The centrality of stupidity in mathematics
mathforlove.com
mathforlove.com
When I started my PhD program, a group of us were given a little talk by the department secretary.
She told the story of how she went to audition for Jeopardy!, a trivia game show. She saw a whole bunch of other people at the audition get really nervous and choke up; her take on it was that they were used to being the most knowledgable in the room -- they were used to sitting in front of the TV screen with their friends or family and knowing every fact, and when they were suddenly confronted with a situation where everyone was as knowledgable as they were, they were suddenly very intimidated.
She, on the other hand, was completely relaxed -- she spent her days working with Nobel prize winners and loads of other people for whom she had no doubt were smarter than her. Being confronted with loads of people smarter than her was a daily experience.
She told this story to us to say, a lot of you will experience the same thing: You were used to being the smartest person in your High School, you were even used to being the smartest person in your classes at the prestigious university you attended. Now you'll encounter a situation where everyone is like you: the best and most driven people in your classes.
You'll feel stupid and inferior for a bit, and that's normal. Don't let it bother you. Eventually you'll notice while that most of these other people have areas where they're better than you, they have areas where you're better. And there will still be the occasional person who seems better than you at everything: that's OK too. You're not the best at everything, and you don't have to be.
To be clear, I'm not saying I was always smarter than people around me, I just felt like I never had to work as hard as I suspected even through my Masters program.
But once you get over the barrier to entry for most white-collar jobs (bachelor degree), what's the point? If you're not getting anything out of the education, you're only borrowing trouble from yourself in the future.
My bachelor's was relatively easy. My masters was MASSIVELY difficult. The PhD was even harder.
Because I sought out those difficulties.
An older use of the word "prove", as in to test, means it says "that's an exception that tests the rule, and finds it is incorrect"
L'exception qui confirme la règle
And there's no ambiguity about it, the exception is confirming that the rule is true.
That's a stupid expression IMO, but I would be surprised if the English expression meant the exact contrary.
So the (existence of an) exception proves (the existence of) the rule.
I wouldn't use the phrase outside of silly internet jokes about 90s popcorn flicks.
If your rule is that carpenters are usually men and someone goes into a carpentry conference with 1000 carpenters and points out that 50 of them are women, you get to say "exception that proves the rule" and be right.
If your rule is that carpenters are usually women and someone goes into a carpentry conference with 1000 carpenters and points out that 950 of them are men, that's not the exception that proves the rule because 95% of the target population doesn't qualify as an exception.
Hmmm... self-referential vibes coming here. It might be that the only exception to the R1 is itself, but then... etc.
It really does get harder to internalize this when it starts to involve real, tangible outcomes like money and job security. No one would reasonably argue that what she said wasn't true on some spiritual or personal level, but it feels like a nothing-burger when people are clearly in a competitive environment, a competitive program, a competitive job, etc.
If I put humility aside for a moment, I'm awfully good in my field, and academia is hyper competitive, and yet I know people who are better at everything that I do -- and just like the secretary in that example, I know some people who I think could do everything that I can better than I do.
But that really is okay, the world has room for all of us and more, and is much better off for having several of us applying our skills and abilities. We are not short on important problems to solve, we are short on solutions and solvers!
I've had the chance to work with world class people in three different fields now. Inevitably, the people who were at the very top (World Champions, folk who literally invented (parts of) the Internet, people creating products for a massive fanbase) didn't "want to be the very best". And they were extremely comfortable with other people being smarter, if it happened.
They inevitably got to the top because every day, they wanted to be a bit better than the last day. "The best" never mattered. "Better than yesterday" did. And when they were at the top, they were the most kind and helpful people you can imagine - uplifting others whenever they could.
And they certainly didn't want to be "like no one ever was" - they invariably had a deep appreciation for the ones who came before them, and they copied everything that seemed worth copying.
The people who don't heed that advice, who don't give that advice, who relentlessly claw? Perpetual second tier, without exception.
As a 50 years old person that some time ago was one of these brightest in class I can say that for most of us, people, it is:
You're not the best at anything, and you don't have to be.
"In a way, meeting those people was liberating. I will never be a world champion at anything, so I might as well play for the love of the sport."
It stuck with me and has become more meaningful as I get older and a bit slower.
I went to ICFP in 2019, and I can say with a high degree of confidence that I was the dumbest person there [1]. Everyone was speaking on four-syllable mathematical notation that I had never heard of, and talking about intricacies in GHC that I wasn't really familiar with, and different aspects of type theory that were completely foreign to me.
It was very humbling; it didn't depress me or anything, but made me realize that there's a lot to learn and improve on, and the people there were actually extremely nice and gave me some pointers so I can get incrementally closer to being as smart as they are.
I think 2024 Tom would be the second dumbest guy in the room if I went again.
[1] Knowledge-wise, I have no idea about IQ or anything.
And this acclimation is also similar to https://en.m.wikipedia.org/wiki/Mathematical_maturity, how you talk with math people the from math village and feel you are stupid because you're not familiar with the symbols, the vocabularies. But when you have learned the language, you will see that you become better and better at learning, you have the means to gain more means--forming some sort of a positive feedback loop.
I'm a college dropout who has managed to work professionally, in software, in RF / embedded development, medical robotics, all over the web, and most recently in AAA games. I've called PhDs colleagues, people with multiple Masters, people with decades of industry experience at the top of their fields...
