Ego and Math [video] | Hacker News Reader

Ego and Math [video]

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240 pointsandersource97 comments

"Utility had a strange backseat for me"

I kinda took that to the extreme when I was young. Used to loathe anything practical - experiments, programming, applied math etc cuz you know they weren't "pure" and engaging enough. I would also have a hardtime processing/registering something if I'm not able to derive it analytically from first principles. It felt like cheating if I have to use a formula without fully understanding how it was derived haha.

I was the other way around. Only when I found utility in something could I finally grasp the subject properly. I remember how I was taught derivatives and integrals in HS, I knew how to do them but I was confused as hell. I asked the professor and once explained some uses it all clicked into place.

Same. There's an infinite things to learn, if you can't argue for the usefulness of any piece of information and if that use isn't related to my own goals I will stop you from communicating said information

Same. I really struggle to learn anything which I can't see a practical use for.

Just as an example, in high school learning trigonometry was really difficult for me, like why would I even care about finding an angle in a triangle, etc.?

Only once I studied physics or game dev, this has started to become relevant, and then studying it got SO MUCH easier.

It's mindboggling to me that every teacher doesn't just debut the subject with videogames as a reference.

"Alright everyone, let's make a video game character out of triangles".

"Let's make a little cannon that you can change the angle of. How do you calculate the angle? Funny you should ask.."

"Now let's learn how you'd make the fireball move up and down as it travels. That's a sine wave!"

Every single student understands the basic concept of a game visually, even if they don't play them regularly. It's just a perfect frame of reference and context for applying the concepts in 2D, and then in 3D. And it's so easy to help the students understand how easily those concepts get extrapolated to other things (engineering, sports, whatever).

Back when I studied these videogames were much simpler. I was explained instead calculating areas and volumes for various functions and that was enough for me to get it. The thing is that not everyone was confused and some can take in theory without a practical application. They’re different modes of thinking and I appreciate both, I just happen to fall in the practical group.

Totally! One of the first thing I did after learning Newton's law of gravity, was to write down a small simulation of planets in orbit and how they "dance" around each other. This little exercise totally blew my mind and the code was really simple to code.

There's probably an untapped opportunity here, but ed-tech is such a difficult industry.

I recall encountering a simple domain-specific PL in school in the late 1980s that allowed physical systems to be easily modelled.

Ed-tech can easily make smart kids smarter but that is a difficult sell to the virtuous.

I'm sure someone actually working in ed-tech will correct me, or perhaps even laugh me out of the room, but I still believe in what I figured out around highschool: that edtech, particularly "educational games", have it all backwards.

Kids aren't stupid. If you take the usual boring curriculum with choreful exercises, and try to "make it more fun" by half-heartedly sprinkling in some colors, characters and cheesy stories, it will backfire spectacularly - kids will see you're just trying to trick them, and not even putting much effort into it.

The right way is the reverse: you need to make something honestly, inherently fun, but design it so that it educates users/players as a side effect. Take Kerbal Space Program: it's not designed to be an educational game, but it's fun, and models real-world physics well enough that you get 12 years old researching and understanding the math of orbital mechanics, all because they'd like to do better than "point roughly half-turn ahead of the Moon and go full throttle", and they'd like to not run out of fuel on the way. Or, look how Minecraft is tricking kids into learning electronics, boolean logic, low-level programming, etc.

(I'd mention Factorio, but I think it's a wash - any gains society gets from the game educating kids are cancelled out by the amount of productivity loss the mere exposure to this game inflicts on software devs.)

(EDIT: or, remember Colobot? A very simple third-person perspective game that had you find and refine resources to build robots, which then you used to kill some big bugs. The twist being, instead of controlling the robots like in a shooter, you had an option to program them in a Java-like DSL, inside the game. It was a great way to organically learn programming. The IP owners later made a "fork" of the game, Ceebot, that was pretty much the same, except it focused on teaching you to program robots instead of having fun exploring and shooting stuff. Predictably, that simple change of focus made the game flop.)

It doesn't even have to be a game: leave a kid in front of Google Earth, and they'll learn geography much faster and much more thoroughly than they would from a globe or a book. Not because the software is better at teaching, but because the kid is just messing around with a virutal model of Earth, and learning stuff along the way.

Etc. Etd.

