The analytical type of thinking that proof-writing is certainly useful, but you can make much the same argument of many other curricula, and besides, it's not like most intro calc courses even do any proofs. The vast majority of them, I would assert, are simply pre-med weed-out courses.

I still remember freshman year, showing up to the standard intro calc course, and dropping it as quickly as I could in favor of my uni's equivalent of Math 55 (i.e. the hardest u/g intro math course) because of how asinine I found the content...

But don't deny yourself an understanding of the meaning of limits. Almost all mathematics before calculus leaves you with a misimpression that neat formulas exist to solve problems. In reality, you've learned to draw straight lines with a ruler, and maybe a few curves with a compass. Before Calculus, you might actually believe that numbers that can be expressed as the ratio of two integers are typical, and that numbers like pi and the square root of two are "irrational" rarities (and until calculus, you probably don't know about Euler's constant unless it was introduced in precalc as another one of those odd and rare numbers).

Look out at nature, where are the triangles, rectangles, and circles? Maybe a wasp nest? Nah, not really. Try to draw a cloud, a tree, a tiger, or a human face. How useful is that straight line or compass? How useful is a line at all, other than to hint at something you can't actually draw, maybe by implying it exists as an ever vanishing limit from above and below? Math required calculus the instant humans decided to describe the world as it is, rather than by the limits of what we impose on it.

Also - in stats, how do you know what the area is under the probability density function?

My claim is that, given the finite amount of time allotted to mathematics in a secondary school curriculum, it's better to spend that time learning about medians/means, regressions, statistical tests, and so forth, instead of memorizing power laws and derivatives of trigonometric functions.

I've used one once. I wanted to make a volume control that I thought sounded subjectively even across the range and gave the right amount of control. So I used a polynomial regression calc on a random website and gave it some data points I wanted it to go through.

But that's not actually doing a regression, just knowing it exists and computers can do them.

Is there any math that has any use whatsoever unless you know a whole lot of it and plan to do some technical projects? I thought the whole point of learning any math(Even arithmetic, since phones exist now) is just so you can learn other more advanced math, and maybe someday be a solid state chemist or something.

Statistics lets you read a scientific paper, but you don't actually need to understand it, unless you're actually going to be reviewing their raw data. Most everyday people just trust the p values and move on to wondering about confounders they forgot.

Seems like you could cover all the statistics people will actually use it about 4 hours.

Perhaps because they never took statistics.

If you want people to understand that numbers are concepts / descriptions of something potentially infinite, and that it's ok to totally work with non-finite describable numbers without ever really writing out their values, just fucking start with that. I honestly think the start of most undergrad math curriculums should be a logic & foundations class with proofs, sets, number theory and the whole "numbers are more logical concepts, not really specific values" vs. calculus. Then teach calculus, with a big dose of 'what is infinity, really?'.

I think it's actually a great disservice and the hand wave that most calculus math classes do about those exact concepts really fucks over a lot of people. It makes calculus a weed out class because the fundamentals are not explained properly so a good amount of people just go into it as yet another thing they have to ape without real understanding. And for the people who don't work well with things they half understand, they really struggle, like I did. It really made my computer science degree a lot worse, because of the instance of starting all undergrad math with hand wavy calculus, for 3 or 4 classes.

I even wrote blog articles about it, it was probably the worse part of my computer science degree, and if math education was done differently where they didn't handwave, I probably would've had a much better time.

https://www.thoughtfunction.com/2019/11/doing-my-compsci-deg...

https://www.thoughtfunction.com/2019/11/what-i-wished-i-knew...

Most people don’t need mathematical statistics, but far more will find meaning/interest in the immediate applications of lighter statistics than the another-math-class-full-of-equations that calculus feels like to many.

Anecdotally, most universities are scrambling to add lighter data science courses to their humanities majors. These all teach basic stats, and many ignore the low-level calculus required for those methods (again, speaking of the humanities versions here).

I was a math major and didn't understand calc all that well until I took advanced calc, and I didn't understand statistics all that well until I took the upper-level mathematical stats courses.

This does not only extend to literature. I have had similar experiences with religion (which I thought of as utterly useless and potentially dangerous, as it "deactivates critical thinking and creates sheep-people which will follow whatever their shepherd/priest/guru tells them"), creative arts (both painting as well as music), and the basic sciences (biology, chemistry, physics).

Now, I don't genetically engineer the stuff in my garden, but understanding Mendel was useful. I don't speak in iambic pentameters, but I can appreciate when it is being used as a stylistic choice. I am in no way a church-going, devout Christian, but I have found meaning in some of the deeper wisdom enshrined in the Bible (and the Koran, and the Bhagavad Gita, and about half a dozen Sutras.)

Would I have come to that if I didn't have primers in school? Maybe. But the primers certainly helped.

On the other hand, I didn't learn plumbing in school, or laying electrical wires, or "doing my taxes", but these are things I can simply have someone do for me who is a lot better equipped and trained to do so, or - for the small stuff - I can figure them out on the fly.

I'm willing to believe that some people like Shakespeare, but it's a small minority. You can tell by the number of people who read Shakespeare for pleasure - is that number larger or smaller than the number of people who read JK Rowling? Why should we teach the entertainment that a small number of people prefer in schools? I believe the only reason we actually do is tradition.

You mention that you can simply have someone learned in plumbing or sundry skill do those tasks for you. I can do one better. I can simply have no one read Shakespeare for me and I can not read it at all and nothing is lost. That is, of course, because unlike plumbing or laying wire there is no reason to need Shakespeare.

It's good that you enjoy Shakespeare, but some people enjoy plumbing. Plus, plumbing has a practical purpose, unlike Shakespeare. There is no real reason to teach Shakespeare, other than tradition, and people trying to seem smart or educated. There are many other subjects that make a much stronger case for deserving to be in school curriculum.

I read a quote somewhere that goes something like "A society that separates warriors and scholars will have an army led by fools and thinking done by cowards." Similar logic, with different vocabulary, applies, I think, to a society where scholars can't do manual labor.

On the other hand, less-than-popular culture still is the foundation for our popular culture today - the amount of times "Romeo and Juliet" or "Macbeth" has been adapted, referenced and deconstructed in (popular) media is astonishing, and without knowing the original, you wouldn't fully get the modern references.

You don't need to love Shakespeare (I don't exactly, though I like to see the "Scottish play" every few years) to reap these benefits.

