of making the concepts understandable and moreover, easily digestible, by which I mean not only are the ideas logically follow line-by-line, so to speak, but that we can extract deep thoughts and intuitions from them that remain with us beyond the text.
I appreciate that the book also offers the historical and mathematical context to understand the ideas that Gödel used to prove his theorems, including the mentions of the work and concerns of other actors in this play. It is always interesting to see how the fields of study develop as a human activity.
Quoting the authors, Gödel’s work “is an occasion, not for dejection, but for a renewed appreciation of the powers of creative reason,” and this book is a short and satisfactory account of how Gödel was able to achieve his proof.
The book is short, but it packs a lot, and both Nagel and Newman does marvelous job of making the concepts understandable and moreover, easily digestible, by which I mean not only are the ideas logically follow line-by-line, so to speak, but that we can extract deep thoughts and intuitions from them that remain with us beyond the text.
I appreciate that the book also offers the historical and mathematical context to understand the ideas that Gödel used to prove his theorems, including the mentions of the work and concerns of other actors in this play. It is always interesting to see how the fields of study develop as a human activity.
Quoting the authors, Gödel’s work “is an occasion, not for dejection, but for a renewed appreciation of the powers of creative reason,” and this book is a short and satisfactory account of how Gödel was able to achieve his proof.
It's funny a coworker asked me if I liked the book and I went I guess. It's hard for me to talk about a Math Philosophy book as like a must read. It's a mind fuck of a book that helped with my understanding of GEB. So much so I want to reread GEB for the 3rd time
Clear and concise explanation of the two incompleteness theorems.
Dear lord this was a difficult read. I've done Physics at university, but pure mathematics of this nature was never my forte, so despite the books brevity I found myself having to reread quite a few sections. However, this may be owed to the fact I only managed to find time to read it on the train to work. The train journey is only ten minutes.
Definitely worth a read though, I will hopefully read it again and this time with more time and attention.
Definitely worth a read though, I will hopefully read it again and this time with more time and attention.