I think it's a tough sell to adults, particularly parents and educators - that if you want to motivate kids to learn, you need to... stop trying to motivate them to learn. Give them something that's honestly fun, involving or benefiting from real-life knowledge and skills, but actually trying to teach them - and then trust that they'll pick that knowledge up on their own.

They call these kind of games "chocolate covered broccoli" and I totally agree.

I think games, have lots to teach, but that most of the time they are a catalyst for learning or inspiration to learn, but on their own, they will rarely actually teach you. It's hard to put the finger on it, as for example, I'm not a native English speaker, but I learned and practiced most of my English from playing video games, and they were the catalyst to make me WANT to learn English, but they didn't exactly *teach* me English.

Another part of it, is I bet if you sample today's scientists and engineers at places like NASA, you'd probably find that a lot of them loved watching Star Trek/Star Wars as kids. So while sci-fi hasn't taught them how to work with Schrodinger's equation, it probably had a major part of what sparked their motivation to get started. Games probably do that too, and then some, thanks to interactivity.

Thank you! Not only I 100% agree with you, you've also managed to provide a few terms and phrases I've been missing, which could've cut my previous comment down to 1/4 of its size, without loss of meaning. Specifically:

- "chocolate covered broccoli"

- "catalyst for learning"

- "inspiration to learn"

> I learned and practiced most of my English from playing video games, and they were the catalyst to make me WANT to learn English, but they didn't exactly teach* me English.*

English is my second language, and I've also learned most of it from video games. Mostly from exposure, but initially through focused effort - I still vividly remember that time when I was maybe 10 or 12 years old, when I made screenshots from loading screens in Star Trek: Generations, and printed them out on paper, one by one, directly from MS Paint, to take back into my room and meticulously translate the story text on those screens, looking up every single word in an English->Polish dictionary. I also remember keeping that dictionary around when playing Fallout 1. The need to understand the stories and dialogues in games is what bootstrapped my English.

> I bet if you sample today's scientists and engineers at places like NASA, you'd probably find that a lot of them loved watching Star Trek/Star Wars as kids. So while sci-fi hasn't taught them how to work with Schrodinger's equation, it probably had a major part of what sparked their motivation to get started.

I agree. And Star Trek is, in fact, what got me interested in STEM. I owe my entire career and most of who I am as a person, to early exposure to captain Picard and the adventures of Enterprise-D.

(A lot of my early STEM self-education was driven by trying to understand the so-called "technobabble", which - at least in TNG - actually made sense. Probably because, in those days, they had proper scientific advisors.)

> Games probably do that too, and then some, thanks to interactivity.

Yup. I mentioned KSP for a reason - not only have I read the accounts of parents impressed by how much advanced math and physics their 8-12 years old kids can pick up, just for the sake of getting better at the game, but myself I also learned these things for the same reason. While Star Trek is what got me interested in space in the first place, KSP is what got me to finally grok how orbital mechanics and rocketry work in reality. It also made me no longer able to fully enjoy any space travel fiction, except for diamond-hard sci-fi.

I should probably give KSP a try again. I guess there's an initial threshold I got to power through first, as I got a bit exhausted after the first mission hehe.

I'm actually working now on a game of my own, with themes of science, and it's indeed a game-first approach rather than an educational game, but I do hope to maybe inspire some ideas and motivation with at least a few players.

I totally believe there's a lot of untapped potential in this area, and advancing towards cracking learning motivation + capabilities could have a huge impact.

> I should probably give KSP a try again. I guess there's an initial threshold I got to power through first, as I got a bit exhausted after the first mission hehe.

What made all the difference for me was a mod (Kerbal Engineering ...something?) that calculated ∆v for each stage as you were building your rocket. Coupled with a ∆v "subway map" of the game's solar system, this solved the problem of running out of fuel half-way through the mission. I eventually learned how to do the math on my own, but I would've given up long before that happened, if not for this mod. It's been some time since I last played KSP, but I hear that this functionality is now built into the stock game.

Good luck with your game! Give me a shout if and when you need someone to play-test it :).

They put something like that in ksp2. very useful, even if the numbers aren't 100% right.

Very glad I'm living in a post-ksp world, even of I'm not playing it, for the real life rocketry it enabled.

I would love to have some sort of statistics on what the proportion of this feeling is. My suspicion is that the practical approach is probably about 90% of the population (who is willing to learn math at all). Would be helpful in trying to figure out how to tune learning programs. (I say this as one who is perfectly content to learn the theory directly and with little-to-no practical motivation, but my impression is I'm very much in the minority on that.)

