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

351 points by itronitron 3y ago

itronitron 3y ago
Previous discussion from 2022... https://news.ycombinator.com/item?id=32530035 https://news.ycombinator.com/item?id=32530035
- ianai 3y ago
  They talk (rightly) about mathematics replacing infinitely small but not 0 numbers with limits there. Infinitesimals were later reintroduced with rigor through https://mathworld.wolfram.com/NonstandardAnalysis.html https://mathworld.wolfram.com/NonstandardAnalysis.html I have long thought such would be easier to work with on modern computers.
  - monista 3y ago
    Note though, that nonstandard analysis isn't compatible with more "intuitionistic" https://en.wikipedia.org/wiki/Axiom_of_determinacy 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.
    - zozbot234 3y ago
      On the contrary, in an intuitionistic setting you can use https://en.wikipedia.org/wiki/Synthetic_differential_geometry https://en.wikipedia.org/wiki/Synthetic_differential_geometr... and https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis which 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.)
    - yuppiemephisto 3y ago
      Can do it in pure ZF: https://www.youtube.com/watch?v=YE68yptMMeQ https://www.youtube.com/watch?v=YE68yptMMeQ https://u.cs.biu.ac.il/~katzmik/spot.html https://u.cs.biu.ac.il/~katzmik/spot.html
  - scythe 3y ago
    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.
    - ducttapecrown 3y ago
      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)
      - ianai 3y ago
        "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?
        karatinversion 3y ago
        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(...).
      - scythe 3y ago
        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.
  - adgjlsfhk1 3y ago
    Infinitesimals are commonly used in scientific computing (see forward mode automatic differentiation).
contrarian1234 3y ago
“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
- gleenn 3y ago
  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
- tux1968 3y ago
  > 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.
  - ajkjk 3y ago
EsportToys 3y ago
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 https://esporttoys.pages.dev/2022/11/21/damped
- pastage 3y ago
  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.
- wiredfool 3y ago
  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.
  - JadeNB 3y ago
    > 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.
thorin 3y ago
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.
- ajkjk 3y ago
  Physics doing the work of actually teaching math is a universal embarrassment to math departments. Or, well, it should be.
  - f1shy 3y ago
    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”
    - DoughnutHole 3y ago
      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.
      - ajkjk 3y ago
        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 mathematicians are wrong to 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.
zzbn00 3y ago
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.
- deleted 3y ago
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- zozbot234 3y ago
  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.
  - zzbn00 3y ago
    "The course is turned into a game of skill, where manipulative ability outweighs understanding"
asplake 3y ago
(1997)
Xcelerate 3y ago
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).
- sesm 3y ago
  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'.
dustingetz 3y ago
they shipped the org chart
deleted 3y ago
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quacked 3y ago
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.)
- acjohnson55 3y ago
sixo 3y ago
The kind of people who fix fundamentals are not bothering to become math professors.
samvher 3y ago
> 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?
- tihal 3y ago
  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.
- xaellison 3y ago
  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
jampekka 3y ago
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.
- marcosdumay 3y ago
  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.
  - starcraft2wol 3y ago
    You need analysis to make sense of infinite series, indefinite integrals, exponential functions, etc.
    - Chabsff 3y ago
deleted 3y ago
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red_admiral 3y ago
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.
maweaver 3y ago
> 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.
- BeetleB 3y ago
  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).
- mrkeen 3y ago
  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...
  - ducttapecrown 3y ago
    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.
hiAndrewQuinn 3y ago
>LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS ARE THE BOTTOM LINE Yes! Jesus. People so often forget you can literally take variables and just fill in easy constants to get a sense of how they work. I can't believe this isn't, like, the first thing they teach in DE.
- selimthegrim 3y ago
  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
  - TwentyPosts 3y ago
    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.
    - selimthegrim 3y ago
      They switched to a more forgiving section that didn’t do variable coefficients
aj7 3y ago
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.
- GeorgeTirebiter 3y ago
  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'?
  - jimmaswell 3y ago
    In the olden days, a lot of problems were solved numerically by methods such as weighing a printed graph of a function
- hyperthesis 3y ago
  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 the only way 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.
  - aj7 3y ago
    Mine too. It’s a hot competitive field to have the best solvers with the best gui.
meroes 3y ago
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.
- kriops 3y ago
  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)
  - gbjw 3y ago
whatever1 3y ago
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.
- SpaceManNabs 3y ago
  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.
  - heisenzombie 3y ago
    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.
ducttapecrown 3y ago
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.
__rito__ 3y ago
Here's the most intuitive introduction to Differential Equation that I have ever seen: https://www.complexityexplorer.org/courses/31-introduction-to-differential-equations 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, I HIGHLY recommend 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.
- JadeNB 3y ago
  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 http://math.bu.edu/odes).
  - __rito__ 3y ago
    Looks really good. Thanks for sharing it.
not2b 3y ago
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.
- eternauta3k 3y ago
  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.
fredgrott 3y ago
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:Geometric_Calculus_Family_Tree.png 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.
dang 3y ago
Related. Others? Lessons I wish I had learned before I started teaching differential equations [pdf] (1997) - https://news.ycombinator.com/item?id=32530035 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 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 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 https://news.ycombinator.com/item?id=11207183 - March 2016 (118 comments)
jojobas 3y ago
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.
grog454 3y ago
DiffEQ was probably my favorite math class in college. Eventually I created https://play.google.com/store/apps/details?id=simplicial.software.Systems https://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.