I always feel like the odd-man out, and, while it used to bother me a bit, I'm pretty content with the fact that I probably always will. I frequently feel like I'm inferior to my colleagues, because of the sheer depth of their knowledge as it relates to whatever the particular domain is.
But I have the sense that I'm doing something right because I get great reviews, I frequently find moments where I can teach my colleagues something they didn't know, and they come to me for help, advice, and say good things about me (and vice versa).
But it is still a very odd feeling and I think it'll be with me for however long I work in this industry.
He’s really talking about curiosity. Calling it stupid is a cheeky glass-half-empty framing. But one of the big problems with math education is that we force students to struggle to a far greater degree than other subjects, under a belief that personal struggle, the “trying”, is the only way to truly get it. This is a pervasive cultural belief that extends to work and money too. And we force it from the beginning when kids are very young, without taking the time to develop their curiosity, and without setting up the system to gracefully nudge people who for whatever reason don’t see why they should try so hard.
Personally I suspect there are lots of things that could help motivate many more students than we do using our current system of demanding kids “try” to grok abstract rules using Greek letters. A combination of more visual storytelling, math history, and physical less abstract problems, along with a grading and progression system that ensures kids get it before moving to topics that depend on having got it might help a lot of people; too often kids are pushed to progress without ensuring they’re ready, and once that happens, “trying” is a fairly unreasonable expectation.
Think about how you learned your first language. Your mom taught it to you by rote repetition. She didn’t expect you to come to any of it on your own, and you weren’t expected to struggle with grammar or understand the rules or judged for getting them wrong, you were just gently corrected when wrong and celebrated when right. I don’t know if first language learning is a good way to learn math, but it is obvious that we have alternatives to today’s system and that today’s system isn’t serving everyone who’s capable of doing math.
so it probably means 'search really hard' rather than 'search again'
The first job of a researcher is to understand what others have done, before attempting discovery. Failure to do that critical step means it’s not considered research. The second job is to build on the work of others. And the third job is to communicate those results to others. Discovery is the seeking of knowledge, which is education. Framing it as self-education is feeding a narrative of research as being an individual sport, but in reality research is entirely a collaborative team sport with incremental dependent results.
One of the central problems of our time in research and academics, I think, is an incentive to focus on areas that are well-established because we know they are likely to produce results that we have confidence in (according to whatever inferential criteria we use). The idea of it being ok to not know something a priori, to have lack of confidence in it, seems sort of discouraged in the current climate, because it's too risky.
The problem, I think, comes from the weaponization of "stupid" against people. The XKCD of the lucky 1000 plays a good role here. If you are constantly deriding others for stupid takes, then anyone that derides one of your stupid takes will hit hard. And that seems to be getting worse.
I disagree.
How can you be curious without something you don't understand?
The point, as I see it, is that if you find yourself in a position of complete understanding, then you must also have a complete lack of curiosity. If you think you are in that position, then the way to revive your curiosity is to deconstruct your position of expertise, i.e., recognize your position of stupidity.
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Curiosity is step 2. Stupidity is step 1. Learning moves us from step 2 to step 3. The important thing to recognize is that step 3 is actually a new instance of step 1. Expertise is the base case of this recursive tree traversal: it's how you stop the learning process.
Good question. You might have discovered my point: curiosity comes with stupidity, implicitly by definition, right? I think that’s what you’re saying too. Maybe you don’t disagree after all?
You can’t have curiosity without stupidity, as you rightly point out. Ignorance is probably a better word than stupidity. Using “stupid” is imprecise and was used here for a bit of surprise and humor.
You can have stupidity (ignorance) without curiosity. When that happens, perhaps the expected result is no progress developing new understanding nor lessening of ignorance.
Given that curiosity implies ignorance, and that ignorance alone is not sufficient for learning, what justification is there for claiming curiosity and ignorance are separate steps or separate things when it comes to education or research? I’m suggesting they are two sides of the same coin, they must both exist before learning happens, and neither one can come before the other. Calling it curiosity instead of ignorance or stupidity is perhaps a kinder framing, especially for people who might not immediately get the self deprecating humor of “stupidity”.
My overall point is that the end of education is expertise, which itself is a form of ignorance. We generally consider stupidity and expertise to be antonyms, but they often exist as two opposing perspectives of the same experience.
The "being curious" thing is independent of the "feeling stupid" thing. It definitely exists, but it's absolutely not the same thing.
Looking at the rest of your comment, maybe we have a lot in common, but I definitely disagree with a lot of what you've written here, so no doubt our experiences are wildly different. Perhaps over a coffee[0] we could talk constructively about education, math, struggles to understand, and work ethics, but suffice to say here that even if we do have substantial common ground, I think we might have very different points of view.
[0] Other beverages are available.
In that case, what is being taught is actually different, it's the similar in some senses to the difference between teaching someone to operate a machine, versus teaching them how to maintain, fix, and possibly improve said machine.
I've said for a long time that if only we could identify early the students who will not go on to study mathematics at an advanced level then we could better benefit them by having a substantially different curriculum taught in a completely different style.
But even then, the willingness to be confused by something, and resilience in the face of being made to feel stupid not by the teacher[0] but by the current (temporary!) lack of understanding, is really, really powerful.
