I was lucky to have a high school math teacher who derived calculus with us. Understanding calculus made understanding physics so much easier. Being able to solve some physics problems using calculus seemed like magic.
Of course, I've never had to use any of that knowledge since, but I'm glad I went through the process to acquire it.
nathan_compton · 4h ago
I think its nuts anyone even bothers to teach physics without calculus.
sn9 · 3h ago
I'd rather they be exposed to physics without calculus than never be exposed to it at all.
Calculus is an elective course in most schools.
trod1234 · 4h ago
I was lucky enough to have several teachers that were able to undo a lot of the damage bad math teachers imposed on me (on a lag) and continue on from Calculus to the foundations of mathematical systems (Abstract Algebra).
I've also used that knowledge quite a lot since, both in reasoning about problems and its also been beneficial in the confidence of accuracy in the methods of problem solving.
Bad malicious teachers nearly tortured math out of me. Without a substitute teacher who retired due to malicious politicking of his pears; but who had a phd in Mathematics Education, and was able to narrow in on exactly what I was taught that was incorrect (3 classes prior to the one being taken); and the patience and technique he used to destructively correct the false teachings while alleviating the operant conditioned anxiety; I wouldn't be where I am today.
There's quite a lot of malevolence in the world based in blindness, fortunately there are also some good souls out there helping elevate others.
jameskilton · 4h ago
Article has been hugged so commenting more along the comments here.
Pharmacy school teaches Calculus. Why would that be? Do you need to run derivatives and integrals to fill prescriptions?
No. Teaching maths, particularly calculus, teaches people how to 1) not make mistakes and 2) catch your own mistakes quickly. Vitally important skills for someone filling out live-saving medicine.
jacobolus · 4h ago
The point of learning mathematics is not to "not make mistakes". More strongly: the misconception that mathematics or mathematics education is about getting "right answers" as quickly and accurately as possible is a disaster for learning.
A calculus class should ideally be making someone think much harder than that. Calculus is about understanding the relationships involved with continuous quantities and modeling the way things move and change. It is a basic prerequisite for understanding biochemistry and statistics, essential background for understanding pharmaceuticals.
tptacek · 4h ago
I'm pretty sure this is a retcon. There are subject-matter-specific reasons PharmD's learn calculus. They don't use it day-to-day, but lots of STEM curricula (a) don't make sense without calculus and (b) don't lead to jobs where you're integrating by parts every day.
9rx · 3h ago
> Teaching maths, particularly calculus, teaches people how to 1) not make mistakes and 2) catch your own mistakes quickly. Vitally important skills for someone filling out live-saving medicine.
Learning calculus achieves the same effect, though. It is not the teaching that is important (although some may find it useful).
The question makes as much sense to me as "why teach literature in the age of typewriters?" Not that the analogies are perfect, but the idea that it's not worth learning something because a related technology has advanced significantly is a non sequitur.
There may be good reasons to learn or not to learn calculus, or literary theory, or anything else, but the existence of some related technology isn't it. I'd go so far as to suggest that perhaps calculus is even more important for some folks to learn in the age of AI (e.g. applications in neural networks), and we don't know who those folks will be in advance.
alphazard · 4h ago
I wouldn't recommend a traditional calculus course to anyone. There's no reason to do derivatives or integrals by hand, and that's most of the course.
The practical applications of running differences and running sums can be taught to people with minimal programming experience and without algebra.
I've never done an integral by hand as part of any productive activity. Monte carlo integration and loops for multiply-and-add have proven incredibly useful. Why not teach those directly?
Actually, Iverson et al. wrote dozens of math textbooks using array languages!
tptacek · 4h ago
I'm early into Calc II right now (MathAcademy's equivalent of it), having started 6 months ago at a D-student's level of Algebra II, and I'm curious what the "right" calculus to learn would be.
It's pretty clear to me as I work through problem sets that I'm never going to do any of this hand-computation in reality, in the same way that nobody computes eigenvectors by finding the roots of a characteristic equation. It's still fine by me, for 2 reasons: (1) because I'm doing this to replace the New York Times Crossword with something productive, and it's great for that, and (2) because every time I get annoyed at like messy trig derivatives with double-angle substitutions and stuff, I instead pivot to learning how to solve it with Sage Math, and so I get better at that instead.
But if there's a smarter sequence, I'm super interested!
jacobolus · 4h ago
For a more conceptual introduction leaning on using computers, whose goal was getting STEM students up to speed to understand the context of work in their various fields, you might enjoy https://www.science.smith.edu/~callahan/intromine.html
I'm gonna push back and claim that learning calculus the traditional way is still worth doing.
Not only will you be even more capable of picking up solving things numerically, but you'll also have the prerequisites for studying physics or probability or machine learning or Knuth's Concrete Mathematics. It opens doors to new intellectual vistas.
Solving things analytically (when possible) also can reveal more about the nature of the problem than doing so numerically, and can give the same satisfaction as finding an elegant solution in code.
