The intuition behind the definition there (it's a definition of conditional probability; the only assumption is that this correctly captures the informal idea of conditional probability) is: P(B) tells you how many worlds/how much probability mass is in the blob where B is true, measured so that if the blob contains everything then it has size 1. P(A|B) means you take the weight of A but constrain yourself to only look at worlds where B is true; so you look at the part of A that intersects B and measure it relative to B so that if it had B's size it would be 1. This is why you take P(A&B)/P(B).
Here's a table representing what's going on:
A -A
B w x
-B y z
where w = A & B, x = -A & B, etc.
Note that for example that probability of B = entire B row = w + x.
Now A|B means we keep only the B row and see what the chance of A is relative to it. This is why it's (A&B) / B - it's the upper left square over the upper row, because when we keep only the upper row we now need to compare squares to the new total of B. Likewise B|A means we keep only the A column and look at the relative change of B, so we get (A&B) / A
I find it easier to think in terms of odds, which are when you write only the relative probabilities (e.g. 1:2 odds and 2:4 odds both mean the same probabilities of 1/(1+2) = 1/3 vs 2/3). Here's a description of the intuition behind the proof, written in odds form:
Wordy description of odds form proof
The prior odds is the ratio of the A column to the -A column. The likelihood ratio is (B|A) : (B|-A), that is, each part is the "vertical factor" of the top row of a column compared to a whole column.
The first part of the likelihood ratio tells us how many B worlds each A world produces, while the second part tells us how many B worlds each -A world produces.
To get the total produced B worlds we multiply the prior (how many worlds we start out with) with the likelihood ratio (how many B worlds each one produces). Since we observed B, the posterior odds of A vs -A is the ratio between the "sources" of the B worlds, and so the product is our final odds.
Formally, we have
A : -A * (B | A) : (B | -A) = (A & B) : (-A & B)
and likewise
(A & B) : (-A & B) = B : B * (A | B) : (-A | B)
but since B : B = 1 : 1, this is just the desired (A | B) : (-A | B).
Example: Suppose that for every person with cancer there are 100 people with it. Then our prior odds are 1:100. If you get a cancer screening that will detect cancer 100% of the time if it's actually there but has a 10% of detecting it when it's not there, then the likelihood ratio is 100:10 = 10:1. The posterior odds of having cancer after you get a positive test result is then 1:100 * 10:1 = 1:10, which equivalently means you have a 1/11 posterior probability of having cancer.
Speaking for myself, 9/10 if I have a commitment the next morning, I won't stay up late on my computer because... I know I have a commitment at a set time
I don't have the social media probrem, but I do have this problem. It's like there's a part of me that only cares about what happens in the immediate future (seconds to minutes) and has veto power over my actions, so that "go to bed now for benefit later" or "start some work now even though it'll be aversive for the first few minutes until you get into the swing of it" or "start reading a new piece of fiction even though it'll typically only start being fun after a few minutes" are all sometimes hard to get myself to do.
Can I get a description of the subjective experience of unfun superstimuli? Usually for me, I'm having fun - but clearly many others don't but do it anyways.
For some reason I seem to be much less susceptible to consuming unfun superstimulus, which is weird given how I am generally impulsive/lazy/low executive function/etc and even used to have harmful compulsions.[1] If I didn't already know better I would predict I'd have a problem with unfun social media!
Short form video generally is boring, it don't want to see more unless I feel like the next might be good.[2] I generally prefer text over video. The content is banal and so doesn't hold my attention for long. When I was much younger I'd scroll reddit for memes but those (along with video games) became less interesting. It's like when people talk about celebrities I don't care about, or sports, except at least then there's some substance to engage with and they can talk in more depth about it.
Here are some other examples of superstimuli, most of which I enjoy; I also don't feel that my joy in other stuff is dampened (a common concern of superstimuli):
I'm glad the world has makeup and that I get the opportunity to see prettier people (both in meatspace and on TV).
Probably this makes me feel worse about my appearance via ingrained standards, but internally it doesn't feel that way (it internally feels bad if I look anti-pretty, good if I look pretty, and while I wish I looked super pretty it really doesn't feel like it makes me feel worse about myself), and it also doesn't diminish my enjoyment of romantic or sexual partners.
Likewise I really like staying up late, even thought it has bad consequences. It's like (a weaker form of) chilling with friends when you have work to do - it has a cost but it's part of what makes life fun.
Obviously many don't feel this way, I hope someone describes what they hate about it and how they are drawn in anyways. It's unfortunate that most who feel like this and write about it assume it's universal, as they had the perfect opportunity to explain it to me.
From when I was younger and depressed: banging my head on tables and finding it difficult to stop despite the experience sucking. It's like scratching an itch that actively hurts to scratch, even knowing that you'll make things worse by doing so and not wanting the wound to grow.