I was going to say that the curriculum is tuned in favor of those who can just learn by theory, but then I realized that's not even true. It's tuned in favor of those who will simply swallow it without any idea what it is for; it is neither contextualized in terms of what it is practically good for, nor is it contextualized in terms of theory. It's just... there.

I’d be curious to see that as well. I loved math until it became too abstract for me to grasp so I lost interest in it. And that worked pretty well as a self selection for the field, well, a large part of it. I wouldn’t want to be in the academia anyways…

> I really struggle to learn anything which I can't see a practical use for.

That's a close-minded, ignorant world view. Much of the world's most important advancements were made before any practical use could be seen. Why do you think that way?

> Why do you think that way?

Probably the same reason that you're such an ass (genes).

Sorry man. Just asking an honest question. It's interesting to me that one can hold two opposing ideas and see no issue:

- History has demonstrated clear value in discovering and understanding concepts that have no practical use today

- One should not care to understand things that have no practical use today

Seems bizarre to think both things. That's why I asked.

You are shadowboxing - fighting an argument nobody is making. Someone is describing their personal experience of the world, not arguing that this is the best way to think about the world. It's an opportunity to learn about the ways that people learn things differently, if you can be curious and kind about it.

You're right. I could've been kinder. Apologies.

It's just the way my mind works and motivated. Motivation is a very elusive feeling that I did not find easy ways to manipulate. It's not as if I'm totally blocked from learning stuff with no clear purpose, but it will require much more mental capacity that is often difficult to muster in the day-to-day routine. Another example, is I did try to learn what I perceive as totally theoretical math such as "prove that there are infinite primary numbers" which was a nice idea to entertain, but it didn't really make me want to dig in further. On the other hand, learning about linear algebra in the context of machine learning, suddenly got Linear Algebra a lot more interesting and easy to learn.

Makes sense. Somewhat related --- I find procrastination to be a very similar feeling. I know what I should do, but I feel compelled not to do it, for whatever reason.

I think procrastination and what you are describing are slightly different, though, because procrastination stems from stress and emotions for me, whereas with what you describe, it doesn't sound like you have to be stressed to experience it.

> Much of the world's most important advancements were made before any practical use could be seen.

In a sense, yes. But usually this was kind of accidental - as in, people making those breakthroughs weren't doing it because they loved manipulating abstract symbols, or believed that someone, somewhen will find it useful; rather, they had some immediate-term reason for doing the work - a problem to solve, a person to impress, or just doing it for shits and giggles - and only later it turned out their work was the key to something transformative.

I have a similar "mental make" as GP too. Over the years I realized that for me, it's not about practical use to me - it's about knowing why something was invented, what problems the inventors were trying to solve. Learning the historical motivation "grounds" the concept for me, and makes it much easier to understand.

Wonder how many people here have similar story.

In primary and secondary school, I had troubles with math - mostly caused by me not doing homework exercises and generally avoiding work (probably an early indication of an issue that took 20 more years to diagnose...). It all changed when I got interested in gamedev - suddenly, I've caught up with most of the material I was bad at, quickly learned trigonometry beyond the secondary school program, and then some basic vector and matrix algebra - and I distinctly remember it all starting with a simple problem: how to make a sprite rotate and move in circles?

Couple decades later, I still have a kind of theory+applications mindset: I always seek to generalize and abstract, but I feel lost when presented with a new abstraction without any context. Over the years, I realized I learn and understand things most effectively by seeking out answers to the question: why?. Not in the sense of, "what will I ever use this for?", but in the sense of "why was this invented?", "what were the problems people who invented it were trying to solve?". I trace the topic back in time until I find the point where the "why" and "how" are both apparent, and then go forward from there.

Same here. A lot of concepts seemed didn't matter if cannot be reflected in real world. That has changed for me though. Abstract things and first principle, zero knowledge actually quite interesting to me now.

Congrats. I’ve made some progress on the abstract axis but still with some faint endgoal to make some progress in the practical realm I feel more grounded in.

I think the parent comment + yours (and others off parent) provides a perfect encapsulation of one of the dimensions of teaching / learning: what's often referred to as "style"*. One way to summarize, specifically, might be something like "inductive" vs. "deductive".