You can definitely go an entire programming career without ever using it, but if you ever do run into a problem it solves, having this tool available to you is only a benefit.
tptacek · 2h ago
We're not at odds! I think all I'm saying is that when manually working out integrals gets frustrating, there's a learning-mode escape hatch to just figuring out how to solve them in something like Sage; when the frustration subsides you go just get back to doing the manual stuff. When I'm in Sage I'm still learning stuff. I'm never abandoning analytical work.
I would feel real weird if there were things (in Calc II problem sets) I could solve in Sage that I simply couldn't do by hand.
I don't feel weird that there are things I can do, but will get wrong a bunch of times if I try to do them, and can quickly bang out in Sage. That seems fine to me. For a lot of these subjects, I don't care about automaticity, just intuition.
It's like a lot of linear algebra: being able to quickly do things by hand is kind of silly, because for real world problems (at least in data science) hand solves aren't even really feasible. But learning to do it by hand is important for building intuition.
sn9 · 1h ago
Oh sorry I wasn't pushing back against you specifically haha. Just those pushing the idea that learning calculus the traditional way is useless since we have computers.
I've been going through Math Academy with a IPython REPL open, too, and I've noticed that I need to avoid using it unless the problem specifically tells me to use a calculator or the implicit skill review in the problem gets skipped. Even writing little functions for myself to one-shot a problem means I'm missing chances to actively recall the steps.
Actually, given what you've said, you'd probably enjoy working through Sanjoy Mahajan's books on Fermi problems and estimations (the books are CC-licensed you can just download them): https://mitpress.mit.edu/author/sanjoy-mahajan-9006/
drivebyhooting · 4h ago
Many times you can speed up your code and boost accuracy by using higher order methods. What you describe, if I understand it rightly, is 0th order.
glial · 3h ago
So much mathematical modeling is based on systems of differential equations. It's difficult to understand the conceptual approach without some basic understanding of calculus.
You CAN model predator-prey dynamics or disease spread using Monte Carlo techniques, but you can't read the historical literature without some grasp of differential equations.
alphazard · 3h ago
Differential equations is kind of the same situation. The important intuitions are more likely to come from repeatedly applying a function to an input and seeing where that takes the simulation, than by doing DiffEQ practice problems.
There will always be some brilliant people that breeze through a hard course, and effortlessly acquire the intuition, and then wrongly attribute their understanding to the course. That doesn't help the rest of us, who are much more likely to build intuition by guessing about the outcome of a program and then checking if we are right.
nathan_compton · 4h ago
Weird. Calculus is fun.
fxwin · 4h ago
not to anyone? really?
loudmax · 3h ago
I'd never recommend someone not learn calculus, but the average non-technical high school or college student would probably benefit more from learning statistics instead.
Learning calculus is very valuable to a relatively small number of people. We should absolutely make calculus available to any student, regardless in advances in AI. But a lot more people, particularly non-technical or non-math type people, would benefit more from experience with statistics than calculus. Statistics, combined with an introduction to simple programming, should be part of the basic high-school curriculum.
lispisok · 2h ago
Stats without calculus is like physics without calculus. Sure you can teach high schoolers to memorize displacement and velocity equations without calc but actual derivation and understanding requires calc.
sn9 · 3h ago
Both statistics and calculus are usually electives, not requirements.
rs186 · 4h ago
Is AI even relevant here?
Mathematica can do calculus and linear algebra better than most if not all college graduates since decades ago, but colleges are still teaching those courses. That should explain enough.
rightbyte · 3h ago
I used Wolfram Alfa like 15 something years ago to explain book exercise solutions step by step in some algebra and differential equation courses.
0xTJ · 4h ago
It is shocking to me that people would seriously discuss not teaching calculus just because LLM tools exist. Computational math engines didn't make understanding how you solve an integral obsolete, but they can make certain tasks faster and less error-prone.
This feels like a "tech bro" idea from someone who has never touched a SEM field (STEM minus the T).
deadbabe · 4h ago
How many people remember how to solve an integral?
justonceokay · 4h ago
Not remembering U substitution of the top of your head is different than not being exposed to it. I remind you that Int(x^3 * 5x^2 + 7) == x^4/4 + 5x^3/3 +7x + C. Just from that alone I bet a lot of memories of integration slot back into your head, and you would know where to look up the parts you forgot.
You can’t “brush up” on something you never learned
rangerelf · 4h ago
It doesn't matter if you can't solve a randomly-appearing-in-your-newsreel integral; it matters that you have the background knowledge of what an integral is, that there are rules to solving it, and you can read up on the rules and understand them.
For the [current] layperson, each of those things I mentioned I might as well be speaking in Martian.
dvrj101 · 3h ago
if you actually spent good amount of time in mathematics during academia, you have developed neural networks for logical reasoning and problem solving but they get activated in life situations giving an edge compared to others.
It's not necessarily just about remembering every rule and trick you can use to simplify and solve integrals. Calculus is fundamental to understanding problems, from basic exercises in a first-year undergraduate physics course to entire fields.
