Lessons I wish I had learned before teaching differential equations [pdf] (1997)

“Why is it that no one has undertaken the task of cleaning the Augean stables of elementary differential equations? I will hazard an answer: for the same reason why we see so little change anywhere today, whether in society, in politics, or in science“

I see a lot of parallels with the Physics department and I think the reason is much more depressing. Both fields embrace a sort of masochism and active desire to keep knowledge impenetrable b/c it acts as a mechanism to feed out the dummies. The system acts an an informal IQ test - that maintains the prestige of the departments. If you're pigheaded and clever enough to get through the masochistic torture is that their undergrad textbooks then you're probably pretty clever and so the prestige of the degrees is maintained.

There is also a certain veneration of the establishment and traditions. I remember on my first day of Classical Mechanics the 50+ year old teacher beamed with pride when he told us he used the same exact textbook when he was in school. As if it was written by the Shakespeare of textbook writing and nobody has managed to surpass it in decades. It' frankly should be an embarrassment

As he observed, other departments will step in an do much better. The best linear algebra class I had was a graduate course in the electrical engineering department

I had a lot of hope for things like Khan Academy, but the issue is video is not text and it's hard to iterate and improve on. I really wish textbooks with open licenses would take over and they could be reworked and improved year after year by different people

What's the problem with a good book, age? Baby Rudin's first edition is 70 and the latest one is from 1976. It's still widely used and will be for a while.

Honestly your problem was that you didn't know any Classical Mechanics yet and you were assuming that the volume of recent developments made old books obsolete. Maybe in Biology, in Physics getting to recent developments would mean that you're familiar with Goldstein, Landau's Vol. I... Abraham-Marsden? Arnold? Those are old.

Often newer editions actually worsen textbooks and then only a few contemporary books become references in the long run. It's always been like this, there's tons of great books from the 70s that aren't used today and could definitely do. At least they're not ~1,000 pp. of waffle, which is what you usually get for your first textbook on anything nowadays.

You're conflating things. This issue isn't that Classical Mechanics has somehow evolved or changed. It's that people continue to find new and better ways to explain and illustrate concepts. At lest personally I don't know of any field or book where I've felt "Hmm, this is basically perfect and I can't imagine a better way to explain these concepts".

Have you looked at for instance Khan Academy's Grant Sanderson (aka 3Blue1Brown) Math videos? it's really apparent there is a LOT of room for improvement in pedagogy.

As the linked PDF illustrates, most people are teaching along a set formula and sequence of concepts. Good teachers will try to tweak and iterate on these formulas and evolve a better curriculum that sinks in better for students.

Naturally as time goes on, if each author has to start from scratch, then it becomes harder and harder to beat "the best book on BLAH" from the last 100 years. (Though I refuse to believe it's a monumental task to write a better textbook than Rudin)

If you have open copy-left books, then in theory people could start with a Rudin, fork it, tweak it and improve it. 70 years of improvement could yield some amazing forks!

"Often newer editions actually worsen textbooks"

That's typically because they select a random new author to in-effect update their copyright date.. and the new author is rarely of the same caliber as the first

There are excellent textbooks on Classical Mechanics, probably because it's a crystal clear subject and you can give a detailed account of the essentials in a single volume without handwaving. Of course everything can be improved, but it also can be muddled. If it works think twice before fixing it. Kind of what happens with Rudin and introductory Real Analysis.

On the other hand, there's for instance Optics where you basically have to condense an encyclopaedia and there's always prettier pictures. Or Thermodynamics, Fluid Mechanics etc that can be taught in different ways depending on the curriculum.

There definitely should be pedagogical considerations in higher education, that's lacking because it's usually an afterthought. And it also should be very clear to people getting into higher education that at some later point pedagogy must end and you have to be capable of working your way through the material.

I think you have the perspective of somehow who's succeeded in Physics. In your typical introductory physics class less than half the students will walk away with a very solid understanding of classical mechanics.

To my mind, if the textbook was actually excellent then that would be 80%+. We're nowhere near there. I think there is LOT of room for improvement

But sure.. Thermodynamics.. things could be worse :)

Sometimes things are just hard because they're complicated and you need to buckle down and learn your multiplication tables. But at least in my own life experience, the vast majority of the time things are a problem because their poorly explained - often by people that poorly understand it themselves.

Once you truly understand something inside and out - and look back on it - it all generally looks relatively simple. But it takes a special talent to be able to go back and reexplain it from the naiive perspective

I almost wrote above that, to my knowledge and IMVHO, no one has succeeded at writing a book on Thermodynamics yet. I self-censored because that would be too flippant, wouldn't it? Lmao

I took thermodynamics over 30 years ago. I remember having the feeling of learning a different language using a text book whose explanations also needed translation. I remember the book explained Entropy using chaos theory or randomness and talked about popular philosophy during the 19th century. After a bit of mental torture, I realized by Entropy they really mean Thermodynamic Stability. It is just that heat usually dissipates when materials touch, they were using words like chaos or randomness to describe the process. But their description was vague and poorly conceived.

I think Thermo can only be taught if you have a solid foundation in statistics. And stats... is not really taught in the US? I tried to selfstudy a bit, but the textbook situation with stats make math look amazing. For very intro practical things, there are stuff like John Taylor's book... but past that anything rigorous - I actually have no idea how people learn anything

To my mind, if the textbook was actually excellent then that would be 80%+. We're nowhere near there. I think there is LOT of room for improvement

I think you overestimate the capabilities of students entering university (even 20 years ago), and underestimate how poor high schools can be in preparing said students.

I went to a mediocre university. A 50-80% drop out rate was there for both physics and EE - I don't know how it compares to the other engineering. And I did not even consider it challenging. Almost all the classes were a breeze for someone like me who was well prepared going in. At least in that EE department, the teachers were very dedicated to teaching. They would allocate 3-9 hours a week for office hours, and the pace they taught as was slow (probably only covered 70-80% of the material that is covered in a top university).

Students were given lots of chances.

The reasons they drop out are:

- Poor preparation at the high school level

- Poor discipline. A lot of students didn't transition well to independence, and didn't have an authority figure (e.g. parent) controlling their schedule.

- Realizing too late what it means when courses are built on top of other courses. Thus you'd have people getting an A in Calculus I, but almost failing Calculus III because they didn't realize they needed Calculus I beyond the course.

- In high school you can get far with a cursory understanding of the material. At university, you could get a B, or even an A, with that approach for introductory courses, but that approach will start trending towards an F in junior/senior level courses.

Sure, I agree with you that pedagogy can be improved, but I expect that 80% would at best become 60% if all you focus on is pedagogy.

In your typical introductory physics class less than half the students will walk away with a very solid understanding of classical mechanics.

That's probably true.

To my mind, if the textbook was actually excellent then that would be 80%+. We're nowhere near there. I think there is LOT of room for improvement

In my view, that's probably false. I don't think the problem is masochism, gatekeeping, and people holding on to old textbooks. I think the problem is thatclassical mechanics is actually hard, at least for most people. If you come in to beginning classical mechanics wanting to have learned it, rather than wanting tolearnit, no textbook can save you. And I think that many people come in that way. They want it out of the way as a prerequisite for something else, rather than really wanting to know it for itself.

Why should pedagogy ever end? That's like saying at some point in health care medicine must end and you're responsible for your own treatment. What are the professors for, doing research and abusing their grad students?

The idea is becoming intellectually independent, arriving in the stage of self-pedagogy if you like. Peer learning when there's the chance.

You can't realistically expect that there will always be someone up the ladder to explain things to you. I mean, who explains stuff to the professors if it worked like that?

You seem to have a very individualistic notion of education.

Even at the research level we are not independent islands of learning and discovery. People collaborate, some pickup certain concepts better than others and vice versa. So we teach and aid each other.

It seems you're firmly against this notion? Or if not please clarify your position?

I mentioned peer learning, collaboration is that.

I think everyone should be capable of working alone as well, and that has been the general assumption around as far as I've noticed. Of course collaboration is usually way more productive and also unavoidable.

But we were talking about education. Theses are individual for a reason.

When you get to university, the lesson is very much that you have to learn things yourself. I found that the more decorated the professor, the worse he was at explaining anything, due to some mixture of being unable to go back to a state of ignorance and being in a seat where his main responsibilities are elsewhere (grant applications!). I'm talking about 1or2-to-one tuition here, done several times a week to kids who did very well in high school.

Yes, you have to shed the expectation that others will teach you, I agree with that. In the end, people slogged through by doing a bunch of reading from various sources. It is maybe the main lesson of university for everyone: you're not in high school anymore, you won't just learn whatever the guy says while talking to you. It's quite the shock if you had actually good teachers at school.

The thing is though, you can still demand good teaching materials. Textbooks have to explain things in the clearest way possible. They shouldn't be confusing, especially considering they end up being the main source for just about everything. In this modern world where there are online lectures and textbooks, there's no reason we can't all have the very best explanations of every relevant concept. Yes, of course as a student you still have to put in the time, but the materials ought to be the very best.

Have you looked at for instance Khan Academy's Grant Sanderson (aka 3Blue1Brown) Math videos?

I have. I went through Khan Academy, Brilliant and 3Blue1Brown. After spending more than 100s of hours I started getting the feeling that these are all good for elementary level math.

But for any serious math (think real analysis, complex analysis, group theory and beyond), all these platforms did was leave me with a warm fuzzy feeling of having learned something cool but in reality that warm fuzzy feeling was not good enough for solving actual exercises that come in textbooks or really deeply understand the material.

I've given up on these online learning media. Back to textbooks. The difference is like night and day.

Note that 3B1B often warns you that his videos bring perspective for a book/class that you're doing or are already done with. And Khan Academy's focus is for K-12.

If you want anything past Analysis 1 I think you'll find that universities guard their content.

If you want anything past Analysis 1 I think you'll find that universities guard their content.

Not so; there's an absolutely vast amount of freely available undergraduate mathematics resources available at all levels. Honestly, so much that it makes it confusing to choose and not get distracted by the options -- perhaps AI-mediated distillation could be helpful in the future.

Really? Can you link some for Analysis and beyond?

I wanted to find good analysis video lectures from a real university complete with problem sets, homeworks and their solutions. I couldn’t. I think MIT OCW now has one analysis course like that, but it’s relatively “recent”.

What about these amazing resources from Daniel Murfet at University of Melbourne:

http://therisingsea.org/post/mast30026/

They have videos as well as everything else. I'd love to study them with someone/some group of people one day.

But, you don't need videos if there are carefully-written course notes PDFs.

Try Oxford:https://courses.maths.ox.ac.uk/course/index.php

E.g. two random Analysis-related courses (second more advanced than first)

https://courses.maths.ox.ac.uk/course/view.php?id=65

https://courses.maths.ox.ac.uk/course/view.php?id=4988

And there are tons of others, but with videos is a bit harder.

Berkeley exam papers with solutions:https://tbp.berkeley.edu/courses/math/113/

Fantastic, thanks!

I think you'll find that universities guard their content.

Hmm. All the way back to when I was in college there was advanced content available from the Open University. You had to be awake at 2am and it was in black and white, but it was there.

I've given up on these online learning media. Back to textbooks. The difference is like night and day.

Are there people who think this is an "either/or" choice, as opposed to a "use both" thing?? I ask, because it's pretty well established that learning is enhanced by use of multiple media types and it seems self-evident to me that books and videos are complementary.

it seems self-evident to me that books and videos are complementary

Can't speak for others but for me it is more about efficient utilization of time rather than complementing multiple learning methods.

I've found that time spent in learning math from videos have poor return of investment. That time is better spent re-reading a chapter or that thing that I couldn't fully understand the first time and doing more exercises.

Fair enough. For me personally, I find great value in jumping back and forth between different modalities, where the different presentations reinforce each other. But what works for me may not work for everyone, and vice-versa.

I think you're just parroting things you've heard other people say. 3b1b's videos are universally agreed to be excellent, and it's baffling that you think it is a choice between watching them and using textbook and doing the exercises. Anyone with the intellectual capacity to study that sort of material is not going to have a hard time comprehending that they are intended to be complementary, as Grant Sanderson makes very clear at numerous points.

I didn't mean to say videos are better - just as far as I can tell that's where the most creative new teaching techniques are on display. I'd definitely prefer they were in the written word. Especially if they were an open collaborative effort. Books are great for flipping back and forth with. You have an "ah-ha" moment and skip back several pages and reread something you misunderstood on a first read-through. It's somehow clunky and takes you out of the flow when you do it in a video.

Critically, you can read/listen to something and come away with the false impression you understand it. Sitting down and doing problem is .. not always fun.. but can be critical for the concepts to sink in. I think this is the main point of what you're saying

I could see in the future it being something like watching a video and then doing a programming exercise

Have you looked at for instance Khan Academy's Grant Sanderson (aka 3Blue1Brown) Math videos? it's really apparent there is a LOT of room for improvement in pedagogy.

There is a study showing that you actually understand material better, if you use the most primitive methods: chalkboard and a lecture. Because you are forced to visualize the material yourself, instead of being presented with a ready-made animation.