School is not about what people like. If you go by that measure, there's no real need in math, because you will find a lot less people doing calculus for fun, as compared to people who play fantasy football. Of course, you can't really understand the ideas behind fantasy football if you don't know maths, but who cares, it's not popular...

Please note that I did not claim physical labour has no value and one should not know about it. I said that it's relatively easy to pick up, and that it may be more worthwhile to get a professional when the job demands it (a professional, I would like to add, who learned his craft after school, ideally in some kind of apprenticeship). In my country, to get a driver's license, you need to know basic first-aid procedures (and yes, there's a one-day course you need to attend). That doesn't make you a neurosurgeon, and if you have persistent headaches, it's probably better to ask a professional than to depend on me with my first-aid course - or some guy who "learned about surgery techniques" in high school.

What about meta-mathematics? Topology? Logics? History of mathematics? Philosophy of mathematics? Combinatorics? Number theory? Discrete mathematics? Graph theory? In the post, the fieds under "electives" are by far the most interesting ones, IMHO.

And I fully agree, in-depth knowledge of probability theory as well as descriptive statistics and of course the application to systematic and sound decision making is absolute key, and ought to be taught to anyone from medic to policy makers (scary: Gigerenzer showed that medics tend to be confused about the difference between P(A|B) and P(B|A) - the very people whose job it is to diagnose whether you have cancer or not!).

Honestly all of those feel more niche than calculus. I agree with you and joatmon-snoo on the usefulness of statistics and would probably support bumping calculus in favor of statistics, but meta-mathematics, topology, logic (which bleeds into meta-mathematics), combinatorics (which is kind of covered by stats), number theory, discrete mathematics, and graph theory are all much less useful even in adjacent STEM fields (discrete mathematics and graph theory matter more in CS, but far less for day-to-day programming). History of mathematics is effectively an entirely separate discipline and philosophy of mathematics has meta-mathematics/mathematical logic as a prerequisite.

Calculus unlocks much of physics and engineering (and lots of stats!). Large cardinal theory does not unlock any other field to the best of my understanding.

There ought to be some way to encourage late high school/early college students to "survey" the field without necessarily taking full courses in these topics. This could also give some earlier understanding in how the different fields relate - for example you could present a toy graph theory problem within linear algebra as a matrix problem, later presenting the same problem in graph theory section and walk through it using graph representation. I think high school courses struggle with memorability precisely because most units are taught basically in a vacuum.

Regardless though, the point of the survey course wouldn't be to remember details so much as to find topics of interest for further study/be generally aware of their existence in case a relevant problem comes up in the future.

And, from that perspective (with which I agree), calculus itself is just another instance of trying to turn a non-linear problem into a problem in linear algebra!

Linear algebra became a lot more interesting once you had cheap computers and Matlab.

The Math in my 4 year engineering degree was structured something like this:

First Semester Year 1 two math courses: Calculus 1, Linear Algebra

Second Semester Year 1 two math courses: Calculus 2, Sequences and Series (this one was probably least useful all I remember from this 15+ years later is Taylor Series and Binomial Theorem)

First Semester year 2 two math courses: Differential Equations, Statistics for Engineers.

From second semester year 2 onwards there were no more discrete math classes this was where the degree really specialized into various engineering streams, Mech Eng, Chem Eng, Civil etc. Had their own courses. I studied Materials Engineering some courses were shared with other eng students (For example I shared Thermodynamics with Mechanical Engineering students) but others such as Non-ferrous metallurgy were pretty deeply specialized.

A lot of subject used built on earlier math (Fluid Dynamics was backed up by lots of differential equations for example, stuff like Gamma Function would come up in a lot of places. Solid Mechanics had a lot of integrals second moment of area etc.), Linear Algebra I can remember from Fracture mechanics and crack propagation (Stress and Strain tensors etc.)

Have a Master's in Material Science and still don't remember a lot of Linear Algebra specifically.

I have some appreciation for calculus now but I really did not enjoy it much in high school or even in college. It turned me away from learning more math for some time which is unfortunate - linear algebra isn't my favorite either but I liked that much more off the bat, so I wish I had some exposure to it in high school. Then again, maybe the high school teaching style is what made me dislike calc to begin with.

And yes probability and statistics are fundamental. I was shocked a bit when I learned it was not taught in highschools world wide (i.e. not in the U.S.A.). But then again I had gotten numb with the current average level in the taught topics people arrive at undergrad at.

Note there is a lot of interconnectivity. To understand a new concept you might need concepts in another. E.g. number theory and probability.

The true foundational classes in the typical undergraduate mathematics curriculum are logic and abstract algebra. People rarely start with them, because the usual way of teaching mathematics is applications before foundations. You learn linear algebra before abstract algebra, proofs before formal logic, and axiomatic probability before measure theory.

And there is definitely such thing as too much linear algebra. Once upon a time, I wanted a decent mathematical background for theoretical CS and continued (at least) until the first graduate-level class in most major topics. Graduate linear algebra was "foundations without applications" for me, as I've never worked on anything building on it.

In the first year (non US) you learn linear algebra, some real analysis and how to write proofs as well as important basics. In your second year you can choose all these electives, which then don't have to spend time introducing natural numbers, induction etc.

If anything should be dropped from a highschool level its all of the insane memorization you have to do for some of the lower level math classes - I found that totally useless. You then learn some basic calculus and realize "I just wasted so much of my life" and never need to memorize those things again.

In the land of the blind the guy with one eye walks everyone off a cliff.

What happens is you learn numerical approximation methods and then recreate the tables with human 'computers', like we used to do before electrical computers came into play. Or create mechanical analog computers like they did in the 1800s or greek times.

How would the typical person go about calculating the normal distribution by hand? Are we going to call everyone blind because they haven’t memorized e to arbitrary precision?

IDK, what does R do?

I also think you could and should start by teaching probability and statistics and using that to introduce calculus as and when it shows up naturally, rather than teaching calculus in the abstract first and then showing the applications much later.

What sort of memorizations do you have in mind that you no longer need to memorize once you know calculus?

To work around the fact that many of us were still in precalc, our teacher just taught us the power rule, the relationship between slopes of tangent lines, etc., without diving into the "why" of why those things worked.

That said, yeah maybe for the most basic of university of physics the whole "derive it on the fly" strategy works, I guess? But when you get to more advanced courses like mechanics of materials, you'll do yourself the favor and take to memorizing at least a few of the commonly used equations.

