When Will I Use Math? | Hacker News Reader

When Will I Use Math?

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62 pointsacangiano45 comments

I solve partial differential equations on parallel computers. This is a key competitive advantage for many industrial applications including aerospace, reservoir, reactor, and data center design. It is also crucial for medical applications, climate, and many disciplines of computational physics. There are nowhere near enough qualified people in this field. If you are qualified, you can more-or-less have your pick of jobs anywhere in the world.

I use every mathematics and physics course I took until part way into my undergraduate studies on a daily basis. In contrast, I believe that I have never had an English course that improved my writing (they emphasized the wrong things for technical writing). Indeed, it would be difficult to find a single thing I learned in any course outside of the hard sciences that has ever added value to my contributions.

It is a disservice to people going into fields like biology or social sciences to pretend that mathematics does not play a pivotal role. Many, perhaps most, professional grade tasks in those fields involve designing studies to test a hypothesis or quantify an effect. This requires statistics, and not just an introduction. In the state I grew up in, Fish and Wildlife had one biometrician. He was a coauthor on every paper published by biologists any office in the state because he was needed to design the study appropriately and analyze the results. With one exception: my father has a Masters in biology, but his coursework was primarily in statistics, and he can design his own studies. Just barely, it still takes a significant amount of reading for each study.

Math matters.

On a related note. Much of applied (primarily solid state) physics, materials science, and the engineerings are now designing new materials for use in many applications. Given the vast possibilities, limited funds, and recent development of powerful mathematical models for simulating systems, many companies and research groups have gone to large scale simulations as the primary means for testing new materials systems and only after discovering the possibilities are these constructed for testing. This means a firm grounding in mathematics (and more specifically these techniques: molecular dynamics, density functional theory, etc.) is headed from "crucial" towards "prerequisite" for people to work in these fields.

Off topic, sorry. Can you elaborate on what "this field" is? Do you mean deep maths training in PDEs, or is a specialty within computing you mean?

Broadly speaking, the field is Computational Science and Engineering (CS&E). It occupies the space between applied mathematics, engineering, physics, and software (high and low level). CS&E picks up around where physicists write the equations they would like to solve and handles aspects like discretization, solvers, optimization, and uncertainty quantification. Software is needed for each of these steps, and it usually needs to be high performance, scale on diverse parallel architectures, and be extensible to accommodate new problems. Something that I find especially interesting is adjusting the discretization so that it is better suited to the hardware that is available (e.g. reducing memory bandwidth requirements in exchange for more flops which are relatively cheaper) while maintaining scalability and robustness.

Looking at the plenary talks [1] for the upcoming ICIAM meeting might provide some perspective on the field (perhaps slightly biased towards applied mathematics).

Part of the difficulty in finding qualified people is that there are few universities with courses aimed at the core of CS&E. Instead, each discipline teaches a subset as viewed from their own corner. The whole is definitely greater than the sum of the parts. The status quo curriculum is a recognized problem that will hopefully be resolved in the next few years.

[1] http://www.iciam2011.com/index.php?option=com_content&vi...

You got me interested. I'm personally studying theoretical physics and have studied all computational courses they give at my university. Seeing these computational tasks from other angle would certainly be pleasant experience. Can you recommend any comprehensive book about CS&E or link to some university's course listing (assuming such list has recommended readings for those courses)

Hmm, I can't think of a single general book that could be considered general CS&E. There are many great books in, e.g. numerical linear algebra, domain decomposition methods, finite element or finite volume methods, etc. While these are quite valuable, I never felt like they really changed my perspective on the field. So instead, here are some less conventional starting points.

Any talk by David Keyes. This one is mostly directed towards a physics audience (it's not specifically about ice sheets). http://www.columbia.edu/~kd2112/IceSheets09.pdf

This one is directed at a computing audience http://www.cespr.fsu.edu/lighthill/keyes_publiclecture.pdf (http://www.youtube.com/watch?v=RYxNVX9SYtk (skip ahead a few minutes to skip the introduction)).

This review paper has more background from an implicit solvers perspective and lots of citations for more background. http://www.cs.odu.edu/~keyes/papers/jfnk.pdf

An open source library. I learned a lot from experimenting with methods and reading PETSc source code. (And soon started developing the library too.) http://mcs.anl.gov/petsc

Algorithms like multigrid, fast multipole, and WENO that are unreasonably better than naive alternatives (for appropriate problems).