[0] Teachers who unnecessarily make students feel stupid should be prevented from ever, ever going near students again. Ever.
> If we get the chance some time, it would be super fun to discuss math & education over coffee!
Where are you based? I travel a lot ... my email is in my profile, and you could always register with my "Meet With Me" system to get a heads-up if I'm going to be in the area.
Yeah I'd very much agree with this. Actually strong agree with all your points there, it’s well aligned with what I think I was trying to say.
I’m US based, currently mostly Utah, sometimes California.
But I'm unlikely to be in the US now for a bit, but if you ping me an email then I can tell you how (if you choose) you can register with my "Meet With Me" system.
Or not ...
Regardless, yes, I think we do (or would) agree on most of these issues.
People who are struggling with math, even in elementary school, will often say things like, "That's so simple, why couldn't I get it? I must be stupid!" I've heard this over and over again in a wide variety of contexts. Including from my own children. We honestly believe that if it is simple, it should be easy to understand. And so the experience of having struggled with something simple, leaves us feeling stupid.
Even people with an incredible amount of math success, wind up feeling this. And pretty much universally have also developed coping mechanisms for it. But the experience itself is pretty much universal. My son felt it in middle school when he was struggling with long-division.
Now does a poor education system make this experience more likely? Does it serve us poorly? Absolutely! I consider it a crime that Singapore has developed a better way of teaching elementary math, and we have not adopted it. Singapore is now moving to #1 across different subjects according to PISA, and Western education systems aren't even curious about how.
However that is orthogonal to the key point here. Which is that math tends to be simple in a way that our brains aren't built for. And when we're confronted with how hard it is for us to learn something simple, it is easy for us to feel stupid. This can be very demoralizing. And it is very helpful for us to learn to accept and deal with that feeling.
I think (like with others in this subthread) that I have mislead you or am being misunderstood or both. I’m familiar with the feeling of stupidity in math, and I’d agree that it’s useful when/if harnessed. I’m calling the process of accepting it and dealing with it “curiosity”, partly since if curiosity is missing then people tend to feel shame and anxiety with their stupidity and tend to avoid math and give up on learning it.
I don’t believe there is such a thing as right and wrong here. I’m making a point of view framing distinction, not disagreeing with the article or the quote in the article. I think that “stupidity” isn’t the best word choice in general, even though it might work for math researchers in this case. That word comes with many overloaded and negative meanings that aren’t accurate to what the article is really trying to say.
Curiosity is a tool for dealing with it. But it's but one tool. And we need a whole toolkit.
1. I don’t think the author was just saying that math is hard or “a struggle”, I think they were specifically pointing out the unusual nature of math(s) as a collection of meaningfully novel cognitive tools rather than facts or recombinations of existing tools. AKA “feel stupid” means “feel embarrassed you don’t understand earlier because now it’s obvious”, not “take a long time to understand” or other synonyms for struggle. We all agree on the vague shape of the proposed improvements to pre-graduate math education, I would guess!
2. That’s not how first language acquisition works, at all: the rules of grammar—not to mention etiquette-are far more complex than most laymen imagine, and intentional parental involvement via correction or the occasional picture book is absolutely the exception, not the rule. This is the core insight driving Noam Chomsky’s lifetime of scholarship, and I think he would agree that childhood linguistic development is more similar to mathematics education than practically any other activity, if we’re talking about “feeling stupid” like the author is.
i think we can do better than we are doing at math education, much better, but there is no way to learn math, or anything else, without diligent effort. it won't happen by passive absorption. you can listen to people speaking spanish all day every day for years without learning more than a few words of spanish if you don't make any effort
curiosity is one possible motivation for making that effort, but the immediate result of the effort is, at first, failure. that's true of language, it's true of playing the guitar, it's true of programming, it's true of throwing clay on a potter's wheel. that failure feels like being clumsy, weak, or stupid, depending on the form it takes
and that's what the article is talking about. trying to do things that are beyond your mental ability makes you aware of, and frustrated with, that mental ability. that's not curiosity, it's feeling stupid. it's also how you increase your mental ability!
I disagree. In fact, I found that often the better and more didactically streamlined the exposition is in a book, the less deeply I end up learning the material. It is precisely the personal struggle, having to make my own sketches and derivations, starting out with a misconception because of bad phrasing in the book and having to explore that misconception until I find what I misunderstood etc. makes the knowledge stick much better because it now feels my own, like an intimate friend.
Spoonfeeding may get people quicker to the point of solving the standardized quiz at the end of the chapter but that's not the same as learning and understanding. Another instance of metric-chasing in action.
It's a bit like how I learned MS Office or Photoshop by trial and error as a kid, or programming by mucking around trying to make a website do what I want. And you bet it was a struggle. Struggle but with reward at the end. Compare that with a handholding tutorial where you do learn how to do whatever the tutorial makers had in mind, but it won't generalize as much. Sounds totally dry. I loved computers, but hated school lessons that tried to teach us MS Office in the handholding spoonfeeding way. It's the death of the subject.
It's a safari in a safe car, looking at the animals through binoculars vs running around in the jungle in your own adventure amongst the beasts.
But alas, I never thought as a kid that I didn't have time for other things. I was always into something.
> He’s really talking about curiosity. Calling it stupid is a cheeky glass-half-empty framing.