I'm trying to remember when I had one of those last!

I actually don't love the idea of cheat-sheets but maaaybe if it's a cumulative final I could see it being helpful? If you're taking chapter exams on some material, I think you ought to have worked enough problems so that formulas/constants are drilled into your head. But I guess too, where do you draw the line at engineering appendices? I sure don't have the MoI of every common body memorized.

Well, sure, but, if you don't, then what's the point of punishing you? As a teacher, frankly, I'd be happy to have my students bring in any static resources they wanted to consult—I say 'static' to emphasise not, e.g., consulting a cheating site, although it's fine with me if they've pre-compiled solutions in advance to any problems they think might be interesting or important—except that (1) I think that would encourage bad study habits, and, more importantly, (2) it would be unwieldy in a packed classroom to try to have adjacent people juggling multiple textbooks, notebooks, etc.

In fact, I loved the freedom to give extended-time, fully open-book, open-note exams during the fully remote classes. I wish I could still do that; if cheating weren't so endemic under those conditions, then I would.

It's strongly self-selecting and as a result can afford to cover a truly insane amount of ground. The overwhelming majority of people who take it drop out (the usual dropout rate from people who take it in the first week is probably > 90%), but the people who stay almost all get As. And you will need to be almost entirely self-motivated because a lot (maybe most) of your waking hours will be thinking about math.

There's a very small minority of students for whom this is an optimal way of learning. For most students this is the quickest way to make them run screaming away from mathematics even faster than they already do.

Not at all, you are only prohibited from taking "freshman courses"[1]—that's what 55 is supposed to cover (definitely not all of the undergrad math curriculum), though some professors go beyond that. Many students go on to take at least some undergrad courses, with those in the 140's range being mathematical logic gems with no real counterpart in the graduate department.

[1]Students from Math 55 will have covered in 55 the material of Math 122 and Math 113. If you have taken 55, you should look first at Math 114, Math 123 and the Math 131-132 sequence. https://legacy-www.math.harvard.edu/pamphlets/courses.html

The Math 140 series have only become more serious courses in the last 10 - 15 years or so IIRC. The 240 series was generally where to go for serious mathematical logic courses (and generally is where you would go after e.g. a first 140 series course in set theory anyways).

Strong agree, but engineers probably need both. I'm currently watching a course on causal inference, and the tools are very much calculating gradients. And even if you just use someone else's MCMC, even in the models a differential equation or integral can randomly appear usefully.

In retrospect I should have taken a stats class in high school when I had that 1 hour gap for 1 semester, just to build a better intuition around the basic concepts.

They don't seem to have a problem intuiting what a plotted PDF is showing. I think that's because in some sense it can be read analogously to a histogram. Of course, they don't have the tools to generate or manipulate one. But that's honestly not something that applied social science researchers have to do often when using traditional methods.

I think math people may have a view of this question that is skewed in an interesting way.

Statistics is very useful and its common techniques are not difficult to apply.

But they seem to be very difficult to apply correctly. We have entire academic fields that are built mostly on the spurious application of statistical methods in contexts that make the whole project invalid.

And this sort of "techniques without theory" approach is what's being advocated for upthread and represents a failure mode that math people are unlikely to consider -- because they know that part of being able to apply a technique is being able to tell whether or not applying it is valid. Math people are unlikely to fall into the trap this approach sets. But the same people who want this approach are also likely to end up being hurt by it.

The only reason I took calculus is because it was the next math course after pre-calc and I still had a year left in high school. I didn’t even realize that there was a type of math that could be used to exactly calculate quantities related to continuously changing processes, but I was absolutely fascinated by how many real world problems calculus could solve - I realized that the problems given in algebra were contrived out of necessity to resolve in a neat manner. Now I had access to an entirely new vocabulary that allowed me to describe the world as it is, not as it needed to be to neatly fit in a 10th grade word problem.

Until that point I was interested in psychology (and had a vague notion that I’d drop out of school to make a career in music somehow), but I immediately dropped everything to take as many math and physics courses as I could. 14 years later and I work in a very math heavy engineering oriented field.

I know my story is probably atypical, and I have no clue what I’d be doing right now if I hadn’t taken calculus, but it’s one of those things that I look at in hindsight and think that I was that close to giving up on STEM, but for being forced to take Calculus. Instead, I earned my Ph.D. in applied math, and there’s not a day where I don’t use calculus of some kind.

And the basics of physics makes next to no sense without calculus, and even parts of intro chemistry make more sense after having taken calculus.

I remember in one economics class, which didn't have calc as a pre-req, the prof said "alright, for those of you who have taken calculus, this is an integral, you can now leave the classroom and come back tomorrow. Everyone else has to stay."

The ideas of calculus just seem so fundamental to me. It is sad that the American schooling system is so slow, and expectations so low, that it isn't taught to everyone. Meanwhile in other countries, everyone, artists to engineers, learns calculus.

It’s all integrals over pdf’s. A lot of integration by parts and other things in the core curriculum.

There's nothing a basic Stats 1 course needs calc grounding for (perhaps Riemann sums under a normal curve but that's more for deriving than the concept itself). As it stands now, AP Stats is used for kids that don't want to progress to an AP Calc, but want a math or need to hit a school or county requirement.

> The analytical type of thinking that proof-writing is certainly useful, but you can make much the same argument of many other curricula, and besides, it's not like most intro calc courses even do any proofs. The vast majority of them, I would assert, are simply pre-med weed-out courses.

A statistics course would be a much more brutal weed-out course.

Stats is not a generally useful skill (the concepts are, but can be taught in a data science course) but understanding how to work with data is.

Discrete math seemed to be the most applicable math class to my CS curriculum. I never took graphics so didn’t use linear algebra, and definitely never touched anything related to DiffEq.

I took calc 1 and stats 1 & 2. Much preferred the stats and it set me up for understanding all kinds of science lingo in articles and papers. I also indirectly use stats fairly often at work.

I'm all for giving tasters of as many different "beautiful ideas" as possible in school, but I think we should be elevating practical statistics into the top tier of subjects that we require kids to go through.

Because America has straight lines everywhere and no curves, unlike other places.

Calculus is absolutely required for most engineering tasks.

Maybe for CS students there could be more emphasis on statistics as well, without diluting the calculus.

Infact, where I studied, calculus was a prerequisite for certain statistics and probability classes. For good reason.

All fifty states or just continental?