As my experience has ... accumulated ... through the decades, I've come to feel that these sorts of differences / preferences likely don't have much impact on ultimate (potential) "level"**. And, I think you see this and related notions of "what mathematics 'actually is'" echoed (in a very fractal-like way, +1 to the universe in achieving a consistency we'll never rival) across the development of individual mathematicians as well as through the history of mathematics [1-6].

These distinctions are important in "pedagogy" - can be very helpful for teachers and students to be aware of and work at, especially at the more "basic" levels. This can make a massive difference in how an individual's arc unfolds - with extremes of "F this subject" vs. "I'm willing to accept low pay in exchange for torturing myself with this material for the rest of my life!" But, aside from trying to be mindful of the differences - and all involved, ideally, trying to USE awareness of knowledge and "EQ" and all of that in making the mutual learning enterprise work for everyone involved, many other aspects of the differences can just be outlets for time-wasting if focused on IMO (/ experience).

* AFAIK, not really my field though and it has been ~15 years since I did any significant reading / study in the area - for the sake of 'full disclosure'

** The effects end up more in details of notes, problems and areas people are drawn to more or less, etc.

[1] https://terrytao.wordpress.com/career-advice/theres-more-to-...

[2] Polya's "How to Solve It", in particular, I think of (from the intro): "The title of the very short second part is 'How to Solve It.' It is written in dialogue; a somewhat idealized teacher answers short questions of a somewhat idealized student.") - many options for accessing / buying, but, for this text, it's in the (unfortunately images) here - https://math.hawaii.edu/home/pdf/putnam/PolyaHowToSolveIt.pd...

[3] https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&d...

[4] https://www.maa.org/sites/default/files/pdf/upload_library/2...

[5] https://en.wikipedia.org/wiki/Galois_theory#A_non-solvable_q...

[6] https://en.wikipedia.org/wiki/Hilbert%27s_program

... and, so many more, of course...

Same. I studied derivatives and integrals twice. First time in HS, and it was just a bunch of random rules I had to learn by heart to be able to do these calculations that were required to pass the tests.

Second time was during the first year at university. Here the professor explained that derivatives are needed to calculate the speed with which something changes. Since we were studying economics, interest rates and growth rates made great intuitive examples. Or that integrals permit us to calculate the area under a graph, thus making it possible to calculate for ex the total expected value of something over time. When I saw the utility out of this mathematical tools, studying how to apply them became of great interest!

Wow, my thoughts word for word

I'm still like that, albeit with some plasticity to avoid dying on my lonely rock.

I've come a long way I guess in the sense that I've learned to adult my way through things that don't necessarily excite me :|

Programming in particular was a gamechanger for me and helped me see and appreciate the beauty in practical problem solving using simulations etc.

Can you describe what do you mean by simulations in this context ? you explore various solution configurations ?

Yep kinda like that. I call it Answer guided monkeying around :) I try and see if I can arrive at the answer using Monte Carlo simulations and then try to work around that. Often times they give valuable insights and help uncover symmetries etc that aren’t obvious.

hmm seeking symmetries, the best kind of fun

> I would also have a hard time processing/registering something if I'm not able to derive it analytically from first principles.

This really resonates with me. I always had a really hard time with anything where I just had to memorize formulas, but I didn't have any issues if I could derive it myself. For this reason I actually struggled a lot more with algebra in HS than I did with calculus in college. I don't know if it's just the teachers I had growing up or if it's a more broad issue with how the curriculum is structured, but I didn't even realize you could derive things from first principles until I took calculus in college.

That section was great. As soon as he said "interesting" I said "hard"! This is a real gem, there's a lot of wisdom in this short talk.

...Used to loathe anything practical - experiments, programming, applied math etc cuz you know they weren't "pure" and engaging enough. I would also have a hardtime processing/registering something if I'm not able to derive it analytically from first principles. It felt like cheating if I have to use a formula without fully understanding how it was derived...

Hello, 'undergraduate me'.

"haha" indeed. The universe is still experiencing California-splitting [1], planet-slapping [2] spasms of laughter at my ... stupidity [3] (speaking only for myself, here, of course).

[1] https://www.bbc.com/news/world-us-canada-48921915

[2] https://en.wikipedia.org/wiki/Tunguska_event

[3] https://archive.org/details/novicetomasteron0000mori_w1f1

Exactly my experience! Can tell you how often I was on the brink of failing school/college because I wanted to derive as much as I could from first principles - under time pressure in an exam! I did myself no favors.

I now find I learn better by being the opposite - finding a problem to solve and using math as a tool.