It probably makes sense to use visual aids for students that just can't grasp the concept, but I believe this will only help in elementary math.

1. I think the research shows that increasing the cognitive load increases the retention. So in general, when it is harder to learn something, you retain it better.

2. When I struggle with books it is because they do not present the motivation behind what they are doing. Videos and "more popular" articles can both provide the big-picture motivation and overview. Sometimes, you have to construct a motivation for yourself, based on what you read. That's hard. Maybe you even invent something new in order to understand a concept better. This approach is slow, though. It's easier if someone explains to you why a certain concept is "hard" or a point of view from which the concept is "easy".

3. I think students who build on a partial understanding are not going to have a better time with videos. They are in greater need of learning how to learn something than they are of facts, but school does not teach that skill (afaik).

I've suggested to college students that they leave their laptops behind and attend lecture with pen and notebook, and take notes. It makes things a lot more sticky in the mind.

And do the homework problems. You'll never understand the material without doing the problem sets.

people continue to find new and better ways to explain and illustrate concepts

I strongly disagree. All the best math and CS books I have are old.

New books about old topics tend to be less informed about the context and core ideas that led to their development.

Then you are objectively wrong. Look at Grant Sanderson's explanation of introductory Linear Algebra. Very obviously it adds something good to complement the best paper textbooks on the subject.

Hmm - I do wonder if for very particular things in physics their heyday has come and gone? Were there more ridiculously talented inviduals deeply steeped in classical mechanics and discussing amongst themselves in the past? I feel modern physicists move onto high-energy phyics or low-energy physics research pretty early in their careers...

I had a 1930s edition of a "radio physics" textbook as a child, and because there could be no prior assumptions about exposure or familiarity, it was filled with very complex ideas explained so cohesively and coherently that, well, I could understand.

The author knew that this book was for people who might be as involved in the business of radio as its science or engineering, so they wrote as much about the application across every industry, breaking down the systems to the component level and manufacturers, deployment, and ordering, as they did the design and theory.

I learned how to evaluate a textbook from its structure and style. It certainly wasn't designed for discrete lessons, and the professor would have needed a diverse and practical understanding to teach it effectively.

complex ideas explained so cohesively

I wonder if the same thing isn't happening with computer science. When I started studying the topic in the late 80's, I was part of the earliest generation that actually did, and everything seemed to be explicitly written with the goal of making sense. Some things (like recursion and pointers) were fundamentally complicated, but they were made as simple as they reasonably could be.

My son is studying computer science in college right now and I look at the way they present the material and it often seems designed to confuse - I'll read it over and then explain it to him the way _I_ was taught it and he'll say, "oh my gosh, why don't they explain it that way?"

I feel like teaching is mostly a one size fits all endeavor, but people learn and think in different ways, just like a processor is optimized for some operations but not others.

Take simple arithmetic like 12x17. Some people do the long form multiplication (carry the one..), some people say it's 12x10+12x7. Some remember 12x12 from times tables and go 12x12+12x5. Some people make it 24x8+12 => 48x4+12 => 50x4-8+12 etc. Some do it on the abacus in their heads.

All valid, though some are slightly more optimal than others. Good teachers empower alternative solutions and try to help people connect what they already know to what they already understand.

As someone who only learned multivariable calculus successfully via independent study of exterior differential forms as a result of off-handed comments from the differential geometry professor who taught my first-year calc course, I wholeheartedly agree.

Oh, and 12×17 = 10×17 + 2×17 = 10×17 + 2×10 + 2×7 = 170 + 20 + 14 = 204.

Do you have any ore information on that book? I would like to dig it up at a local library. Is is this one perhaps?https://archive.org/details/dli.ernet.16412/page/n9/mode/2up

I'd like to know which book it was.

The problem is that there is no good textbook for Classical Mechanics out there. At least not on an introductory level.

It's funny that for the 19th century and the first few decades of the 20th one physicists were so eager to simplify and generalize their knowledge. With the side-effect of learning quite a few surprising things from the work. And yet for almost a century the goal is explicitly the opposite. (It's almost like if Academia is in crisis...)

I learnt a lot of physics from Marion's 2nd edition (not SR there though). An older and completely forgotten fine textbook is W. Hauser'sIntroduction to the Principles of Mechanics. Then you jump into Goldstein (the 1980 one, again not SR there). It's a good idea to buy any Schaum book from Spiegel about this too, also for vector analysis if you can't take a course on that.

Yeah, in retrospect, I've set myself for failure with that universal claim.

But the proof of failure is instructive :)

While not exactly "introductory" in terms of mathematical prerequisites, Spivak'sMechanics I[1] is an interesting take on the subject, with extensive historical references if you're in to that sort of thing.

[1]https://archive.org/details/physics-for-mathematicians-mecha...

Taylor’s is fairly “introductory” and it’s fantastic. Very well written.

No Bullshit guide to Math and Physicshttps://minireference.com/

(serious reply, if it's insufficient for a freshman course i propose following up with Feynmanhttps://www.amazon.com/Feynman-Lectures-Physics-boxed-set/dp..., any objections?)

Your response does a pretty good job demonstrating the internal challenges universities face. The interest in making a change was motivated by unmet needs, and it's not going away until those needs are met. The only way to deal with people who won't meaningfully engage in the problem solving process is to go around them.

A textbook should be provided for reference. Copying its contents on a blackboard isn't teaching. You still have to design your course. There's goals to meet, you have to evaluate where your students come from and your job is getting them there.

Besides pedagogy, in college you have to respect your students as studying adults and give them a proper bibliography, emphasizing references for independent study if they don't like your lecture notes, nor your approach, nor whatever.

I understand what a university is, I also understand and am qualified in secondary education and it would be incredibly depressing turning colleges into extended high schools because of business models. That would be exploiting students, I never agree with that.

I actually completely agree with everything you wrote. I'm also very familiar with the ongoing battle between education and training.

What I'm talking about is expanding options to meet additional education needs. Since universities are a shared resource, any solution must be carefully designed to preserve the ability to continue providing existing services. That's difficult to achieve, so I understand the obstructionist response.

All I'm saying is that if you position yourself as an obstructionist, don't be surprised when you're treated like one.

I'm very adaptable nowadays. It's just that I think I know which things should work, but I'm not fighting society, particularly not on this.

The thing with giving the public what they want and being too much of a pragmatist is that we've seen it before.

Consider Western universities in the 17th century, they were still there churning out degrees, but modern science, mathematics and technology developed elsewhere.

The old one-size-fits-some approach is as pragmatic as it gets. Society's education needs have grown beyond what the old model can adequately service. We need new solutions that don't cause regressions on the old solutions.

You're right to protect your existing solution against regressions, and there's value in revisiting old topics in the new discussions, but you're not going to constructively contribute much if you're unwilling to engage with why so many people feel the need for something different in the first place.

The problem is that the pedagogy, as in, the best most effective way to teach people things,shouldimprove massively in 70 years. Whether it actually does, that is another question.

Should it really though? Maybe pedagogy is just hard and we can't expect constant progress.

Pedagogy, like any other field is constantly developing. We are continually learning new and better ways to teach materials and thus I think the continued evolution of teaching materials has the potential to be a good thing (and I've created curriculum for professional learning, for grad school and for bootcamps).

It is true that just because a book in newer it is not necessarily pedagogically better

It is also true that a poor selection of content or understanding by the author could doom a book even with better pedagogy.

All that said, I love the idea of OSS books/exercises for teaching - I don't know if a sufficiently engaged and competent (domain + pedagogy) would evolve around and/all of them, but it'd be a fine experiment to try!

It would also be great training material for LLMs to help them to tutor using more thoughtful metaphors and examples.

I think that researching pedagogy is very difficult, and, like many scientific fields, it is hard to reproduce results found in papers. (I am not an expert in this area -- I am just a former teacher with around 10 years teaching experience.) One of the main things I notice is that standardized test scores are not really improving. I think that high school students today would score about the same as high school students in the 1980's if they were given the same multiple choice tests. This implies to me that the field has not advanced a lot. I do think that LLMs and other computer based teaching could help.

The problem is, like many social sciences, people refuse to apply the scientific method to pedagogy and would rather lean on vibe-based "theories".

I took Lebesgue integration and some pretty high powered group theory classes in the 1989s and am retaking them now, and I have to say the presentation now is much better. I think having Terrence Tao blog about how stuff really works makes a difference, at least for lower level grad classes. :) and trying to present groups as the widely useful abstractions they are instead of just a cute self contained theory makes a difference.

Interesting to note we haven’t got a text book for these classes just lecture notes and a number of text books recommended if we want additional presentations.

I had a lot of hope for things like Khan Academy, but the issue is video is not text and it's hard to iterate and improve on

When the entire video is generated by an AI, that dynamic changes. It's no issue to improve "the script", and have it seamlessly regenerated. We can't be that far away from this capability, and it could have a profound impact on available learning resources.

The problem isn't writing a better text, it's having departments willing to teach it, or to try alternate approaches at all.

You have to have a pretty helpless attitude to blame someone else (teachers) on your inability to learn from a good textbook.

How are you ever going to learn anything truly novel in the world if you always delegate teaching to a teacher?

I shouldn't need to read a 500page textbook to understand a class that a teacher has been 'teaching' for 20 years. He's there to give me an understanding of the subject. I have at least 5 classes each semester which are very technical and difficult, it's impossible for me to read a textbook on each while also reading a 20-30 page lab instruction each night, all written in a technical language which I barely understand. I can try but so many times constant consumption of this literature lead to burnout

The entitlement in your comment is staggering...

Why such a desire to be spoonfed, whining about having to put in the work?

Am I not supposed to expect a quality lecture from a university professor? I have had such a bad experience with certain lecturers, I'm certain what they were saying wasn't incorrect but the way they were speaking made it impossible to follow. It wasn't just me who thought so. I admit I've had learning difficulties my whole life but I believe it is actually impossible to learn all the material from the classes with all the details from books. You can pick a field or two which interest you the most and delve deep into the details. An extremely fundamental topic at my uni is called "introduction to circuit analysis". It was taught by a professor who was so bad at teaching it there was no point in coming to his lectures. You would learn more from watching youtube videos. Everyone was aware of the state of that class. Even our lab professors would mock his teaching style. Now if every class was like this we would be lost. I'll give a recent example of another lecturer who I also consider bad, but fortunately he is teaching an easy subject. First class a kid asks 'what's the difference between a router and a switch'. The professor answers 'a switch works on layer 2 and a router works on layer 3'. In my opinion this is a non answer. Yeah he said the truth but it didn't mean anything. If that student didn't know the difference between a switch and a router then surely he doesn't know about the ISO OSI model. Before you think he probably didn't have time to answer more specifically to make sure he understands. This guy notoriously keeps digressing from the subject. As that classes subject is one I am familiar with I knew the way he taught it was impossible to make sense of because he never answered in a concrete way, everything felt so drawn out and lacking a focal point, it's hard to describe. It took him four hours to go over the ISO OSI model and honestly while I wasn't paying attention to every single word he said. With all the digressions and constant comments about stuff you just couldn't follow his train of thought. On the contrary electromagnetism class was absolutely amazing. It's still a difficult subject, at least for me because I've always been pretty bad at physics but the lad who taught it was brilliant. He made sure we all understood, he started with very basic concepts that everyone knew, just to be certain everyone is on the same page. His presentations were concise, easy to follow with links to read up/watch stuff that you didn't fully get or wanted to learn more about. They had plenty of visualizations with gifs of stuff. It's a pleasure going to his classes. So I think that I should expect the lectures to give me an understanding of a subject. If I had to read a book for every class there wouldn't be enough time in a day for me to catch up.

no one is there to give you anything

they are there to give you a chance

maybe it is impossible for you to do this, but it’s clearly not impossible for others

that challenge is valuable information

maybe a different area is a better pursuit for you

or maybe a better way to view learning

good luck

We call this "University"

"Technical and difficult": there are both important concepts and broad strokes that your teacher should be good at explaining, and important details that should be explored in a textbook because they need good pictures, data, editing etc. more than good explanations.

Taking a longer time than normal to learn a topic and prepare university exams might be a good alternative to burnout; or maybe your lifestyle and/or intellect are incompatible with studying.

You are just wrong. There is a lot more that goes into teaching a subject better than 'generating a better textbook'. There are thousands of hours of work that need to happen to figure outhowto make the subject make more sense. It is not a question of 'writing down known result sin a different order'. The subject needs to be reworked in its entirely. Advanced results need to be made elementary. Simple formulations of complex ideas need to be discovered, articulated, and developed into exercises. Out of every available model the best models need to be picked and refined and tested. Etc. And then you have to get everyone on board with the project!

Almost none of the work is 'rewriting the book.'

The OP referenced the Khan Academy as an attempt to displace the ossified educational institutions we have. But it has its own limitations and challenges, some of which will soon be rendered moot by the coming wave of AI technologies, perhaps giving it the final push it needs to truly take hold.

Can imagine a world where it doesn't matter where or how you get your education, all that matters is that you can have your knowledge verified by an accredited _testing_ institution. This would open education up to a world of creative competition for students, and allow individuals to find a learning paradigm that best works for themselves.