I'm questioning your math pedagogy.

Calculus is the single most important math anyone, of any field, can learn as it's the first "practical math" you actually learn. Life behaves like calculus and in order to think about real life you need the concept of limits, derivatives, integrals, differentials, etc. It's patently absurd to say this should be replaced by statistics, which done to any rigor requires up to 2 years worth of calculus (through diff eq.) to even appreciate.

I'm shocked that you're a math major and didn't take away the biggest thing from learning analysis - the ability to think clearly through a problem and prove it correctly. While you may not be asked to vomit cantor's diagonalization onto paper for an interview the ability to think about problems you learned from doing these proofs translates to so many different fields, jobs, and life skills that I take the complete opposite view. If you want to understand anything you need to learn how to proof. I don't care if you're a nurse or an accountant. A rigorous proof based math course will change your life.

If by "learn statistics instead of calculus" you mean being able to mindlessly vomit today's new machine learning paradigm without understanding a thing then I think I can understand where you are coming from. Otherwise, I think this is some absurd parody of someone who studied math.

Otherwise it just gets absorbed and forgotten like another 'useless information' class. Ask how many CS majors post college use or remember calculus vs understanding stats for the seven millionth badly designed A/B test they ran at work today.

Probability theory is applied analysis. For example a measure density is a derivation of a measure with respect to another measure (Radon–Nikodym derivative). Or how do you often assign probabilities to measurable set: Lebesgue-Stieltjes integral.

So you better have a very good understanding of calculus before starting with probability and statistics.

I was a math major and this is not exclusive to calculus or analysis. If anything, algebra classes in things like group theory were much more instructive on this point. Everything from my third and fourth years (when I specialized into combinatorial optimization) was built on a cornerstone of proofs and theorems.

I do think logic is important, and teaching that instead of focusing on math proofs might be a good alternative.

For science-oriented high school curriculums, calculus is necessary, and should only be optional in a few cases where the student already really knows what they want to do.

By the time a student wants to specialize at say 20, they should have already had some foundational ability for that specialty, otherwise they wouldn't be able to compete internationally. For people who don't end up using calculus, it's a waste, but there is a trade-off. As a result, much of the high school curriculum is to build a broad foundation to prepare for multiple possible specializations.

Maybe it would help them not get scammed by keto diets and unsafe drugs and such though.

However, one thing that has been VERY applicable is proofwriting. Although math proofs are far more rigorous than most real world stuff, the discipline I learned in writing proofs has carried over into pretty much everything from programming (will this algorithm work every time?) to executive decisions (why, specifically, should we believe X?). Obviously in the former case I wind up doing actual proofs, and in the latter I make strong arguments based on logical consequences of established or presumed facts, or find flaws or gaps in arguments that are being considered.

I really wish I’d spent a lot more time on proofwriting than say, vector calculus.

Of course you may want specific math to solve real problems, and that’s a real need too! Not to diminish your point at all, just advocating for proofs to be seen in a practical light.

* Axioms

* Substitution

* Modus Ponens

* Universal Quantification

Induction or proof by contradiction are just special cases of this.

But yeah, geometry for introducing proofs is difficult, because it is so easy to confuse visual intuition with proof. At the very least, you need a capable teacher who knows the difference. But nobody expects children to understand it all from the get go. A healthy struggle to disentangle intuition and proof, and then to entangle them again later on once you know the difference, that's the path to understanding mathematics.

Dumb question: what’s the best way to learn proof writing — let’s say for “fun”?

The working title is "Practical Math for Programmers," and the idea is to build a collection of 60-75, 3-page long, _compelling_ demonstrations of mathematics used in production settings, biased toward stuff a generalist programmer might find useful. Not going into proofs or foundations, but providing lots of references to further reading.

I'm aspiring for it to be like a Hacker's Delight, or Programming Gems, but just for math that is genuinely useful.

Read more here: https://buttondown.email/j2kun/archive/a-week-of-book-writin...

Sign up to hear when it's released here: https://jeremykun.us11.list-manage.com/subscribe?u=99aa071e9...

However, I also have a fundamental objection. I don't see how you can be an intelligent tool user without at least a little curiosity about how your tool functions. Maybe you can apply your tool, even be highly effective, in certain instances. But this is inherently brittle knowledge. When the parameters of your problem change and you don't understand your tool well enough to adapt, you're lost.

"Math for people who just want to use it" is very broad. What do you want to use it for? Physics, biology, chemistry, computer science? Sociology? Economics? There might be some shared stuff, but for all of these disciplines there is a vast space of mathematics that might be relevant.

I think Eliezer Yudkowsky's idea of a book (series) covering "The Simple Math of Everything" is fantastic. I would love to read that book.

https://www.lesswrong.com/posts/HnPEpu5eQWkbyAJCT/the-simple...

1) The problem with math in school is that there's not enough "real math".

2) Relatedly, insufficient exposure to "real math" in compulsory schooling is also (a major part of) why people think they don't like math.

My suspicion (again, without the actual background to make this claim with any authority) is that they are dead wrong on point 2—the "real math" parts probably contribute strongly to most folks' dislike of the subject, and the parts the mathematicians didn't like are probably relatively popular among people who don't go on to become mathematicians. This puts point 1 on some shaky ground (though it could still be true and well-justified, for other reasons).

1. Students in math courses and their parents grow to prefer(through the overall institutional constraints) to have a simple exercise that guarantees them credit - while actually doing math is a matter of crossing the Rubicon into tough puzzle-solving, and it needs some guidance for unexceptionable students to start enjoying.

2. Math teachers, particularly in the lower grades where qualifications are lower, have a harder time teaching concepts than they do exercises. And they are also incentivized to hand out a grade, preferably one that satisfies the parents.

So no matter how the high level is set up, everything converges into giving the kids a worksheet to "plug 'n chug." Which is just a confusing, badly paced grind, and therefore an easy reason to hate math. Either you get it completely and are just sitting there chugging through the problem set, or you have no idea what's going on and it's due tomorrow so your grade rests on something you feel defeated by.

I actually think that for the parts that are currently treated as rote memorization work, the curriculum should lean into it and treat it like learning the alphabet, with worksheets where you literally fill in the dotted lines repetitively; hand them out to everyone as a portion of the homework. And then the logic and critical thinking aspects need to proceed like a philosophy course, with interaction through a step by step process, not "get the answer in the back of the book". This element is something I've long thought could be automated in some degree with computer systems that let you play with the concepts, and therefore correct your thinking.