You'll (probably) never apply the ability out the kinetic energy vs. time of a ball rolling down a hill, but these exercises build understanding of the tools. Derivatives are everywhere in a fundamental electric circuits course, you need to have an intuitive understanding of basic calculus. The relationship between current through and voltage across ideal inductors and capacitors are directly described in the language of calculus, even if you're not "using" the calculus substitutions you learned each time you analyse a circuit.
And good luck getting through a couple weeks of an introductory quantum mechanics course without using calculus as a fundamental building block. You can solve many of these problems with computers, but it's not going to build intuition on how to approach future problems. (I don't mean this as a joke or picking an arbitrary complicated-sounding topic; this is a core course in some engineering programs.)
Many engineering problems have nice closed-form equations (at least to get approximations). Obtaining those equations often involves calculus, and someone has to do that in the first place.
(I'm giving examples from the lens of my education, but each field of science, engineering, and mathematics will have their own context, and will vary from little-to-no calculus to being all-calculus.)
yoyohello13 · 4h ago
Why learn anything? I for one want my brain to be completely empty, devoid of any thought or knowledge. Then I can simply pay OpenAI to think for me.
Honestly, the trend toward anti-intellectualism in the world is very disturbing to me, and it seems AI is enabling this kind of contempt for knowledge even more.
somenameforme · 4h ago
It's not anti-intellectualism so much as people forgetting the reason we learn in the first place. It's one of the biggest downsides of going from a world where education was largely optional to one where people are largely shoved onto a treadmill from birth to college.
So many people never stop think about why we do the things that we are simply expected to do, instead of doing other things.
muldvarp · 4h ago
I genuinely wonder what even is worth learning in the age of AI. It just feels like learning stuff doesn't really matter anymore. Unless you're an expert in some field, a novice with access to an LLM will usually produce better results than you.
softwaredoug · 4h ago
The most important skill with using AI is critical thinking. And that's really only honed by learning and questioning a lot of subjects.
I don't use Calculus in my day job, but it was still valuable for me to learn to hone my ability to think
florbnit · 4h ago
> And that's really only honed by learning and questioning a lot of subjects.
Sounds like a job for AI.
bregma · 4h ago
Where will the experts come from in the future if no one learns and we just accept whatever the Markov chain generates for us from the Bayesian database of old information?
tapoxi · 4h ago
Good question, but that isn't incentivized by capital. We're in an era of massive student loans and unaffordable housing. Unless there's a clear and direct incentive to learn, people will take the easiest path. Life is tough enough as-is.
muldvarp · 3h ago
The number of people that can actually dedicate their life to producing new knowledge is tiny. It's not really that motivating of a reason.
sifar · 2h ago
However, we don't know beforehand, which of those people would produce new knowledge.
muldvarp · 2h ago
I'm not arguing to not teach children stuff. All I'm saying is that LLMs made me lose motivation to start learning stuff that I'll never be an expert in.
HPsquared · 4h ago
You still need a decent mental model and mathematical fluency to critically assess the outputs.
muldvarp · 4h ago
Currently. Although producing something yourself (especially if you're not an expert) doesn't guarantee high-quality output either.
q2dg · 4h ago
Maybe the question isn't "why" but "how"
cubefox · 4h ago
By letting the pupils do the "homework" under teacher supervision.
cultofmetatron · 4h ago
you literally cant do regression analysis without calculus.
y-curious · 4h ago
Like, on the backend? You can't run code without electricity, but FAANG doesn't teach that in their onboarding
tptacek · 4h ago
No? You literally can't derive regression procedures without it, but you don't need to. You can just do least-squares.
srean · 2h ago
Along the happy path yes.
But when things break and you have to understand why, or you have to fix it, or even describe the problem in some detail that someone else is able to guide you to fix it, it sure helps to have an idea of how it works.
tptacek · 2h ago
So then my only rebuttal here is I think the original claim was overstated.
9rx · 4h ago
Not teaching calculus does not imply not learning calculus.
tonetegeatinst · 4h ago
I mean I think the bigger issue is that, at least for me, unless I am regularly using a skill I learn I forget most if not all of it.
I trig, and calc 1 yet I hardly remember how to solve most problems because its not something I use regularly.
Same with subnetting or remembering certain programming languages.
I literally don't use these things 99% of the time and so I forget them. Sure I understand some of the basics and I could probably pick it back up way faster than someone who knows nothing but I am human.
yoyohello13 · 4h ago
I was recently studying some basics of quantum computing and had to re-learn a bunch of linear algebra. I thought I had forgotten everything, but when I sat down to review the material I picked it up very quickly (much faster than when I was in undergrad). It was like riding a bike. I think we think we forget, but the knowledge is actually dormant and much easier to get back.
So I don't think forgotten knowledge is wasted, it's still valuable.
j45 · 4h ago
Calculus is critical.
One of the few moments in university was learning how so much was actually Calculus.
Whether it was physics, chemistry, etc, the formulas I was given ften had a calculus version.