This might work well for knowledge/IC-based attainment. There are still other groups, who are possibly the majority when combined:

Some people are there for "the experience", no matter how much it costs, nor how little benefit it results in. They will still be paying $80k for an experience that qualifies them for nothing, even though they had that qualification already before they started.

Some other people are there for the network, and/or will learn leadership skills in that environment. They can't build/learn those things in an online environment.

Agreed. One size does not fit all. It's just the hope that options like Khan Academy are made truly viable for those who are well served by such options.

Then you could simply read the script — no need for the video at all.

For those who prefer to read the script, of course. For those who benefit from a much more multimedia presentation, the generated video and associated generated items, will be preferable.

The video won't be generated by AI, because there will no video.

The mid-game is personalised, conversational, interactive tutoring.

The end-game will be deciding how much humans need to know about any of this, because AI will be far better at 99% of it.

If this seems unlikely, consider that Cauchy, Gauss, Laplace, and so on were all super-elite 0.001%-ers in terms of ability.

No one knows whether AI will eventually have the creative ability to work at their level of insight.

Personalised training could select for prodigies, and - whatever we decide to do with them - we'd have more super-talents than we do now.

Currently a lot of talent is wasted because of political and economic inefficiencies.

there will no video...

There will likely be video. Visual presentations can be an important part of learning. Perhaps it will be generated JIT, but likely AOT compilation of video will represent a useful savings in processing power. Students can always "put their hand up" to interrupt a presentation for a little one-on-one time with professor AI.

Both fields embrace a sort of masochism and active desire to keep knowledge impenetrable b/c it acts as a mechanism to feed out the dummies. The system acts an an informal IQ test - that maintains the prestige of the departments. If you're pigheaded and clever enough to get through the masochistic torture is that their undergrad textbooks then you're probably pretty clever and so the prestige of the degrees is maintained.

I have been a on the mathematics faculty, including at some decent places, for almost 20 years. I have never met a single university mathematics teacher who thought this way. To the contrary, we are delighted when someone shows even a spark of interest or aptitude (hopefully both). Granted there are high bars for reaching professional competence, but that's intrinsic to the subject—and there is a welcoming place, in math probably more than in any physical science, for amateurs as well as professionals.

As he observed, other departments will step in an do much better. The best linear algebra class I had was a graduate course in the electrical engineering department

It's worth noting that this isn't necessarily because EEs are better teachers—although of course in any particular case they might be—but because they can give you a course more focused on your interests. Math departments wind up teaching many courses populated largely, if not entirely, by non-math majors, and we cannot be discipline experts in every field of application in which students might be interested—nor, even if we were, could we simultaneously teach one course in a way that appealed simultaneously and particularly to the diverse applications needed byeverystudent.

I don't think the line of thinking is directly "lets make this painful on purpose" but there is a sense of - everyone has to go through the same gauntlet and some people are just not cut out of math/physics/etc. It's just the way of the world. There is not direct desire to make the experience better/smoother/easier than it was your generation.

I don't know how it is in the Math department, but in Physics there is almost a sort of hazing that goes on, where some subjects are just known for how grueling they are and how you just have to go through it

"To the contrary, we are delighted when someone shows even a spark of interest or aptitude" Not to read too much into it, but this kinda hints at the problem. You're delighted at the students that have passed your IQ test. There is generally very little care given to the 70%+ of students that aren't making the cut. The true horror is how many students are in the class and not getting it. And the teachers are not freaking out

From my university experience (which was a while ago) it was abundantly clear that most students hate their math/physics classes, were bored out of their skulls and the teachers are only interested in the engaged students that are getting it.

What you should be "delighted" by is when you find a new way to explain something that resonates with most of the class

I don't know why you're catching downvotes, I think you're completely right. In my engineering curriculum in some cases the teachers were openly hostile to students that had a hard time understanding the material.

Professors of undergraduates don't seem to think from the undergraduate perspective. Most undergrads have only ever known school, and are just following directions while fumbling their way to their first interview. From that perspective it doesn't make any sense why some classes have to be immensely difficult and high-stakes, and others can be a little easier. From the graduated perspective, you can see "the big well-intentioned lie"--in truth, no field can be condensed into 16 weeks, even if you're studying it and nothing else 14 hours per day. The more difficult a class is, the closer it is to the truth that the class itself is a carefully structured playground, and the real field is more dizzyingly wide and complex than any student can imagine. However, I don't see why this lesson has to be so painful to students.

In my engineering curriculum in some cases the teachers were openly hostile to students that had a hard time understanding the material.

While there's no excuse for it, I think faculty see so much apathy on such a regular basis that sometimes it's easy to mistake sincere struggle for a lack of desire to engage. For precisely that reason, I try very hard to recognize and reward my students who are willing to take the time to work with me, whatever their existing comfort or proficiency level with the subject is, but I know that there are times (probably many more than I'd like to imagine) when I don't rise to that ideal.

Professors of undergraduates don't seem to think from the undergraduate perspective. … However, I don't see why this … has to be so painful to students.

I think you might underestimate how painful it is to faculty, too! Most of us are in this for the love of our subject, and many are even in it for the love of teaching, and it's painful to have something that's so beautiful and beloved for us be the cause of suffering in others.

Anyway—while I completely agree that it would be better if the learning experience could be the joy it should be, and free of the pain that it often carries, I do wonder if some of this might be intrinsic. There are certain topics that are just difficult, and on their first encounter with which most learners will find themselves lost and confused. But those topics have to be, or are believed to have to be, understood to be successful in the field, so that loss and confusion will have to be felt some time. Isn't it better if that's in the classroom rather than on the job?

The more difficult a class is, the closer it is to the truth that the class itself is a carefully structured playground, and the real field is more dizzyingly wide and complex than any student can imagine.

This is a beautiful description. I know as a teacher that I try to communicate this to my students, but also that I surely fail much more often than I succeed.

This is a beautiful description. I know as a teacher that I try to communicate this to my students, but also that I surely fail much more often than I succeed.

Thanks. There's an element of hopelessness in trying to explain the immensity of human knowledge to someone who's lived their entire life captured by mandatory schooling. It's just not an interesting thought to most of them. Their worldview has been so artificially limited that attempts to explain the limitations of their circumstances just appear to be more limitations. "Guys, there are more than six hundred thousand mathematicians working right now in the US, and they're working with concepts first thought of as long ago as 3000 BC and maybe earlier!" "Okay, will that be on the test?"

Isn't it better if that's in the classroom rather than on the job?

If only we were having this conversation in a bar instead of a text forum. That's a huge question and it's clear you're actually interested in talking about your thoughts on it. It's also a subject that interests me.

When you say:

But those topics have to be, or are believed to have to be, understood to be successful in the field,

I think that's where the big disconnect comes from. What's the point of school? Is it to know enough to be useful at a job, or to plumb the depths of knowledge? Is it some third thing? Ask any recruiter and they'll tell you the new hires aren't prepared to actually do anything useful, and ask any advisor and they'll tell you the new graduate students aren't prepared to actually do anything useful, so we can at least conclude that there's some kind of disconnect going on. Students spend thousands of hours and hundreds of thousands of dollars doing something that does not actually adequately prepare them for what people want them for.

I have a hundred different ideas about how to address this. One thought I've been mulling over recently is that a redefinition of grades is in order. From essay that we're commenting on:

"My colleague’s error consisted of believing that the more testable the material, the more teachable it is. A wider spread of performance in the problem sets and in the quizzes makes the assignment of grades “more objective.” The course is turned into a game of skill, where manipulative ability outweighs understanding...

In an elementary course in differential equations, students should learn a few basic concepts that they will remember for the rest of their lives, such as the universal occurrence of the exponential function, stability, the relationship between trajectories and integrals of systems, phase plane analysis, the manipulation of the Laplace transform, perhaps even the fascinating relationship between partial fraction decompositions and convolutions via Laplace transforms. Who cares whether the students become skilled at working out tricky problems? What matters is their getting a feeling for the importance of the subject, their coming out of the course with the conviction of the inevitability of differential equations, and with enhanced faith in the power of mathematics. These objectives are better achieved by stretching the students’ minds to the utmost limits of cultural breadth of which they are capable, and by pitching the material at a level that is just a little higher than they can reach."

At the undergraduate/introductory level, the only important question is "what awareness do you have of the breadth of this field, and what mastery do you have over the concepts that most agree are its "most fundamental"? You and I both agree that class is a "playground", and any "grade" is in fact meaningless. Rather than assigning As, Bs, Cs etc. as a percentile of subjective completion of arbitrary problem sets, As-Fs should be assigned at the discretion of the instructor as a holistic assessment of oral and written examination, completion of problems and problem sets, participation in the class, and general wisdom.

Students would of course resist this. Theywantto be graded on impartial, meaningless criteria. That's how they're taught from third grade, and it's the method that allows for the least possible interaction with the material. The only reason they want these grading criteria is so they can plan to spend as little time as possible on the class. This method of approaching learning simply has to be broken at every level of education. You shouldn't even have a GPA until college.

At the undergraduate/introductory level, the only important question is "what awareness do you have of the breadth of this field, and what mastery do you have over the concepts that most agree are its "most fundamental"? You and I both agree that class is a "playground", and any "grade" is in fact meaningless. Rather than assigning As, Bs, Cs etc. as a percentile of subjective completion of arbitrary problem sets, As-Fs should be assigned at the discretion of the instructor as a holistic assessment of oral and written examination, completion of problems and problem sets, participation in the class, and general wisdom.

I'm not completely sure I agree that this is the solution—if we're re-inventing grades anyway, then I'd like to do something more radical than using the same old A–F and just interpreting them differently (although, even if given free rein to do whatever I liked, I don't know what Iwoulddo!)—but I definitely agree that grades, and the standard approach to them, are the most pernicious part of "education" (in the sense of the current schooling system). If there were any way to get away with it, then I would be happy to—indeed, I would prefer to—have all evaluative exercises be diagnostic and informative, only for the students' benefit, and to assign no grade at all, or an A for everyone; but this seems incompatible with a modern university structure (and anyway is essentially forbidden by university administration).

I mean, have you ever taught a class? Do you know how it feels when we make sure a topic is covered both in class and recitation with plenty of time for questions and the students still stuff it?

No, I've never taught a class, so it's possible that the experience of teaching as a professor would irrevocably change my opinion. However, I have spent a lot of time tutoring math and directing bands and choirs, so I'm familiar with the frustration that comes from explaining concepts and then assigning necessary work with plenty of time to do it and then having people still show up unprepared.

I believe that there is a negative feedback loop in the current model of schooling: unprepared students produce jaded instructors produce unprepared students. The big problem with the current lecture method is that any interruption of the momentum of the course for the students' own personal benefit comes with a social cost, so it's better to just shut up and pretend you know what's going on. Furthermore, many lectures build on themselves, so if you misunderstand a concept at minute 3, by minute 20 you're checked out and by minute 50 you're clock-watching. That's why so many math lectures are silent with the exception of the occasional interjection from a star student.

Combine this with the fact that most students are insufficiently prepared for course material in the first place and you end up with modern STEM college: kids who don't get the material slogging through piles of completion-based assignments and exams and putting in the minimum amount of work to get the degree, after which most get the exact same job they'd have gotten if they'd worked harder anyway. They're almost incentivized against deep examination of any one topic, because all time spent working on one assignment incurs a cost against other assignments, or against leisure.

There are so many issues with the way education works at scale in the first world that going down any pathway would take a thousand words, so I'll sum up by saying that I believe that you have a point, and I also believe that the majority of undergraduates are underprepared, entitled, have underdeveloped work ethics, and lack both the discipline and drive necessary to really get something out of their education. However, I still think that the extreme burn-out classes cause more detriments to higher education than they bring as a whole.

You're delighted at the students that have passed your IQ test. There is generally very little care given to the 70%+ of students that aren't making the cut.

That's because happy students are all alike, but every unhappy student is unhappy in their own way. It's just not feasible to care in depth about why any one of the 70%+ is not learning effectively. They're probably missing some prior topic that's effectively a prereq for the class, but that's something that should be addressed by the student themselves.

What you should be "delighted" by is when you find a new way to explain something that resonates with most of the class

You are right, and I am! I don't think that's incompatible with also being delighted when someone shows interest or aptitude. In fact, the synergy is the best part, when someone shows interest or aptitudebecausethey are willing to put in the work to follow the pathway that I have tried to open up for them.

Both fields embrace a sort of masochism and active desire to keep knowledge impenetrable

I have a PhD in particle physics, and I've been a researcher at CERN with colleagues from multiple countries.

And I cannot understand what you are saying there.

> If you're pigheaded and clever enough to get through the masochistic torture is that their undergrad textbooks then you're probably pretty clever and so the prestige of the degrees is maintained.

I have a PhD in particle physics, and I've been a researcher at CERN with colleagues from multiple countries. And I cannot understand what you are saying there.

Hrm.

???

For what it's worth, I was a researcher at CERN with colleagues from multiple countries, where Iturned downan offer to do a PhD in particle physics :-)

(A foolish career decision in retrospect.)