So you're saying you know nothing yet are sure "experts" are wrong, based on no evidence. OK.

Use it for what? That is the question. If you pursue the what, you will inevitably be exposed to genuine ways that mathematics may be employed by it.

The academic standard for "learning math" is like "learning programming" by reading the C++ language/STL spec from front to back. No one productively learns programming that way, and even if someone did, they would hardly be well off when faced with a real-world production C++ codebase that follows $BIGCORP's inhouse programming style.

I think statistics is by and large the most proportionally underrated subject proportional to its utility. A good command of stats and probability expands your power to use data to reason about answering questions. The channel author, Ben Lambert, has an alternative playlist where he uses some of the techniques taught in this playlist to solve problems in econometrics. However, a lot of what is taught here builds a great foundation for other domains, on everything from bioethics to data journalism to computer vision.

Another great channel that focuses a bit more on the machine learning side of things is StatQuest with Josh Starmer: https://www.youtube.com/c/joshstarmer

You have generations of teachers who barely know math and view it as a punishment, teaching kids and instilling the same views in them.

And then you have outsider still saying "can't we have condense it and simplify it further so we won't have to learn all these useless abstractions" and the curriculum bends further this way. But these actual situation of math is that not understanding what's happening is the thing makes it an empty and unpleasant activity.

Edit: Also, yeah, 90%-99% of math can be accomplished with some math software. It's just for the remaining small percentage of stuff you need some understanding and for a small percentage of that you need lots of understanding. So most of this seems useless but 99% correct is actually not enough in some significant number of technical situations, etc.

This needs to be explained further during education and motivated appropriately. We have a short-term utilitarian perspective, and we need to take a step back at times and recall that it takes time and lots of sculpting to transform a wood log to a art piece.

As you can jog everyday for fun and/or for the challenge you can also jog to improve your physical health. And not doing proofs is like declaring a guy can weight-lift by just watching videos on youtube and never lifting a weight. Or a guy can "code" without writing a line of code.

https://web.stanford.edu/~boyd/vmls/

(I'd even replace Strang's "linear algebra" recommendation with this book.) Imo, proofs are useful in so far as they are enlightening (e.g., the proof that a problem has a minimum is often useful in so far as it tells you how to solve it!) but in many cases they are less so.

Math is pretty fun, though, proofs and all, and I'd recommend trying your hand at it as a cool little side hobby! It can often help with "clarity of thought" :) (In many cases, proofs are just one or two lines that tell you something interesting, too, not page-long arguments that are mostly definitions chasing.)

I find proofs and identities very hard to read. I assume it's a bit what having dyslexia feels like. I have to turn them, glyph by glyph, into something more algorithmic to make any sense of them. The only math-for-fun I've ever enjoyed are simple recreational math puzzles—proofs, reading or writing, are torture. I actually enjoyed the parts of math classes that mathematicians insist are bad and are the reason kids don't like math—the parts heavy on memorization and drilling the application of an algorithm—far more than anything that came later. Perhaps not coincidentally, those have also proven to be by far the best bang-for-the-buck of all my time spent in math classes over the years. I use that stuff every day.

It's called "engineering mathematics." (The books by Stroud would be an excellent choice here.)

I've forgotten 90% of the math stuff I learned

I wish for a service just like them but with many more difficulty levels and less (non-spaced) repetition.

The thing that makes it VASTLY better than most self study math programs or books is that there are hundreds of exercises that you can do, and see if you got the right answer. If you didn't, it will in most cases explain how to do the problem so you can try again with a completely different problem, so you're not just memorizing the answers.

Another thing that makes it great is you can do a little bit a day, start and stop, and come back to it and it will remember your progress and where you left off.

Khan is also a gifted teacher. Unlike a lot of math teachers, he has great pronunciation and handwriting and you can watch his lessons as many times as needed.

I just tried clicking on some linear algebra topic, but it seems there are just videos

https://www.khanacademy.org/math/linear-algebra/alternate-ba...

https://www.khanacademy.org/math/linear-algebra/vectors-and-...

Even though I've learned linear algebra decades ago, Andrew Ng's example of using a matrix to encode 5,000 images then doing linear algebra on it blew my mind. I've since used that perspective in many other fields. Not once have I applied a proof to solve a programming problem.

I've thought of publishing, i.e.blogging, examples that I've come across but that would just be a mish-mash of stuff I've read elsewhere with no overarching theme/framework. Besides, someone else must have done this, no?

EDITED: Used the correct book title.

There are people in this very thread insisting that proofs are extremely useful in programming. I dunno if I just picked up the same skills elsewhere (Logic? Philosophy? Just... IDK, thinking and developing an absolute shitload of heuristics through years of experience?) or am entirely missing out and in fact don't have a clue how to program, but I don't see it (outside some rare niches where it probably is useful—coq exists, after all).

Sure, the word "exhaustive" can apply both to accounting for all (reasonably) possible problems in a block of code, and also to proofs, but the former doesn't feel at all like working on proofs, to me, to pick just one example (and some posts have seemed to imply that accounting for e.g. edge cases is exactly one case in which experience with proofs come in handy, but man, they feel like very different and barely-related activities to me).

Someone without any background in the subject would probably find Dafny interesting.

For example, I might not remember how to code bisection search but I can figure out the loop invariant and from there it's easy enough to code a working binary search. And even if you have right bisection search down to muscle memory, you can modify the loop invariant to create left bisection search if you need it, while if I tried to keep two binary search algorithms in my head I feel like it would be more error-prone.

Those are indeed very useful for life/safety critical systems such as aerospace, air traffic control, medical devices, etc.

That said, you're absolutely correct that more justification and motivation is important. So much of math can be taught with problems from physics, computer science, etc. Perhaps a good book for you would be Concrete Mathematics by Knuth? I haven't read it but people swear by it.

The catch is that I myself don't actually understand this well. It's just "What I learned googling stuff while working as a programmer. I've had help reviewing it but there could be errors.

I'm trying to cover all areas of math that a nontechnical person, or typical non math focused programmer would need to know, but I treat actually doing any of it by hand on paper as an arcane thing for the really dedicated, so there's not really any excercises.

Instead of actually teaching a real understanding of math, which I can't do, because I don't know it well, I just explain what people who do understand it use it for and why you might want to go actually learn it.

I also have any historical math related stuff that I find interesting.

