Almost all fields of pure math use a much stronger set of axioms called ZF and essentially everyone also accepts the axiom of choice (making it ZFC). The axioms in ZF are reasonable but the axiom of choice is surprisingly controversial for an axiom. There are some unintuitive consequences of the axiom but even more unintuitive consequences without it or with the negation of it.
Number theory uses a much smaller set of axioms.
It should be stated that most mathematicians don't really mind the logical foundations of their work when they are actually working in the same way that most programmers don't worry about assembly language or transistors.
I have no php experience so I have absolutely zero conception of what the language used to be like, just what it looks like now. I can't help but see it as some kind of frankenlanguage put together with pieces of java/python/javascript and a ton of dollar signs.
In other words, it seems too similar to languages that I already know to be worth learning. Lisp was fun to learn because it is an alien language. C was worth learning because of its usability and centrality to modern computing. The only benefit I can see from learning php is being able to understand more of software history. Cool but I'd rather learn another lisp.
Does anyone disagree with my analysis? I wish the language well but I don't see what it offers me. Obviously it offers a lot to the people who see JS as franken-php and so forth.
My belief is that people believe whatever they want to believe. If you are liberal it is because you want to be liberal. Alternatively you might just not want to be conservative. Same for the other direction. Naturally formed beliefs are just an afterthought. It goes without saying that this might not apply to everyone or to every belief.
For the record, I recognize that perhaps I believe this belief strictly because I want to believe it. I also think many people don't want to believe it.
I'm not sure it's that simple. As Mr Graham points out, we are shaped by our echo chambers, the chorus of voices that shout out against our fellow "heretics". As children, those chambers come from our families and religious / community institutions, as dictated by our elders. As we reach our rebellious phases, sure - we scrabble around and find beliefs we want to believe in - but the effort expended in doing so is difficult and the reward is little. From a social perspective, a little deviation acts as a differentiator - we are made interesting within our peer groups by our rebellions - but too much heresy makes us "other". As we get past that, self-realisation comes in waves; nestled between desires of belonging in a new group, we find moments where we are invited to re-evaluate some deeply held truth. Rarely is the re-evaluation self-driven, and almost never is that belief so different that a nearby echo chamber is not crouching nearby, waiting to nurture our newfound "chosen" belief in it's warm, corroborating belly.
For the record, I want to believe your belief. I recognise that this autonomous self-truthing could be the pinnicle of what we as humans could become. I also think the very concept of society becomes a meaningless nothing if it were possible.
FWIW, I think that a huge amount of political alignment in the U.S. derives from aesthetic sensibilities.
The thing last year where Trump ordered McDonald's for the college football team that was visiting him because the White House cook staff was not working due to the government shutdown is basically the perfect example of this. The left though it was the most gauche, trashy thing that ever happened, as in, "I can't believe someone would server toxic trash food to a guest in their own home." And right wing people mostly though it was totally normal, as in, "Of course you would order McDonald's if something came up and you couldn't serve a home cooked meal to a guest."
Feels like topsy-turvy land. The right, as the defenders of monarchy and aristocratic taste, should obviously be horrified that the King is eating fast food, and the left, as the voice of the people and universal equality, should be delighted that the Chairman is eating the food of the common folk.
Blindfold chess? Just search for it on youtube. Most GMs can do this easily. Some simultaneously play 10+ sighted players whilst blindfold. Even when not blindfolded they often look away from the board whilst calculating so that the current state of the board doesn't distract them.
It's not exactly blindfold. With blindfold, the blind player still knows the opponent's moves...They just don't have the luxury of being able to _look_ at the state of the board (although top-tier chess players don't need that luxury).
This is a little different, but I think/speculate they would still have enough heuristic knowledge to crush most people.
I'm glad the label worked for you but it did not work at all for me. It kind of shoehorned me in to a personality type and limited my social options.
In all honesty I look back at my "smartness" as a kind of attention-seeking behavior where all I really wanted was the attention you got from getting the highest grade in the class or being the only one from your school to go to the ivy league.
To me, the smart label just means you are willing to sacrifice more than others to study/learn/build. Everything I have learned about people since starting college has told me that this is a net negative.
Believe me, it’s much worse being considered not intelligent instead. I knew so many people growing up that were orders of magnitude more intelligent and accomplished than me (and I still do).
What is everyone’s favorite translation of Meditations? I bought a cheap copy and the translation is unsurprisingly hard to read and distracts from the text’s content.
Tangentially related: it's a pity there aren't more websites like this one: [0]
It's nice to compare different translations in-line. The last line of section 3 is an excellent example of how different translations can be: [1]
Enchiridion is well worth a read (I gather Carter has the best translation but I've gleaned this from places like Reddit so pinch of salt!), and I also enjoyed Derren Brown's Happy [0] which draws heavily on stoicism, summarising a lot of famous works on it as he goes.
1) The translation by Gregory Hays is the most accessible. Get the "Modern Library" hardcover edition. Has a nice introduction and notes.
2) The translation by Martin Hammond in "Penguin Classics" is also very good. Has detailed introduction and notes.
3) The translation with detailed notes by A.S.L.Farquharson is considered one of the definitive ones. Get the hardcover published by "Everyman's Library".
4) Finally there is a good translation by Robin Hard in "Oxford World's Classics". Robin Hard has also translated Epictetus "Discourses" and hence you may find his work unifying.
I did a pretty deep survey of the various translations (there are a lot) and landed on Maxwell Staniforth's as the best balance between readability and faithfulness to the original language. It's a fairly straight translation rendered in modern-enough English, unlike some of the older ones which add difficulty due to the age of their language, pointlessly, since it's a translation anyway and they're (obviously) so much more recent than the original that they are in no way contemporary so there's no claim to be made on that merit, as one might for, say, a 19th century translation of Jules Verne.
I think the only hardcover is from Folio Society, so medium-pricey as fine books go, but his translation's also the one used by at least some editions of the work from Penguin Classics, so, cheapish paperbacks. I'm having trouble verifying which editions of theirs are his translation, but it may be all of them.
Number theory uses a much smaller set of axioms.
It should be stated that most mathematicians don't really mind the logical foundations of their work when they are actually working in the same way that most programmers don't worry about assembly language or transistors.