I think they are saying that if you were clever enough to do physics at that level, you wouldn't understand the problem being discussed. So when you said you don't understand, you gave a live demonstration.

When you emphasised your physics credentials as if to say the above didn't make senseevento a physics-smart and well-credentialed person, that pattern-matched even more strongly with the idea that those who go far through higher education institutes don't relate to the problem so don't tend to work on improvements.

What I meant is that I think I know something about academics in science in general, and in Physics in particular.

Getting my PhD was not easy for me, as it wasn't for the most people I met, because none of us is a so-called genius. Rather, all of us had to put a lot of effort and discipline in order to move on.

And never did we met any artificial obstacles or deliberate difficulties. Physics is hard; it is as hard as many other academic fields. You need focus, you need discipline, and yes, you need passion to be able to keep focus and discipline.

But you will be welcome if you try, and you will receive a lot of help. Maybe you will find out it was not the right choice for you and you will change your target, but you will not be screened out and rejected just because.

Just out of curiosity, what was that classical mechanics textbook?

Pretty sure it was An Introduction to Mechanics Kleppner, Kolenkow

The book was probably okay-ish, the point is that surely someone could have made some improvements in the past few decades

Doesn’t it use cgs only (I think the new editions finally bit the bullet and switched to mks)

Goldstein?

Curricula are hard to change for the same reason widely-used apis are hard to change: it’s the downstream dependencies. There’s a shared understanding of what “took undergrad differential equations” means, and breaking from that has a cost. If you teach differently, your students will be at a disadvantage because they won’t know the things futre courses, tests and employers expect them to know. Maybe that will be outweighed by the objectively superior education you’ll have provided with your bold new approach, and maybe not.

Not that we should give up and resign ourselves to outmoded received pedagogy, but these are hard breaking changes and we should set our expectations accordingly.

Otoh biology classes don’t suffer from this as much. Textbooks are routinely updated to reflect new understandings. Professors seem to have a lot of choice with what they cover, often highlighting their own subfield where they can speak more to it. Physics classes in comparison felt like a factory. You’d also pay twice as high a lab fee as any biology or chemistry lab to play with 40 year old brass weights and frictionless cars.

One argument TFA makes is that a lot of the material is not ever subsequently used.

This masochism also applies to classical music (eg violin). Decades of rote before you’re allowed to express your own musical thoughts.

There are plenty of alternative teaching methods for instruments. (TBH, I don't even know what you are referring to by "classical music", but yeah, some places can be quite oppressive.)

I was going to make a variant of this comment. $FAMOUS_COMPOSER's $REALLY_HARD suite is right there, you can get the sheet music and it clearly says "play this note, then play that one, then stop for a specific duration...and so on". Thatdoesn't$EXPLETIVEmeanthat if you watch Khan Academy's video on how to play notes[1..n] you can play the music according to the composer's specification.

I think people who are frustrated they can't grok diff eq in a n-week course are likewise deluded. There is such a deeper meaning those symbols on the page, and I suspect there is simply no way to spoon feed it.

Both fields embrace a sort of masochism and active desire to keep knowledge impenetrable b/c it acts as a mechanism to feed out the dummies. The system acts an an informal IQ test - that maintains the prestige of the departments. If you're pigheaded and clever enough to get through the masochistic torture is that their undergrad textbooks then you're probably pretty clever and so the prestige of the degrees is maintained.

In a twist of irony anyone could have predicted: this selects for the middle of the bell curve, and not the right-hand tail.

I feel as though there are a set of personality qualities that must not be present in order to achieve anything worthwhile. Wasting time, frivolous activities, and fostering an outsized ego are just a few.

Yeah, the high-capability outliers are used to coast in any subject and may or may not have time to adapt to the laborious requirements before failing.

Also, anybody with "street-smarts" can see the cost-benefit ratio and perceive the benefit is kept much smaller than some other, easier subjects, while the cost is artificially inflated.

What it does select is people that really love the subject. And is at least a little bit smarter than average.

given the current political environment i think we should protect and promote old textbooks as much as possible. they’re lean, devoid of ceremony, makes no assumptions about race/gender/political orientation/socioeconomic status of the students as they work their way through the foundations of the subject. for a modern example of mumble jumble junk, with so much presuppositions about your intelligence, see the rust book[0]. it’s a tough book to study because it wastes so many words saying nothing at all. many people have used it yet rust lifetimes and borrow checker semantics is as arcane and elusive as the holy grail.

[0]:https://doc.rust-lang.org/stable/book/

The system acts an an informal IQ test

Undergrad textbooks in Physics may be somewhat challenging. But the graduate texts (at least in theoretical Physics) tend to be MUCH harder, especially if your undergrad degree didn't include several courses of abstract algebra and topology.

Ideally, physicists should have Geometric Algebra (Clifford Algebra) as part of their undergraduate classes. But at least when I went to Uni, there was no space in the undergrad tracks for this level of math.

There is nuance to this.

It's not so much a habit of being deliberately arcane as, in my opinion at least, badly attempting to keep the sense of the sublime that engineering departments often lose touch with.

I went to a pretty shit physics department, but there were glimpses of beauty. The engineering department was more professional, better run, definitely more fun if you like the thrill of actually making something, but as engineers theory is just a means to an end, so I would've been bored on some fundamental level.

Aside:

As a (I suspect) dyslexia diagnosis in-waiting I am a sucker for really good, crisp typesetting. Old books often struggle with that (e.g. particularly curly non-latin characters are almost unreadable for me), but reasoning is timeless.

Landau & Lifshitz is old, ugly, a bit terrifying, and yet timelessly brilliant. Difficult, but in a physically challenging way rather than the more modern, Grecian-thinking, rigour-by-nomenclature style found in modernity.

Khan Academy is excellent for Calculus however their material for Differential Equations is rather minimal since they don't really go into university level material.

Perusing online resources for Differential Equations shows that there doesn't seem to be much agreement on how to define the core subject, and there is no grounding on 'how we got here' to provide a path forward to people just starting out. My experience so far, at least.

Came here to quote the same exact part. The author is hilarious and savage but it seems appropriately. Amazing prose and sad state of affairs

I have a lot of expertise in this domain:

I had a lot of hope for things like Khan Academy, but the issue is video is not text and it's hard to iterate and improve on.

Not the issue.

I really wish textbooks with open licenses would take over and they could be reworked and improved year after year by different people

The issue.

There actually isn't even a good open content license, analogous to the GPL-style licenses. Improving on video means having access to the source files. Ditto for interactive activities. Khan Academy is designed to look as open as possible, while withholding just enough and being just mean enough with license to make any sort of reuse a hopeless endeavor.

With the proper piece in place, video is very possible to iterate upon.

Makes me think of Jackson's electrodynamics - yit always seemed to me a hazing ritual. I remember a story of someone asking him a question from the book and he basically said "how the hell would I know?"

I enjoyed the book and class myself but it did not make me a better physicist, just a better mathematician

I don't think it's that at all. The real cause is just some entirely institutionalised academics wrote the textbooks and everyone else is copying it rote as if it's the way of doing everything because that's all they know.

You need people from outside the echo chamber. I found The Open University to be a considerably better education provider on that basis than the red brick I attended quite frankly. The material and tuition is far far far better.

Incidentally your point in Khan Academy is spot on. That's basically OU but the material is miles better and it actually leads to a qualification.

Example free course:https://www.open.edu/openlearn/science-maths-technology/intr...

I really wish textbooks with open licenses would take over and they could be reworked and improved year after year by different people

Could anyone explain to me how they think this might work in practice?

I am presently producing an undergrad textbook in quantum theory. I have two motivations: 1. IMO the "qubits first" (ie teach finite-dimensional QM before wave mechanics) approach to introducing the theory is superior (basically only Feynman did it of all the "classic" books) and 2. I'm involved in third world education and I want the book to be freely downloadable.

Now its alotof work despite having taught the course multiple times and produced comprehensive lecture notes etc. Once its done I am sure I will not have the time to keep updating it, expanding on the problem sets and so on. A former student on the course is helping with the conversion and he will be a co-author, but like me he sees it as a service not at all about producing a product. So I think we're both very open to the idea of such "open license".

Given all that here are the kinds of questions that immediately arise:

- Mechanically how should one make the book available for such re-working? Put the source files on github? (Not something I've ever used, but I know roughly how it works).

- Via what mechanism does someone get to be credited for work they might do on better versions?

- Who decides what is the current "definitive" or "best" version? I will have a separate website for the book so I guess new versions can be announced there. But one way or the other I won't be involved forever.

- QM is fraught with crackpots, people who have whacky ideas on how to explain things and so on. Can they be prevented from "taking over", rewriting large chunks into (what I would view as) nonsense and so on? Note that presumably my name would still be associated with the new versions, so the issue is primarily not lending credence to stuff I fundamentally disagree with, not that they shouldn't be allowed to go do their thing.

- We will make a POD service available for purchasing hardcopies, the (expected to be small) royalties from which would be donated to third world physics/math education. Is there some license that can ensure any subsequent use of the material is also similarly non-profit?

I can see some (though not perfect) analogies with open-source software, so perhaps someone here has useful ideas about this kind of thing already...

Both fields embrace a sort of masochism and active desire to keep knowledge impenetrable b/c it acts as a mechanism to feed out the dummies.

Did you go to a top university? And did you study physics in grad school? I find this statement amusing given how easy the undergrad curriculum was compared to grad school physics.

In my undergrad, physics was challenging only in that you needed a good command of the mathematics. If you had that, the actual instruction (and textbooks) were of average difficulty.

But again, experience may vary from university to university. Certainly I can see professors who could have made it much tougher if they wanted to.

As for the rest of your comments in this thread: Sorry, but to me this is another HN thread where people insist itcanbe taught better, and teachers are being irresponsible in not finding such approaches, but with very little actual proof that it can be as good as imagined. It's not like you have concrete examples of better pedagogy to pointed out.

"Both fields embrace a sort of masochism and active desire to keep knowledge impenetrable b/c it acts as a mechanism to feed out the dummies. The system acts an an informal IQ test - that maintains the prestige of the departments. If you're pigheaded and clever enough to get through the masochistic torture is that their undergrad textbooks then you're probably pretty clever and so the prestige of the degrees is maintained."

So true, in many areas. I know a couple of young people that got destroyed by this kind of meat grinding machinery. Typically applies to every medical studies in France, where many kids (with parents that have the money to pay) are going to Spain, Romania or Belgium to study in a less oppressing environment and better teaching methods.

If when I started learning about calculus at 14/15 someone had explained why we were doing this it would have made it so much less confusing. Explaining in terms of speed/distance/acceleration seems to make complete sense now (one of several potential examples). However discussing a function and talking about splitting in into small (infinitesimal) strips of delta quantities and a list of equations / proofs made it completely dry and uninteresting. I don't think I had any ideas one what was going on until it was mentioned in physics several years later.

Physics doing the work of actually teaching math is a universal embarrassment to math departments. Or, well, it should be.

I remember my first calculus exam at the University: I made a lot of explanations and justifications based of physics. Of course I did not pass. As I asked why? It was all correct, right? The prof. said “is perfect reasoning, but this is the math department, physics is one floor below”

He’s right - intuitive physical reasoning is useful (and often valid in the applied sciences) but it isn’t rigorous mathematical proof.

Some areas of mathematics are applicable to physical realities, but they aren’t defined by those realities.

Yeah it's not rigorous. The tragedy is that the organizations responsible for teaching math, like how to do it and how to understand things mathematically,for other purposes, would rather waste their time on rigor.

No problem with there being a rigorous math department. For mathematicians. But, quite literally, no one else cares. Mathematicians and everyone else are at crossed purposes, and the mathematiciansare wrongto choose their purposes over their students' needs.

Uh... Not that I'm defending the previous poster. Just griping about rigor in general. Physical arguments for mathematical problems are rarely sound, and I'd assume that an undergraduate who thinks they have made a sound one is wrong until reading it to see.

No problem with there being a rigorous math department. For mathematicians. But, quite literally, no one else cares

As someone who did an undergrad in mathematics, I have to disagree with you here. People think math is about numbers and computation, but that's like saying literature is about letters and composition. Fundamentally, mathematics is the study of things that are provably true. Without rigor it's not math.

Study of things that are provably true.

Why would you study such trueness if whether you know it is true or not does not have any added value in the physical world?

I don't even understand the question. Is there in truth no beauty? Why do anything that doesn't add value to the physical world?

It seems you did understand the question and I thank you for your answer, it is helpful.

That's just false. What physicists (and engineers, etc) are doing is also math and it's way more useful and insightful.. and (imo) the world would be a better place if that's what everyone else was also learning in college. The idea that math isonlyabout rigor isn't intrinsic; it's a historical accident that people seem to not realize is optional. It is also about understanding things and being able to wield concepts to accomplish goals or convey understanding to others.

Novelists and lawyers both do writing, but I wouldn't call their output the same thing. You might call legal writing more useful or novelists more insightful, but that's a matter of opinion. One is not intrinsically more valuable than the other.

