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How to Understand Things

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Re: How to Understand Things

#71

Post by jddj »

drorco wrote: post #50
knzhou wrote: post #41

Earlier quoted context omitted.

> says you need to master each step on the track before moving onto the next one. this is poisonous and borderline evil What’s wrong with it? You do need to understand calculus before classical mechanics, classical mechanics before quantum mechanics, quantum mechanics before quantum field theory, and quantum field theory before the Standard Model. I’ve seen tons of people disregard this and the result is always confu…

Some people like to work their way in reverse. I'd often pick a really complicated subject I'm after like "stellar fusion" and then work my way downwards and learn whatever I need to learn in order to understand it. If I had to start from differential mathematics, without knowing why I need it, I'd probably give up.

Are there any notable teachers who take or have taken this approach?

I can't think of any from my education, everything was bottom up building. But when I self-teach (as it naturally emerges from pursuing projects related to work or hobbies) it's often top down digging.

Re: How to Understand Things

#72

Post by tashi »

BeetleB wrote: post #53
afarrell wrote: post #20

> This quality of “not stopping at an unsatisfactory answer” deserves some examination. > This requires a lot of intrinsic motivation, because it’s so hard; so most people simply don’t do it. It also requires self-confidence, persuasiveness, and social power. Without these traits, your attempts to really understand something will be dismissed as "overthinking things" or "trying to understand the universe". Those arou…

There are both extremes, and best not to be at either one of them. Example: I forget the name of the principle, but in mathematics the statement "P implies Q" is considered true if P can never be true. As an example, let P be "George Washington was a woman" and Q be "Queen Elizabeth is a man". Then the statement "If GW was a woman, then QE is a man" is considered to be a true statement. I have a friend who refuses to…

A real logician can correct me if I'm wrong, but I think those statements are considered "vacuously true" for the same reason we say the empty set is a member of every set and x^0 is 1: it makes the rules of calculation simpler if we define it that way than other ways. But you can't use these convenient definitions to learn anything new.

You can keep multiplying your equation by x^0 and union-ing your set with the empty set and or-ing your proposition with "If eggs are diamonds then fish can talk" all the livelong day, but you never gain any more information.

Re: How to Understand Things

#73

Post by sega_sai »

There is one point there that I thought was interesting, regarding asking questions, and not being afraid to look stupid and about many senior/distinguished people doing that. Being in science, I noticed that it gets easier when you get more and more stature. I.e. if you are a professor, I think it is very easy to ask a postdoc/student after the talk -- sorry I don't understand it, or sorry if it is a stupid question. And the reason is 1) you know much better the limits of your knowledge, so it is unlikely a stupid question 2) You may be less dependent on the opinion of people surrounding you. While if you are say a grad student, when you ask a question, there is a higher chance of looking stupid, just because you may know significantly less and also you may be more afraid of people around you because of their higher career status.

Because of all this, I find it a bit disingenuous when senior people say to younger people that you should always ask questions and not being afraid of looking stupid. But being curious and asking questions is good overall. It is just not always easy.

Re: How to Understand Things

#74

Post by BeetleB »

tashi wrote: post #72
BeetleB wrote: post #53

Earlier quoted context omitted.

There are both extremes, and best not to be at either one of them. Example: I forget the name of the principle, but in mathematics the statement "P implies Q" is considered true if P can never be true. As an example, let P be "George Washington was a woman" and Q be "Queen Elizabeth is a man". Then the statement "If GW was a woman, then QE is a man" is considered to be a true statement. I have a friend who refuses to…

A real logician can correct me if I'm wrong, but I think those statements are considered "vacuously true" for the same reason we say the empty set is a member of every set and x^0 is 1: it makes the rules of calculation simpler if we define it that way than other ways. But you can't use these convenient definitions to learn anything new. You can keep multiplying your equation by x^0 and union-ing your set with the em…

I pretty much told him the same: That we declared such statements to be true merely for convenience - it makes the proofs shorter. And that none of the theorems he would normally deal with would have a different "outcome" if we don't accept such statements - they'd still be valid theorems.

But he won't be convinced until he formally studies logic. No idea when that will happen.

Re: How to Understand Things

#75

Post by gameswithgo »

alicemaz wrote: post #13
refrigerator wrote: post #6

Great post. Personal anecdote: I don't think I really understood anything in school, but I was decent at going through the motions of carrying out certain methods and recalling certain facts when I needed to. I went on to study Maths at university, and for most of my first year, I had the same surface level "methods + facts" knowledge that got me through school. After some studying, I could recite definitions and the…

>I suspect lots of people go through the entire education system like this. +1. it took me a couple years after getting kicked out of college to get my head sorted out to the point where I felt like I could "think" again I think one of the most harmful things about schooling is the way it imposes a tracked structure on learning. it demarcates knowledge into discrete subjects and sets up a linear progression through t…

seems like a lot of people are imagining a caricature of what school actually is and blaming it for problems they would have had anyway.