I can't tell you how to derive or prove Euler's equation , but you don't need to know math to understand the emotional impact from a humanities perspective, and be amazed that all those constants fit together like that, and that someone could discover it.

Ultimately, I think traditional math education has it totally right. My life would be so much better if I knew it, because there's jobs that seem to require exactly what they tech in math class.

It's not directly useful for non STEM types, but the idea seems to be give everyone a head start since so many do want STEM jobs.

I think you really do have to get to the being able to do proofs level to make use of it in the real innovative applications.

The common everyday applications math people like to cite can usually have a dedicated software package. It's not like we still need to add two numbers on paper. If you want to build something, we have RealThunder's FreeCAD.

Excel's Goal Seek and CAS systems do the stuff people say we will use algebra for.

But if your in tech eventually you run into the wall and need to do something like a Kalman filter or calculate stresses in a bridge and you're screwed.

But math is a broad field so you're going to have to pick specific courses. For example, partial differential equations are quite common. But if you learn it in a math department it'll have proofs and full rigour just like a course in set theory. While some techniques for solving them will be covered, we mostly study the underlying mathematical structures, why certain techniques work, etc. If you take it in a physics department they'll teach you techniques and numerical methods to solve a certain class of problems like heat equations or fluid dynamics. But learning through direct applications will inevitably limit you to those techniques while learning things in full generality tends to make it easier to pick up specific techniques when needed.

If you're talking about college level, "math for people who just want to use it", is basically all it is (outside of math departments and perhaps outlier curriculums in some elite places) and that's a problem.

Learning just the applications without picking up the theorems and without a true understanding of the concepts makes more advanced work in whatever discipline one chooses more difficult. Why? Because you'll have to follow mathematical arguments and it's so much easier to do that when you got the background to fall back on.

I think students before college need more focus on mastering the basics. They're rushed so much, and tragically, math is very cumulative. Any one who has tutored before will notice that it's disturbing how many people don't really understand how to manipulate fractions as they enter a calculus course.

It's still a bit much to absorb, but you can't really ever say the author didn't attempt to describe each lesson in real world terms. You memorize or learn to graphically/logically derive a few things, like how log(1+x)≈x for small x or how you can sorta guess for most quantities or plots. (One book example figures the data capacity of a CD if you know its music storage using informed guessing of each conversion factor from seconds of music to bits.)

As another example: powers of cos x from -pi/2 to pi/2... you can sketch cos x and then roughly sketch the second power, and then it's clear that the more times you do that, you get a bit more of a bell curve. One in the middle will always stay one in the middle, but you get less area with each iteration as the rest of the function approaches zero. If you wanted an integral, you eyeball where the plot is at 0.5, draw a full-height rectangle from the left 0.5 value and the right 0.5 value. Because the height is 1, the width is pretty close to the true area. You can decide at this point---if you want more precision---to visually guess if the tails or the main part of the function should have more area and adjust your answer.

There's an Open Access PDF at https://mitpress.mit.edu/books/street-fighting-mathematics

There were lecture videos on MIT's Open Learning Library (and somewhere else before that), but they're currently down.

The class site is https://ocw.mit.edu/courses/mathematics/18-098-street-fighti... as well as http://web.mit.edu/18.098/

You'd have to know each octave doubles in frequency.

Side quest: When you play the bugle, the played frequency increases or decreases by MULTIPLES of the base frequency---NOT powers of 2. Suppose this base frequency is 250 Hz. There is an octave from 250 to 500, but there's a note between the octave from 500 to 1000 at 750 Hz, and a few notes between 1000 and 2000 Hz, which is the part of the musical scale something like Reveille is played. If Reveille jumped from octave to octave, it would just sound like the intro to Justin Hawkin's cover of This Town Ain't Big Enough.

So, if you know transformer hum is 50 or 60 Hz and Queen's frontman starts his singing at 100 Hz, then he can sing up to 1600 Hz, or four octaves. Mentally recalling what his falsetto sounds like, you can imagine a really high-pitched guitar solo an octave above this, and you can still imagine what an octave above that would sound like. (Maybe you're getting close to dog whistle territory in your imagination.)

This, then, is 6400 Hz you are imagining. The top of each sound wave to the top of the next is 6400 Hz. To record this, you'd need the top AND bottom of each sound wave, because the speaker cone moving from maximum to minimum displacement is how the sound is made. If you want to make sure you aren't accidentally recording the middle (zero crossing) of each wave, you can even take three or four or five samples per sound wave instead of two. It's a lot of thought, but you can reasonably decide that 25000 Hz is a good sampling rate for capturing much of the range of human hearing. Going too far beyond that, you're wasting storage space.

A CD holds a bit more than an hour of music, or 3600 seconds. If you've listened to Dire Straits, Eagles, Cyndi Lauper, Metallica, David Bowie, Led Zeppelin, ELP, or nearly any other band, you're probably aware the recordings have independent left and right channels.

Finally, each sample is going to be somewhere between "speaker fully retracted" and "speaker fully extended". With 5 bits, this gives 16 "stops" from the middle point to fully extended. But we know that music can get really quiet when it fades out, and a lot of volume knobs can go from zero to thirty and sometimes higher. When you have the volume at one, you can still tell the difference between loud parts and quiet parts, so you'd need an extra 5 bits just to get good dynamic range at loudest and quietest volume settings, or 10 bits. What happens when you double this? If you have 20 bits, you are probably close to wasting bits. You have a million places where the speaker coils can move to. For a speaker that moves a few millimeters, this means 20-bit resolution allows steps of a few nanometers. This is the scale of computer chips and color wavelengths. If you took the color blue and shifted its wavelength by a few nanometers, it would still be practically the same shade of blue! Without knowing about bit depth, you can reasonably assume 16 bits is good because it's a power of two and will give a lot of dynamic range. 8 would be too low. 32 is just wasteful.

With 32 bits, a speaker capable of moving 1 cm end-to-end will have 10 carbon atom diameters of linear resolution. The ears are impressive, but I don't know they can differentiate the air displacement of (speaker cone area) x (ten carbon atoms). Even having 0 to 100 on the volume knob, this leaves 25 bits of range at each volume setting. This is audiophile (and arguably, snake oil) territory.

