Math professor here. If you want to learn math, then for the most part I recommend choosing time-tested avenues, using popular materials.
There are two reasons for this:
(1) Popular materials are usually popular for a reason: they reflect an approximate consensus, across a significant fraction of the mathematical community, that their approaches are more-or-less the best.
(2) If you learn the same way everyone else does, you'll have an easier time talking to others and finding materials on the internet.
I know some very innovative books which I highly recommend, for example Visual Group Theory by Nathan Carter:
But the innovation is pedagogical, in what Carter chooses to emphasize and how he presents everything. At the book's core, Carter agrees with everyone else about what the foundations of group theory are and should be.
Even Sheldon Axler's Linear Algebra Done Right (another excellent book), with its hilariously provocative title, only differs in its choice of emphasis and order of presentation. His choices are quite compatible with everyone else's.
I'll be trying to solve some problem, half-remember an email conversation from several years ago on something relevant, and want to look it up.
This feels like the most natural thing in the world to me, and it's not like the ability to save emails is new. Why, exactly, would a forced change of habits be for my own good?
>Why, exactly, would a forced change of habits be for my own good?
At the personal level, it wouldn't be. It makes a lot of sense, and I do the same with Fastmail.
At the corp level where it's often in M365 cloud, you've got hard limits from Microsoft on one hand (100GB primary mailbox - period), and corporate data retention limits on the other. Legal often has strong opinions on how long you are allowed to retain emails which you may or may not be able to personally override. Could be just a few years, which forces a different strategy.
I'm not sure on the details of Google, but one imagines corp workspaces have equivalent interests.
I don't recall any kind of retention limits at Microsoft, at least not for engineers. My mail archives went all the way back to my hire date even 15 years later.
When I worked at Capital One there was a policy of automatically deleting everything that had not been subpoenaed as soon as it was legal to delete it. Usually 3 years or so. Retaining longer was viewed as creating legal risk for future lawsuits. They didn't want to leave evidence lying around if they could help it.
My company moved to a 3 year retention for legal purposes a few years ago. Somewhat annoying from a nostalgia point of view when I’d get mails pop up from 2095, but everything I need has been in jiras for the last 10 years.
No, not conversations, actual data. Think reports, invoices, large PDFs, etc. Emailing files to yourself, that sort of thing. Then they end up with multiple PSTs.
Some drivers seem to resent the idea that they should have to share the road, or slow down for anyone. Even if cyclists do everything right, they're still slower than cars, and so will present at least a minor inconvenience for drivers.
In Canada the fight has gotten nasty, with governments in Alberta and Ontario putting forward legislation that could remove existing bike lanes.
While I think cycling is great - environmentally, for health, apparently for mental health - bikes and cars don't mix unless they are going approx. the same speed in 1-2 lanes.
Driving a car, bicycles are hard to see - I wouldn't be surprised if visibility in cars is specified to be sufficient to see other cars. Bicycles appear out of nowhere and disappear. Also, cyclists - no better or worse than their automobile counterparts - don't always drive well, and they do things that cars don't such as weaving through small spaces between cars; running lights as if they are pedestrians, but on the road; appearing from sidewalks and other places - really anyplace. I don't object to creative driving - as I said, (city) drivers aren't much different in their way - but it makes bikes unpredictable and hard to see. Then there's the speed difference - bikes much slower than traffic are as dangerous as cars driving that speed (again, except I can see the cars). As long as there's one lane - and if cyclists 'own the lane' and don't let cars squeeze by - it's safe: you can see the bike; multiple lanes and the bike ends up in blind spots, weaving back and forth itself, etc.
I read that in (Belgium? The Netherlands?) the law is that if there is a small (10 km/h?) difference in speed between cars and bikes, they cannot share the road.
Cyclists never do everything right, though. Contested stop signs are a prime example. For every cyclist who stops properly, 99 blaze through with attitude. They are lawless, and cause safety issues for drivers who have to deal with it.
You’ll also see them run red lights, cut off pedestrians, bike right into oncoming traffic (in the same lane, no less), cut across three lanes of without blinking. All in the name of laziness, not safety.
Bikes are different machines with different capabilities and parameters. That they aren't used like cars isn't laziness or even lack of personal safety, but maybe lack of discipline to operate as if it has the capabilities and parameters of a car.
Whatever the motive, it's still dangerous because everything on the road needs to operate in an integrated system of rules. Bikes acting like bikes are unpredictable and using different rules.
But consider the functional differences:
> Contested stop signs are a prime example. For every cyclist who stops properly, 99 blaze through with attitude.