Note: CS&E is significantly bigger than partial differential equations, but PDEs are still a very central component.

Thanks, a concise and extremely informative answer.

I use every mathematics and physics course I took until part way into my undergraduate studies on a daily basis.

I think this highlights an issue I have with the meme that with a major in math, you can do anything. What you can do with a math degree still depends on what you learn aside from pure math (essentially domain knowledge), be it in elective coursework or on your own. "You can be a physicist"? Yes, I could if I'd studied physics more. I would strongly advise other math majors to take a minor or a second major (my other major was CS, where the math background definitely helps, and it would be nice to have even more).

This is an insightful comment.

My first-order take had been that the lucrative jobs that the video advertises are plentiful only if the "math person" is also proficient at some type of computing. Reason is, I've known a lot of mathematically talented people who are not able to compete for these kind of jobs because they have no sense of how to compute with their abstractions.

But I think your idea captures the problem with the video better than the simplistic "lack of computing expertise" does. The budding "math person" that the video targets could do quite well at, say, a biological or finance application, even without being computer-savvy, just by acquiring expertise in these areas.

It's shame that in the entire video, not one concrete example was given about the application of math in a real world profession.

It's mostly just "Yes, math is cool! You can do so much with it!". This is what my teachers always kept telling me, but it never registered because it was presented in a very abstract and meaningless way.

What would have convinced me was concrete applications. Something as simple as defining and understanding the algorithms required to make a simple Pong game work. That would have inspired me.

Yeah that's how I felt about the whole video but I think this is just the introduction. Maybe the future videos will show people at work. They could talk to a video game programmer who takes them throught the process of displaying an object in a 3D space (no code just the idea) and show them it's pretty much the stuff they are learning right now but more advanced. Then interview a carpenter, the carpenters I know do a lot of calculations during a day. It can be considered low level math but I think the point should be to show that there is no safe place from math ;p

If they hope to reach out to people and interest them in math and how practical it is, then not showing examples of this does a disservice to their goal.

It's like playing World of Warcraft for me... I find it incredibly boring to play a game where the majority of what you do is "Gather/Hunt/Mine X units of Y and take it to Z". My friends who are into it say "Yes, but that's just the beginning... the real fun starts at level 60+". That may be true, but I'm unwilling to torture myself through months of "gameplay" just to see if at level 60+ I still find it fun.

If it's not interesting from the beginning, what are the odds most people will stick around?

I agree. I think this blog's target audience (myself included) would unanimously agree that math is important, but the video itself seems to speak to people who already agree with it's assertions. When a struggling high schooler from an inner city school hears a mathematical theorist proclaim "Math is fun!", I believe they're going to be weary of everything else this video is trying to say. Math is not fun for everyone. The individuals presented in this video are not relatable enough for the average math student. With this in mind, an emotional approach to the argument that "math is useful" is ineffective. The argument should be strictly fact based in my opinion. I think computers and the internet are proof enough that math is essential to everyday life.

> I think computers and the internet are proof enough that math is essential to everyday life.

I think that they are not enough. Computers are something you buy, not build. The Internet is something you pay for, like TV, or buses, or school, or whatever. The point is that you, personally, don't need any math knowledge to have and/or use them.

Like OP-1 said, we need examples that have a clear use of maths in it.

Sometimes I try to convince people I know that math, physics and the stuff is important. But I'm running short of real use-case arguments - not the "you'll get better job" ones.

Math is like a mental martial art. It doesn't need a concrete application to be useful to everyone because it shapes the way you think about everything! Besides, applications are generally not why people do math. It's really about the doing.

(That's not to say that practical applications aren't important or numerous, just that they're the most boring aspect of math, once you get to know math.)

As far as I can tell, this is the problem with science eduction. People who become teachers or professors do so because they just love the subject. But, in general, whether you're talking about math or literature or foreign language, the people you encounter aren't going to love the subject, so saying you ought to do something out of love (or fun) isn't a selling point at all. And someone who loves something tends to make it more boring for those who don't, I think; for example, I took a class in complex analysis, which I'm not interested in and the instructor spent nearly all his time on proofs that he found excited which made it even less interesting to me.

I've tutored 100s of people in math and I think that I can say that the argument that math teaches you how to think better isn't practical, that is, people do not practice it. Of course math is very tight logical thinking and you can apply that way of doing things elsewhere in your life. But, to students, math, e.g algebra, is in the box of stuff they don't like and won't use, and they're probably right and they just throw it away when they're done; they don't use it and they don't learn how to think better.

