This book is the reason I was able to pass my computer architecture exams in grad school. Just lugged the 800 page monstrosity around an entire semester and read it every chance I could. A really wonderful book ... I should get a recent copy and re-read it again.
The continuum hypothesis is more like Euclid's parallel postulate than a Gödel sentence - assuming ZFC consistent there are models with CH true and CH false (the cardinality of the continuum doesn't have to be the first uncountable cardinal).
Everything gets qualified with "assuming ZFC consistent" or "assuming Peano consistent" because any inconsistent theory proves any statement. More of a proof technicality than anything too profound.
As someone who took a number of mathetical logic classes between undergrad/graduate I always found this Gödel-sentence argument on the face-laughable, since the logical inconsistency was so easy to repair. My favorite critique of this was Hans Moravec's dialogue between Penrose and a robot AI, having resurrected his brain long past humanity's extinction: http://www.calculemus.org/MathUniversalis/NS/10/10moravec.ht...
Not a historian, but _Why the West Rules - For Now_ by Ian Morris was a good read. Covers the sweep of world history with the audacious goal of quantifying human development at all moments in history. It mainly focuses on the west (starting in the Mesopotamian region) and the east (starting in the area surrounding the Yellow river) and addresses questions like "why didn't China discover America" (basically - too far away). You might also find it interesting!
I made what I think is a good faith effort to review the link you provided, but I don't see any indication of factual errors Howard Zinn makes. It appears the criticism of him is that he is biased, he makes heavy use of secondary sources (a quantifiable claim which as best as I can tell is false given that Zinn's sources are enumerated and mostly primary) and engages in something called "presentism", which is to make moral judgements of historical acts based on modern moral standards, a claim I don't have any particular issue with but even if I did it has no bearing on the factual content of his work.
At no point have I seen anyone point out a factual error that Zinn has made, only that people don't like the conclusions he draws from those facts. Another commonly repeated point is that there are other historical works that do a better job than A People's History. That is almost certainly true as well, but that claim is no more relevant than pointing out that there are better scientific works that do a better job than Einstein's "Relativity: The Special and General Theory".
Having read A People's History, the very beginning of the book explicitly states that he is biased, that bias is something very hard to escape in almost any social study and is present among almost all historical literature whether the author admits it or not, and that the difference between his book and other historical works is that he makes the conscious decision to be biased from the point of view of the people who were conquered as opposed to writing from the point of view of the people who did the conquering.
Certainly Zinn has likely made some errors, but none of them are in the link you provided (as far as I could tell).
This is also very similar to the criticism Noam Chomsky have received. He has been very accurate for decades but is still handled as an outcast.
Come to think about of it actually, it's also similar with the criticism of David Graeber's (RIP) book "Debt", which usually focus on "well, he wrote something really wrong paragraph about Apple, so you can't believe anything he writes really"
This is not the limit of criticism of history scholarship. Braudel's Civilization and Capitalism isn't considered dated because of factual errors but instead because it over-represents a particular narrative and analysis style that fails to paint a detailed picture of things.
History scholarship is absolutely not a list of facts. It is the construction of a narrative from source material. A text that is 100% truthful to fact can still be entirely bogus scholarship (I'm not saying that Zinn's is).
Sure it is a relevant book, especially in leftist history work. The problem scholars have with it is that it is held up as the text by some laypeople. It is not unique in having narrative issues. That's true of all history. It is just frustrating to see people say "well, he didn't get any facts wrong so all criticism is invalid".
No, I would never say all criticism is invalid. I think it's a relevant and important text to read, along with criticism of it. Nothing should be taken in isolation, scholarship is ultimately a conversation at a high level between knowledgeable parties, and it's a pretty dated book in a lot of ways. Nonetheless it was extremely influential and solid as a basic leftist history text, and he even acknowledges its bias in the preface and title as I've mentioned. I don't by half think it's a perfect book and I have a lot of criticisms of Zinn.
I ask for specifics because I know all the general arguments around it. I asked about what was specifically untrue, I have not read solid criticisms of it that cast serious doubt on its factual claims, only that it shows history from a certain perspective, which (to be fair to Zinn here) is evident in the title.
Right, so go check the thread that you were linked to. It's clearly going to be a more holistic treatment than you're going to get in this thread on hacker news, a site for programmers and software entrepreneurs, not historians.