So then, you can say 3600 seconds is pretty close to 3000 seconds, 2 channels is close to 3, 16 bits is close to 10, 25000 Hz is close to 30000 Hz... 3 x 3 x 3 x 10 x 1000 x 10000 ≈ 3,000,000,000. Since a byte has about 10 bits, divide by ten, and this yields a first approximation---based on logic reasoning of what we know---of 300 MB. It's wrong, but it's not "very" wrong. (It's off by a factor of two, not a factor of ten! Not bad for 4 rounded, intermediate conversion terms...)

(The idea is to round each term to a value starting with 1 or 3, because multiplying 3 and 3 is close to 10. The reason 2 is close to 3: 10^(1/2) = 3.16. This states that a good midpoint of 1 and 10 is 3.16, because if you square each term, you get: 1, 10, 100. Now, 10^(1/4) = 1.78. This means that any value less than 1.78 would be closer to 1 after squaring, and any value higher will be closer to 10.)

You can even take the analysis further and back-calculate things like how fast the CD might spin by guessing the track width and bit area, how long a track skip would be, whether the size limitation of the CD is due to optical or material properties, how far the laser would need to be to converge at one bit while being close enough that any deviation in the surface flatness doesn't send the return beam away from the sensor, etc. (This is all the info you'd probably use to begin the approximation if you weren't aware an audio CD holds an hour of music, like if you were asked in 1975 to "back of envelope" whether a compact, non-contact, vinyl-like, LP-length recording medium was possible.)

Today, kids need to be told: Math is nothing like what you learned in high school.

I think K-12 math should be divided into 4 quadrants, taught in a soft of spiral:

1. Arithmetic (symbol manipulation, through algebra and calculus) 2. Computation (numeric and symbolic) 3. Learning from data 4. Theory (sets, proofs, etc.)

Some of these things could be blended with the science curriculum.

Otherwise you will be lost as soon as you leave the textbook territory.

Proofs are just one way to build intuition.

The best way to learn applied maths and get intuition is an “Introduction to Maths for Physicists” 101 course.

> Are there any "math for people who just want to use it" tracks in math pedagogy

It's called Industrial Engineering /s (but only kind of)

It is tame as compared to a pure math or physics B.S. But, you pretty much cover the gamut of every math tool that is useful in the real world. From stats, supply chain, search, calculus to combinatorics and so on. Each concept is squarely grounded in a field where it is used.

I was surprised at how well my mechE (usually the closest undergrad degree to Industrial) degree prepared me for applied math and ML coursework for my CS masters. In comparison, a lot of CS undergrad peers struggled in those courses.

So if you think about "real world usage", you can either use the math results and implicitly trust that they "just work", or you can dive a bit deeper, see if you agree with those results, or at least gain some insights from the proofs that have been presented.

And just to be clear, there is little to no "real world usage" math without academic math.

Well, it is not only about people. You would also want to "convince" Nature (so that the aircraft you designed does not crash right after a takeoff), your own or, often, someone else's wallet (so that money is not spent on nothing).

This might sound a bit dense, but the alternative is what 90% of programmers do every day.

Yup. You need to know what a proof is.

Applying mathematical facts isn't math, it's calculation at best.

That's what most programmers already do. They throw things together in the hope they work without proof they got all edge cases covered.

And it works pretty good, because often the edge cases aren't that bad.

The problem is that the answer will depend heavily from person to person and from field to field and often the most sensible answers require a mathematical maturity that creates a chicken-and-egg kind of difficulty.

Do you care about engineering? Well you'll need some calculus for that. Do you care about prediction modeling? Well there's some stats you'll need for that. Do you care about finding patterns in the world? Well there's abstract algebra for that. Do you care about reasoning itself? Have fun with mathematical logic. But because humans have different motivations, there's no one-size-fit-all motivation-based approach.

This is especially painful for mathematics because I think most people who have learned some amount of pure mathematics will relate heavily to what helpfulclippy says in a sibling comment: "However, one thing that has been VERY applicable is proofwriting. Although math proofs are far more rigorous than most real world stuff, the discipline I learned in writing proofs has carried over into pretty much everything from programming (will this algorithm work every time?) to executive decisions (why, specifically, should we believe X?)."

The skills of rigorous and abstract thinking that pure mathematics provides is both nearly-universally helpful, but also simultaneously as a result very difficult to motivate. "This will help you think better across everything you do" is lofty-sounding, but generally not a convincing sell unless someone is already curious. But it's true that being able to wrap one's mind around pure abstraction (after rattling off a rigorous definition for an abstract question: Question: "But what is X really?" Answer: "X is just that. No more, no less.") has ramifications for all that one does.

And the most painful part of all of this is if you try to start by teaching the wonders of pure mathematics instead of all the messy, boring rote stuff, students' eyes are liable to glaze over even more because of the aforementioned chicken-and-egg issue with mathematical maturity.

In this way it's similar to trying to motivate someone to read and write. The key that unlocks that interest for everyone is going to be different and it's very hard to explain the near-universal benefits that reading and writing bring to one's way of thinking (but I can always just have a computer transcribe it or read it aloud to me!) without some inherent curiosity in them.

I'm a little snarky, but you have a broken idea of what math is. It's not even your fault. I don't even claim an unbroken idea for myself, I went through public education too, though I do think it's less broken. Somehow compulsory education has managed to get near universal basic literacy, but seems to have failed on whatever equivalent some sibling comments have hinted at exists for math or at least mathematical reasoning. A lot of algebra work taught for junior high can be understood as just a foundation to be able to understand later things (though you can of course use some of it directly as taught without having to learn more for every-day things like some boy scouting activities, or helping with putting together a garden or a fence, or programming -- and some of course is entirely useless). But instead of pushing algebra even earlier, states are instead moving to push it even later. (Let alone trying to spread awareness of even a hint of the subtle divide between more general algebra and analysis that a lot of STEM undergrads don't even really get a whiff of except maybe knowing it's often said to be a thing.)

To try and be more helpful, I'll suggest you don't actually want to learn math at all. So don't! At least, not directly. Instead, find something you want to learn more about in science, engineering, or technology/programming, and dig into it until you start hitting the math being used. For many things, especially at the introductory level, it's fundamentally no more complicated than being able to read a junior-high-school level equation. Occasionally you'll need to know about some functions like square root, or sine, or exponentiation, or some other new functions that will be explained (like a dot product) in terms of those things. When you don't understand something, you may need to find an outside reference (or a few) for it, if the book itself doesn't cover it enough or to your liking. Even then, you can often find outside presentations of that thing which are still motivated by the general field and are thus not proof-heavy.