I know for me, I had a lot of trouble with math until I found something that I liked that used it. It wasn't until there was a definite, concrete use for it that I took it up. Math seems kind of like shopping for hammers. If I need to build a house, then I'll go get a hammer, but if I don't, I'm not going to stand mesmerized on the hammer aisle.

I think the question is badly formulated. "When" will you use math implies that, no matter what, you will end up using it. This is simply not true. The vast majority of jobs out there, from flipping burgers to fixing toilets, involve nothing except the most basic arithmetic.

I think a much better answer is: if you learn math, you will have plenty of opportunity to use it. It's the only way to become qualified for some very good kinds of jobs.

The vast majority of jobs out there, from flipping burgers to fixing toilets, involve nothing except the most basic arithmetic.

I remember working as a busboy in a diner in Alaska in the early 90's and applying queueing theory and Little's Law. Miraculously, I efficiently had tables ready for the waitresses when they needed them, and they liked me for this and gave me great tips.

A girlfriend of mine told me about doing the same sorts of optimizations intuitively while working as a take-out order taker at McDonalds. Her manager loved her, because she always cleared out the queue quickly. She later went on to medical school and is now a doctor.

I also remember trying to order from a McDonald's some years later on a road trip and noting that everyone behind the counter was consistently waiting 2 or 3 on line on the same station. It took me a full 15 minutes to get my order!

If you want to see the result of people being sheep and losing out because they aren't applying such knowledge, just go and take a drive during a busy time on a Houston freeway when people are merging and taking offramps. People consistently make the worst choice. A small minority do egregiously bad things.

I should have been more clear and said "require" instead of "involve". Yes, math can improve almost anything, but you can do an acceptable job at McD with no math knowledge. Still, it's a McD job, and if you had learnt some math then you could have found something a lot better.

Yes, math can improve almost anything, but you can do an acceptable job at McD with no math knowledge.

The optimally suboptimal queueing at that one place was not acceptable! Still, as my ex demonstrated, you can manage the same optimality at that job just be being observant and using common sense. The human brain is a pretty powerful general optimizer. Math is most useful in situations where data is not so accessible to a casual observer. The problem in those situations, is that the optimization itself then becomes harder to observe/explain/comprehend. There is a similar problem in large enterprises with automated testing and refactoring.

Still, it's a McD job, and if you had learnt some math then you could have found something a lot better.

Well, in Homer Alaska at the time, from my vantage point, it would've been some more machismo so the right person would give me a job as a forklift driver, some cooking experience, so I could get a job doing that, or maybe some more muscle -- all those would've done a lot more to get me a higher paying job than math.

"some more machismo [...], some cooking experience [...] or maybe some more muscle -- all those would've done a lot more to get me a higher paying job than math."

In the short run.

Well, for that, I had the prospect of saving enough money from the diner busboy job so that I could drive back to the lower 48 and grad school.

I wonder how much money someone could make as a consultant, teaching McDonald's employees the basics of queue theory as applicable to their jobs? Try talking to a few McD franchise owners.

I don't think it's the responsibility of the people who work in customer service to be applying math to their job if it's not required. Sure it may help if the organization doesn't use math to optimize operations, but for non-math jobs I think it's the responsibility of the corporation to enforce any applications of math.

Some clear, strong answers to the question in written form,

ftp://math.stanford.edu/pub/papers/milgram/milgram-msri.pdf

with a lot of interesting information about mathematics education.

(Hacker News may not style this URL as a link, but I've been there and it works.)

That's a great find. The first quote I feel directly related to the Software Dev explosion since it is so hard to find a good programmer. I feel the reason is deeply rooted in sucky education, and a lack of mentors.

I usually refer people to read What Is Mathematics For? published on the Notices of AMS. http://www.ams.org/notices/201005/rtx100500608p.pdf In short, most people will not use math more advanced than arithmetic, unless they are mathematicians/physicists.

I remember asking this very same question to my Math teacher when I was 12, during middle school.

She paused 5 seconds and then replied: "It's exercise for your brain, it makes it more elastic!".

Eventually, I became an engineer (and learned the hard way why math matters), but I really hope she dosen't teach anymore.