Likewise I struggle to do work in a way that feels somewhat similar. I would think social media for others is analogous, but also when I describe my difficulties getting myself to e.g. fold my clothes, people tend to not relate.
My twitter feed sometimes has cool stuff, but othertimes is filled with boredom, so sometimes I find myself scrolling past boring stuff hoping to hit something cool even when this expectation is unreasonable. But if my feed never had cool stuff then I think I wouldn't have the problem.
You'd be better off with memorizing chemical names + structures and reactions than the periodic table. Likewise I doubt that the world map and the monarchs are a good way to do geopolitics and history - for the former, various statistics or historical events or even just "Country X is in a civil war right now, dating back to year A, while Country Y is allied with Z and enemies with W".
I'll add that for physics, you should choose some reference numbers to memorize, in good units that make remembering and calculating easier. This will unlock the ability to do quick Fermi estimates and give you an additional understanding of the subject.
I broadly agree with the argument of the post, but having good taste as to what to memorize is important. Also, if you're using a spaced repetition program anyways you should put some of the concepts down along with simple facts.
As for why this "morally" checks out: moment of inertia of a single point mass at radius r from the center is I = m r^2, and in this case the energy of course has to be the same as 1/2 m v^2 as it's a point. Distributed objects have lower moments of inertia but if you look at the formulas for a bunch of cases (as mentioned uniform solid sphere is 2/5 = .4) you'll see that generally the moment of inertia factor isn't super small unless it's a lot denser closer to the center. For example, the Sun is 100x denser at center than its average and has a (predicted) moment of inertia factor of .07, while the Earth is only 2-3x and has a (confirmed with measurement) factor of .33. So your ball would basically have to be as center-heavy as the Sun to ignore it safely.
...only four people out of how many estimated the speed from the video? Here I was chastising myself for thinking of essentially Robert's approach second (after about a minute or two) instead of immediately, and am surprised so few measured speed at all.
Tangentially, I'm not sure if I'd have thought of ramp flexibility even after observing a different speed than expected from energy conservation - I might have just chalked it up to rotation or friction. But to not measuring the velocity...!
Once in high-school I successfully predicted from theory alone the height required for a free falling toilet paper roll to hit the ground at the same time as a roll that has a sheet held to a wall, but even just my chance of a math error is high enough that I wouldn't trust it. And I consider that more effort than measuring speed, despite being comfortable with the math.
On the other hand, gain of function research is both probably not very useful and unnecessarily risky. Many think it likely that COVID was caused by it; even if you don't, it seems likely that such research could cause similar pandemics.
(I don't think this is the place to argue about whether COVID specifically was caused by gain of function research, so suggest that replies to this comment not be about that)
Additional evidence: I don't feel companionate love as strongly as you do (I expect I feel love closer to the average amount, skewed a bit on the low side), but still have the same negative sentiment towards Green. Internally to me the Greeny feelings don't feel like love, but maybe most Greens do in fact feel them similar? For me it feels more like "story-thinking" or "narrative fit". I think fiction is probably the most Green feeling thing I do, and it feels to me like people apply it to the real world and don't feel the sense of "uhh, this is Real Life, not a book".
See Paul Krugman
As a physics enjoyer, I am reminded of a quote from Feynmann (or someone else?) that agreed with my personal experience: the strength and weakness of intuition is how malleable it is. The strength being that you can make the math structure become intuitive. It's not just that you assume A and see whether you actually can get B, but rather you assume A, get some weird result, and then stare at it until it makes intuitive sense. Maybe you then land upon a new concept. Alternatively you might crystallize some intuition you have to see what the true nature of the thing really is - this can be best seen in mathematics itself, where the definitions encode express many of the "True Names" of concepts. This is a lot more than just falsification, you do a lot of locating and understanding of ideas this way.
I don't know enough economics to give helpful examples, as all the ones I have are ones where I already knew the math (and so can't compare clarity after learning the math) and that could plausibly be qualitatively understood without it.
I can however think of the VNM theorem as an example where the basic theorem itself is the concept, that is that you can get utilities via preferences on lotteries by having "strength of preference" refer to what you'd choose under uncertainty.
Regardless, quantitative statements are still pretty important though. To be able to predict the size of the effect, or to have your theories make claims about the sizes, matters a whole lot (including qualitatively - it's important to know which consideration are negligible and which ones dominate).
Off the top of my head, the Lerner index is the marginal profit as a percent of the price, and the theory is that a monopolist maximizes profits when this is equal to -1/elasticity of demand. Without following the calculus you won't arrive at this direct relationship between elasticity of demand and the profit percent.
Also if you count voting theory as economics, there are some rather counterintuitive results like Condorcet paradoxes (which I once came up with an example of independently in the context of intransitive dice, but wouldn't have been able to do so if I didn't know about directed graphs - the point here is the help mathematics provided) and Arrow's impossibility theorem and other no go theorems.