One of the most important characteristics to succeed in maths is the ability to acknowledge things you don't understand, to fail, and to persist in spite of failure. Trying really is the only way to understand some hard things, because there are some things that are conceptually extremely difficult.
He's not talking about relative stupidity where there are other people you feel are smarter, he's talking about stupidity on an absolute basis. You don't know. You don't understand. But somehow you have to find a way to carry on, and then later on, looking back once you do understand, you're baffled by why you didn't know/understand or couldn't see some crucial things. You have climbed up a ladder and pulled it up behind you and it's hard now to imagine what it is like to be on the ground.
It's not about curiousity. Of course you have that - if you didn't you wouldn't be there in the first place.
Feeling stupid is when you're confronted with an external you cannot understand why it works.
Curiosity can help push through feeling stupid, but you can be curious about something you won't struggle understanding it at all.
Sounds a bit like Kernighan's lever: "Debugging is twice as hard as writing the code in the first place. Therefore, if you write the code as cleverly as possible, you are, by definition, not smart enough to debug it. Yet." [0]
[0] https://www.linusakesson.net/programming/kernighans-lever/in...
To be able to truly learn any given concept means being capable of answering a practically infinite number of different questions about that topic. The process of teaching is essentially trying to uncover which questions the student can't answer. The challenge, of course, is that the student doesn't know what questions they can't answer, because the questions haven't occurred to them. That is, until they start testing themselves, to see if they really do understand the concept.
Problem sets in textbooks are the canonical way of addressing this teaching challenge, but there are only so many pages in a textbook, and there are other concepts that need to be taught, so the scope of the problem sets are necessarily finite.
How many times have you nailed all the problems in a book, only to discover that there was some aspect of the topic you didn't understand, despite getting all the right answers?
Fortunately, what I have retained is the ability to pick up almost any subject and learn about it on my own. That's more important than anything they can teach you in a classroom.
Or so I'm told.
The Master shows the Gate, but it is the Student who has to walk through it.
To show one the Right Direction and Right Path, Oral Instructions from a Master are necessary but Mastery of the Subject only comes from one's own Incessant Self-Cultivation.
There is also a great inspirational story in the Mahabharata of "Ekalavya" (https://en.wikipedia.org/wiki/Ekalavya) who became an exceptional archer through self-training.
Eklavya inspires a life-long learning philosophy and his presence seems to be a celebration for the masses. In this EklavyaParv, the motto is 'You Create Yourself" and the legend of Eklavya is a testimony that is forwarded by many thinkers as well. The discipleship that Eklavya represents is the best for a student and enables one to be the creator of one's own destiny.
Adapting to current times, "The Master" can be a "Good Book" and you can have "Many Masters" but the effort and learning has to happen within the Student.
Source: I self-taught myself Martial Arts (JKD, Karate, Taijiquan) from books when i was young. Decades later when i did join a dojo to study under a Master, i was one of the top students with good skill and power.
This is very interesting and impressive. How common would you think a complete self-study of martial arts actually is? I've always thought this cannot be done alone -- or, it would be extremely easy to get lost or head to some wrong direction, eventually harming yourself mentally or physically. Akin to how a common suggestion about yoga or meditation (I used to exercise vipassana daily for quite a while) is that the first, basic principles should be taught by a good master. Possibly due to personality type, I've always wanted to challenge this assumption, though.
While learning martial arts on your own, what did you do to overcome more serious mental blocks or standstills (provided you had any)? Did you ever feel that "books are not enough"?
The key aspects in learning were Overwhelming Drive (the motivation was to learn to fight like Bruce Lee :-), daily practice of basic blocks/punches/kicks, slowly progressing through the movements of the shadow-boxing/Kata routines given in the books and no self-questioning/self-doubt/no-comparing with anybody else. I also had a good friend who was also very interested in Martial Arts (he eventually joined a Shotokan Karate school) and so we would practice/encourage each other. One defining lucky moment was coming across E.J.Harrison's "The Manual of Karate" (which was a translation of a Japanese text) where the author explicitly states Karate is useless without makiwara style training/conditioning. I took that to heart and made a canvas pillow with coconut-matting, tied that to a tree (a small one which could vibrate and absorb your hits) and would train with full power on it. This worked so well that my basic punches/kicks became more powerful than my Shotokan-training friend who only did non-contact practice. To this day i can generate very good power relative to my size. Reading Gichin Funakoshi's "Karate-Do : My way of Life" was also instrumental where he mentions his two teachers and their instructions; Master Azato would tell him to think of his limbs as swords so that he went through his opponents and Master Itosu would tell him to harden/condition his body so that he could absorb any blow. Next it was Miyamoto Musashi's "The Book of Five Rings" which led me to studying Martial Arts mindset/strategy/theory and validated my approach since he was also largely self-taught in many disciplines (the concept of "Hyoho/Heiho" acquired in the study of one discipline is internalized and then used to study other disciplines effectively/effortlessly).
Today you have far more resources and avenues open to study Martial Arts but the main points i mentioned above must be kept in mind even though many teachers may not teach you those. As mentioned in the quote above, Mastery is always dependent on "one's own incessant self-cultivation".