The only thing I took away from this that wasn't really vague is that you can make money if you have a math degree. As someone who has been tempted to study mathematics but not for the purpose of making money, this is the opposite of what I was looking for.

Not exactly job related but... Just today while talking to my girlfriend about making plushies, we stumbled on the question of how to make ball plushies. Upon realization that you get cones from flat sheets by cutting a triangular wedge, resulting in a cylinder's whose radius changes at a constant rate (and therefore a cone), we realized to get a dome, we would just have to get a wedge who's sides have the right curve (we're still working it out... got kinda distracted, intuitively feels like a tan or something).

Math pops up in the darnest places (queue Donald Duck in Mathemagical Land).

A friend of mine once did the calculus to create his own juggling ball pattern. What he found is that there are all sorts of factors (like the changing stretchiness of the cloth versus direction and details of doing the stitches) that were too hard to calculate, which left lots of room for engineering. (Making a lot of examples, and tweaking the design.)

Domes have non-zero curvature. Cones have zero curvature. You can't do anything as simple as a cut to get a dome starting from flat material.

This seems to be an organization rather than just a one-shot video. Does HN perhaps want to distill it's criticisms and suggestions and forward them to weusemath.com to try and actually be helpful? The goal is certainly laudable.

I think using maths isn't about applying equations that one has learned, or something like that.

Learning maths changes the way one thinks. But the benefit of this is very hard to explain to someone who doesn't know.

This video would be a lot stronger if they had only 1 or 2 academics out of the dozen or so people, rather than the opposite ratio.

Q. "When will I use math?"

A. In business, whenever you can find a use and start a company to pursue it.

Note:

(1) I know some math: My Ph.D. dissertation was on the applied math of stochastic optimal control. The work was advanced enough mathematically to address measurable selection, solved a real problem, and made some contributions to making such calculations faster.

(2) I'm pursuing a use of math in business. My current work is an information technology startup based on a Web site. The key to the promise of the work is some original math I did based on some advanced prerequisites.

So, I know something about applications of math.

Jobs? Early in my career, I knew some computing, was at GE which was shrinking (again) in computing, sent some resume copies, and in two weeks got seven interviews and five offers. The computing worked for a job.

Broadly the video is nonsense, contemptible, deceptive, misleading, badly supported nonsense. Students need to be warned.

Wall Street? In graduate school, Wall Street was one of the career directions I had in mind. So, I got a relatively good background in stochastic processes. Later I wrote Fisher Black (as in the Black-Scholes option pricing model) at Goldman Sachs and got a nice letter back saying that he saw no opportunities to apply math on Wall Street.

I interviewed in computing at Morgan Stanley, showed some nice applied math I'd done, and mentioned that I wanted to work on automatic trading but got no interest.

I got a call from Google, and they wanted only C++ programming.

I got a call from Microsoft and explained some work I'd done, and published, that would help Microsoft but never got past the first 'phone-screen' girl.

Early in my career, there was some interest in applied math, but this interest was essentially only for work paid for by the US DoD and for the Cold War. The US DoD has done a lot of good applied math, but those successes have had essentially zero influence in business.

Can look at the Web sites of 500+ US venture capital firms, nearly all in Silicon Valley, near Boston, or in New York City, look at their investment interests, and never see the words math or mathematics. Chris Sacca dropped out of a math Ph.D. program and may have the best background in math of any venture partner in the US. His biography may have the only mention of math on a venture firm Web site.

The interest of math in information technology venture capital is somewhere between zero and deeply negative. E.g., I got back from Ron Conway's son at Andreessen Horowitz

"your company is outside our area of expertise".

By "expertise" he had to mean the math, not the Web site.

Yes, in topics central to Hacker News, there should be interest in math for 'machine learning, data mining, big data, ad targeting' etc., but there is not. Instead, the world would rather stumble along with intuitive approaches to these topics instead of solid (powerful, valuable) math approaches.

Here's the main point: In business, the US is still organized mostly like the factory floor of 100 years ago where the supervisor knew more and the subordinate was there just to apply muscle to the ideas of the supervisor. And that 'inequality in technical knowledge' continues all the way to the CEO and COB who want to believe that they know the most.

So, no one in middle management wants to bet their budget or career record on some project using some math they don't understand; they don't get bad marks for such projects they do not pursue.

Essentially only a CEO could sponsor a project with some significant math, and CEOs rarely sponsor projects directly.