351 points by itronitron 3y ago

Previous discussion from 2022... https://news.ycombinator.com/item?id=32530035 https://news.ycombinator.com/item?id=32530035

ianai 3y ago
They talk (rightly) about mathematics replacing infinitely small but not 0 numbers with limits there. Infinitesimals were later reintroduced with rigor through https://mathworld.wolfram.com/NonstandardAnalysis.html https://mathworld.wolfram.com/NonstandardAnalysis.html I have long thought such would be easier to work with on modern computers.
- monista 3y ago
  Note though, that nonstandard analysis isn't compatible with more "intuitionistic" https://en.wikipedia.org/wiki/Axiom_of_determinacy 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.
  - zozbot234 3y ago
    On the contrary, in an intuitionistic setting you can use https://en.wikipedia.org/wiki/Synthetic_differential_geometry https://en.wikipedia.org/wiki/Synthetic_differential_geometr... and https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis which 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.)
  - yuppiemephisto 3y ago
    Can do it in pure ZF: https://www.youtube.com/watch?v=YE68yptMMeQ https://www.youtube.com/watch?v=YE68yptMMeQ https://u.cs.biu.ac.il/~katzmik/spot.html https://u.cs.biu.ac.il/~katzmik/spot.html
- scythe 3y ago
  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.
  - ducttapecrown 3y ago
    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)
    - ianai 3y ago
      "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?
      - karatinversion 3y ago
        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(...).
    - scythe 3y ago
      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.
- adgjlsfhk1 3y ago
  Infinitesimals are commonly used in scientific computing (see forward mode automatic differentiation).

contrarian1234 3y ago

“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

gleenn 3y ago
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
tux1968 3y ago
> 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.
- ajkjk 3y ago

EsportToys 3y ago

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 https://esporttoys.pages.dev/2022/11/21/damped

pastage 3y ago
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.
wiredfool 3y ago
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.
- JadeNB 3y ago
  > 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.

thorin 3y ago

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.

ajkjk 3y ago
Physics doing the work of actually teaching math is a universal embarrassment to math departments. Or, well, it should be.
- f1shy 3y ago
  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”
  - DoughnutHole 3y ago
    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.
    - ajkjk 3y ago
      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 mathematicians are wrong to 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.

zzbn00 3y ago

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.

deleted 3y ago
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zozbot234 3y ago
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.
- zzbn00 3y ago
  "The course is turned into a game of skill, where manipulative ability outweighs understanding"

asplake 3y ago

(1997)

Xcelerate 3y ago

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).

sesm 3y ago
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'.

dustingetz 3y ago

they shipped the org chart

deleted 3y ago

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quacked 3y ago

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.)

acjohnson55 3y ago

sixo 3y ago

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

samvher 3y ago

> 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?

tihal 3y ago
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.
xaellison 3y ago
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

jampekka 3y ago

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.

marcosdumay 3y ago
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.
- starcraft2wol 3y ago
  You need analysis to make sense of infinite series, indefinite integrals, exponential functions, etc.
  - Chabsff 3y ago

deleted 3y ago

[deleted]

red_admiral 3y ago

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.

maweaver 3y ago

> 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.

BeetleB 3y ago
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).
mrkeen 3y ago
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...
- ducttapecrown 3y ago
  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.

hiAndrewQuinn 3y ago

>LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS ARE THE BOTTOM LINE Yes! Jesus. People so often forget you can literally take variables and just fill in easy constants to get a sense of how they work. I can't believe this isn't, like, the first thing they teach in DE.

selimthegrim 3y ago
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
- TwentyPosts 3y ago
  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.
  - selimthegrim 3y ago
    They switched to a more forgiving section that didn’t do variable coefficients

aj7 3y ago

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.

GeorgeTirebiter 3y ago
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'?
- jimmaswell 3y ago
  In the olden days, a lot of problems were solved numerically by methods such as weighing a printed graph of a function
hyperthesis 3y ago
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 the only way 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.
- aj7 3y ago
  Mine too. It’s a hot competitive field to have the best solvers with the best gui.

meroes 3y ago

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.

kriops 3y ago
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)
- gbjw 3y ago

whatever1 3y ago

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.

SpaceManNabs 3y ago
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.
- heisenzombie 3y ago
  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.

ducttapecrown 3y ago

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.

__rito__ 3y ago

Here's the most intuitive introduction to Differential Equation that I have ever seen: https://www.complexityexplorer.org/courses/31-introduction-to-differential-equations 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, I HIGHLY recommend 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.

JadeNB 3y ago
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 http://math.bu.edu/odes).
- __rito__ 3y ago
  Looks really good. Thanks for sharing it.

not2b 3y ago

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.

eternauta3k 3y ago
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.

fredgrott 3y ago

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:Geometric_Calculus_Family_Tree.png 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.

dang 3y ago

Related. Others? Lessons I wish I had learned before I started teaching differential equations [pdf] (1997) - https://news.ycombinator.com/item?id=32530035 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 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 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 https://news.ycombinator.com/item?id=11207183 - March 2016 (118 comments)

jojobas 3y ago

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.

grog454 3y ago

DiffEQ was probably my favorite math class in college. Eventually I created https://play.google.com/store/apps/details?id=simplicial.software.Systems https://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.