South Indian Martial Art "Kalarippayattu" - https://news.ycombinator.com/item?id=41540103
On Patanjala Yoga Sutras - https://news.ycombinator.com/item?id=41538322 and https://news.ycombinator.com/item?id=41554764
In my previous comment i pointed to the essence that you need to focus on whether you are studying under a Master or by Yourself (harder). For actual techniques you can choose books by noted masters (eg. Mas Oyama, Masatoshi Nakayama) in the style you are interested in on Amazon. For insight/details into Martial Arts mental training and theories see the works translated by William Scott Wilson (https://en.wikipedia.org/wiki/William_Scott_Wilson).
I don't think a student needs to always answer a question "on the spot". Being able to find an answer in a reasonable amount of time and explain an answer would be in my opinion more valuable. So then it's more about "how efficient can the student give the answer to the question" (answer on the spot, spend one hour, spend one week, etc.).
I had a junior helpdesk employee that I was training/mentoring years back. He was 20 years old, fresh out of tech school. He was good at what he did, but he only did things he knew how to do. When he didn't know something, he'd ask me. Which is great. I'd say "Well, this sounds like DNS, it's like a phone book..." or "That's an APIPA address, it must not be getting DHCP. The computer shouts out to the network asking for an address..." and so forth. However, he kept asking the same questions.
After a few months in one of our monthly meetings he kind of broke "I don't understand what I'm doing out there, you need to train me! I need to be trained!" Completely perplexed I asked him what he was talking about. "You just answer my questions, but you're not training me!" I realized he was expecting me to learn him the answers to everything. I had to explain to him that the responsibility of learning was actually on him. "This isn't school, there's no study guide. We have documentation and Google. It's your responsibility to read it and make sense of it."
I told him that I can give him all the puzzle pieces but I can't put them together for him. To be fair, helpdesk is kind of about making things work and remembering the quick fixes and tricks for things to close out your tickets.
So I said, "Ok, I think you need a project. What do you do at home for fun?"
"Well I play a lot of video games."
"Perfect, we're setting up a Minecraft server". He laughed.
I said "No, I'm serious. We're using like 5% of this massively overblown server that was sold to us. Maybe this will help you put the pieces together."
I gave him a restricted vSphere account for his DMZ'd VM, sent him a guide and unleashed him.
"Well, I've never done this before..."
"Exactly. That's how you learn my dude."
"But..."
"RTFM"
"This VM doesn't do anything."
"Right, it needs an OS."
"We'll how do I install one?"
"Here's a guide."
"I installed the OS, how do I get into it?"
"SSH"
"No I mean the desktop."
"There isn't one."
And so he learned that a computer isn't the Windows desktop.
"I can't SSH in, it says connection refused."
"Right, that's the firewall."
"Well what do I do?"
"Google UFW"
"I can't SSH in anymore, it says connection timed out."
"Can you ping it?"
"No."
"Check the IP address in vSphere"
"It changed..."
"Why?" I asked.
"DHCP...! That's what a static IP is for!"
From then on he finally understood that learning actually takes a little effort and curiosity AND yes, it's OK to Google things. He had this idea that he had to know everything, memorize everything, and looking things up was "cheating". Not knowing something and feeling dumb is actually where learning happens rather than pure repetition.
About a year later he thanked me and said that he completely misunderstood my motivations initially and that he thought I was brushing him off and being lazy, when in reality I was giving him the opportunity to learn by not feeding him every detail. He felt like he was failing because he didn't know all the answers and said that he looked back at himself a year ago and couldn't believe what he was doing now and how far he'd come. "I had no idea what an IP address was but now I understand how the packets move through the switches, request an address..." etc.
We both ended up quitting and going our separate ways as the IT department there was an absolute shitshow. He's now a sysadmin and we chat now and then and he's mentioned that he's actually glad he learned in such a fucked up environment because you were absolutely forced to understand due to all the ridiculous hacks and workarounds that had been piled on over the years. Nothing could be taken for granted.
I learned in a similar way and I think trial by fire may be one of the best teachers. "Smooth seas never made a skilled sailor."
Learning only happens through a mish-mash of Trial-and-Error, Trial-by-Fire, Questioning, Curiosity, Reading/Copying/Mimicking, Thinking, Reflecting and finally Doing. All of the above are needed in some measure.
The trick is to do the above without losing our self-confidence in ourselves (we are guaranteed to feel "stupid" during the learning process) that "we can grok it" at some level and over a period of time. The problem today is that there is so many aspects and so much to learn about any one thing that students are trying to move very fast to learn everything which is an impossibility; they need to ruthlessly cut down on all inessentials and learn to focus on only one or two core things i.e. "sift the wheat from the chaff".
Very little of anything worth knowing, in practice, can be given this account or a reliable one at least (physics sure,.. teaching?). Say, after decades of teaching, an exceptional teacher is not going to be able to (in general) report their methods in terms of the explicit accounts of methods as established in books. These are highly varied anyway, and full of rival theories.
Indeed, a person who could give such a count is most likely to be a poor teacher by comparison: since all their labour has been in the creation of these accounts, not in teaching (or far less).
You cannot do both. You cannot both acquire a vast depth of expertise that grounds good judgement (risk/reward, problems that arise in practice, context-sensitive question, intuitions for failure/sucess, etc.) -- and develop baroque accounts of that knowledge (its origins, remembering which papers you read, remembering all your projects, all the theories developed by academics, their history, and so on).