The idea that math is only about rigor isn't intrinsic; it's a historical accident that people seem to not realize is optional

I mean, it is by definition. To a mathematician, doing calculations is not mathematics, no more than spelling is doing poetry. Which is not to say that doing calculations is without value! I think what we have here is (ironically) an unrigorous definition.

doing calculations is not mathematics

I hope you realize that physics isn't about calculations either? You can put numbers or types in the equations, just like you can put numbers or types in the expressions or results you get in math.

I quite strongly disagree with this. Mathematicians aren't rigourous just for the hell of it. We have examples where a lack of rigour caused a bunch of people to waste a bunch of time working of stuff that ultimately didn't work at all.

For example the Italian school of algebraic geometry caused the waste of legitimately decades of work by many people because the foundations weren't right:https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge....

Part of the purpose of a maths degree is to teach the students the rigour required to be a mathematician. If you don't want to learn that rigour then thats completely fine, you can use physical or intuitive arguments, but the place to go and do that is in the physics or engineering departments.

I agree mathematicians would bewrongif they were forcing their rigour on physics students or whatever, but I think they're emphaticallyrightto teach it to maths students.

On the other hand, if you want full and complete rigor that means formalizing things in a computer-based proof checker. Now, that can often reveal ways to refactor the argument into a cleaner, more elegant, more intuitive proof, which is why full formalizations are often regarded as publishable work - but the other side of it is that dotting all the i's and crossing all the t's does take a whole lot of effort since some published proofs are not nearly as clear as they're supposed to be.

I agree that some published proofs are not as clear as they're supposed to be, but I disagree that formalizing things in computer-based proof checkers is the way forward there. Computer proof checking is cool, but its an augment to what we already do and not not a substitute for it.

The purpose of a published paper is to explain things to other humans, while that of a computer formalization is to explain things to computers. Just as it is generally not easy for a computer to understand a published proof, it is usually not easy for other mathematicians to understand the code you feed to Lean or whatever.

The stuff you need to focus on to get a computer to accept your proof is usually quite different to the stuff you want to focus on when explaining things to colleagues. Roughly speaking we care aboutwhyyou're doing things, while the computer cares about the intricacies ofwhatyou're doing.

If some mathematics are not applicable to physical realities, to what are they applicable? Is there any un-applicable math and if so, why does it even exist?

Up until about World War II and use in cryptography, number theory was pretty much an un-applicable field of mathematics.

Pretty amazing that someone discovered and explored number theory without knowing if there would be an application to it. Hopefully they lived to see its usage.

Number theory was a big deal already in ancient greece and probably got there from Babylonia, so it's probably safe to say that they didn't live to see the public key crypto usage. :)

But (integer) numbers and their behavior had huge significance in the ancient worldview, and still do even in our days if you look deep enough. And the Pythagoreans et al applied number theory in e.g. music.

I may be wrong but this seems inaccurate. In WWII, the cryptanalysis of the Enigma (https://en.wikipedia.org/wiki/Cryptanalysis_of_the_Enigma) was related not to number theory, but to group theory. Number theory became relevant to cryptography only since the 1970’s/1980’s when Diffie and Hellman published their paper on public-key cryptography. Although number theory was related to cryptography in the past (e.g. in shift ciphers and block cyphers), it was the Diffie-Hellman paper that placed number theory in the central role.

Maybe depends on what's meant by application, but I'd say a lot of, probably most, research-level math has no (clear) practical applications and most research is not done from the PoV of applications or relationship to physical reality. Math is study of formal systems, sort of "what can and can't follow from some set of formal rules".

It's more a surprise that some rulesets map to physical world as well as they do [1].

For some there may be very important ones in the future like e.g. number theory got in cryptography. But for example just knowing whether P=?NP or if the Riemann hypothesis is true probably has no "physical reality" direct applications.

[1]https://en.m.wikipedia.org/wiki/The_Unreasonable_Effectivene...

So if the application is not visible today, it might appear tomorrow. In fact that makes sense.

Fiction writing exists, should it not be enjoyed? Sometimes mathematicians invent things before scientists discover how to apply it.

The biggest example I can think of is Fermat's Last Theorem. Uncounted mathematicians devoted years of their lives to the theorem, and the developments they made along the way may have some applications. But I have yet to hear of any physical truths derived from the fact that a^n + b^n != c^n for n > 2. Most math is like this - beautiful, and useless.

The motivation of most non-applied mathematicians is aesthetic, much like the motivation of someone in the arts. It's done for its own sake ("mathematical beauty"), not because it has any practical application.

Mathematics might turn out to be "useful" more often than the average novel - but that doesn't mean the math was done with any use in mind. For example many developments in differential geometry made in the 19th century were inapplicable to anything physical for decades until Einstein applied them to create general relativity. Plenty of math becomes useful long after its development, and plenty more never finds any practical application.

Then shouldn't the professor have said that it isn't rigorous? Saying it's perfect implies that it is rigorous, just that they took issue with the association.

And they were right imho. Having a clear notion of a subject is good for both physics and mathematics, but what's good for physics or engineering, is not enough for math, in math, you also need rigorous reasoning, or you'll fall into subtle mistakes.

Fundamentally, withoutsufficientrigor there can be no proofs; "maybe" and "someone says so" aren't good theoretical building blocks.

Many math teahcers (and likely mathematicians) could not care less about the real world applications of math. My college calculus teacher had no interest in physics or science. She liked math in the same way someone would like doing a crossword or Sudoku. A puzzle game with a well defined rule set and the challenge of finding answers.

I cannot overstate how destructive this is, because it basically presents math as this arbitrary logic game rather than as a fundamental language of the universe.

I cannot overstate how destructive this is, because it basically presents math as this arbitrary logic game rather than as a fundamental language of the universe.

You are making a clear philosophical assumption yet don't realize it. Is math really the 'fundamental language of the universe'? or is it an arbitrary logic game that, in its most common interpretation, describes the universe well? Given that we have no full formulation of the universe in terms of mathematics (no theory of everything), the claim that mathematics as it is today is the 'fundamental language of the universe' seems laughably wrong.

On the other hand, there are thousands of formal mathematical systems that don't describe anything 'real' as far as we can tell. I mean, the entirety of the lambda calculus was developed before there were any computers that could realize any it in any physical system. Yet, its development has proven incredibly important in the entire foundations of mathematics (many theorem provers, etc are based on some variant of lambda calculus).

I'm not denying that some people learn calculus, algebra, etc (i.e., standard high-school/early college maths) with physical systems in mind. However, mathematics is much more than that and limiting the field only to that which we can sense is actually detrimental to mathematical progress as a whole. It seems best to me to encourage purely mathematical thinking itself in the hope that the arbitrary systems humans create may one day be used to describe something. History is littered with 'useless' subfields of math later becoming fundamental to the economy (number theory and elliptic curves to cryptography, lambda calculus to computation, group theory to quantum physics, etc)

Personally I don't care that it's aboutphysicalthings. But I care that it's aboutdoingthings: understanding concepts intuitively, solving problems, being adept at thinking mathematically. The thing that is not useful is tedious definitions and tedious proofs.

A simple example is: nobody needs aproofof how differentiation or integration work. They need to know how to do it, think with it, apply it to problems, and generalize it. The proof is a tool for not being wrong, and that is a perfectly fine thing for mathematicians to do and a waste of everybody else's time (unless they really need convincing).

It is an absolute travesty how many people enjoy math until they get to their first college class which is taught by a disconnected mathematician that ruins the subject for them. At my school everyone stopped at multivariable because that's the level at which mathematicians systematically ruined math -- by teaching and testingthe wrong stuff. Personally, I survived by the skin of my teeth, and then later found physics andactually learned multivariable calculusand learned to love math again.Despitethe efforts of the people who were paid to teach it.

Personally I don't care that it's about physical things. But I care that it's about doing things: understanding concepts intuitively, solving problems, being adept at thinking mathematically. The thing that is not useful is tedious definitions and tedious proofs.

That's great. Not everyone does. Thinking simply via formal systems is not inherently inferior to 'doing' something with it. Both have their uses. In particular, as I said, quantum mechanics would not be where it's at today if it weren't for many pre-quantum mathematicians pushing around symbols on a page.

I'm very much not arguing that nobody should do pure math. I'm arguing that teaching pure math to undergrads who are trying to learn applied math is a tragedy and ruins math for alotof people.

I think you're generalizing far too much from limited experience. Not all mathematics, even applied mathematics, corresponds neatly to intuitive physics models you build naturally by being an animal that interacts with the world. Quantum mechanics was already given as an example, but I had the opposite experience with cryptography. So much of the training as a CS student was along the lines of "accept that this works" and then practice by implementing something, and sure, it worked, but at no point did it ever become obvious why any of it should work. Until I finally went back and spent a solid year and a half learning group theory and algebraic number theory in a more rigorous way, figuring out all of the proofs we had skipped over, at least trying it myself, but giving up and reading someone else's proof if I couldn't figure it out within a day or two. And in this case, the abstraction definitely came before the application. This is math that was thought to be compeletely useless for centuries, but people worked on it anyway just because they loved math. Lo and behold, it made the Internet possible, so I'm glad someone did it and loved the tedium.

nobody needs a proof of how differentiation or integration work

This "nobody" includes a lot of industry workers, e.g. those in finance and economics, not only "pure" mathematicians.

I would not call this "destructive." Solving problems is fun in its own right, and things would start making sense in their interconnection, regardless of whether they can be applied to sciences or not.

It is destructive to people who do not find abstract math problems inherently fun, which is (I would guess) the great majority of people. They come to the conclusion that they hate math or that they are bad at math, even though they would have grasped it better and enjoyed it had it been presented to them differently.

If your goal is teach math to the bulk of people, including those who will not go on to be mathematicians, it makes sense to tie math to something that the bulk of people can relate to. And most people DO enjoy thinking about physical objects and physical space (because we ourselves are physical beings who evolved to interact with our physical environment), so this is a really great starting point for introductory math for the average person.

If your goal is to only teach the subset of the population who prefer highly abstract puzzles, and to alienate all others, then our current methods are working fine I guess. But I don't think this is a good goal for general math education.

I think you put your finger on a challenge fundamental to teaching itself. Speaking as a (former, at least for now) teacher, the concepts that I find most intuitive were always the most difficult to teach, whereas skills I struggled to master were the ones I was best able to instruct. For problems in the second category I could diagnose the gaps in my students' understanding or application - likely because I'd faced them, or similar, myself - and strategize an approach that would help. Conversely, the subjects that I "got" without effort, left me floundering (at least at first) to find an alternate way to present to students.

(I suppose that's where the terrible canard of "those who can't, teach" comes from. There's a kernel of truth to it, of course, but the more generous and relevant point is that success through effort is better preparation for teaching than success through natural ability alone.)

My guess is not that your teacher would disagree that math is "a fundamental language of the universe", just that she spoke it so fluently that she wasn't well able to relate to people who don't.

It's worse than that. I remember that when I was doing physics aged 14/15/16, they weren't allowed to assume that we were taking the maths course that included calculus, so while you could see these repeating patterns crop up (half-thing-squared, you say? How peculiar) they were effectively prevented from taking the time to explain it properly. It would have made thingsso much simpler.

I had a professor freshman year of college who offered the question “How many times can you split the distance between me and you in half?”. Ultimately I arrived at the conclusion infinity which forever more made the general idea of calculus stick.

> I don't think I had any ideas one what was going on until it was mentioned in physics several years later.

This is where kids with an engineer parent have an advantage. Such a parent can offer examples of how some of the math is used in the real world. Once relevance is established it's OK to point out that this stuff is often buried in software but someone had to put that math into code form so we can all benefit from it. "You may never use it, but you use tools that use it under the hood" can be motivating when they're wondering if it's all just weird busy work.

I don't know how or where you learned calculus, but when I learned it and when I taught it, the classes were filled with physical world examples to motivate and train the material. That said, my experience as a student reflected on as an instructor, was that a lot of the explanation and example stuff didn't have a lot of fertile ground in my mind yet when I was first learning the material. I'm sure it helped, but there were definitely times when as a grad student I reviewed the foundational material and thought "This makes so much sense, why didn't they teach me this when I was in high school?" only to realize that almost certainly they had taught me that in high school and I just didn't have the mathematical maturity yet to retain it. I think I was also hampered in my chances of learning the conceptual fundamentals because I was able to do most of the work through a solid ability at algebraic manipulation. While solid skills in algebraic manipulation is quite important, I do think it would be a good idea to restructure those classes so that a solid conceptual understanding is also more necessary to pass the class. Of course, easier said than done.

However discussing a function and talking about splitting in into small (infinitesimal) strips of delta quantities and a list of equations / proofs made it completely dry and uninteresting.

This was exactly my experience as well. I spent almost an entire first semester with a C average, mostly because I couldn't grasp what I was doing or why I was doing it. I still remember my "Eureka!!!" moment of sitting in the school library and working on a series of problems about water escaping a pool through an every-widening hole in the side. At that point, it all made sense, and I a) finished with mostly A's and b) was the only one in my class to score a 5 on the AP test later that year. I eventually went on to major in EE with a focus on Signals Processing (including grad school), so I always find it ironic that I went from not getting it all, to basically 8 straight years of nothing but Calculus.