Bicycles both stop much more quickly than cars and take more effort to restart. Restarting from a stop and accelerating to full speed takes energy and wears on tired muscles - and it's not just one intersection but 100 in one ride.
So many times I've seen bikes approach the intersection at moderate speed. That's dangerous in a car - you might need to stop short, you might hit someone or something with your 2,000 lbs metal object which could cause serious harm even at slow speeds. On a bike it's fine - you can easily stop your 200 lbs object, which is also much smaller and more maneuverable and thus avoids collisions easily, and which does little harm at slow speeds.
So the bike does the bike thing, but the car sees a car thing: The car see the bike moving at a normal rate, and assumes it will act like a car and drive right into the intersection. The car stops and lets the bike go first.
> run red lights
At lights, bikes are like (very fast) pedestrians. On foot, at least in the US and many parts of the world, if the road is clear people don't wait for the light, they just cross. Functionally, there's no reason for bikes to do differently. That's dangerous to do in a car because their size and lack of maneuverability makes them big targets and makes accidents hard to avoid, and because they cause serious harm even at slow speeds.
> cut across three lanes of without blinking
Again, bikes are much smaller (able to fit in small spaces) and much moremanueverable. It makes some sense for a cyclist; it would be far more dangerous in a car.
I do understand it takes effort to stop and re-accelerate, but isn't making effort (exercise) part of the purpose of cycling in the first place? My main objection is to cyclist failing to stop at pedestrian crossing lights: as a pedestrian, when I see a green light, I expect to cross without having to dodge moving vehicles on the road. I do check the the road to make sure that cars have stopped, but cyclists (who rarely stop at crossings, at least in London) are harder to see, as they are smaller and often obscured by the traffic.
They really don't. Even if you slam brakes and OTB on a bike you will still fly further ahead than a car doing double your speed will travel after applying brakes normally.
This is the insanity of running stop signs on a bike - you can't stop, you cannot swerve nearly as quickly as a car and you will take much more damage when T-boned than a car driver would yet you believe it's safer because:
I think that's just wrong. Bikes stop on a dime (unless going high road-bike speeds), and are much more maneuverable due to their mass and two wheels, and are effectively much more maneuverable because their dimensions make it much easier to avoid objects and fit into the many more spaces than cars can fit in.
In my experience as someone who rides a bike with hydraulic disc brakes and drives frequently, they really don't stop very quickly. I pretty much always am more confident I can stop my car faster. The bike may have a lot less mass but it also has a lot less traction and it's much easier to lock up a wheel and skid instead of stopping quickly, especially in poor conditions/on road paint. I do cycle fairly fast, but this is on a mountain bike on the road, certainly not fast compared to a fit person on a road bike.
I will give you that they're much more maneuverabile and accidents can often be avoided by putting the bike into a space in the road that a car couldn't go into.
Maneuverability of a bike suffers from the same traction issue. At the speed when you can't plant your foot to pivot around bikes need to go much slower than cars in the same corners to not skid out. Check out descends on any road race, bikes are braking to 20-30 mph in sone corners, which cars pass at full speed. Not to mention that a car can skid fairly easily on either axle, while on a bike very few can get out of the rear wheel skid and the front wheel skid is the game over.
> The bike may have a lot less mass but it also has a lot less traction
I've long wondered about that tradeoff: I've heard some (non-physics-aware) people say that heavier cars/trucks stop faster because they have more traction, but obviously there's a big tradeoff in that equation. As a guess, on a moving wheeled vehicle, 1 kg adds forward energy proportional to velocity, and downward energy constant and independent of velocity? And maximizing that tradeoff would seem to be an engineering goal in order to minimize muscle/gas expenditure. (I'm sure it's well-known but I'm too lazy to look up the details.)
That ignores road/tire and air resistance which are proportional to velocity. And I suppose a key question is the degree to which traction depends on vehicle weight: maybe it depends more on tire and road characteristics, for example. Certainly ice or some kind of low-grip tire would be a big factor.
And that ignores locking up the wheel, as you pointed out.
You could just get on a bike and test for yourself your theory. People who ride bikes tell you they are not good at stopping quickly, perhaps you should ask yourself how you became so knowledgeable without riding?
When I go to research lectures, I sometimes hear that in response to audience questions, although not especially consistently. Some speakers do this more than others, I don't think anyone does it all the time.
It was so long ago that the specifics have faded, but I remember I was coached to use a variety of positive responses. "That's a great question," yes, but also things like "I'm glad you brought that up," and "I was hoping someone would ask about that!" It wasn't my cup of tea, too artificial, but the advice was contemplated.