In venture funded information technology entrepreneurship, the 'technology' is supposed to be just routine software that could be understood by a bright, self-taught middle school hacker and, thus, can safely be ignored as crucial 'intellectual property' by any Ivy League history major, Harvard MBA venture partner.

Broadly, bringing high technical expertise as an employee to a business is a fool's errand. Yes, there are a few exceptions, e.g., some technical topic deep in the CIO's organization, but generally the employee is supposed to know less, not more.

An employee who, on the side or whatever, starts a good project based on math will find that their management doesn't like the project. If the employee proposes the project more broadly, then they will be told not to communicate outside of their management chain or face getting fired.

Once as a professor, hired in optimization, I wrote a research paper in optimization based on some math, with theorems and proofs. My department reviewed the paper and concluded that it was "not publishable". It published right away, without significant revision in the 'Journal of Optimization Theory and Applications'. At Yorktown Heights, we were doing some work in server farm and network monitoring of health and wellness using artificial intelligence. I found a much more powerful approach with some original work using some advanced topics in stochastic processes, and the lab concluded that the work was "not publishable". The work published right away without significant revision in 'Information Sciences'. The reason: The people I was working for didn't know math and didn't like it.

Really, if math is to have a significant role in a business, then the main mathematician has to be the founder, CEO of the business, e.g., Jim Simons, Andrew Viterbi. Otherwise, f'get about it.

Broadly, math is by far the most exacting and difficult academic subject, is rarely pursued by students beyond what is needed for K-12 teaching, physics, engineering, or basic statistics, and is just not known, liked, wanted, or even tolerated in society more broadly.

For a good career, mostly an electrician's license is a better foundation than a Ph.D. in math, electrical engineering, or anything else.

My view is that math is by a wide margin the most powerful and valuable topic for the future of information technology entrepreneurship for at least the first half of this century, but to make this promise real the mathematician has to be the founder, CEO of the company, and the math will not be an advantage in seeking equity funding.

To make this promise understood, about the best I can do is be successful financially, buy a yacht over 200' long, and give some interviews on the yacht cruising in Long Island Sound.

Broadly our society does not value, want, like, or tolerate math.

An interesting comment. I wonder if the YC funding process is at all an exception, a funding process in which mathematics background might be an advantage.

Some of the qualifications of some of the YC partners are exceptional for venture partners: You might be correct.

But, to add a little detail, it appears that information technology (biotech might be significantly different) venture funding has been reduced to a large font on one side of a 3 x 5" index card: For a seed round, see a demo of the software. For a Series A, see 'traction', that is users and revenue both growing quickly. For a Series B, see good accounting numbers. For a Series C, are basically just buying part of an on-going company. Could teach a dog that 3 x 5" card in a weekend.

The Series A criteria are so uniform that my guess is that they are 'suggested' by some of the more important LPs.

But, for anyone with a serious background in math, science, or engineering, (information technology) venture funding is a shock because, if only as a special case of the 3 x 5" card, the partners will essentially never take seriously a technical review of the project on paper. This is a shock because essentially all serious work in math, science, and engineering is done just on paper, not with prototypes or 'market ready' products.

My joke about venture capital is that they would have said, "You build and test one and get one more ready for delivery, and we will buy half the gas for the Enola Gay".

But the bomb was a big waste of money? The bomb cost about $3 billion in 1940s money (see Richard Rhodes), and that's $3000 for each of the 1 million US soldiers that might have been killed or injured in an invasion of the home islands of Japan, that is, a grand bargain just as money.

So, (information technology) venture capital just won't judge based on looking in detail at the core 'secret sauce' intellectual property and, in particular, won't consider math.

So, if the math helps with the traction or accounting numbers, then fine. The entrepreneur can tell the equity funders that the project is just 'software' and otherwise keep what is crucial in the secret sauce just between his own ears. If he has an algorithm that shows P = NP (not what I have) and is blazingly fast on the full range of NP-complete problems, then don't tell anyone and just let the software run and do logistics, manufacturing scheduling, etc.

In my project, the math is the crucial work and the main promise of a valuable company quite beyond just some Series A 'traction' numbers. But the only person in business who can see this promise now is me, and this is at least a surprise.