> I would also have a hardtime processing/registering something if I'm not able to derive it analytically from first principles.

I still find it easier to understand something if I understand it from the ground up instead of in an ad-hoc way. For example, I found it easier to reason about probability once I had seen a rigorous definition for what a probability distribution is. I guess the reason is that it gives me a way to sanity check my intuition.

I still struggle with the fact that in software development, you get hundreds of technologies thrown at you and you barely have any time to understand them all fully. It makes me sometimes feel not very confident in what I do. I feel that I could understand e.g. Kubernetes better, if I had real in-depth (not just superficial) knowledge about networking. A lot of the time I'm just missing crucial information like "what problem are we trying to solve?", "why does this technology work the way it works?", etc. Something like Kafka is another example.

I had a hardtime remembering if I didn't understand.

But I tried an experiment at uni (once...) to learn a subject solely by rote. I could typically recall about 7 out of 10 items. I was astonished I could do it at all. Unfortunately, I lost all my usual strengths in that subject: unable to generalize, unable to justify, unable to adapt.

Re: software technologies: non-leaky abstractions are the way to understand without details. Algebras are a great example: arithmetic, concatenation, boolean, Kleene, relational. Although they still leak (overflow, PCRE, etc), an idealized core plus ad-hoc crap beats all-crap. jq has an algebra of , and | operators (though it doesn't call it that)

\aside git: I wonder if git internals can be interpreted as an algebra (absent in UI)? TBF I tried to design a programming language around git internal operations (tree navigation and construction), but it was tedious to use. Maybe something analogous to a declarative SQL over relational algbra would solve this?

I enjoy his 3B1B videos, but this talk did not resonate with me at all. I was good at math as a kid, but no part of my motivation came from "a desire to be seen as being good at it." If anything, many people looked at me kind of funny for being good at math, so I learned to play down my ability when necessary. Maybe it's just because I grew up in an earlier era, but being nerdy was definitely not cool when I was young.

I grew up in the eighties when being nerdy wasn't cool, but math was something that people recognized as real, not a nerdy invention, even if it was weird and nerdy to enjoy it. Other nerdy pursuits like D&D or fantasy (or, at the time, computers) were seen as escapes for people who couldn't face the difficulties of the real world and had to invent easier worlds to live in where they could pretend not to be lame. By contrast, math was real and hard. Every kid in school had moments when they wished they were better at math. I was a weirdo and an outcast, but I was a weirdo and an outcast who had an ability that people recognized.

Being better at math to make up for being socially useless in every other way didn't take me very far, though. Once I got to a top ten PhD program and was surrounded by people who were just as smart, some of them much smarter, and I faced the likely reality of ending up at minor university cranking out trivial results to get tenure, permanently outed as a mediocrity, making minor contributions that did nothing to advance the real work done by brilliant people, I couldn't face years of hard work for that outcome. Now as a programmer I have zero prestige and negative social cachet, but I get to do useful work on educational software used in primary school classrooms.

Programmers don't have negative social cachet anymore.

I imagine telling people "I build software to help young children learn" will get a nearly universal positive reaction.

Not from parents who've seen all the "educational software" and who want their kids to study and practice instead of more time staring at distractions on screens.

"Educational software for primary school" is a field far more crowded with trivial results than pure mathematics is.

More videogames aren't what kids need to help learn better.

That's not the kind of software I work on, but if you go to a classroom, you'll probably see that when students are practicing via a "video game," they still get all the elements of instruction that we did when we were kids, with the addition of immediate feedback and better teacher awareness. When I was in school, kids would work on paper without getting any feedback until the teacher wandered by their desk to look over their shoulder, which could be the whole class period if the teacher spent time with other students first. Teachers would have to be pretty sharp and active to notice during class time that half the class was struggling with a certain kind of question; they might catch on later when they graded assignments, or they might not. I saw a teacher glance over a screen of updating results and within minutes interrupt practice to reteach an idea. Again, not my software, but I wouldn't dismiss the value of it without seeing it in action.

Teachers these days aren't that different from decades ago. I'm sure there are plenty of bad and lazy teachers, just like when I was a kid, but the ones who drive the adoption of software are engaged, hard-working, and interested in results.

Is there a way to filter simple gamified education tools vs the type of tools you allude to? I understand not wanting to dox yourself, but as a new parent it's difficult to vet education tools before my child is already enrolled and using potentially poor education tools.