It helped me open up to taking math and stats courses I never would have as a comp sci student, which in turn gave me a different perspective than just taking cs courses alone.
trod1234 · 4h ago
Anyone have a mirrored link? Looks like the site is timing out now.
Onavo · 4h ago
...how do you plan to do gradient descent if not without calculus? Seems like OP is trying to lock an entire generation into proprietary tooling.
lokrian · 4h ago
The question I keep wondering about is why teach anything, or what exactly is worth teaching.
rangerelf · 4h ago
Because, nature in its infinite wisdom, gets rid of what's not used.
You don't use your muscles? They atrophy. You don't make an effort to travel without a gps regularily, to force your brain to remember your way around naturally? Your spatial memory atrophies and becomes useless [here: https://www.nature.com/articles/s41598-020-62877-0 ]
People don't need to learn math anymore, hence, no more calculus lessons? People are literally becoming idiots who can't calculate simple change at the cash register without pulling out their calculators.
It's exercise. It keeps the brain itself from atrophying. It stops you from becoming a "wetware LLM" that's just parroting whatever echo of a thought (natural or otherwise) goes through it.
pavel_lishin · 4h ago
Reading Pump Six will provide a partial answer.
wrs · 4h ago
Anything where you care about getting the right answer?
deadbabe · 4h ago
How to use an LLM
ryandv · 3h ago
Don't. Calculus is going to become as obsolete and antiquated as studying Euclid's Elements and doing your own compass and straightedge constructions, or using an abacus.
LLMs have made obsolete the study and practice of the mechanics and minutiae of assembly language, memory allocation, system calls, hardware interrupts, process scheduling, cohesion, coupling, design patterns, code smells, test-driven development, "clean code," etc. There is no reason for anybody to learn any of this when the LLM, pretrained on all the literature backing these concepts, can emit code at 10,000x the speed that already embodies all of these lessons.
There is no reason to suppose mathematics is any different. You don't break out your pen and paper to compute change at the checkout aisle at the grocer's; you don't even have to punch numbers into a calculator, or indeed be numerate at all - just scan the barcode. Prepare for a world where we have machines that make calculus and programming as easy as scanning a barcode.
Look forward to the future described in Stephenson's Anathem where cutting-edge spacefaring nuclear technology is wielded by the literally illiterate, and the only people capable of reading books on sheets of dead trees are effectively monks living in Neo-Luddite monasteries.
inetknght · 3h ago
> Calculus is going to become as obsolete and antiquated as studying Euclid's Elements and doing your own compass and straightedge constructions, or using an abacus.
What do you think is going to replace calculus? Until something better comes along, calculus is among the best introduction to higher theories of mathematics.
> LLMs have made obsolete the study and practice of...
It sounds like you're trying to say that Calculus will be replaced by LLMs. In fact it's the reverse: LLMs can get better by studying calculus and the higher math things that calculus enables.
> There is no reason to suppose mathematics is any different. You don't break out your pen and paper to compute change at the checkout aisle at the grocer's; you don't even have to punch numbers into a calculator, or indeed be numerate at all - just scan the barcode
Buddy, barcodes can be wrong. If you don't know how to do math yourself, you won't know when a sweet potato gets confused with a differently-priced yam. Or your new tablet with 5 year warranty gets scanned as a server CPU with 1 year warranty; then when your tablet dies at 4 years 11 months, your warranty was never activated, and your tablet is reported stolen.
> Look forward to the future described in Stephenson's Anathem where cutting-edge spacefaring nuclear technology is wielded by the literally illiterate
You may be right on that. But, in that world, the "literally illiterate" people will always be outshone by people who actually know how things work.
ryandv · 3h ago
So are you bullish or bearish on the prospect of LLMs replacing programmers? The industry is breathless with hype and media pieces on how traditional programming is obsolete; do your criticisms of my position also apply, were software engineering, rather than calculus, its subject?
inetknght · 3h ago
> are you bullish or bearish on
"bullish" and "bearish" are words that are useless in this context. Use them for stock markets where those words belong.
> LLMs replacing programmers? The industry is breathless with hype and media pieces on how traditional programming is obsolete
I believe LLMs fill a small (over-hyped) niche currently, but that niche will grow over time. I don't believe LLMs will, within our lifetimes, fill the specialized roles of high intelligence. It certainly will (and in many cases already does) fill the roles that don't require lots of thinking and intelligence.
Could LLMs replace programmers? Definitely, right now. Could LLMs replace developers? Currently, it could replace some of the worst developers, and help but not replace some of the better developers. Could it replace engineers? Definitely not right now, and probably not for the next several decades.
Is that a bad thing? To a lot of people, it is, especially in short term measured by job losses. But in the long run I believe LLMs are transformative in nature. Programming and development will be transformed to use LLMs as the tools that they are. Anyone who doesn't adapt to using an LLM will end up being a loser in their industry or be required to retrain to another role. That's generally good for the economy as a whole even if it's not great for the individual person.