I'm really excited for the movie and the casting looks incredible so far. I wonder how the 2 movie split is going to work narratively. The book itself has kind of a perfect 3 act setup but I'm not sure breaking the story at the end of the book's part 1 (or halfway through part 2) is going to feel very satisfying.
> The book itself has kind of a perfect 3 act setup
Standard 3-act structure has a natural breakpoint at the midpoint twist of Act 2, not that my recollection of Dune is detailed enough to confirm that it maps naturally to 3-act structure in detail.
I think you could stop as a cliffhanger when Paul and Jessica escape from the worm, and are confronted by Stilgar. Then you start with a knife battle for the second movie, and who doesn't like a knife battle?
No, that would be like moving Luke vs Darth Vader fight to the beginning of episode VI.
This actually reminded me of a colossal mistake made by Hobbit pt.2 movie, when they ended the part with a dragon flying away to burn the city after the cringiest of chase scenes.
They could have easily end the part with slaying a dragon, and then show it again in part 3. All they needed was to find a different POV character!! One dragon kill, shown twice.
I'd guess comments like this come from people thinking the primary complexity of Redux is the data store, rather than the "all changes to your state are represented as plain objects". The Redux FAQ goes into this: https://redux.js.org/faq/general#when-should-i-use-redux
Can't blame them missing the forest for the trees when you spend all your time writing the reducers and connectors vs the actions themselves.
Note that our new official Redux Toolkit package [0] simplifies most Redux use cases and logic, including eliminating the need to write action creators and action types by hand and allowing "mutating" immutable update logic in reducers. The React-Redux hooks API [1] is also generally simpler to use than `connect`. We're now recommending that people use RTK as the default approach for Redux apps [2], and I'm working on a new set of tutorials that will teach RTK and React-Redux hooks as the standard approach [3].
I've been pursuing mathematics as a hobby for the last 2 years or so. I got a mathematics major in undergrad so my motivating factor was mainly to explore some areas that I hadn't done coursework on, primarily algebra and number theory. (I focused more on logic in undergraduate/grad.)
I really enjoy how the subject is divorced from a lot of the modern attention demands and encourages more of a 'zen' thinking style.
As others have highlighted, it can be difficult. I work full-time as a software engineer and at the end of the day there's usually not much left in the tank in terms of "creative work". The morning is usually more productive for me - generally I'll spend 10-15 minutes on the commute in reading over the proof of some lemma or working through some computational exercise.
Things that have helped me:
- Focusing on a particular problem area rather than just "mathematics". The classical problems of Gauss and Euler tend to be more my speed than the modern mathematical problems of Hilbert or beyond. What started my journey was looking into the insolubility of the general quintic polynomial equation, something you learn in high school as a random factoid but has a lot of depth.
- Studying from small textbooks that I can fit in a backpack, so I can "make progress" during my commute. Dummit + Foote might be a great algebra reference but it's just too bulky to transport.
- Limiting the scope of how I think about the activity - my goal isn't to master these concepts on the level of a mathematics graduate student, it's more on the order of Sudoku. If I don't get something, that's okay. People spend their whole lifetimes learning this material and I'm just trying to fit this into whatever creative time I have left after the full-time job is done.
Even if the number field is finite, there are an infinite number of polynomials for it. For example the finite field F2 has two elements {0, 1} but an infinite number of polynomials 0, 1, x, x + 1, x^2, x^2 + x + 1, x^2 + 1, «x^2 + x, and on and on. Some of these have nontrivial factors (x^2 + x = x(x+1)) and some don't (x^2 + x + 1 has no factors beyond itself and 1).
(The other answer to your question is more complete but also a bit more advanced, figured this was worth surfacing.)
This is not correct, there are a finite number of polynomials for any finite field. This is using the usual definition for polynomial equivalence, where two polynomials are equal if they take the same values over all points in the domain.
You are talking about the polynomial ring, which is not a field.
The finite field F2 has four polynomials: 0, 1, x, and x+1. Other polynomials do not exist, because x^2=x.
So there are no (distinct) constants other than 0 or 1 and no multiples or exponents of x other than 1.
These facts might be obvious to someone who understands the jargon and theory of the mathematics in question, but probably need a bit more clarity when the target is the general populace.
Generally given a polynomial in R[x], you want to be able to evaluate it not just when x is in R, but when it's in any R-algebra. So we wouldn't automatically identify x^2 and x.