I'd approach it the other way around: find good teachers, and trust them to choose the tools they use.

I also grew up as a nerdy kid long before being nerdy was cool. All the nerdy kids I knew took a lot of pride in being smarter than the cool kids. Being good at math was just an extension of that. I actually didn't know playing down how smart you were was a thing until much later when some of the popular kids who I had just assumed weren't that bright went on to become engineers, or doctors, or one who went on to get a PhD in biochemistry. I know a few of them still and they all talk about playing down that side of themselves in order to blend in. It makes me feel pretty silly about how smug I was about my intelligence back then.

Being open about your motivations to yourself and others and identifying how that can help or hinder you is the real takeaway here, not the particular motivation discussed.

I personally appreciate the candor and his own story of growth in this subject.

The video resonated with me. Looking back I think that I studied pure math in further education because I thought that others perceived it as the hardest subject you could study and therefore would think highly of me for studying it. I think that motivated me much longer than Sanderson too, as I think it's the main reason I started my PhD in a topic that probably wasn't the most interesting to me. Along the way I developed an appreciation for the innate beauty of the subject but these days I find it much more rewarding to work on something that's useful to someone else no matter whether it's particularly easy or hard.

The sad thing about wasting your youth trying to be seen as smart or successful is that later in life you'll probably have much less freedom of choice in what to work on.

I opened the video in the background and immediately recognized the person: the author behind the 3Blue1Brown YouTube channel. It has a long series of video regarding various mathematical topics which are rather accessible.

My favorite by far is a "proposal" for an alternate notation which makes much more sense and, if adopted, would make mathematics way less intimidating (Triangle of Power (2016), 3Blue1Brown - https://www.youtube.com/watch?v=sULa9Lc4pck ).

I'd give him a Fields medal (or at least an honorary mention of some sort) :-]

Note that the alternate notation was suggested by someone named "2'5 9'2" on the Mathematics Stack Exchange [1], and not by 3Blue1Brown.

Obviously, this should not take away from the amazing educational work that 3Blue1Brown has achieved, but the honorary mention would probably suffice :)

[1] https://math.stackexchange.com/questions/30046/alternative-n...

He mentions in the video that he saw the idea in "a math exchange post" and he also has a link to the exact post in the video description. Doesn't that count?

It does count as properly attributing the idea, yes. It doesn't count as having it. Possibly the Fields Medal suggestion assumed it was his?

The point was that it's just not his invention. He attributes it extremely clearly, no one is accusing him of theft. But you don't get math prizes for presenting someone else's ideas.

Fortunately, you do.

https://en.m.wikipedia.org/wiki/Leroy_P._Steele_Prize

I agree with the comment that says this removes the personality of each function

Why doesn't subtraction need personality?

I don't think there are many deep theorems about subtraction, but there are a lot of very deep theorems about powers (and polynomials), the exponential function and logarithm functions (especially in complex analysis).

> immediately recognized the person: the author behind the 3Blue1Brown YouTube channel

Yes, that is noted by the 3B1B in the title

But yeah the asymmetry of operators in math is exhausting.

I'm a fan of his work but that's just not what Fields medals are for.

I'm a math professor with about a decade of teaching math. On the list of things that make math intimidating, for undergrads at least, the notation for powers, roots, and log, is very low. The "proposal" also ignores that

1. The kth root of x is often denoted x^(1/k);

2. We have convenient shortcuts for the square root and the natural logarithm;

3. Parentheses become a mess;

4. The notation for squares, cubes, etc. is deeply entrenched; does anyone really think that write "x triangle 2 above" (yup, it's a mess to write in ASCII) instead of x² or x^2 would make mathematics less intimidating to everyday people?

5. Having symbols, subscripts, prescripts, and superscripts above the symbol all strewn together is much more intimidating to anyone.

6. How do you nest them? Try to write down log_a(log_a(x)) to see what I mean.

I enjoy 3B1B's videos in general, but this one really only makes sense if you don't think too much about it.

I'm no math professional, and my struggled around some math topics was rarely notation, but 1) a bit of hidden information/culture (which can show in notation too) 2) a misalignment on what the topic was trying to achieve.

The last bit which I can't describe clearly is 'maturity'. Sometimes an idea just eludes you for years, until it doesn't. Changing maths would probably not have fixed this.

Oh, and the underlying human feelings behind the problem solving made by others. It seems that a lot of maths is lowering energy required to express or find a solution, no matter what subfield you work on, that seems to be the goal.