You could manually do a lot of things that a CNC mill can do. Or you can learn how to use a CNC mill. Or you could pay someone else to use their CNC mill. But a CNC mill is not a great thing to use for cutting raw lumber or assembling furniture. You "could" use a CNC mill for it, but it'd be extremely inefficient. You could use it to manufacture furniter pieces, but it's not going to assemble them. LLMs are kind've the same thing for computers.
> do your criticisms of my position also apply, were software engineering, rather than calculus, its subject?
I think yes. Software Engineering (as an Engineering discipline) is used where it matters (eg, regulated aviation, safety, etc). It's not used much (if at all) for basic websites or even most games. But when it is used for these things, it's a much better product. That will pretty much always be true even with LLMs guiding the developer (not engineer).
For your example I replied to: I don't know much about nuclear reactors. But I know more than, say, 80% of the general population. I know enough to know that they can be dangerous but some designs are less-dangerous than others. Could I build a reactor myself? Probably, while the average person wouldn't even know where to start. I wouldn't do it without a lot more knowledge though because anything I come up with would be miles behind current state of the art in safety and procedure. I probably could identify nuclear fuel ores, but I have no idea how to refine it safely. I don't really care to learn that, it's not my passion. But I also know that refining it is dangerous if not done "properly" and so I wouldn't even want it as a hobby.
Contrast that with robotics. Robotics can certainly be dangerous, but not nearly as dangerous as refining nuclear fuel. I know how robotics work, and how to work with them with some modicum of safety, and so I do that as a hobby. That doesn't mean I would encourage anyone to come along and just use an LLM to build robots because they'd be missing out on sooooooo many fundamentally important concepts and would have to rely on the LLM to tell them what's dangerous or what's not and how to implement safety features and procedures.
ryandv · 2h ago
> What do you think is going to replace calculus? Until something better comes along, calculus is among the best introduction to higher theories of mathematics.
> Programming and development will be transformed to use LLMs as the tools that they are. Anyone who doesn't adapt to using an LLM will end up being a loser in their industry or be required to retrain to another role [...]
Mathematics will be transformed to use LLMs as the tools that they are. Any mathematician who doesn't adapt to using an LLM will end up being a loser in academia or be required to retrain to another role.
The study of calculus will be replaced by tooling that carries out the mechanics of differentiation, integration, finding extrema, etc, making such education obsolete.
> You could manually do a lot of things that a CNC mill can do. Or you can learn how to use a CNC mill. Or you could pay someone else to use their CNC mill. But a CNC mill is not a great thing to use for cutting raw lumber or assembling furniture. You "could" use a CNC mill for it, but it'd be extremely inefficient. You could use it to manufacture furniter pieces, but it's not going to assemble them. LLMs are kind've the same thing for computers.
You could manually do a lot of things that a calculator, CAS, or LLM can do. Or you can learn how to use a CAS system or LLM, or pay someone to use their cloud service. But an LLM is not a great thing to use for deciding research programs or assembling a proof. You could use an LLM for it, but it'd be extremely inefficient. You could use it to formalize certain lemmata or minor results, but it's not going to compose them together into a larger theorem.
> Could LLMs replace programmers? Definitely, right now. Could LLMs replace developers? Currently, it could replace some of the worst developers, and help but not replace some of the better developers. Could it replace engineers? Definitely not right now, and probably not for the next several decades.
Could LLMs replace arithmeticians or calculators (in the original sense of the word, as a human occupation)? Definitely, right now. Could it replace mathematicians? Definitely not right now, and probably not for the next several decades.
You distinguish between Software Engineers(tm) and lowly developers; I point out that you can draw the same distinction between mathematicians and lowly arithmeticians or "calculators" (in the original sense of the term as a human occupation), or between calculus and analysis.
Leave the solution of this particular integral or the differentiation of this particular function, the mechanics of calculus, to the LLMs and the machines. Mathematicians need not concern themselves with such trivialities, and will prove results for all continuous functions in arbitrary metric spaces, or something similarly general and insightful.
Jtsummers · 2h ago
> The study of calculus will be replaced by tooling that carries out the mechanics of differentiation, integration, finding extrema, etc, making such education obsolete.
What makes you think this time is different? We've had tools for those things for decades now.
ryandv · 1h ago
Exactly the things that differentiate LLM tooling from traditional vim, IDEs, valgrind, gdb, linters, the coreutils, ...
Jury's out on whether or not that collection is non-empty.
Of course, I've never had to use any of that knowledge since, but I'm glad I went through the process to acquire it.
Calculus is an elective course in most schools.
I've also used that knowledge quite a lot since, both in reasoning about problems and its also been beneficial in the confidence of accuracy in the methods of problem solving.
Bad malicious teachers nearly tortured math out of me. Without a substitute teacher who retired due to malicious politicking of his pears; but who had a phd in Mathematics Education, and was able to narrow in on exactly what I was taught that was incorrect (3 classes prior to the one being taken); and the patience and technique he used to destructively correct the false teachings while alleviating the operant conditioned anxiety; I wouldn't be where I am today.