Although I studied calculus in a pure math perspective, I knew enough physics to tie the two concepts together and it may have benefited me some. For the most part, though, all I needed to know was "integration = area under curve" and "differentiation is computing the slope at a point on a curve" and most of the first 3 semesters in calculus made sense from that perspective.

However, my concern with your approach is that those examples help only if you are interested in physics (or whatever field those examples come from). I worked at a math tutoring institute for a few years and often saw that physics examples in textbooks like Stewart's confused a lot of students who were not that interested in physics. Not only do they have to learn the math, they now have to learn physics concepts just to understand the examples! There were plenty of students who could do differentiation/integration, but struggled at the questions involving physics.

Likewise, the department had a separate calculus course for people in finance, economics, etc. That textbook had applications in finance. All the tutors struggled to help those students because we had to learn basic finance concepts to understand the problems. And on the flip side, the students ended up useless at solving calculus problems on their own - but they could do the ones involving finance concepts.

There seems to be something similar in other branches of mathematics too (and lots of other fields). A clear example I recall was studying Fourier transforms in math and I couldn't make any sense of it. To me it was just some by-rote algebra with integrals of exponentials of complex numbers.

But then I happened to do some audio signal analysis, and when I saw a magnitude spectrum of a waveform, it was instantly obvious what's going on (and understanding the phase part after this was no problem). Such practical examples seem to be almost banned from university math, I'm guessing to make sure everything is very abstract and rigorous (e.g. you have to work with finite length and discretized signals where the maths don't strictly apply). And after getting this intuition the formal math started to make sense too.

But when one begins to teach, it becomes quite easy to see why things are like this. All of these things are so obvious to the teacher that it's hard to understand how one thinks before these are obvious and the standard notation/vocabulary is typically a good way to work with these, but only after understanding the stuff.

What I try to do is to probe out something that the student already knows, which may be from a totally different field, and find a simple example in the new topic so I can say that "this is exactly the same thing, but with this different notation/abstraction". This very often causes the things to "click" for them.

This is of course very hard to do with a textbook or a mass lecture. And probably the main thing why we need humans to do teaching instead of just giving out material.

I'd say that the Fourier Transformation doesn't belong in the calculus program at all.

It should be taught in linear algebra, where it does not only make sense, but is a non-trivial example of application that the textbooks have so little of.

You need analysis to make sense of infinite series, indefinite integrals, exponential functions, etc.

At the end of the day, analytical Fourier analysis doesn't get nearly as much practical use as numerical approaches, and you don't really need THAT much background to make sense of DFTs.

I know that, for myself, revisiting Fourier Analysis after going through DFTs in my numerical analysis classes made a lot more sense, and I kinda wish I had started with that angle in the first place.

Analytical fourier analysis is needed for many other fields of math though. For example in statistics it's very central for dealing with density functions. I'd guess in many fields of physics it's also very crucial.

But for a lot of fields you just need the DFT and the calculus stuff can be mostly a distraction. How I finally figured out FT is something like the infinitesimal limit of DFT.

If there's one place for Fourier analysis to be is on linear algebra.

The calculus course is way too bloated for historical reasons that haven't mattered for more than a century. Pushing everything into that context only serves to make the contents hard to understand and seemingly useless.

Why should it be in linear algebra? You mean as in doing linear algebra with functions in infinite-dimensional spaces? What I recall mostly (not) learning from linear algebra were various classifications of matrices and their properties (which made very little sense to me at the time too).

I'm not exactly sure where FT should belong in the math syllabus. It's heavily related to trigonometry of course but it is an integral transform, so needs a bit of calculus. Although discrete versions are probably easier to grasp with just multiplications and sums, and there is quite rarely much actual integrating as in find-the-closed-form-antiderivative going on.

Maybe trying to have too "unified" syllabus for engineers/scientists is not the best way in practice. Instead there could be more discipline-specific teaching (something TFA hints to too). With ML (and 3D graphics earlier) something like this seems to be happening to linear algebra. The "mathy" linear algebra (that I studied at least) is mostly about properties and decompositions of (complex) matrices, but in ML/3D engineers very rarely use such things (but do need stuff outside "traditional" linear algebra like tensor products, projective coordinates and rotation groups).

In fact, there is a lack of linear algebra in infinite-dimensional spaces on the general curriculum, and it is an important subject for physics and a few engineering areas besides the relevance for mathematicians. It wasn't clear to me how the dependency between calculus and algebra was supposed to be, but your comment makes it crystal clear.

The discrete transformation is quite fitting for finite-dimensional algebra, and honestly I can't understand how anybody every thought teaching the continuous transformation first and the discrete one never was a good idea. The only explanation is that since everything is thrown on the calculus package without any consideration, there's only time for one, so people kept the most general one.

Infinite-dimensional spaces do indeed come up in e.g. Gaussian processes and the kernel trick. I have to admit I've never really understood them the rigorous form. For ML for example it would be likely more useful than e.g. most matrix decompositions.

I'd guess math uses continuous forms because it's where the mathematical tools are and many things tend to get simpler in mathematical sense when you let something go infinitesimal or infinite. This could maybe be different if digital computers would have been invented before calculus.

I've learned to appreciate that mathematicians think of maths quite differently to engineers or scientists. To them the interest is in the "mathematical objects" and their (provable) properties, not the applications or relationships to "the real world" (and this is probably a good thing in itself) and for engineers/scientists it's the opposite. Maybe something like how linguists vs novelists approach language.

It also comes up a lot in the foundations of RL - the basis of how it is justified (or in some cases proven) to work is contraction mappings and functional operators.

Is there a good book for learning ML Linear Algebra? I've taken linear algebra math courses in uni and most of it went in and out one ear.

I'm looking for a book that helps me understand ML algebra, so that when I read research papers I'm not just lost and nodding my head aimlessly. Such a book may not exist, maybe it's a group of books, but if ANYONE would have pointers in this direction, I would be in your debt

It's hard if you don't know any practical examples. Anecdote: I taught a college math course for a semester while I was between jobs. I shared an office with some other teachers including a bright grad student who was TA'ing differential equations.

My degree was in physics, and I had worked in industry. I told him that I wished the math courses had included some engineering applications of differential equations. He looked at me with a straight face and said: "There are no engineering applications of differential equations."

What is TA'ing?

TA = Teaching Assistant, usually a student who helps the main teacher of a course by teaching some units of it, evaluating tests, etc.

TA'ing = being a TA

Lol. That's every teacher I had, includingin university. It wasn't until I dropped out and years later resumed reading physics and maths for my own interest that I properly understood three point of some of the maths they were teaching.

Such practical examples seem to be almost banned from university math

For background, I studied theoretical math. I am also amateur electronics engineer.

I agree, theoretical math students and staff are a bit dismissive about everything else. At least where I studied. Where I studied, we would regularly joke about other students claiming that math is hard saying, "what they are learning isn't even math, it is just a bunch of formulae and rote learning". And this included physics or applied math students. Don't even get started about CS students...

It is unfortunate, because math concepts were frequently invented to solve a real physical problem. Or they were invented and we found them extremely useful for a physical problem. Historically people would not put much distinction and would usually do both physics and math and there would be a lot of cross-pollination.

Just read a bit about how Einstein came up general relativity -- it is fascinating story about how Einstein wasn't good at math but needed a problem solved so he reached out to some friends and those friends basically gave him some private lessons until Einstein clicked and figured it out.

For me, I pretty much understood Fourier analysis before I started doing electronics. But the usefulness and practicality of it only clicked after I moved to higher frequency problems and started using frequency domain when working with my circuits. You don't get much significance of it just studying theoretical math.

I agree, theoretical math students and staff are a bit dismissive about everything else.

I think this does go somewhat both ways. E.g. in the current discussion there are quite a few people complaining about "useless maths". I think a big problem is that different areas of studies don't appreciate that they are, well, different, and it's not trivial to transfer ideas across fields.

Just read a bit about how Einstein came up general relativity

Also an interesting story about special relativity: Minkowski (Einstein's math professor) proposed very early on to Einstein that special relativity should be formulated geometrically (with what's now called Minkowski spaces), but Einstein deemed it too mathy and unintuitive, only to accept later that it was the right way to go. Without the geometric formulation it's very possible Einstein wouldn't have developed general relativity later on.

But the usefulness and practicality of it only clicked after I moved to higher frequency problems and started using frequency domain when working with my circuits.

One argument for teaching FT more abstractly is that the "useful parts" are quite different in different fields. E.g. in many applications of sound and acoustics (and probably in EE) the main interest is in the spectrum. But in e.g. statistics the main thing is the convolution theorem and the spectrum has practically no use.

Previous discussion from 2022...

https://news.ycombinator.com/item?id=32530035

They talk (rightly) about mathematics replacing infinitely small but not 0 numbers with limits there. Infinitesimals were later reintroduced with rigor throughhttps://mathworld.wolfram.com/NonstandardAnalysis.html

I have long thought such would be easier to work with on modern computers.

Nonstandard analysis is one of those things that sounds nice in the abstract, but even though I first learned about it more than ten years ago, I have yet to see a worked didactic-level example of how you would, for example, obtain the derivative of sine using nonstandard analysis, or derive the natural logarithm.

I literally saw the sin one yesterday, so I'll rewrite it!!

Let e be an infinitesimal.

We write st(x) for the function dropping the infinitesimal part of a number.

Then f'(x) = st(1/e (f(x+e)-f(x)))

Now use the angle sum identity and cos(e) = 1 - e^2, sin(e) = e. I don't know how to justify these values other than the power series identities for sin and cos...

So st(1/e (sin(x+e)-sin(x))) = st(1/e (sin(x)cos(e)+cos(x)sin(e)-sin(x)) = st(1/e (sin(x)(1-e^2)+cos(x)e - sin(x))) = st(1/e (sin(x) - sin(x) - e^2 sin(x) + cos(x)e)) = st(e sin(x) + cos(x)) = cos(x)

"cos(e) = 1 - e^2, sin(e) = e. I don't know how to justify these values other than the power series identities for sin and cos..."

Isn't that from the definition of cos and sin, even geometrically?

The version of rigorous infinitesimals I've seen didn't let you have these identities, because e^2 != 0 (because it still maintains the field axiom xy = 0 => x=0 V y=0); but the argument still goes through by carrying the whole power series in place of these simplifications – you can write the higher order terms as e*f(x, e) and drop them when you evaluate st(...).

You can get the relationship:

1 - x^2 < cos(x)^2 < 1 / (1 + x^2)

from ordinary trigonometry, although you can't just do this with nilsquare infinitesimals; you need a more sophisticated setup.

https://alok.github.io/2022/09/12/y539043/

Note though, that nonstandard analysis isn't compatible with more "intuitionistic"https://en.wikipedia.org/wiki/Axiom_of_determinacy(in place of axiom on choice), which free you from Banach–Tarski paradox and have some other appealing properties.

On the contrary, in an intuitionistic setting you can usehttps://en.wikipedia.org/wiki/Synthetic_differential_geometr...andhttps://en.wikipedia.org/wiki/Smooth_infinitesimal_analysiswhich aren't nearly as easy to phrase classically. (Sure you can phrase things in terms of toposes, but that's just a roundabout way of saying that you're operating in a constructive setting.)

Can do it in pure ZF:

https://www.youtube.com/watch?v=YE68yptMMeQ

https://u.cs.biu.ac.il/~katzmik/spot.html

Infinitesimals are commonly used in scientific computing (see forward mode automatic differentiation).

How math relates to the physical world is not at all clear. Like yes 2 apples clearly makes sense, but non physical quantities like 10^241 of anything start to break down the relationship.

It’s not the abstraction that gets me usually, it’s the lack of an explicit relationship between math and physics. I’m sorry but personally I am not satisfied by what I learned in my math and math/science classes on this. It started for me day 1 of high school physics and I’m now graduating in 1 month in undergrad applied mathematics.

I felt reinforced when I read that Plato clearly delineated the two “realms”. I was hopeful a single math teacher would discuss this idea for one lesson, but it never happened. Plato was a champion of math who recognized the division.

I’m not one of those disbelievers in complex numbers or anything like that, but if some model formulates precise physical scenarios, we should be able to have ample cross-over language. Even the explanation for why honeycombs have six sides: there is always some kind of fumble from going from the mathematical reason to the physical reason. It’s the paradigmatic example of mathematical explanations for physical phenomena, and between two heavily studied fields, and yet we fumble it continually. What is it about the mathematical hexagon which determines the physical honeycomb? These questions are at the height of philosophy of mathematics, e.g. Mark Colyvan, yet are still disputed and we act like it’s all so obvious. I get it works, but don’t tell me it’s not mysterious, because 6 years in now I’m not at all satisfied and have little confidence it will be mentioned in future math/science classes.

Can you elaborate on why you describe 10^241 as non-physical? Just because it's too large to describe anything physical?

Anyhow, Aristotle had another take on the problem of universals, and taking his view that every concept abstract or otherwise can only exist as part of our physical world (I'm not a philosopher, and there are variations on both themes that different people subscribe to), then your supposition would be nonsense in itself. Stated as a question: What is your reasoning for preferring a Plato-like model of the world vs. a Aristotle-like one?