1. The kth root of x being x^1/k would be written as this:

  k     1/k
  △x = x△

2. I don't think these are as important. Also, ln x / log x is still the same or more symbols than `e△x`.

3. Not significantly more than parens in exponents. You also get rid of one level of parens from log.

4. Notation change is often a huge hassle, this is absolutely true.

5. Isn't this a problem for current notations as well? Especially if you ever want to put a complex expression for k in the k'th root notation.

6. log_a(log_a(x)) would be `a△(a△x) `.

Still, I don't personally like the symbol. The biggest problem to me is that it requires smaller letters (subscripts/superscripts) all the time, which makes it more annoying to write than the regular notation for the base of an exponentiation and the argument of a log or root. Complex expressions in small letters are very annoying to me, and this notations makes it necessary to use them in all cases, where the normal notation at least has some cases where this is not needed.

The answer by user 'Blue' on the same thread seems more practical to me. Their answer[1] uses notation that doesn't translate to this messageboard.

Perhaps a reordering of their method, using the existing caret notation:

  b^p = r :: the result from base b with exponent p

  ^pr = b :: the base giving result r from exponent p

  br^ = p :: the exponent yielding r with base b

[1] https://math.stackexchange.com/a/1158802

For me, the intimidating bit and source of "math anxiety" is that there's only one right answer. You either get it or you don't. At the time, I had a fear of failure and this caused a lot of stress, especially at the chalkboard in front of the class.

I preferred humanities, where there was wiggle room and you could bullshit your way around the gray areas. That all ended when I became a dev, where failure is nearly constant so there's no time for feeling bad about it.

mathematicians: Humanities are stressful because you don't know what's the right answer!

Good academics fear the difficulty of good humanities. Bad academics enjoy having the excuse.

This is why "humanities" in general have a terrible reputation, but the individual people who've done great work are respected.

As someone who has studied both humanities and "hard" subjects, I'd definitely say that I respect really good researchers in the humanities just as much as I would a Fields medalist, but there's a much lower bar and there's a lot of shoddy research being done (as well as really bad students who just coast by somehow, something which is much harder to do in math-heavy subjects).

Then there's also the problem that there's not even a clear consensus on what great research is in a "soft" field. I might find someone highly accomplished, but someone else might think the opposite. With maths, either someone proves a theorem or they don't, there's almost no middle ground (Mochizuki notwithstanding).

Composition of mathematical notation is always terrible. Where do I write the -1 on the square root to indicate that I would like the preimage? Even worse, Where do I write this on a trig function?

> On the list of things that make math intimidating, for undergrads at least, the notation for powers, roots, and log, is very low.

I guess whoever reaches undergrad math courses already passed this hurdle. It would be interesting to know if this makes a difference for school children being introduced to the subjects, like ~8th class for roots, or ~10th class for logarithms.

I think his key point applies to many other fields. I'll summarize it as "evaluate the work you do, at least in part, on its utility to others".

I've heard of devs who were asked to solve simple problems, but went out to choose exotic and complex approaches because that tech is the latest new hotness (though not well tested). I'm sure there are other examples.

This comment immediately clarified a thought I was trying to form after watching the video, which is: math is inherently part social activity, like almost everything else we do. Grant first described the ego thing in embarrassed negative terms like ‘childish’, but it’s a good idea to recognize that this self-deprecating framing isn’t the only way to look at it. Math is a language we learn and use to communicate, and being good (and being seen as being good) at speaking the language is usually an important step to taking part in the conversation.

In case you're hesitating to click on a video link: it's only 10 mins long.

Few minute read instead: https://youtubetranscript.com/?v=z7GVHB2wiyg&t=183

On the other hand, seeking prestige at the expense of personal satisfaction, say, by conducting research in an "interesting" or "important" area, may be seen as an altruistic means of furthering progress in that field, and seeking personal fulfillment through the knowledge that one is helping others may be seen as a form of self-indulgence.

That’s the beauty of helping others! It’s a form of self-indulgence which is entirely ethical. Normally “self-indulgence” carries with it a negative connotation. In the case of altruistic behaviors, it’s actually a positive thing

I have an incredible feeling that this quite short commencement speech is quite complete in its treatment of the subject of one’s relationship to career choices. Utility, originality and personal appreciation of tasks are key parameters in order to find a fulfilling job.