However sometimes the best explanation may still be found in a "pure" book just about the thing, and if you can get over whatever problem you have with proofs you can learn to see how they can be used to build your understanding of the thing in smaller pieces, not just as tools to say whether this or that is true or false. In other words, proofs can serve the same function as repetitive problem-solving exercises, and are often given as exercises for that reason.

I'm a fan of the Schaum's Outlines series of books just for the sheer amount of exercises available in them, I just wish I had better self-discipline to actually do more exercises. Though they maybe aren't the best resources for a brand-new introduction to something.

To give a small example, maybe you're interested in game programming, and eventually want to dive into studying 2D collision detection more specifically so you can implement it yourself instead of using someone else's library, so you might stumble on a copy of "2D Game Collision Detection: An introduction to clashing geometry in games". Its explanation of the dot product comes early (its whole first chapter is on basic 2D vectors), consisting of 2 diagrams and two code examples (the first mostly defining dot_product(), the second using it as part of a new enclosed_angle() function) and some text all over 2.5 pages. It gives things in programming notation instead of mathematical notation, apart from some ² squared symbols occasionally. It gives a few equivalences like a vector's dot product with itself is its length squared, shown as dot_product(v, v) = v.x² + v.y² = length², without proving them, and points you to wikipedia of all places if you want to know more about how that or another detail are true. Why learn it? It's used immediately after in explaining projection, and then later in collision detection functions. Generally that book is structured as: learn the bare minimum of vectors, use them to implement collision detection for lines, circles, and rectangles.

I'm not saying this is a great book but it's representative of what you'll find that I think you're really after, which is motivated use of some bits of math. If you don't like that book's treatment of vectors, there are a billion other game programming books that cover the same thing as a sub-detail of their main topic, and maybe even better for you because it'd be grounded in e.g. a graphical application you've already got setup and running to see results rather than a standalone library. Or there's special dedicated math books like "Essential Mathematics for Games and Interactive Applications". Or you can go find dedicated "pure math" books on linear algebra if you want. Or maybe your junior high / high school math education was good enough you can more or less skip most of this and move on to something more interesting, like physically based rendering (https://www.pbr-book.org/) which also of course has vectors and dot products with brief explanations. Or maybe you don't care at all about game programming, and want to learn about chemical engineering, or economics, or the mechanics of strength and why things don't fall down, or...

I do not, which is why I explicitly acknowledged that what I want is not "real math". I don't care about math for math's sake, even a little. Not my thing, never will be.

Basically I want to learn to apply useful results from hundreds-of-years-old "advanced" math the same way I learned math in grade school: memorization, pattern recognition, heuristics, intuition, and drilling, all with a focus on application. Keep the proofs far, far away unless there's some excellent reason I have to know them (and perhaps there's actually no way around that, but I suspect the current situation has more to do with the interests and world-view of people who design math curricula, i.e. mathematicians, than strict need, if you're mainly focused on application). Ideally, almost every single problem set would consist mostly of so-called word problems, drawn from realistic circumstances.

For instance, let's take a very common "application" of statistic: given a test that has false positive probability p, you can apply the test n times, and the false positive rate of the composite test is p^n. (For e.g. this is used to analyze bloom filters, or the Miller–Rabin primality test, or hash tables). However this is only true if the tests are independent. If you have to apply this result to analyzing (for e.g.) some new data structure you wrote, you'd first have to prove that the tests are independent. And maybe it's just me, but I find that if I use heuristics to check if some events are independent, I really often get it wrong.

Another example: the uniform limit theorem (https://en.wikipedia.org/wiki/Uniform_limit_theorem) is quite useful, but to properly apply it, you have to understand the difference between uniform convergence and pointwise convergence, and maybe it's just me, but I found the difference very unintuitive when I first encountered them (in a standard proofs-based analysis class). Even now, if you gave me some random series of functions, I can't really imagine using heuristics to check it uniformly converges to its pointwise limit, I'd want to try to write down a proof to be sure. So this useful tool is gated behind understanding some (to me) subtle proofs.

You might want something less applied, but I highly recommend Burden & Faires "Numerical Analysis" and Trefethen "Numerical Linear Algebra". The various interpretations provided by Trefethen for understanding a matrix, eigenvectors, singular value decompositions, etc. are amongst the most insightful descriptions I've come across.

Drilling proofs is a valid kind of drilling and can be an effective way to learn something. Not necessarily the only way, sure. But there's nothing fundamentally different or "more real" about drilling proofs vs drilling grade school multiplication problems. You'll memorize things, you'll see patterns, develop heuristics, gain intuition.

Application can sometimes be tricky; did those grade school multiplication drills have application? Are they granted more application by phrasing things in terms of word problems around counting apples or whatever rather than the compressed a times b = blank expression? Well, sometimes the application of proofs will be more direct, sometimes less, and can be phrased better or worse, more realistically (and necessarily more complexly) or less so, like any other exercise, whether it wants a proof or not. CS proofs about big-O complexity are applicable to analysis of algorithms, which is pretty important if that's your focus. Though most problems you could find to drill specifically on big-O (as opposed to other parts of algorithm analysis, like recurrence relations) would likely take the form "find the complexity of this" or "given the complexity is such, estimate..." rather than "prove that...". There are many things no one knows how to prove that are still an area of study, clearly proofs aren't the be-all-end-all. Anyway, the mental processes involved between something like "find x, the hypotenuse of the triangle" and "prove the Pythagorean theorem" often aren't that different. There are multiple ways to prove it, you could drill on them.

And technically, computer programs themselves can be thought of as proofs (Curry-Howard correspondence) so if you've ever written a program that terminates you've written a proof... Proofs don't necessarily have the form or flow "by axiom 1, axiom 2, theorem 34, modus ponens on this, proof by contraposition on that which we'll name lemma 8, and by induction over the integers here, we have proved blah, QED".

And if you grant simple algebraic symbol manipulation as something you would do to solve a word problem, well, that itself is a style of proof. (There's a whole automated proof engine written entirely on the basis of substitution, the same process you use in a simple algebra problem of substituting x + 3 = 10 with x + 3 - 3 = 10 - 3 and reducing to x = 7.)

But fine, no proofs, not even in disguise! What is it that science, engineering, and technology focused subject books that use math only as needed without bothering to prove things when unnecessary (some having exercises and drills of word problems from realistic circumstances) don't do to solve your craving?