Re: How to Understand Things

#76

Post by knzhou »

alicemaz wrote: post #70
knzhou wrote: post #41

Earlier quoted context omitted.

> says you need to master each step on the track before moving onto the next one. this is poisonous and borderline evil What’s wrong with it? You do need to understand calculus before classical mechanics, classical mechanics before quantum mechanics, quantum mechanics before quantum field theory, and quantum field theory before the Standard Model. I’ve seen tons of people disregard this and the result is always confu…

calculus is required to understand classical mechanics, but intuitions about classical mechanics can inform and accelerate how you learn calculus. I actually tutored calc 1 a little bit in college and most of the time the people I was helping could do the calculations asked of them fine, but felt lost and confused because they didn't know what the output actually meant . everyone learns linear algebra before abstract…

While I completely agree with your general comments, I still want to emphasize how hard this ideal is to implement in practice.

Yes, every subject is linked to every other subject. If I'm tutoring introductory physics, I will use examples and analogies from math, computer science, engineering, biology, or more advanced physics, depending on the background and taste of the student, and it works fantastically. But if I'm lecturing to a crowd, this is impossible, because the students will differ. If I draw a link, some people will think it's enlightening, some will think it's totally irrelevant, some will think it's boring, some will think it's so obvious it's not worth saying, and most will just get more confused.

The same thing goes for the top-down "working backwards from cool results" approach; it's supposed to bottom out in something you know, but whenever you teach multiple people at once, everybody knows different things. The bottom-up linear approach is useful because it gives you a guarantee that you can draw a link. If I'm teaching quantum mechanics I expect to be able to lean on intuition from classical mechanics and linear algebra. If I didn't know the students had that, I would draw a lot fewer links, not more.

Similarly, "if people learned X in school, then Y would be easier to understand later" is true for almost any values of X and Y, because of the interconnectedness of knowledge. But if you ask any math teacher, they'll tell you the school curriculum is already bursting at the seams. You can't just add logic and set theory to existing school math without taking something out. In the 70s we tried taking out ordinary arithmetic to make room for that. It was called New Math, and everybody hated it.

Re: How to Understand Things

#77

Post by aarghh »

sega_sai wrote: post #73

There is one point there that I thought was interesting, regarding asking questions, and not being afraid to look stupid and about many senior/distinguished people doing that. Being in science, I noticed that it gets easier when you get more and more stature. I.e. if you are a professor, I think it is very easy to ask a postdoc/student after the talk -- sorry I don't understand it, or sorry if it is a stupid question…

> Because of all this, I find it a bit disingenuous when senior people say to younger people that you should always ask questions and not being afraid of looking stupid. But being curious and asking questions is good overall. It is just not always easy.

I agree with your general point, but a minor comment here: in many cases the professor/leader asking those (stupid) questions may be to create psychological safety for a more open conversation. It is less threatening to the person being questioned; it also encourages other people to participate.

Re: How to Understand Things

#78

Post by Benjammer »

alicemaz wrote: post #13
refrigerator wrote: post #6

Great post. Personal anecdote: I don't think I really understood anything in school, but I was decent at going through the motions of carrying out certain methods and recalling certain facts when I needed to. I went on to study Maths at university, and for most of my first year, I had the same surface level "methods + facts" knowledge that got me through school. After some studying, I could recite definitions and the…

>I suspect lots of people go through the entire education system like this. +1. it took me a couple years after getting kicked out of college to get my head sorted out to the point where I felt like I could "think" again I think one of the most harmful things about schooling is the way it imposes a tracked structure on learning. it demarcates knowledge into discrete subjects and sets up a linear progression through t…

>stack up prerequisite knowledge before they're even allowed to touch it

The way you phrased this reminded me of A Mathematician's Lament by Paul Lockhart[0]

[0] https://www.maa.org/external_archive/devlin/LockhartsLament....

Re: How to Understand Things

#80

Post by wenc »

RichardChu wrote: post #31

I worry that the way modern society is structured disincentivizes deep understanding. 1. Industry cares more about concrete results, quick execution, and bias for action. 2. Academia cares more about positive results, quantity of published papers, and small achievable experiments over big experiments that might fail. Where are the institutions that care about deep understanding?

I would say the IAS is one, as are other pure research institutes like Perimeter.

I would also say that a deep understanding isn’t usually necessary to make progress in many fields of human endeavor. Fields like engineering work on the basis of empiricism — theoretical understanding usually comes later. I could be all wrong here but my gut feeling is that the majority of great breakthroughs in engineering have come through tinkering and luck rather than a principled application of science — the latter is used more for refining the execution.

Even in recent times, neural network models have been shown to work without any deep understanding apart from the basics. It’s only recently that a lot of new theory have come out.

Deep understanding is a worthy goal in order to deconstruct things to learn how to make them fundamentally better or to build a foundation for further progress.

But as humans, we do very well surviving in a world that we largely do not deeply understand for the most part — I’d argue we are able to do so through heuristics (see Gigerenzer). We definitely should not let the lack of deep understanding prevent us from taking the initiative to do things.

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