(If it needs to be said: This is not a gotcha or anything - your opinion intrigued me, and I'm curious as to your reasoning)

Well, based on a quick Google, there are approximately 10^186 Planck-length volumes in the known universe. So 10^241 is not physical in the sense that there is no way to 'realize' a set of 10^241 things.

Another related notion is that there is no way to 'realize' an irrational 'number' -- sqrt(2) can be defined but we can never draw a length of sqrt(2).

It really depends what you consider 'real'. It is actually pretty straightforward to realize a representation of 10^241 (you just did). Why does that make it any less real than the digit 2, which represents two things, but is actually just itself one thing.

Another related notion is that there is no way to 'realize' an irrational 'number' -- sqrt(2) can be defined but we can never draw a length of sqrt(2).

Again, it's unclear what anyone means by things like 'sqrt(2)'. Why is drawing a thing of length sqrt(2) any different than drawing a thing of length 1? If I draw a thing of length 1 and say it's a line of length 1, then why is that different than my drawing the same line, claiming its length is sort(2) and then pointing out that it's actually now impossible to mark where 1 would appear along its length?

It really depends what you consider 'real'. It is actually pretty straightforward to realize a representation of 10^241 (you just did). Why does that make it any less real than the digit 2, which represents two things, but is actually just itself one thing.

One is a map, and the other is the territory. Both 'real' in some sense but a map without the territory feels less 'grounded' (pun?).

Again, it's unclear what anyone means by things like 'sqrt(2)'. Why is drawing a thing of length sqrt(2) any different than drawing a thing of length 1? If I draw a thing of length 1 and say it's a line of length 1, then why is that different than my drawing the same line, claiming its length is sort(2) and then pointing out that it's actually now impossible to mark where 1 would appear along its length?

Perhaps a more precise way to describe the situation is that one can define one or the other as a base 'unit' but you can never get one from the other (they are 'incommensurate', as the greeks would say). Irrational numbers can be defined but not 'realized' (or 'constructed') in the same way that the rationals can.

Irrational numbers can be defined but not 'realized' (or 'constructed') in the same way that the rationals can.

Once again, I'm not 100% sure what you're saying. If you have something that's of length 1, then you can easily construct the line with a ratio sqrt(2):1. Draw another line of length 1 (use compass and straightedge) at a 90 degree angle. Repeat 4 times until you have a square. Now draw the diagonal. You have successfully constructed the square root of 2 in your own setup.

There are numbers that cannot be constructed geometrically using only a compass and straightedge (e for example). However, again these are no less 'real'. Drawing lines is not the only measure of real. There are other methods. In computing, we often say a number is computable if you can define a function that, given any rational number can tell you if the number it represents is greater than or equal to the rational number. The square root of two and e and pi, etc are easily representable this way. There are some numbers that are not, and perhaps these can truly be said to not exist.

However, the field of computables is closed anyway, so it really doesn't matter if you don't want to believe the reals exist.

Yes, apologies -- using the term 'construct' muddied the point as sqrt(2) is a 'constructible' number as you point out. The term 'real' is what's at issue here and I am arguing for a distinction between a 'map-like' real and a 'territory-like' real, the latter of which has some sort of spatiotemporal grounding.

There are some numbers that are not, and perhaps these can truly be said to not exist.

So then we have a real issue because the vast majority of the real line is composed of these uncomputable numbers which you've suggested don't exist.

Sure, but no one uses the reals anyway for 'constructible' things. As I pointed out, the computable reals themselves form a closed field and are the things you would find when describing real life.

As for the 'problem'... I personally don't view it that way. In my opinion (and it's just that, since there's no mathematical 'truth' here), I don't believe non-computable reals exist in any meaningful way. I believe this is similar to how we talk about a 'program that can check if another one halts'. Anyone can make that statement and claim that such a thing exists, but it's not at all clear that such a thing exists. But that's a lot different than saying there are X particles in the universe, thus the number X + 1 does not exist. Because x + 1 does exist and you can write a turing machine that can compute it to any precision (or a lambda calculus function that'll give you the next church encoded representation of it, etc).

My point is two fold. Firstly that there are certainly numbers that are greater than the total number of 'stuff' in the universe. Secondly, that there are some numbers that cannot be described in any meaningful way. These can be said to not exist (my belief), but others disagree.

Speaking of 2nd-order linear ODEs w/ const factor (a.k.a. mass-spring-damper systems), I wrote a blog post[0] deriving all possible general solutions in a concise matrix that makes it easily implementable in code.

The following is the complete solution in Lua:

   function sprung_response(t,pos,vel,k,c,m)
      local decay = c/2/m
      local omega = math.sqrt(k/m)
      local resid = decay*decay-omega*omega
      local scale = math.sqrt(math.abs(resid))
      local T1,T0 = t , 1
      if resid<0 then
         T1,T0 = math.sin( scale*t)/scale , math.cos( scale*t)
      elseif resid>0 then
         T1,T0 = math.sinh(scale*t)/scale , math.cosh(scale*t)
      end
      local dissipation = math.exp(-decay*t)
      local evolved_pos = dissipation*( pos*(T0+T1*decay) + vel*(   T1      ) )
      local evolved_vel = dissipation*( pos*(-T1*omega^2) + vel*(T0-T1*decay) )
     return evolved_pos , evolved_vel
   end

[0]https://esporttoys.pages.dev/2022/11/21/damped

My grad school was basically working through F=Ma-Cv-Kx in all it's expanding varied glory, working up to full blown finite element analysis.

Ultimately it all comes down to choosing the most convenient basis functions for the questions you're answering.

My grad school was basically working through F=Ma-Cv-Kx in all it's expanding varied glory, working up to full blown finite element analysis.

If I'm understanding correctly, you want another = there for a standard damped spring: F = ma = -Cv - Kx.

F is a forcing function, not the resultant force. It’s often arranged this way with all the derivatives on one side (as opposed to having the resultant force, ma, on one side of the equality by itself) so that it matches the general form of a non-homogeneous second-order linear differential equation.

(At least I assume this is what the original commenter meant!).

Ah, that makes more sense. Thanks!

This is a good example. That was math heavy, it took me too long to parse for basic understanding and honestly I am not sure I got it right. The LUA code was easy to understand even though those variable names force you to read all the code, and guess what is what.

It has been a long time since I did diff eq at work, and I agree with the PDF the more I knew the less I understood. I dont know why I need to have the math tainted by the unclean reality to understand, and if that hinders my understanding of them.

Just like solving analytically integrals, (O,P)DEs require tremendous skill of pattern matching. You need to know all of the available tricks, and pick the one that seems to fit. You test it and it just works.

No wonder there is super-high barrier of entry. And that is why us mortals are just using numerical methods instead.

This is why I found this article kinda pointless.

At the end of the day, most practitioners will work with just a bag of tricks, so if that is what the class teaches most of the time, I really don't mind at all. Yeah there is deeper theory to changes of variables and all that, but for an intro class that is aimed at a wide audience (physicists and any sort of applied math), then yeah some light theory and a bag of tricks is perfect. The students that need more will learn that theory later anyways in a second or even third class.

You don't have to sacrifice that much rigour either.

Not every class needs a narrative either. I agree that concrete examples to go with the abstract is more needed in mathematics, but differential equations isn't a victim of that.

I see other comments say that differential equations are victim to :

why we are we learning this?

I don't think this really applies to differential equations at all to be honest. And people say the same complaints about learning scales in music or wtv. Just learn shit and figure out meaning later works too.

I think the underlying question here is: "Is it better to spend lecture time learning the bag of tricks, and then the people who need more can learn theory? Or is it better to teach theory and intuition and then people who need to do it can later look up the bag of tricks?".

Reasonable people can probably disagree, and this might be biased by their personality and learning style.

I personally forgot every trick from my bags mere moments after walking out of my graduate exams. Later, when I got a job that required me to actually /do/ things, I (re-)learned the theory and tricks I needed for the task at hand. The courses that have had a real impact on me were not the "bag of tricks" ones, they were the ones that planted a small grain of understanding and/or appreciation in my brain. I think these are like seed crystals: They make later learning orders of magnitude more efficient and are far less tangible but far more important than "bags of tricks" learning.

I completely agree that theory is incredibly important to undestand and retain things. I just disagree that DE doesn't teach the necessary theory, at least the ones at HYP (for physicists) didn't.

Only a small subset of O/PDE's can be solved analytically in the first place, and few of those are anything but theoretical curiosities. Linear w/ constant-coefficients are the one class that's of big practical interest, which is why OP argues that a well designed course should be focusing on these.

I went to grad school for chemical engineering a decade ago and part of my frustration with the courses is that the math was never rigorous enough; when I asked for clarification, the inconsistencies were glossed over and hand-waved away.

The linked article mentions differential forms, which is a perfect example. In engineering courses, these are often introduced without any rigor or formality. They just “appear” as a way of rewriting the equation. What is a “differential”? Who knows. Is there some kind of axiomatic basis for how these symbols can be manipulated in a consistent way? Who knows. But here are the steps to solve the equation; good luck on the test.

Another example is the quantum chemistry class I took (my first introduction to quantum mechanics). The professor mentioned something about how you can take a measurement and collapse the wave function of an electron; the probability of measuring its position at any location in space is non-zero. I followed up with some question about an experiment to confine the wave function collapse to certain regions of space which could be used to transfer information faster than the speed of light, which I thought wasn’t possible. My question was mostly met with a blank stare and something along the lines of “this course doesn’t cover that” before the professor moved on.

The most egregious example was a grad-level course on statistical mechanics. The professor mentioned something about the overall wave function of the system being a Slater determinant of individual wave functions. A student (not me) asked why we have wave functions for individual particles and a separate one for the overall system, and which was more “correct” fundamentally? The professor replied that the wave functions for the individual particles were more fundamental, and the overall wave function for the system was just an approximation. I replied, sure, the Slater determinant is a way of enforcing certain constraints that must be met for the whole system, but one of the defining features of quantum mechanics is that the function describing the state of the entire system is generally inseparable; without this we couldn’t have entanglement. The professor dismissed my response as incorrect and said students shouldn’t challenge professors on topics they don’t know anything about. This was a guy whose research career was built on dozens of computational chemistry papers that heavily utilized DFT by the way (and by “research”, I mean plug a file with atom coordinates and atom types into the DFT software, run the software, and publish the results).

How big a system can be while maintaining a quantum entanglement (also called 'quantum coherence') - is still an open question. A very important one, because it affects quantum computers. Experimentally researchers see, that a big enough system can't maintain quantum coherence for long. If you want to know about how this process is described mathematically, search for 'quantum decoherence'. If you want to know about possible physical interpretations, search for 'objective collapse theory'.

DFT uses the Kohn-Sham approximation which is single particle and is a ground state theory. There’s definitely been research along the lines you bring up in quantum chemistry but not in DFT. You want to look at the Levy-Lieb density and maybe some of Mazziotti’s papers

My experience as a math major is that most non-mathematicians, including engineering and chem/phys professors, have a working understand of math and don't think too deep about it.

Is there some kind of axiomatic basis for how these symbols can be manipulated in a consistent way?

Yes.

This was a guy whose research career was built on dozens of computational chemistry papers that heavily utilized DFT by the way (and by “research”, I mean plug a file with atom coordinates and atom types into the DFT software, run the software, and publish the results).

Again, large portions of academia simply stick to their small subfield, publish there, and don't think much about the broader implications, if they think about them at all. This is why so many discoveries are made by new entrants to a field with a slightly different background.

For those not in the know: DFT = Density Functional Theory

What can we expect students to get out of an elementary course in differential equations? I reject the “bag of tricks” answer to this question. A course taught as a bag of tricks is devoid of educational value. One year later, the students will forget the tricks, most of which are useless anyway.

That's exactly how I was taught differential equations 20 years ago. And as expected I have totally forgotten all of them. I do wish I'd gotten some deeper lessons from it. Even a concept or two that stuck with me would be better than remembering I learned a bunch of math tricks a long time ago that I can no longer do.

The further I got into maths (not very far mind you, but I think I at least did a few differential equations,) the more it felt that maths really was just a bag of tricks, moreso at the advanced end.

* Keep deriving circular functions until they cancel out. * Completing the square * Use the quadratic formula to solve degree-2 polynomials ... No formula for degree-5 or higher. * Use the Laplace transform here... for reasons...

Many famous mathematicians have quotes along the lines of "it's not the result that mattered here, it's the new method/trick that is cool!" and essentially it's because solving math problems requires a bag of tricks and a new one can open new doors in every field it hasn't been applied to yet.

How often have you used them. It's been decades for me as well, but I still remember how to solve basic linear differential equations as I needed to use them often (in other courses).

LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS ARE THE BOTTOM LINE

Yes! Jesus. People so often forget you can literally take variables and justfill in easy constantsto get a sense of how they work. I can't believe this isn't, like, the first thing they teach in DE.

OK, then wait until you run into the students who somehow passed algebra two but don’t understand the difference between a constant coefficient and a variable coefficient

Clearly they should not have passed linear algebra without this understanding, and you're free to fail then if they didn't personally manage to correct for it by the end of the term.