The next question (which is a great one, from what I understand) is: Why do LLMs use these phrases so much if humans rarely use them in written form? Maybe a fair portion of training data comes from lecture transcripts, where such responses are common when responding to direct questions? And/or system prompts are just instructed to be like that?
> Why do LLMs use these phrases so much if humans rarely use them in written form?
As far as I understand, it's due to RLHF. The reviewers the AI companies use don't necessarily know what kind of question is a good one, so when the LLM answers "That's a good question!", they tend to rate the answer higher because they like being flattered. Proxy models that are themselves trained on RLHF inherit this pattern. Similar effects contribute to sycophancy.[1]
> I’m skeptical that there’s anything special about high trust communities other than a higher baseline of morale
Strictly speaking I'd agree with you -- but I would consider a higher baseline of morale to be itself quite special! Especially when it is shared amongst the entire community.
I went to Rice which had a similarly strong honor code, and it absolutely inspired pride. In me, and from what I could tell in many of my classmates.
Is it propaganda? In some sense, yes, the only way to maintain such a culture is to repeatedly insist on its importance to prospective and current students. But if so, then it is self-fulfilling propaganda, and in my opinion the honor code made my experience richer.
This mirrors my experience at Stevens. The professor would not babysit us during exams and that really did inspire pride. Also the exams were often brutally hard which inspired despair.
This is a great personal experience to share on HN. It makes me wonder: What makes an honor code work (or not)? In your example (Rice), what did the university do to promote the honor code? And why was it so culturally impactful upon you?
I will never forget being in high school and seeing so many classmates cheat on homework and take-home exams, yet raised their hands with ease to give the honor code pledge. It was a farce. Please don't read my personal anecdote as doubt that honor codes can work.
These are very good questions, but it was a long time ago and I'm afraid I don't remember well enough to answer.
I'm not even sure I could have answered you at the time. In my memory, it was mostly students promoting the honor code. But I have to imagine that the university was quietly doing things to keep this going.
I live in South Carolina, about half an hour from Congaree National Park.
It's not a place you'd drive across the country to see, but it's beautiful and highly worth a visit. There are some freaky trees -- with knobs of roots that stick out from the ground, like nothing else I've seen. The place is kinda creepy, as if you're about to be attacked by zombies, and I love that.
There is a two mile elevated boardwalk which takes you around some of the most scenic areas of the park, and further hiking trails which branch off of that. There is also a river that goes through the park, and it's quite atmospheric in a canoe.
I knew this guy was not worth listening to when he wrote "a solid five hours away from absolutely anything else you might want to see or do". I live in Charleston, two hours away and at the heart of the Lowcountry, one of the most historically fascinating and beautiful places in the world. And a couple of hours in the other direction are the Blue Ridge Mountains!
Charleston is an awesome city. Loved going there for work. I highly recommend Big Bad Breakfast. I believe it started there. They have this croissant-based french toast thing that is absolutely out of this world. It's 30% butter, but 130% worth it.
I’m from the Lowcountry, live in the Upstate now. Kayaking through Congaree is one of my top memories as a Boy Scout. Incredibly beautiful in a way you don’t get most anywhere else. Feels like you’ve travelled back in time to some Jurassic age going through the tree covered channels. Just keep an eye out for the snakes!
When I started my Ph.D. program, there was a weekly seminar that I started going to. At first, I instinctively left each week immediately after the talk was over. But I noticed that a lot of people would hang around afterwards and chat. Even though I found it a bit awkward, I started following their lead -- a habit I'm very glad I developed!
Personally, I've enjoyed classes in yoga and in group fitness.
This has several advantages: exercise is obviously good for you, and you can meet people while you're there. But more subtly, once you show up, someone is constantly telling you what to do. For me, this has meant much less of a drain on my mental energy and discipline than if I tried to work out on my own.
There are two reasons for this:
(1) Popular materials are usually popular for a reason: they reflect an approximate consensus, across a significant fraction of the mathematical community, that their approaches are more-or-less the best.
(2) If you learn the same way everyone else does, you'll have an easier time talking to others and finding materials on the internet.
I know some very innovative books which I highly recommend, for example Visual Group Theory by Nathan Carter:
https://bookstore.ams.org/clrm-32/
But the innovation is pedagogical, in what Carter chooses to emphasize and how he presents everything. At the book's core, Carter agrees with everyone else about what the foundations of group theory are and should be.
Even Sheldon Axler's Linear Algebra Done Right (another excellent book), with its hilariously provocative title, only differs in its choice of emphasis and order of presentation. His choices are quite compatible with everyone else's.
https://linear.axler.net/LADR4e.pdf
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