I appreciated the article for emphasising memorising definitions and statement of theorems... But not for proofs. For proofs, a general outline would be sufficient.
For proofs, I find it a good idea to memorise (or at least implicitly retain) the reason a result is true. So, yes, an outline, but minus any of the implementation details of the proof. I kind of think every book in the definition-theorem-proof style should really be definition-theorem-reason-proof.
The reason part being essentially a one or two line natural language summary of ‘why the proof works’ — something that is almost always possible and is enlightening and conducive to efficient memorisation, but that for some reason is very rarely written down explicitly.
I think a better word is "motivation" -- why we chose this option at this juncture instead of many other options. Yes, it's a "reason", but "reason" already means something else.
The "Reason" as result is true is that it follows from the previously established axioms via logical reasoning.
Motivation is important too, but it’s not what I meant. A very simple example would be
Theorem: Every subspace Y of a second-countable topological space X is second-countable.
Reason: Intersecting each set in a basis for X with Y yields a basis for Y.
Proof: [formal symbolic stuff involving open sets and unions, and mentioning cardinality, etc.]
(I’m not claiming ‘reason’ is the best word for this — it probably isn’t. But it’s not the same thing as motivation.)
> The "Reason" as result is true is that it follows from the previously established axioms via logical reasoning.
One could argue this is not the reason a result is true; it’s the reason we know it’s true. The fact that true statements follow from established truths by logical reasoning is more a property of the formal system (which hopefully is sound and consistent) than it is to do with the notion of truth itself.
It depends if you want to be able to prove new things by yourself or not. If you want to do it, then you definitely need to understand /recall all of the whys of every section of the proof. They are all there for a reason. If you don't, you just want the intuition of why the whole theorem is true.
You should definitely memorize most of the "basic" (and short) proofs in some field you are super interested in. The intermediate and advanced proofs, only the outline is sufficient.
> For him, sitting down for twenty minutes is a much more consistent tool for maximizing energy - even compared to sleep or coffee.
Firstly, I'm glad JS (and OP) see very positive effects of meditation but I'm highly skeptical meditation is more powerful than coffee (assuming sleep has diminishing returns). I doubt this is generalisable to most of us.
> Even within my new sessions, I can already feel a difference: after ten minutes, I reach a flow state that just feels great.
> Whereas before I’d be prone to fall into vicious spirals of self-consciousness and unease, a streak of meditation would allow me to calmly, warmly, and directly engage with people. If an awkward moment arose, I wouldn’t feed my internal fire with negative self-talk, but rather look outwards with an internal smile, wait for the moment to pass, and find a clever prosocial solution. But again, that explanation understates the magic.
Secondly, again, I'm happy OP found the answer to the greatest wall for new meditation practitioners: "how do I know if meditation is working???". I haven't found it. Brain states stuffs are more art than science, I believe.
Not a guitarist (take my comments with a pinch of salt).
Your teachers are both right: Either advices (1) or (2) works as long as the practice is hard.
That said, while (2) may sound easier because the approach pursues "comfort in the mind" over perfection... this is still hard because because by definition: you still need to get from uncomfortable to comfortable!
A similar example in bodybuilding: muscle confusion [1]. To build better squats, one requires both compound(2) and isolation (1) exercises
I have mixed feelings. I loathe instapoetry as much as any modern poetry reader... However, I also hate that every modern poetry demands readers to read "good" poetry first to understand why instapoetry is bad (spoiler: instapoetry lacks depth).
What's the alternative of long-term bond investments? Republican-style tax cuts? If I understand the article correctly, Carmel's Republican mayor Jim Brainard broke off from his party's motto.
> The prevailing model of austerity, low taxes, and limited government in many Republican-led states and cities has often failed to improve the quality of life. Meanwhile, Democratic-led cities struggle to effectively address issues like homelessness, housing affordability, and public safety despite greater willingness to invest, just not in actual services and infrastructure.
No, there's nothing wrong with bonds, if the thing they get invested into has good positive ROI.
The elephant hiding in the rural balance sheets is boring economics, meaning that there's usually no such high-ROI thing within city limits, because to get good and affordable services the city needs economies of scale, and that requires both sizes and density.
This basically translates to things that these cities could do: rezoning and waiting, joining up with neighboring cities for deals (but this is usually already the case, school districts and water districts).
Homelessness and public safety is worse in urban cores, and this would require a lot more spending to fix. [0]
(These cores are usually poorer, have funding issues, crowding out effect of rich suburbs means that the police force of the urban core needs to sort of match the salaries, etc. [1])
Also ideological differences lead to worse outcomes in public safety definitely. (Again, especially in these metro areas where - as other comments have mentioned - it's relatively easy to push poverty and crime around.)
It's not a uniquely American problem, but the combination of wealth and income inequality, extremely fragmented hyperlocal institutions, gang and gun violence, plus senate veto make progress extremely hard. And in practice it serves as the perfect fuel for political radicalization.
As a long-time Anki user (and memory enthusiast), I would like to offer some caveats for those transferring actors' memorization techniques to study techniques (programming languages, chemistry, math etc...)
> This deep understanding of a script is achieved by actors asking goal-directed questions, such as “Am I angry with her when I say this?”
1. Do embed emotions. For example: Go's CamelCase for exporting variables implies capital letters want to be loud.
2. Do invoke experience. Don't merely read and write code, teach it and activate multiple sensory inputs and multiple contexts!
> Deep, elaborative processing enhances understanding by relating something you are trying to learn to things you already known
3. Do chunk it. Instead of creating new standalone memories, chunk it with other programming languages' syntax.
In summary, while engaging with the material is important, it is not sufficient for passive tasks (how many times have you read a book and forgotten the main points?).
> There are doubts that industrialization can create the game-changing benefits it did in the past. Factories today tend to rely more on automated technology and less on cheapworkers who have little training.
> Multinationals like Goldman Sachs, Victoria’s Secret and the Economist magazine have flocked to the city and set up hundreds of operational hubs — known as global capability centers — to handle accounting, design products, develop cybersecurity systems and artificial intelligence, and more.
What I wanted (from Freakonomics) was to peer through "the hidden side of everything"... What I needed was the serenity to accept that causality is too damn hard.
I appreciated the attempts by Steven Levitt and Stephen Dubner to communicate with layman non-economists like me. I think The Economist article was, as always, too harsh and nitpicky. I think Freakonomics holds a special place in the "intellectual" and "rationalists" community. I cannot verify the flaws in techniques and conclusion as stated in this article but I would still recommend reading Freakonomics first over the dry economics textbooks or MIT OCW courseware.
The trouble with doing that for Freakonomics is that the work on abortions reducing crime, which has been proven wrong, is the first chapter and the centerpiece of the book. It's the thing that they use to exemplify the "freakonomics" approach in the rest of the book.
do you have links to the papers disproving that? That result is kind of the basis for the moral argument for allowing abortion for consensual fetuses for me. Without that result the cost/benefit looks terrible because the deflationary death spiral of the below replacement birthrate it helps cause is really, really bad.