If knowledge is only, as academics say, just their own sort of accounting -- then one would feel stupid all the time. Since almost nothing can be thus accounted for.. and yet the world is replete with highly practiced experts in a very large number of domains.
A teacher of topic X is not an expert in topic X. They are an expert in “teaching topic X.”
But at the end, there is a question about sharing a graham cracker, which I am 100% sure a child of that age understands. They want at least the same amount of graham cracker as the other person. The kid also gets that one wrong, at the cost of his own bottom line. That really sold it.
I'd be convinced that the kid doesn't get it if they swapped crackers and then it stopped being fair.
Fair enough. It's definitely missing the opposing case where 1 graham cracker each is split on only one side and therefore the situation goes from fair to unfair, even though it's the same amount of graham cracker.
> I'd be convinced that the kid doesn't get it if they swapped crackers and then it stopped being fair.
I wouldn't count on that. I think the kid (and most adults) would claim it's fair if they thought they could get away with it. The deep intuition for "fair" that I expect from children would be derived from past experience negotiating with peers, not from any kind of moral theory.
Also children are brought up with super obvious problems like “what object fits into which hole?”. I feel like some of these tests measure less the child’s understanding of the given problem per se and more whether they have previously been introduced to trick questions/illusions.
And even controlling for all that, you’re totally right. Even adults get confused by mass, weight, volume, apparent size etc. sometimes. The kid doesn’t even intellectually know those concepts. His only input here is by sight, but his answer may be different if he got to hold both objects and feel their weight.
The way I currently think about it is that a learning space is a sort of skill tree (poset), and the easy concepts/skills are the ones where we can learn all the prereqs, and then just combine them (join reducible elements), whereas the tricky concepts/skills (the ones which make us feel stupid) are the ones that only have a single prerequisite, so we can't just combine things we already know, but have to do something novel[2] in order to acquire them (join irreducible elements).
[0] and both of them were probably confident all along that they'd make it, the former because they had already sketched out a few likely paths in their mind, the latter because they've always managed to muddle through before
[1] furthermore, having travelled from A to B multiple times, it's difficult for a teacher to empathise with those who are not following
[2] to https://news.ycombinator.com/item?id=41566945 I'd say maybe that's why traditionally we've waited until people are in their late teens/early twenties, significantly ego-invested in research as a career path, and have an experienced mentor, before we throw them overboard into the lake of obligate stupidity[3]?
(there are exceptions: Feynman habitually tested himself by attempting to self-derive [practising research mode] before allowing himself to read expository texts [entering spectator mode]. Somewhere he[?] claims something along the lines of him not being that smart, just that people were impressed after they asked him questions for which he could give answers he'd already spent hundreds of hours thinking about)
[3] everybody genius until it time to do genius shit, yo
The lawyer friend who dropped out went into a field where her "bag was secured", to use a contemporary phrase. The author acknowledges that she was capable; perhaps the root issue wasn't "feeling stupid", so much as "feeling like I'm going to be broke even if I crack this nut."
To an extent the techniques are still woefully primitive too. The standout for me personally is the calculational proof. It’s arguably the biggest advance in how math is done since the equals sign, but despite that it’s still rather uncommon. I suspect it will be another generation or two before it really catches on. Thankfully mechanical checking will drive adoption.
If you want to be taught by a hallucinating crack head that gets only 60-70% at best of what it says right, then yes. As a qualified mathematician, I would suggest that the best path to teaching is small steps in a properly defined hierarchy of knowledge and practice practice practice.
Most of the teaching is seen as rubbish because people didn't get enough practice further down the tree to be able to do it instinctively so higher level concepts can be retained.
That description is only a slight exaggeration of some high school math teachers.
The ability to explain what I know already, hand-wave at what I think I understand about my question, and then get a description that meets me where I'm at is invaluable.
Sure, occasionally Claude will tell me (incorrectly) that a CRDT's lattice operation needs an identity function: you absolutely have to go back and forth with wikipedia.
LLMs are not a magic genie or oracle. But if you use them for what they're good at, they're amazing.
We talk about scaffolding and the importance of pedagogy in math education yet none of that exists at higher levels. In my case it's literally been the blind leading the blind. It's a horrible environment to learn in. I say this as someone who has tackled some really tough material with no issue in the past thanks to having that hierarchy you mentioned. When that doesn't exist or there is nothing else, the process truly stalls. So sadly, I will take the crackhead over nothing.
Then again, maybe I just hate what I'm studying which is it's own problem.
(User23, to what extent can Claude help with finding either door or tree?)
When I realized that, I realized that feeling dumb was actually normal, and that I should embrace it and expect to spend the majority of my career in that state. Not only did this dissolve my embarrassment, but it made me seek out ways to thrive in uncertainty and chaos -- which skills have been to my advantage for many years.
It is uncomfortable to admit you don't know things, or you don't know the best way to proceed, or you don't understand something. The temptation is to downplay that, to pretend you understand, to retreat toward the things you understand well. But poking at the unknown is how you get smarter, and ultimately how you solve problems. It takes courage, especially in a crowd, but it is also what solving problems normally feels like.