They switched to a more forgiving section that didn’t do variable coefficients

I had him for second term calculus, which was the first math course I took in college. So I get to listen to him speak it in my minds eye —- his voice is unforgettable. He was a great teacher. 2. I have an interesting take on this subject. I became an engineer at 50. Out of personal choice. What I found was that engineering had changed and what mattered was your savvy in manipulating expensive programs on the computer. In those programs, differential equations were solved numerically. No one even thinks about solving them any other way. There isn’t any time.

I, too, had Rota for his "exploring higher mathematics" seminar. What a remarkable human being! I retain excitement for hierarchies of infinity to this day.

As to your second point - yes, programs solve DE's numerically. I suppose it's still nice to know a bit about how they were solved in the 'olden days'?

In the olden days, a lot of problems were solved numerically by methods such as weighing a printed graph of a function

Aren't most DEs not solvable analytically? i.e. only a tiny subset are neatly solvable - like precisely manufactured puzzles. In general, for any real problem you encounter "in the wild", numerical methods are theonlyway to find solutions (which are approximate).

DE theory is still useful for designing frameworks within which numerical methods can operate.

At least, that's my understanding.

Nicely written. But isn't a corollary that we should stop exams in Maths (and Physics/etc/etc) degrees and just teach people who want to learn, how much they want to learn?

Exams are for people designing bridges and doing surgery. Basic science just learn and start doing some to see if it is for you.

Exams are ultimately for the students' collective benefit; they ensure that lecturers are teaching effectively, with good learning outcomes. If you're someone who "just wants to learn" and don't care for the rigor of a full Math/Physics curriculum, that's one thing. But if you do care, exams are great.

"The course is turned into a game of skill, where manipulative ability outweighs understanding"

4. Teach changes of variables

This has been something that always seems a bit magical to me, when someone comes up with an obscure change of variables that completely simplifies the problem. I'd love to learn this in a more structured/rigorous way. Does anyone have any recommendations for picking this up?

I think there is less of unifying theory than an assortment of artful tricks - and it's a pretty small assortment, too.

If you want to be able to reproduce every clever mathematical trick of the last 200 years you probably can (I believe in you!). But everyone short of Ramanujan learns them by studying many problems, repeated exposure, and either systematically or organically memorizing them. Afterwards, using them in a problem is more about pattern recognition. This is my experience as a physics phd-master-out ymmv

People might disagree strongly, but my recommendation for you is to look into lambda calculus for this.

There are two important concepts that can be generalized: - Substitution, the method of renaming and replacing variables - Abstraction, replacing a variable with another term (function) that can also serve as output

This should give you a more natural feel to concepts like parametrization or what it means that the solution to one function may be another function. Perhaps you might even look a bit into the simply typed lambda calculus which has given me yet more perspectives on the fundamentals of mathematics.

Nearly every educator I've ever learned from has unconsciously insisted that learning material remain sterile, and never become as funny and opinionated as this essay. For most students, "learning" is outrageously boring until you get out of college (or into graduate school) and suddenly learn through essays and memoirs that what was presented to you for 12-16 years in a bone-dry format as established gospel truth was actually the life's work of sometimes dozens if not thousands of people, each of whom fought through their own careers, formed their own views on society, joked, married, divorced, died, squabbled, sniped, and felt immense passion for the subjects they ended up distilling into textbooks for us. Much of the information we learn as students was at the time of its discovery controversial in the extreme.

A few years ago I experienced a perfect example of this phenomenon in a glassblowing museum in Sandwich, Massachusetts. The museum takes visitors through the history of glassblowing in New England, mostly by static exhibits but also with live glassblowing demonstrations. At one point in the tour I read a placard that said something like "in 17XX, John Newenglishman invented a stamped glass method that allowed for much higher production rates of glass. The already-employed glassblowers resisted this development as it meant infringement on their industry." Then I looked up at the wall and read a quote from John Newenglishman that said something like "after demonstrating my stamped glass method, I hid in my room for weeks... gangs of blower-men hunted for me, no doubt to inform me by force their displeasure with my invention." I am paraphrasing in the extreme, but I was struck by how sanitized the placard was compared to the actual words of J.N. It reminded me of the history textbooks I used to read, which removed all of the flair and human-ness from the teaching of history.

If I could change one thing about modern education, I would somehow disabuse students of the notion that the development and archival of information has ever been "regimented" or "unbiased", and take great pains to expose them fully to the tremendous wit and wisdom of the past authors whose work they pore over. Anyway, great essay, saved.

(As a somewhat related point, I have also consistently found mathematicians and engineers to be much funnier than artists and authors, with the exception of artists and authors devoted to comedy. Somehow the tragic hilarity of writing math textbooks or working in a late stage engineering company never made its way into the material I was reading about math and engineering until I was out of my aerospace engineering degree by about three years.)

Nearly every educator I've ever learned from has unconsciously insisted that learning material remain sterile, and never become as funny and opinionated as this essay.

As a former teacher, I think the reason for this is the educational system as we know exists to (attempt to) have reproduceable learning outcomes, at scale. We try to transmit the hard-won knowledge of centuries of geniuses into the minds of adolescents. It's kind of a weird thing to do. It's actually surprisingly successful when I think of it that way, even though I think that it is not great at transmitting the actual insights behind the knowledge.

I suspect a better way to educate people is the more student-led, discovery driven approach. But that is both seemingly harder to scale and less deterministic. Instead, we play out the same cycle of boring, mildly effective education over and over.

I couldn't agree more with you. I personally will attempt to circumvent "boring, mildly effective education" by homeschooling my children (who are at present still too young to even speak in words), but homeschooling is an ugly compromise to a stupid problem. A true comprehensive education would take place everywhere in society.

Have you ever read Ivan Ilich? I find his style almost too iconoclastic for even my contrarian tastes, and I can't stand his sentence structure. But he makes some very coherent points about exactly what you're talking about. John Gatto is a former teacher who writes in a somewhat less brazen style but keeps a lot of the same opinions. It was actually on HackerNews where I first read "the seven-lesson schoolteacher".

Here's the most intuitive introduction to Differential Equation that I have ever seen:https://www.complexityexplorer.org/courses/31-introduction-t...

Explains DEs from scratch, physical significance, one or two traditional methods, then goes on to numerical methods.

If you ever want to learn DEs, IHIGHLYrecommend this tutorial. This is short, too.

I have never found a resource where the concepts behind DEs are explained better.

This tutorial won't prepare you for a course, or tell you everything there is to know about DEs, of course.

For a course-level approach by some of the early pioneers of the pedagogical approach of using numerical methods to understand differential equations, I strongly recommend Blanchard, Devaney, and Hall's book (http://math.bu.edu/odes).

Looks really good. Thanks for sharing it.

Lots of good criticisms of the traditional way of teaching in here. The only issue I would raise is his claim that Bessel functions should be dropped. In physics and engineering they come up all the time when the geometry is cylindrical.

I suppose he could argue that they could wait for a second course, perhaps.

I agree Bessel functions are not excessive, they are just enough to scare you straight.

And yeah, physics students need to be able to solve QM and EM equations, can't get away from (some) special functions.

The kind of people who fix fundamentals are not bothering to become math professors.

There's a lot going on here. First, one of the problems with engineering - and CS has it even worse - is that new ideas and methods get added all the time, but the length of an undergraduate degree stays constant. Compounding the problem, since schools choose or are made to teach more and more things all the time as well (including computing, and topics downstream from computing like "digital media"), one way to deal with more and more topics is to sacrifice depth/rigour on some of the old-fashioned ones like math, so universities have to teach more and more things that they used to rely on students knowing from school as well. The discussion last year about California math reforms and the open letter signed by Scott Aaronson et al. which I'm sure was discussed on HN as well, may stand as an example of this (yes there's a "social justice" angle to it, but the roots of the problem are far deeper than that).

Secondly, a related problem is that a lot of "classical" mathematics was born as an answer to the big questions of the day, and the way it's taught is often just the answer, without any context about the question, and less still about why anyone would care about the question. For example, there's basic logic courses that show 0th-order logic is complete and sound w.r.t the usual rules, but don't give any intuition that this links back to the Entscheidungsproblem and Goedel's theorem and fundamental questions about the nature of the world and philosophy - the mathematical community cared about this at the time! It would be like teaching Turing's halting problem just by giving a definition, a proof and perhaps an example without any idea what question Turing (and Church) were trying to answer in the first place, or the implications for proof-checkers and code-verifiers these days (Coq, TLA+ and so on all exist so Turing's negative result doesn't imply that all is lost).

For differential equations in particular, Cauchy, Sturm-Liouville and others mentioned were all answering (or trying to answer) particular questions that the mathematical community at the time cared about, and without this context the subject can indeed seem dry. I agree with G.-C. Rota (the author of the PDF) that these are not necessarily questions that today's engineers care about, and so yes we can mention that uniqueness theorems exist and move on to linear systems with constant coefficients. If you can't find space to teach and motivate the question, then sure, you don't need to teach the answer either.

As for "functions" that have integrals but no graphs, I also don't like avoiding them by mumbling something about different types of integrals and measures, as if these objects were some kind of organs for sexual reproduction that we can't name because Queen Victoria would blush. No, an ordinary function is something with signature A -> B, and a density on A is a quite ordinary function f with signature S(A) -> B where S is a sigma-algebra that is a subset of the powerset of A and f has certain properties. Usually, but not always, evaluating a density at a set with a single element just gives you 0 and is not very helpful. Measure theory is useful because we can work out certain properties of these functions without always falling back to the powerset definition - that is we can pretend they are a kind of function A -> B in those cases where it does something useful and doesn't break anything else, as long as we understand what's really going on. Case in point: a distribution on the reals that is 1/2 the uniform distribution on [0, 1] plus 1/2 probability mass on the point 2. The probability of any measurable set X is 1/2 the intersection of X and [0, 1], plus 1/2 if 2 is an element of X. There is no mysterious delta function whose value at 2 is exactly "one half infinity" or anything like that, it's just a function from sets to sets where one particular set with one element happens to have a nonzero value. Once you can do this, you are allowed to write 1/2 * U_[0,1] + 1/2 * \delta_2 if you want to.

This sounds misguided. What do you mean, obsolete? It's largely 17th century maths, 19th century understanding was already pretty good. Have there been a pedagogical breakthrough allowing for igniting a spark in a brain much easier? I don't think so.

What do you mean, reduce to the minimum first order equations? I went to the best technical uni in my country (a country known for quite some orthodoxy in teaching), most people would have had some grasp of differential equations before they left high school. Having the most basic equations revisited with actual rigor was eye-opening to whoever I've spoken to about it.

What do you mean, concepts not tricks? Is looking for a solution in a suitable Fourier representation a trick? Certainly is if you ask me. A mother of all tricks hinting at the spectral fabric of the universe.

Last but not least, he sounds like he has some sort of metric on how well his students apply his enlightened teaching later in life vs orthodox inefficient professors'. I have some doubts on this.

DiffEQ was probably my favorite math class in college. Eventually I createdhttps://play.google.com/store/apps/details?id=simplicial.sof..., originally for a course about computer graphics. I think it lays the foundation for an intuitive understanding of the basic concepts in a way that a text book cannot.

Interesting, I wonder if the same things exist in teaching programming? Well, yes!

Let me give you an example. Both Dart and Flutter borrow from Google's own lessons learned and innovations they did with the native Android OS and widget stack. In fact the testing stack of widgets, goldens, and re-using storyboarding as BDD come directly from the same practices on native Android.

In programming we have a legacy of borrowing from the language and frameworks that came before the current stack that is helpful to know about as everyone uses human knowledge and human language shortcuts to refer to past programming languages and frameworks.

The problem with this area of math is that you need a good basic grasp of geometry as that is how it was built and discovered, see this history chart to see why I state that from wikipedia:

https://en.wikipedia.org/wiki/Geometric_calculus#/media/File...

Of course I always found it easier to think and solve things in geometrical terms and methods. It drove one of my Calculus professors crazy, that and sleeping in class and yet still passing class with an A, getting As on all the tests.

they shipped the org chart

When I was a kid, one of my parents told me diff eq was a course they had not even gone to class for and passed, which set me up for disappointment in diff eq pedagogy for sure. It didn't help when I read this article, and then took sophomore diff eq and it was exactly as described. Later in grad school I'd actually teach the course to my complete chagrin. I think I mentioned this essay to my students, because how could I not? Of course I did not try to redo the diff eq pedagogy, but I did put a lot of emphasis on linearity.

Related. Others?

Lessons I wish I had learned before I started teaching differential equations [pdf] (1997)-https://news.ycombinator.com/item?id=32530035- Aug 2022 (177 comments)

10 lessons I wish I had learned before I started teaching differential equations-https://news.ycombinator.com/item?id=19005798- Jan 2019 (2 comments)

Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]-https://news.ycombinator.com/item?id=15163979- Sept 2017 (108 comments)

Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]-https://news.ycombinator.com/item?id=11207183- March 2016 (118 comments)

(1997)