There's quite a lot of malevolence in the world based in blindness, fortunately there are also some good souls out there helping elevate others.
Pharmacy school teaches Calculus. Why would that be? Do you need to run derivatives and integrals to fill prescriptions?
No. Teaching maths, particularly calculus, teaches people how to 1) not make mistakes and 2) catch your own mistakes quickly. Vitally important skills for someone filling out live-saving medicine.
A calculus class should ideally be making someone think much harder than that. Calculus is about understanding the relationships involved with continuous quantities and modeling the way things move and change. It is a basic prerequisite for understanding biochemistry and statistics, essential background for understanding pharmaceuticals.
Learning calculus achieves the same effect, though. It is not the teaching that is important (although some may find it useful).
Article archive: https://archive.is/2AdUJ
It's very short.
https://en.wikipedia.org/wiki/Tai%27s_model
There may be good reasons to learn or not to learn calculus, or literary theory, or anything else, but the existence of some related technology isn't it. I'd go so far as to suggest that perhaps calculus is even more important for some folks to learn in the age of AI (e.g. applications in neural networks), and we don't know who those folks will be in advance.
I've never done an integral by hand as part of any productive activity. Monte carlo integration and loops for multiply-and-add have proven incredibly useful. Why not teach those directly?
Also College Math with APL: https://archive.org/details/APL_books/Introduction%20to%20Co...
Actually, Iverson et al. wrote dozens of math textbooks using array languages!
It's pretty clear to me as I work through problem sets that I'm never going to do any of this hand-computation in reality, in the same way that nobody computes eigenvectors by finding the roots of a characteristic equation. It's still fine by me, for 2 reasons: (1) because I'm doing this to replace the New York Times Crossword with something productive, and it's great for that, and (2) because every time I get annoyed at like messy trig derivatives with double-angle substitutions and stuff, I instead pivot to learning how to solve it with Sage Math, and so I get better at that instead.
But if there's a smarter sequence, I'm super interested!
For something more traditional, take a look at textbooks by Piskunov, Courant, or Apostol. Spivak's Calculus has excellent problems if you are looking for something more abstract and rigorous (probably better after a first course). https://archive.org/details/n.-piskunov-differential-and-int... ; https://archive.org/details/ost-math-courant-differentialint... ; https://archive.org/details/calculus-tom-m.-apostol-calculus... ; https://archive.org/details/introductory-calculus-book-colle...
Finally, if you want a strategy for those tricky integrals, per se, take a look at Schoenfeld's "Integration: Getting it All Together", https://files.eric.ed.gov/fulltext/ED214787.pdf ; some results of teaching the solution of integrals by this method were presented in https://www.jstor.org/stable/2320344
Not only will you be even more capable of picking up solving things numerically, but you'll also have the prerequisites for studying physics or probability or machine learning or Knuth's Concrete Mathematics. It opens doors to new intellectual vistas.
Solving things analytically (when possible) also can reveal more about the nature of the problem than doing so numerically, and can give the same satisfaction as finding an elegant solution in code.
You can definitely go an entire programming career without ever using it, but if you ever do run into a problem it solves, having this tool available to you is only a benefit.
I would feel real weird if there were things (in Calc II problem sets) I could solve in Sage that I simply couldn't do by hand.
I don't feel weird that there are things I can do, but will get wrong a bunch of times if I try to do them, and can quickly bang out in Sage. That seems fine to me. For a lot of these subjects, I don't care about automaticity, just intuition.
It's like a lot of linear algebra: being able to quickly do things by hand is kind of silly, because for real world problems (at least in data science) hand solves aren't even really feasible. But learning to do it by hand is important for building intuition.
I've been going through Math Academy with a IPython REPL open, too, and I've noticed that I need to avoid using it unless the problem specifically tells me to use a calculator or the implicit skill review in the problem gets skipped. Even writing little functions for myself to one-shot a problem means I'm missing chances to actively recall the steps.
Actually, given what you've said, you'd probably enjoy working through Sanjoy Mahajan's books on Fermi problems and estimations (the books are CC-licensed you can just download them): https://mitpress.mit.edu/author/sanjoy-mahajan-9006/
You CAN model predator-prey dynamics or disease spread using Monte Carlo techniques, but you can't read the historical literature without some grasp of differential equations.
There will always be some brilliant people that breeze through a hard course, and effortlessly acquire the intuition, and then wrongly attribute their understanding to the course. That doesn't help the rest of us, who are much more likely to build intuition by guessing about the outcome of a program and then checking if we are right.
Learning calculus is very valuable to a relatively small number of people. We should absolutely make calculus available to any student, regardless in advances in AI. But a lot more people, particularly non-technical or non-math type people, would benefit more from experience with statistics than calculus. Statistics, combined with an introduction to simple programming, should be part of the basic high-school curriculum.
Mathematica can do calculus and linear algebra better than most if not all college graduates since decades ago, but colleges are still teaching those courses. That should explain enough.
