The centrality of stupidity in mathematics

For every job I've ever had, I've been the "math and functional programming nerd", where I know lots of tricks in Haskell and F# and even concurrency theory within Java. I felt very smart.

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.

My first day at computer science I saw a guy with a huge beard playing Dwarf Fortress, and I was like "oh crap, he's like ten times smarter than me."

> You're not the best at everything, and you don't have to be.

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.

That secretary is probably hard funded, makes more money than 75% of the scientists she supported, enjoys vacations, and will have a comfortable retirement.

> 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.

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.

Well said. This should be part of orientation for every new college student.

So did she get to be on Jeopardy!?

Athletes deal with it too. You go from being the best football player in your school, maybe even the best football player in your college. Then you go to the NFL and you're middle of the pack. Lots of guys get lost at the transition: a first round draft pick gets to the NFL and immediately loses his mojo.

I don't know your background, but speaking as someone who has done a PhD in Pure Math, and working with a lot of people who have done PhDs in Pure Math, I disagree with you. It's very much a case of feeling stupid, and being able to embrace that and live with it.

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.

Well, mathematics at a graduate level really is a subject that can only be self-taught, as are most subjects at the graduate level. Yes, some guidance can be available, but the pedagogical hand-holding that is undergrad is simply not possible. The analogy to language only really applies to mathematics that is well-understood and can be taught this way. In grad school, almost no mathematics you encounter is that well understood, so the teaching methods are absent.

There are two authors here, since the post contains an inset about dealing with your own ignorance by another professor. They aren't saying quite the same thing. The inset is saying that every grad student will confront their "absolute ignorance" and it will be difficult, scary and possibly painful. The author of the post is saying it can be a source of joy. I suppose they can be reconciled. It could also be that so little of our behavior is based on knowledge that the only sane reaction is at least somewhat negative, whether characterized by being overwhelmed, or sad, or detached.

Reframing it as curiosity is a good point, although the essay as written resonated with me because it emphasizes the "productive ignorance" of research.

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.

> 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.

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.

Curiosity is when you're reaching out to find out about a thing (let's call it X) which you want to learn more about.

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.

kids struggle with grammar, pronunciation, pragmatics, vocabulary, etc. they do so naturally, and maybe you've forgotten your own struggles, but they're very real

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!

But he is not talking about education, about doing a course, where "getting the right answer makes you file smart". He is talking about research where nobody knows the right answer.

The struggle isn't necessarily bad for learning. It really is a good way to learn. I like it.

But alas, I never thought as a kid that I didn't have time for other things. I was always into something.

As someone who is currently studing maths I strongly disagree with this

> 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.

I think the framing as "stupidity" is to highlight that you don't always chase creative questions. Quite often, you should chase the obvious or understood points.

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.

> He’s really talking about curiosity. Calling it stupid is a cheeky glass-half-empty framing.

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.

For me the real lesson is that simple is hard.

Unless "as cleverly as possible" is zero.

I've become convinced that, in the end, no one really teaches you anything, you end up teaching yourself. That phrasing is a bit hyperbolic. It's more accurate to say a good teacher only gets you 50%, 60%, maybe 70% of the way there, and it's up to you to get you to 100%.

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?

I'm a recent graduate, but also worked in colleges for a couple of decades. Students I've interacted with almost universally blame themselves first when they don't get something. Only when they see many fellow students in the same boat do they tend to blame the teacher. From my vantage point it seems you're either assuming their frustration with the subject is frustration with the instructor, or overgeneralizing based on non-representative students, a non-representative subject, or... a non-representative instructor. I'm not trying to be a jerk, but career longevity wasn't a great indicator of pedagogical insight or its prerequisite EQ.

A good teacher can offer some tools and methods (as in methods of "learning to learn") to bridge the gap between stupidity and (for lack of a better meaning) "enlightenment" Something especially my math teachers in general lacked..

I disagree. I think good students judge their teachers based on how motivated and passionate the teachers themselves are about a topic. Though I've had classes where I only realized their importance multiple semesters after having them. (Mostly the non technical, business classes)

I have the perfect story to illustrate this.

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."

My view is that the when academics call other academics “experts,” it’s just noting who works professionally on a topic. Usually those people will be able to give a reasonable account of their field. But a lot of the game is reviewing the specific subject matter before a presentation. Or steering a conversation toward familiar ground.

A teacher of topic X is not an expert in topic X. They are an expert in “teaching topic X.”

I think this video, and most such experiments involving children, suffer from a central issue: children are extremely sensitive to what they think adults want to hear. The kid watched the woman manipulate the things. He probably figured it would be rude not to acknowledge her changes. They should figure out a way to have another kid as the experimenter and disguise the obvious test/interview situation somehow. Especially the cracker thing feels sooo odd. I can’t believe he would let a peer get away with it, but who’s he to argue with an adult, much less a stranger?

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 crackers question changes from amount to fairness. It's possible that the kid uses a different rule to evaluate it. Like "a small person needs small portions" or some variation.

I'd be convinced that the kid doesn't get it if they swapped crackers and then it stopped being fair.

I only feel dumb if I don’t know how to start looking at a problem, in some cases because I don’t understand the description of the problem either.

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…

the two types are nice !

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.

But when people quit STEM degrees, they don't say "it made me feel ignorant" or "I didn't have enough knowledge". They say "I was too stupid". The author is expressly trying to address those people, and the people who might be able to intervene in their lives. "Yes! We're all stupid! That's part of what it means to learn and research math and science!"

The analogy there isn't as direct as I'd like, because both engineers found a path between A and B[0], and I'd thought TFA was saying (because my experience has been) the initial feeling of stupidity comes from not seeing any path at all between them[1].

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

> It’s cool that AI is making that accessible.

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.

"Mathematics is the Poetry of Science" is a great title that also explains the phenomenon noted in https://news.ycombinator.com/item?id=41570976

You are likely referring to ignorance: not knowing what you don't know thus resulting in a possible false confidence.

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!

anticipation of cluefulness? epistemic frustration? temporarily embarrassed expertise?

perplexed?

If you think outside orthodoxy, why you care what others think?