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Comment by andai | Hacker News Reader

>mathematics is basically the only scientific discipline that rejected any notion of utility

I think this might depend on the department, but I was at a pure math department last year, and struggling with my Linear Algebra textbook (written by the professor, incidentally, who was not a great communicator).

I consulted the machines, and learned, to my great delight, that linear algebra is used in like 20 different fields in the real world. It's "perhaps the most applied branch of mathematics in existence".

I complained in the group chat, that our didactic materials, specifically tasked with providing motivation and concrete examples, did not contain a single application, of this most richly applied field.

I was promptly pilloried, and shunned.

(Apparently that particular department was the wrong one, to ask a question like that!)

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I’m a physicist, so I’m biased, but my experience of pure maths was about the same. We had to do it, but at no point was any utility actually demonstrated - that was left to the physics professors. It was all just “look at this thing I can do with these symbols” without any actual tangible relationship to anything.

Then again, I remember how we were taught calculus at high school - we were taught how to mechanistically integrate and derive everything under the sun. At no point did anyone think to explain that we were measuring the areas under curves, or their rates of change - it was all just “memorise this operation”. Again it was left to the physics teachers to explain why this was useful, and what we were actually doing.

Poor teaching, if you ask me, and it more often than not left me retrospectively wondering if said mathematicians had actually understood any of what they did, or if they just had little blind symbol manipulation Turing machines in their heads.

>At no point did anyone think to explain that we were measuring the areas under curves, or their rates of change

In my experience you get taught the definition of a derivative of a function at a point is equal to the instantaneous rate of change and that integrals are defined as a Reimann Sum, the sum of the area under the curve. Everything in the class comes from building on top of those definitions.

That you think this way (and if like me, it makes you excited!) I think it's because it has clicked for you.

For many that light bulb above their head doesn't flash on, hence they get to dislike the subject or forget it after they are done with their studies. I was lucky enough to appreciate math that much to redo it in my free time after high-school and make it click for me.

No, my calculus class in HS very literally started with finding the area under a curve “manually” and introduced integration as a generalization of that. I’m not surprised to hear that calculus is sometimes taught very poorly, but it’s not universal.

Teaching only the practical side risks not teaching the subject with the appropriate theoretical depth and the ability to generalize it to other applications. Courses for purely applied fields utilize calculus to solve the current problem and then move on without teaching the finer points. Basing a calculus course on physics alone might be preferable in high school, but would be of disservice to students in university.

Well, in at least some cases, they do understand what they're doing, on a deep intuitive level!

Using spaced repetition to see through a piece of mathematics

https://news.ycombinator.com/item?id=18895613

The title is a slight misnomer in my opinion. The author did make heavy use of spaced repetition of course, but deemed it neither necessary nor sufficient for the result he described.

The actual active ingredient is curiosity to the degree of obsession. Actually wanting to understand what's going on.

You can do that with pure math just as much as with applied.

But it may be harder for many people for such a desire to arise if there is no obvious connection to real-world matters. (I certainly had such thoughts, sometimes! What is this for? What's the point?)

Re: obvious intuitive explanations are ignored.

I remember squinting at the definitions in my calculus textbook for a full 10 minutes before realizing... oh my God, if this had been a 5 second animation, I would have understood it instantly.

I remember being rather upset! "If it's so obvious and intuitive, why didn't they just say so?"

I want to say, "well, unfortunately you cannot put an animation in a textbook." But that's actually false!

Even as a child I made flip book animations. And there was plenty of room here in the margins for several of those :)

> I complained in the group chat, that our didactic materials, specifically tasked with providing motivation and concrete examples, did not contain a single application, of this most richly applied field.

> I was promptly pilloried, and shunned.

Heh. In my day I may have participated in the pillorying.

I do think that there is value/merit in professors mentioning real world applications, where they exist.

What they're sensitive about are the theorems where there aren't real world applications. They don't want to (and shouldn't) justify them.

So even when there are real world applications, the posture is "Who knows if someone is making good use of this in the world somewhere? I don't care. It's not why we learn or teach this!"

Knowledge for its own sake is great, but it's worth noting that many "useless" fields of mathematics turned out to be very practical in the long run.

Number theory was long thought to have no practical application, but now it's the backbone of cryptography. Boolean algebra was developed in the 19th century (George Boole died in 1864), decades before it was used to build computers.

Those "useless" theorems being proved today may turn out to unlock a world-changing technology centuries from now. When the breakthrough comes we'll be grateful for the people who laid the foundations.

No one is disputing that - not even most mathematicians. They just don't want it to be their job to know the useful applications.

Hear me out on this one:

For a lot of math departments, that is exactly why they teach this. Education is rooted in application. We have entire careers that depend on certain aspects of mathematics, so most companies gatekeep that career by a degree. The degree requires the class. The student taking the class may not even be old enough to drink alcohol yet, and they can't possibly be expected to know of all the applications. Knowing and not telling them is doing them a disservice.

I think for many people (myself included) understanding mathematics is rooted in application because it helps bridge the divide between intuition and rote memorization. Without the application, IMO instructors are doing a disservice to their students and pedagogy of mathematics itself. They’re intentionally ignoring a significant fraction of the class, unless they’re teaching some esoteric grad level pure math.