In the end, for me, now, that the venture partners will ignore or just hate the math is no longer very relevant: I have the math and the corresponding software long since done, and it was fairly easy given my background. The rest is just routine programming. Since I'm using Windows, .NET, ASP.NET, and ADO.NET, I've had to learn those topics. This learning has been routine but slow. But a few more Web pages, and the software will be ready for at least first production. Then get and load some good initial 'base data', and I will be able to go live. If the project is good, then there will be good 'traction' numbers. Then I could raise a Series A, but if the project is good soon it should throw off enough cash that I won't t need a Series A. If I get a good business without equity funding, then the venture partners won't be part of it because they refused to evaluate the real promise, the math, and waited until the business was so far along it didn't need equity funding.

Then, just why would I want a Board that hated math?

Net, for this thread, for an information technology entrepreneur, the math just has to be something the entrepreneur keeps between his ears, and that is a piss poor career direction to recommend to young people.

I am more recent in applied math (coming from pure) but my impressions are very similar. For the last few years I keep discovering the same thing over and over again: the level of math in “technology” is abysmal, nobody understands it, nobody needs it. Incidentally, I am writing a blog post about PageRank right now. Short version: mathematically, it’s very unsatisfying.

Have you tried with the upcoming robotics companies? iRobot, Foster-Miller, and Boston Dynamics tend to be outside of the startup bubble but might be more receptive to someone with a background in control theory. My biggest problem when I tried to move in robotics was my lack of a math background.

The 'job' qualification that worked for me was 'computing' the time in two weeks I went on seven interviews and got five offers.

Since then, with applied math, a Ph.D., etc. getting a job has been a scavenger hunt.

A good background in stochastic processes was supposed to be good for a career on Wall Street, but Goldman Sachs and Morgan Stanley didn't agree.

I'm no longer looking for a job. Instead I'm starting a business and now am quite far along. I may have a nice stream of revenue later this year.

Yes, for this thread, a background in parts of control theory might help in work in robotics. But some of the other considerations remain: Will the hiring manager be comfortable with a subordinate who knows more? Will the HR people recognize good qualifications when they see them? Is it really good to recommend to a young person to study control theory on some possibility that it will help getting a job in robotics?

There is a bigger issue that I omitted: Math in the US is still dominated by 'pure math' (some of which is crucial) which as a field is still mostly interested in "the analytic algebraic topology of the locally Euclidean metrization of infinitely dfferentiable Riemannian manifolds" (an old joke, extra credit for knowing the source), etc. and just HATES any suggestion of 'professional' education. Moreover those academic departments don't have even a weak little hollow hint of a tiny clue about applications outside of pure math or math physics. The pure math departments worked hard to push out probability, statistics, stochastic processes, optimization, control theory, continuum mechanics, numerical methods, graph theory, etc. Yes, there have been some exceptions at Brown and Courant. Yes, for some years there were efforts in 'the mathematical sciences'.

So, in particular, in business, they will take seriously professional work by lawyers and accountants but not mathematicians (main exception, actuaries in insurance). Part of the difference is that lawyers have a rule that a lawyer working as a lawyer can be supervised only by another lawyer. Net, math just has no 'professional' status in business, and that is a big problem for math helping a career in business.

Eventually there was the David Report that claimed that pure math just was not well enough connected with applications to be very relevant, and the NSF then started cutting back on grants for US pure math.

Science, engineering, computer science, statistics in the social sciences, etc. have had to do their own math and generally have done it poorly. It's a bad situation.

Net, for this thread, for a person planning their career, the video was not good; for all the potential math has, a person who pursues math in business will be mostly out on their own; the math will be in some secret sauce that neither the customers nor the venture partners know about.

This video is completely worthless for the people they claim to be targeting, the average student. It literally doesn't talk about where math is used in the real world in any meaningful terms to a highschooler.

Right, a student is thinking, "WTF is this quadratic formula for?" might not have gotten much from this. This shows the benefits of the end-game, not the middle point. Bridging that gap for students I think is the hard part.

I learned the quadratic formula in algebra II (the highest math class I've taken) and promptly forgot it. Then a year or so later, I was writing a raytracer and realized that the ray-sphere intersection is just the quadratic formula. I couldn't help but wonder why we weren't given a single use case for it in school.

The only reason I can imagine is that most use cases require other specific knowledge that will also seem like useless stuff to 99% of the students and its hard to find a use case that fits most of their probably interests.

"So in the future, when you're writing a ray tracer..." just doesn't ring with most students anymore than the word problems that required the formula either.

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