But as long as I understand to some degree what we want to achieve, and have some vague idea of what corner I might start in, I usually don’t “feel” dumb even if I know very little about the final solution…
For the first type, I argue it's simply the resistance to a new mental model. The article's example of epsilon-delta language is a perfect example. It's a new way of thinking that takes time (and it did historically) to sink in. Competing on how fast you grasp this new concept is stupid. When the new mode of thinking becomes natural, it won't care how long you took to adapt to it.
For the second type, it's simply an impossible standard to reliably have eureka moments. Clearly, smarter people will have more of these than the average people, but no one can do this reliably. On the other hand, while it takes more work for us mortals to have these insights than a genius, there are plenty of ways to get there that don't require a super high IQ. Teachers should try to foster these moments because they are huge confidence builders, but try to minimise the impact of someone showing off their brilliance.
the author likens your first type to building "mental" roads, which form new pathways of cognition, and takes time, and has emotional resistance, and requires conscious effort and practice to carve out. also to relate the roads to other roads correctly, so the mental map of roads is consistent, and can be traversed.
the problem is that most students do not grasp ideas fully and develop facility with it. when this happens, the foundations are shaky, and facility is lost. then, they label themselves as incapable which leads to a vicious cycle where, the belief of being stupid leads to more stupidity.
the second type is where the roads (ideas) are there, but a route from source to destination is not clear, and the aha moment is when you see the full path in the mental eye.
> Science involves confronting our 'absolute stupidity'.
I understand where the author is coming from, but these are just useless statements. Stupidity and knowing that you don't know stuff are not the same thing. The former involves an inability to understand or learn, whereas the latter involves an acknowledgment of our current state of ignorance and that we can do better.
I don't believe one can be successful in science by constantly feeling stupid and getting used to it. You have to be comfortable with not knowing stuff, but with the drive and self-confidence that you can discover new things and expand your knowledge, which is of course not easy either.
I send this to students who feel discouraged by university-level topics. If math professors find things difficult, then you're probably OK... just keep hacking at it!
We write linq-pad queries against the planetary NSA Meta+C db collected by those neat seeing stones everyone carries since 2008. Its anonymous, its collected unaware, its omni-present, its not deformed by the questionnaire, its obvious the conclusion of the science of neurons and the model of the mind.
And now for a word from our project lead: Peter.
“Best I can do is a star.”
https://scottaaronson.blog/?p=4974
> OK, but why a doofus computer scientist like me? Why not, y’know, an actual expert? I won’t put forward my ignorance as a qualification, although I have often found that the better I learn a topic, the more completely I forget what initially confused me, and so the less able I become to explain things to beginners.
Yet
That sense of feeling of success and failure are so fleeting I learn to let go of both.
They are the fuel that drive my quests but I don't sleep with them. They are both volatile.
Cedric Villani, Field medalist, was saying the same thing. The problem is that not everyone is equally stupid, and if you're too stupid, you won't get the job or be a low performer in your field/team.
You know what it feels like to be stupid? It feels like you are really smart! I feels like you already know all you need to know, about, say, vaccinations, or about hot to parent somebody else's children.
I'm currently a student teacher, and I'm really struggling to get this point across to my students. I'm asking them questions which make them really think, and since no other teacher has done that to them before, they feel really stupid. But they are not being stupid. If they were being stupid, they'd feel like they had it all figured out.
So yeah, there is this emotion, commonly but unfortunately called "feeling stupid", which you feel when you are trying to figure something out. What would be a good name for that emotion???
perplexed?
https://news.ycombinator.com/item?id=41548104
(Mulling/marvelling ATM how wikipedia talkpages, mastodon, o1, or HN are all inadequate for the kind of conversations we all hope for, something about the tension between sublimating koinonia & prowess-seeking)
EDIT Getting a bit selfreferential here too https://news.ycombinator.com/item?id=41567018
EDIT 2: “barely bottled in-game insanity” could work
a) FPD read the Glass Bead Game, and instead of fixating on the game itself as I have done, noticed how the conversations tended toward an ideal (also, IIRC, recommended in the NE): https://franklin.dyer.me/post/123
b) etymologically, companions eat bread together, and symposia involve drinking. Maybe a problem with digital fora is that although we may have moved on from grooming and nitpicking, we still require at least a minial amount of analogue ingestion and imbibing for conviviality? (in the dating context, see https://news.ycombinator.com/item?id=910852 )
EDIT: Re: "barely bottled in-game insanity", either someone needs to tell Stross that Series Landys don't have ignition buttons, or I need to accept the foreshadowing and realise that detail was an early in-story clue of power of eldritch horrors* to transmute and pervert even the most solid basis of innate rural goodness.
* in the world of the Laundry; in our world it was Tata who added the buttons
There is a very boomer joke, probably at least fifty years old by now, which involves a (male, of course) boss who interviews 3 (female, of course) secretarial candidates, and maybe there's some WPM involved, but he also gives them each a pile of cash as part of the interview.
One goes to the track, one invests in the overnight market, etc.
So which one does he hire?
The one with the most wood in front of the hut.
Lagniappe: https://www.youtube.com/watch?v=XSBb8ESt96g
Blonde is americain? The tattooed guy, alpine accent?
https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext...
(I'm guessing North America?)
The singers rotate pretty frequently, but the core of their line up has stayed reasonably constant since they were much younger: https://www.youtube.com/watch?v=LuMUOiaHPA4
Did you get to/understand the bra story from the Radio Melody interview?