This feels like a "tech bro" idea from someone who has never touched a SEM field (STEM minus the T).
You can’t “brush up” on something you never learned
For the [current] layperson, each of those things I mentioned I might as well be speaking in Martian.
https://dibeos.net/wp-content/uploads/2025/08/what_happens_t...
You'll (probably) never apply the ability out the kinetic energy vs. time of a ball rolling down a hill, but these exercises build understanding of the tools. Derivatives are everywhere in a fundamental electric circuits course, you need to have an intuitive understanding of basic calculus. The relationship between current through and voltage across ideal inductors and capacitors are directly described in the language of calculus, even if you're not "using" the calculus substitutions you learned each time you analyse a circuit.
And good luck getting through a couple weeks of an introductory quantum mechanics course without using calculus as a fundamental building block. You can solve many of these problems with computers, but it's not going to build intuition on how to approach future problems. (I don't mean this as a joke or picking an arbitrary complicated-sounding topic; this is a core course in some engineering programs.)
Many engineering problems have nice closed-form equations (at least to get approximations). Obtaining those equations often involves calculus, and someone has to do that in the first place.
(I'm giving examples from the lens of my education, but each field of science, engineering, and mathematics will have their own context, and will vary from little-to-no calculus to being all-calculus.)
Honestly, the trend toward anti-intellectualism in the world is very disturbing to me, and it seems AI is enabling this kind of contempt for knowledge even more.
So many people never stop think about why we do the things that we are simply expected to do, instead of doing other things.
I don't use Calculus in my day job, but it was still valuable for me to learn to hone my ability to think
Sounds like a job for AI.
But when things break and you have to understand why, or you have to fix it, or even describe the problem in some detail that someone else is able to guide you to fix it, it sure helps to have an idea of how it works.
I trig, and calc 1 yet I hardly remember how to solve most problems because its not something I use regularly.
Same with subnetting or remembering certain programming languages.
I literally don't use these things 99% of the time and so I forget them. Sure I understand some of the basics and I could probably pick it back up way faster than someone who knows nothing but I am human.
So I don't think forgotten knowledge is wasted, it's still valuable.
One of the few moments in university was learning how so much was actually Calculus.
Whether it was physics, chemistry, etc, the formulas I was given ften had a calculus version.
It helped me open up to taking math and stats courses I never would have as a comp sci student, which in turn gave me a different perspective than just taking cs courses alone.
You don't use your muscles? They atrophy. You don't make an effort to travel without a gps regularily, to force your brain to remember your way around naturally? Your spatial memory atrophies and becomes useless [here: https://www.nature.com/articles/s41598-020-62877-0 ]
People don't need to learn math anymore, hence, no more calculus lessons? People are literally becoming idiots who can't calculate simple change at the cash register without pulling out their calculators.
It's exercise. It keeps the brain itself from atrophying. It stops you from becoming a "wetware LLM" that's just parroting whatever echo of a thought (natural or otherwise) goes through it.
LLMs have made obsolete the study and practice of the mechanics and minutiae of assembly language, memory allocation, system calls, hardware interrupts, process scheduling, cohesion, coupling, design patterns, code smells, test-driven development, "clean code," etc. There is no reason for anybody to learn any of this when the LLM, pretrained on all the literature backing these concepts, can emit code at 10,000x the speed that already embodies all of these lessons.
There is no reason to suppose mathematics is any different. You don't break out your pen and paper to compute change at the checkout aisle at the grocer's; you don't even have to punch numbers into a calculator, or indeed be numerate at all - just scan the barcode. Prepare for a world where we have machines that make calculus and programming as easy as scanning a barcode.
Look forward to the future described in Stephenson's Anathem where cutting-edge spacefaring nuclear technology is wielded by the literally illiterate, and the only people capable of reading books on sheets of dead trees are effectively monks living in Neo-Luddite monasteries.
What do you think is going to replace calculus? Until something better comes along, calculus is among the best introduction to higher theories of mathematics.
> LLMs have made obsolete the study and practice of...
It sounds like you're trying to say that Calculus will be replaced by LLMs. In fact it's the reverse: LLMs can get better by studying calculus and the higher math things that calculus enables.
> There is no reason to suppose mathematics is any different. You don't break out your pen and paper to compute change at the checkout aisle at the grocer's; you don't even have to punch numbers into a calculator, or indeed be numerate at all - just scan the barcode
Buddy, barcodes can be wrong. If you don't know how to do math yourself, you won't know when a sweet potato gets confused with a differently-priced yam. Or your new tablet with 5 year warranty gets scanned as a server CPU with 1 year warranty; then when your tablet dies at 4 years 11 months, your warranty was never activated, and your tablet is reported stolen.
> Look forward to the future described in Stephenson's Anathem where cutting-edge spacefaring nuclear technology is wielded by the literally illiterate
You may be right on that. But, in that world, the "literally illiterate" people will always be outshone by people who actually know how things work.