Historically, mathematics that initially seemed to have little imaginable practical application later become core to various fields of physics and engineering - non-Euclidean geometry as developed by Gauss-Bolyai-Lobachevsky amd Riemann became the foundation of Einstein's general relativity, number theory became the basis of cryptography (to the likely dismay of GH Hardy), etc.

So keep plowing away, mathematicians, at whatever you want to, and don't be surprised if some applied science type picks up the results and uses them for something in the so-called real world (but don't expect many of us to check your proofs, no thanks, taking it all on faith is the norm).

Yes, math is to physics what art is to design.

It's crazy what these two fields produce, but once I a while something useful comes out.

Grant Sanderson is a righteous dude.

His videos on mathematics are amazing.

Grant's Patreon, for anyone who feels so inclined: https://www.patreon.com/3blue1brown

This reminded me in part of the work of Francis Su on math and the virtues: math and human flourishing https://youtu.be/FTXhj-puDgw

Mathematical practice can be a means of achieving the various virtues, and ‘show up’ (or make us more sensitive to) our vices. Meaning that there is something inherently good in the learning and practice of math, for it to lead to more good and to manifest to us what is bad.

There's a small number of people I know who say in social contexts "I love math" but they come up blank when asked what fields they find beautiful. I find this correlated with narcissism and it makes me believe these people don't actually find math beautiful, but just like the idea of others thinking they do and want to assert that they're the smart person in the group.

Devil's advocate: they liked the little bits of math they were exposed to.

I don't know if there's a field i like. But there's something intoxicating in math. Sometimes it's very strong. I listened to a CS lecture now and normally I find CS a bit boring but as he kept describing aspects of the problem he was facing (finding points in intersecting disks) and as the problem got more complicated I got the itch lol. Sitting in a logic class is more exciting than a roller coaster. It's a bit scary because I don't understand why.

Makes sense, and I think many people fall into that category. Saying "there's something intoxicating in math" and that you enjoyed the roller coaster of a logic class already excludes one from the situation I am describing.

My comment was too judgemental. People should be allowed to say that they enjoy something without any follow-up and without being judged for it. I think that sometimes it just seems like a facade when people say they really like math because when you try start a conversation on the topic its like they're not actually interested in it at all. It gives the impression there's something disingenuous about their proclamation of liking math. But perhaps its just the way I personally have approached it.

Maybe it's the shallowness of pop math? I sometimes find pop math even insulting. Not a grown up response, sure, but how many times can you hear "WaOw there are Bigger infinities???" sure it's nice that they find it cool but when I start to talk about the things that are more interesting I can see the shift in them as they quickly get bored. Perhaps it's the quick realization that no one outside of math actually cares that is insulting, after all those times they hyped us up.

I think you're being a bit harsh. I can barely do simple algebra right now but I like math. I liked my math classes in high school and college because it felt satisfying to both learn and solve problems. It felt challenging in a unique way that my brain had never been forced to do before. It felt good that I was above average at it and it felt good to understand something something that started off looking like gibberish.

You can love paintings without knowing anything about how to paint.

Perhaps I'm being a bit harsh. On the other hand, I feel its more similar to asking someone who says "I love reading fiction" what is a book they liked and them not having an answer.

Yeah but it is the situation. I remember a roommate saying "I love math!". When I talked with her a bit I understood she likes doing solutions of easy integrals\derivatives because they are simple, relaxing, and rewarding for her - her math classes were easy and she could get accolades by just doing those simple excersizes. Ofcourse the distance between that and a proper field of math is vast. But that was math for her.

There are comments on HN I find correlate to blithe superciliousness. A light expression of common sentiment (math is beautiful!) without domain knowledge is somehow a marker of social superiority... yeesh. kandel's sibling comment being the kind of reasonable rejoinder that shouldn't have to exist if there was a good faith follow-up to this sort of conversation, instead of turning up the nose...

You know what, I think you're right. I should be less judgemental. I'm not sure why I thought that was okay. I think some of it comes from the disappointment when someone describes themselves as loving math being part of their personality, but then when I try to engage with that as a jumping-off point or common interest its like there's nothing there.

Props to you. Thanks for the kind follow-up

To me the perfect example of people who claim to "love math" but actually don't are people who wax endlessly about something like Category Theory (or non-classical logic), but they can't even define what a group is.

No disrespect towards any serious scholar of Category Theory or Constructivism.

Can this be used in a way similar to “supercomputers” proposed for Haskel?

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