(AML was sticking to high german for the interview; for a better example of his dialect listen to the intro to https://www.youtube.com/watch?v=zSwQ1icECNg , handily subtitled in what we call "writing german")
[I can't tell but suspect the tractor there, despite being in Deere livery, is a Steyr as well]
Incidentally, Radio Melody also interviewed Gölä*, which gives me an excuse to point you at the yodel-crossover version of "Indian": https://www.youtube.com/watch?v=sLTLlGCknUw
Lagniappe: https://www.youtube.com/watch?v=ltXpMuH8SI4
* this is the local word for, not shit in general, but specifically the slurry with which fields are fertilised. Durrenmatt also uses it in "The Visit" (1956), but names his town Güllen — dialect being oral and not written, orthography often has a few local conventions but otherwise, as english orthography once was, is up to personal taste.
(To my nondefence, dialect means accent in England, e,g, one would say that the young E2R spoke a Saxon dialect)
Anyways, to try & adhere to a proper division of labor (mine lot is to safely dispose of spilled ink) herewith is summarized my latest attempt at designori integration (more to unpack moving forwards)
How to effectively fund research from the bottom-up? To improve on VDH, in the horizonal Union, your research pays the hoi polloi to be entertained! It is said Mondragonese labor hires capital, can we refine (i,e, “sophisticate the sophistry”) this! Proles hire technicians who hire fonders who hire growth engineers who hire capital! Keep candidates few & condorcet becomes ideal!
(Romer & Veblen have a common thread that human capital have a tendency to be destroyed in the OC.. perhaps you’d like to reply to the hot thread on ShowRunning, while i further refine my understanding of designori in the HolyWoods)
Edit: it is my understanding that present day academia favors destruction of human capital.
It's my understanding that the old E2R had shifted from cut glass RP towards a more generic BBC (although not lerped all the way to Jafaican, or even MLE)?
German dialect variance is probably closer to standard english vs irish english (in which "a ride" may be an action but may also refer to animate objects, specifically an attractive person, and furthermore even borrows some grammatical constructions from the gaelic)? Compare the range between SAE vs AAVE.
EDIT: https://www.youtube.com/watch?v=9J9dRmnlWJY is an example of the same interviewer employing a broader (too far east for my ear to place) dialect; presumably in the AML interview he had also been accommodating by hewing closer to swiss standard german.
EDIT2: and for https://www.youtube.com/watch?v=ogpYWKpYVaw&t=40s (with an eastern german singer) he starts in dialect for the introduction to his audience, but then immediately switches to a very standard german for the interview qs posed to her.
> "Sophistication? Don't talk to me about sophistication, love. I've been to Leeds." —HRE
From Mary of Teck (grandmothertongue, not RP) https://www.youtube.com/watch?v=PO9VEfOeOxo&t=2m9s
The article is about "feeling stupid" in hindsight. Because you cannot unsee it anymore. Which makes you wonder about other myriad obvious things you are missing.
"Feeling stupid" can also mean that you get the impression that everyone around you gets it but you don't.
The irony especially in maths - it seems - is that you can feel stupid because you don't get it and then quit but also keep feeling stupid if you finally got it because in hindsight everything falls so neatly into place that you can't imagine that you had so much trouble to get it in the first place!
The article for this thread has:
"First, I don't think students are made to understand how hard it is to do research."
Well, from my career and Ph.D., I found something different:
(1) In pure math, read some of the best materials, look for unanswered questions, and pick and try to answer one of those.
Took me 2 weeks. Paper got accepted in the best journal in the field.
(2) Applied Math. In practice, outside of math, see a need, problem, weaknesses in what is known/done, pick, and try to answer.
Another few weeks, then submitted to a journal and got published right away, no revisions.
(3) Engineering. In some real situation with some big, i.e., expensive, activity, see where a lot of money is being spent, pick, attack, maybe solve, and write it up.
Most of the work was done an airplane flight, and the results became dissertation in an engineering school.
One device: Pick some advanced topic in math, e.g., 'tightness' in probability theory, and use that as a tool to get some new results on the problem picked. There are problems where such a tool could be used but is poorly known by people concerned with the problem.
More difficult problems:
For the work in (1)-(3), still waiting for a check.
Problems:
(A) Getting hired for such work.
(B) When do get hired, get paid enough to buy a house and support a family.
Lessons:
(i) For academics, never saw where the professor got paid enough to buy a house and support a family.
(ii) For business, if do get hired and paid and are doing good work, then maybe, still, are not paid enough to buy a house and support a family. But the owners of the business are making more money from the good work than they are paying.
So, maybe start a business, do some good work key to the success of the business, and keep all the earnings for yourself.
Looking around, it appears that the people who make enough to buy a house and support a family own a successful business, although maybe just a successful LLC (limited liability corporation -- a relatively simple business type) as a physician, lawyer, tax expert, etc.
So, in simple terms, start a business, e.g., initially as just an LLC, and continue with hard work, good ideas, good insight, until have a successful business, and then sell it. Uh, if the money was from successful 'pesonal services', e.g., law, medicine, then may not have anything to "sell".
These days the Internet may provide some good opportunities.
…And me too, especially when I think outside the orthodoxy, which I do all the time.
As I must be wrong of course I don't talk about such stuff, as one would be lauged at and ignored.