Yeah this was my position. I don't mean "Sally has six apples" necessarily, but my brain needs some concrete handles to grab onto so it can actually manipulate this stuff intuitively.

See also: Using spaced repetition to see through a piece of mathematics

https://news.ycombinator.com/item?id=18895613

Where the author describes his mind eventually turning the mathematical objects (symbols) into some kind of mental objects he was able to actually manipulate "directly". As a result of this facility, his mind spontaneously produced many new insights into the matter.

That level of fluency obviously requires a great deal of work, but in my experience it can be approximated and accelerated with analogies. (e.g. see Feynman's teaching.)

> For a lot of math departments, that is exactly why they teach this.

Depends on the course. That's why some departments have separate calculus courses for math majors - because otherwise the whole class will be full of non-math majors (engineers, etc) and focusing on their needs does a disservice to the students in their own department.

> The degree requires the class. The student taking the class may not even be old enough to drink alcohol yet, and they can't possibly be expected to know of all the applications.

If I'm a CS major, and the degree is requiring a class outside of the CS department, you shouldn't expect the professor of the class to know why the CS department is requiring it. It's on the CS department and its faculty to explain it.

The ultimate goal of university education is to raise researchers, who are the people that investigate the knowledge frontier of their field and then advancing it. To do that they have to understand a large part of the existing field so they can communicate with their peers, avoid investigating things that have already been throroughly explored, and draw useful connections to other fields.

Even in more applied fields it can take decades before advances become practically relevant. Restricting teaching to topics that have immediate practical relevance would therefore do students a huge disservice and prevent them from approaching the knowledge frontier of the field.

> The ultimate goal of university education is to raise researchers

The head of my CS department was quoted as saying, "our job is to produce researchers!"

This was in the context of a conversation about why the program wasn't better suited for industry.

It turned out that in our department 7% of students ended up going into research.

So he was openly declaring his intention to disregard the needs of 93% of his students.

(I'm not arguing against research, or on going deep with the theory. I just thought that was kind of funny, and it seemed a bit wrong to me.)

CS is special in that it is about the science, and not the application. They'll teach you mostly theory. That's why some universities/countries have a separate degree called "SW Engineering".

Other engineering departments are not as fussy about the distinction.

I love teaching kids and young adults calculus by socratic method. They get so mad when they figure out you were teaching them math, but they often admit it was pretty fun. Only had the chance to teach like that a few times but it's dynamite when it happens.

I did this when I taught my third grader calculus on the train, when she asked a question about the train accelerating faster sometimes than other times. She loved it, but I was just taking advantage of children's natural curiosity.

Do you have some examples that the adult could instigate, rather than waiting for the child to express curiosity?

I've used filling a tank, balloon, or bucket (rate of flow, can be subdivided to teach limits, and use weird shapes for teaching area under curve and interpolation en route to integrals), or the classic throwing a ball back and forth and trying to describe the shape, the distance it flies, peak speed vs peak height, figuring out how hard you are "actually" throwing instantaneously. Honestly as soon as you start thinking about bulk substances moving around (gravel piles! fuel tanks!) it's easier to find examples than you'd ever reckon. Rate of change is everywhere.

Seems like I start by asking "how do we know how much this tank holds?", or "how fast does this line go up on the side of the tank?" and curiosity goes from there usually.

Yes, exactly, identifying things that change is the key. Thank you!

I thought linear algebra was pretty much the poster child of applied mathematics - the entire field was invented to represent computations in a regularized form to feed into computers. Well not really, but much like Boolean algebra or the Fourier Transform, it was pretty much a curiosity until computers came along.

> but much like Boolean algebra or the Fourier Transform, it was pretty much a curiosity until computers came along.

https://en.wikipedia.org/wiki/Linear_algebra#History: “Later, Gauss further described the method of elimination, which was initially listed as an advancement in geodesy”

That’s an application of linear algebra in the 19th century.

Many important advances in linear algebra happened in WWI to solve optimization problems for logistical and industrial planning. A lot of these applications boil down to high-dimensional systems of linear equations, which back then were solved by hand. Efficient algorithms translate to reduced labor costs.

>(Apparently that particular department was the wrong one, to ask a question like that!)

Yes, the math department.

In any case linear algebra, stochastics, calculus; plenty of engineering and science applications for all these.

despite being theoretical i would have greatly benefitted in learning linear algebra if i had seen even one or two not-obvious applications, like galois fields for reid solomon erasure coding.

As my Linear Algebra prof used to say, basically everything is applied Linear Algebra.

Even when this really isn't the case, the math is preferably reduced to a linear problem whenever possible since dealing with the original complexity is usually so much more painful.

As a friend of mine who also happens to be a math professor once said: mathematicians are like sculptors who marvel about the beauty of their creation, and are kind of disgusted when a physicist comes nearby and says “that's a cool hammer you got there, may I borrow it?”.

I would be flattered , but that is just me

John: "I've just gotten word. They found an application for my theorem. They say they've used it to cure cancer."

Frank: "I'm so sorry."

If it was the students, then students can have things they think are cool or uncool.

If it was the professor, then that would be very embarassing on his or her part.

Typical pure-math linear algebra course has to cover so much material that there's really no time for applications! That's why applied math is typically separate track.

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