"bullish" and "bearish" are words that are useless in this context. Use them for stock markets where those words belong.
> LLMs replacing programmers? The industry is breathless with hype and media pieces on how traditional programming is obsolete
I believe LLMs fill a small (over-hyped) niche currently, but that niche will grow over time. I don't believe LLMs will, within our lifetimes, fill the specialized roles of high intelligence. It certainly will (and in many cases already does) fill the roles that don't require lots of thinking and intelligence.
Could LLMs replace programmers? Definitely, right now. Could LLMs replace developers? Currently, it could replace some of the worst developers, and help but not replace some of the better developers. Could it replace engineers? Definitely not right now, and probably not for the next several decades.
Is that a bad thing? To a lot of people, it is, especially in short term measured by job losses. But in the long run I believe LLMs are transformative in nature. Programming and development will be transformed to use LLMs as the tools that they are. Anyone who doesn't adapt to using an LLM will end up being a loser in their industry or be required to retrain to another role. That's generally good for the economy as a whole even if it's not great for the individual person.
You could manually do a lot of things that a CNC mill can do. Or you can learn how to use a CNC mill. Or you could pay someone else to use their CNC mill. But a CNC mill is not a great thing to use for cutting raw lumber or assembling furniture. You "could" use a CNC mill for it, but it'd be extremely inefficient. You could use it to manufacture furniter pieces, but it's not going to assemble them. LLMs are kind've the same thing for computers.
> do your criticisms of my position also apply, were software engineering, rather than calculus, its subject?
I think yes. Software Engineering (as an Engineering discipline) is used where it matters (eg, regulated aviation, safety, etc). It's not used much (if at all) for basic websites or even most games. But when it is used for these things, it's a much better product. That will pretty much always be true even with LLMs guiding the developer (not engineer).
For your example I replied to: I don't know much about nuclear reactors. But I know more than, say, 80% of the general population. I know enough to know that they can be dangerous but some designs are less-dangerous than others. Could I build a reactor myself? Probably, while the average person wouldn't even know where to start. I wouldn't do it without a lot more knowledge though because anything I come up with would be miles behind current state of the art in safety and procedure. I probably could identify nuclear fuel ores, but I have no idea how to refine it safely. I don't really care to learn that, it's not my passion. But I also know that refining it is dangerous if not done "properly" and so I wouldn't even want it as a hobby.
Contrast that with robotics. Robotics can certainly be dangerous, but not nearly as dangerous as refining nuclear fuel. I know how robotics work, and how to work with them with some modicum of safety, and so I do that as a hobby. That doesn't mean I would encourage anyone to come along and just use an LLM to build robots because they'd be missing out on sooooooo many fundamentally important concepts and would have to rely on the LLM to tell them what's dangerous or what's not and how to implement safety features and procedures.
> Programming and development will be transformed to use LLMs as the tools that they are. Anyone who doesn't adapt to using an LLM will end up being a loser in their industry or be required to retrain to another role [...]
Mathematics will be transformed to use LLMs as the tools that they are. Any mathematician who doesn't adapt to using an LLM will end up being a loser in academia or be required to retrain to another role.
The study of calculus will be replaced by tooling that carries out the mechanics of differentiation, integration, finding extrema, etc, making such education obsolete.
> You could manually do a lot of things that a CNC mill can do. Or you can learn how to use a CNC mill. Or you could pay someone else to use their CNC mill. But a CNC mill is not a great thing to use for cutting raw lumber or assembling furniture. You "could" use a CNC mill for it, but it'd be extremely inefficient. You could use it to manufacture furniter pieces, but it's not going to assemble them. LLMs are kind've the same thing for computers.
You could manually do a lot of things that a calculator, CAS, or LLM can do. Or you can learn how to use a CAS system or LLM, or pay someone to use their cloud service. But an LLM is not a great thing to use for deciding research programs or assembling a proof. You could use an LLM for it, but it'd be extremely inefficient. You could use it to formalize certain lemmata or minor results, but it's not going to compose them together into a larger theorem.
> Could LLMs replace programmers? Definitely, right now. Could LLMs replace developers? Currently, it could replace some of the worst developers, and help but not replace some of the better developers. Could it replace engineers? Definitely not right now, and probably not for the next several decades.
Could LLMs replace arithmeticians or calculators (in the original sense of the word, as a human occupation)? Definitely, right now. Could it replace mathematicians? Definitely not right now, and probably not for the next several decades.
You distinguish between Software Engineers(tm) and lowly developers; I point out that you can draw the same distinction between mathematicians and lowly arithmeticians or "calculators" (in the original sense of the term as a human occupation), or between calculus and analysis.
Leave the solution of this particular integral or the differentiation of this particular function, the mechanics of calculus, to the LLMs and the machines. Mathematicians need not concern themselves with such trivialities, and will prove results for all continuous functions in arbitrary metric spaces, or something similarly general and insightful.
What makes you think this time is different? We've had tools for those things for decades now.
Jury's out on whether or not that collection is non-empty.