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The title of the article is literally "Lessons from My Math Degree That Have Nothing to Do with Math". Of course tedium and frustration are not end goals of mathematics. The author's point is not to e.g. explain the inspiration he felt when he gained intuition of Gauss's remarkable theorem. It's to explain the other side.

Lockhart's lament is thrown around here a lot around here, and it does make a valid criticism of math education. On the other hand, I think we're justified treating math differently than art. We tell students that math will be useful because it's true. Math teaches you that viewing problems more abstractly will make it possible to gain more insights into them, and math provides frameworks for making those insights. This skill takes practice. More than most skills, it's clear when you're doing it well, which makes practicing it all the more effective.

Math is beautiful, sometimes. I see this article saying the mundane parts are worth fighting through, too.

Good point and I gully agree.

It's just that the most people who encounter mathematics in some shape of form (before college for most) aren't even aware of the non-mundane parts.

Sorry, but the person has been doing math alright.

Their only problem is that their interesting post fell prey to the usual "dismissive top HN comment" though...

Math-as-"fun", math-as-art, social-math is just one perspective, albeit the one that dominates academic mathematics. Presuming it's the only way to appreciate math is frankly obnoxious.

Perhaps it's not the only way, but it's a way that people outside of "academic mathematics" usually aren't even aware of (from my experience explaining it).

The way most people know math is the way it is presented in the article: "Math is pain. Studying math is masochistic." (literal quote).

There are many ways to appreciate math, but perhaps the ones described as "masochistic" aren't the ways we should actively spread, in my opinion. And yet, we do.

The pain in math comes from trying to solve problems you know are solvable. Either doing exercises, or deriving a result some article called trivial.

Compare this to doing original work. Here, when you get stuck it is frustrating because you want progress, but there is no reason to presume that what you are doing is feasible. Therefore, it is not painful that you make no progress, for it is not expected of you to be able to progress.

This is of course a big difference between undergraduate mathematics and doing mathematics research.

Nice gatekeeping, friend. I've also personally never heard my math graduate student friends talk like this.

Gatekeeping? Quite the opposite. Rather, sadness.

I think that the system has failed the author, and deprived them of fun.

If I met them, I'd tell them the same thing I'd tell you: come back to math! There's fun and beauty in it, and suffering is not a necessary component of that.

As far as your math grad friends go: the school, most likely, is failing them. Statistically, about half of them are depressed. Most have no job prospects that will allow them to keep doing what they are trained to do (simple matter of #students >> #positions). Ask them how often they considered suicide (not if), the answers may surprise you. Graduate school does make mathematics (amd life) a funless process way too often - and that's a problem. But that's not how things should and can be.

Please read Lockhart's Lament I linked to in order to understand my point better.

What I ultimately am saying is that what is being pushed onto people as math is way to often isn't really, in a way that a cheese product isn't cheese.

I read Lockhart's Lament, I had the opposite reaction, maths is one of the few arts that is actually taught in school. If it had as little practical application as painting or music it would be nonexistent as a school subject.

He is wrong in that painters and musicians tend to get rigorous instruction in the basics before they become independently creative. Picasso's realism phase. Musical theory etc.

>He is wrong in that painters and musicians tend to get rigorous instruction in the basics before they become independently creative.

They still get to do it: they get to paint, play, sing, etc. Most math students don't get to do math. They learn about it until a year or two into grad school sometimes.

>maths is one of the few arts that is actually taught in school.

His point is that what is taught is mathematics as much as the names of notes is music.

Don't forget that the elementary math learned by the billions was ince new and exiting. The proofs are simple enough to stay implicit, but the learning process is creative discovery. Because you need to see applicability of what is learned, in order to remember it. Consequently, many don't even successfully learn about maths. Insofar you are right, but wrong and elitist to generalize.

Some of their language is unfortunate but they pulled through with the sharing part. It is quite hard to share maths with others however. How can I share that feeling inside my brain algebraically closed fields give me?

>How can I share that feeling inside my brain algebraically closed fields give me?

That's kinda like returning home after a long day, because algebraically closed fields are about the nicest structure you can hope to find yourself working in!

The algebraic numbers aren't complete. So you cannot measure the circumference of circles in them, nor does a series like sum(1/k!) converge.

Personally I'd posit finitely generated abelian groups as the nicest structures. Perhaps there is also something to be said for finite algebraically closed fields.

There are no finite algebraically closed fields, but I see your point, it depends on what you have to do really, if you want to do any kind of analysis completeness is great, but being more on the logic/model theory side I like all algebraically closed fields!

Ya'll rock.

That's the reason why NLP is hard. Language is good for communicating things where the two sides already share a model of. It is much harder to communicate a new model than filling out the details of an existing model.

Plenty of math is demonstrable in a tangible way--see the scientific method.

That you can't communicate a particular piece of math suggests that it has no tangible relation to physical phenomena. To me, that's a major red flag.

>suggests that it has no tangible relation to physical phenomena.

A lot of mathematics is that way, and there's nothing wrong with that.

Sometimes mathematical concepts do find tangible physical phenomena that demonstrate them - but sometimes, centuries after they've been invented.

Mathematics is often about what one can imagine. Our imagination can be inspired by the world around us, but it is not limited by it.

It's not a red flag that some concepts don't have short explanations. Sometimes you really just have to know the full joke to get the punchline.

> How can I share that feeling inside my brain algebraically closed fields give me?

Poetry.

  it was correct by construction
  stretchy compactness
  continuity
  we wrote numbers in the sand
  watched the glass dance of entropy
  against tired eyes
  until our vision blurred
  remaining only feeling
  legs crossed in a field
  blade of grass
  simplification is hard work
  it's clear there is no enemy
  but misunderstanding

    it's clear there is no enemy
    but misunderstanding

Has a lot of resonance. Is it a reference to something?

References to Dijkstra "Simplicity is a great virtue but it requires hard work to achieve it and education to appreciate it.", Greek and Indian sand-based mathematics education, and a bit of constructivists and grappling with topology and analysis.

Perfect fit. Did you write this yourself? If no, may I know where it's from?

Wrote it. Used to be a poet. Not my best work heh.

This is great!

Just like JavaScript isn’t real programming, eh?

I’m not perfect at this, but am trying to get better at not telling others what is “real” programming or math or OO or anything.

No, not at all. More taking programming quizzes and never writing a single piece of code on your own ever isn't programming.

That's fair. Are you saying that because the author studied but never practiced, he/she may not have a great insight into what practicing math would be like?

I think I understand what you're saying, and thanks for clarifying.

Yes, that's the point. The problem is that the way math is taught is quite perverse: one can be learning about math, and never get to do it!

"Math isn't for me", "The truth is, math didn’t move me.", etc. - I wish I had a dollar every time I heard that.

Imagine someone saying "Meat isn't for me, it just didn't move me, and I'll never have it again" after having a tofu burger. That's what it sounds like.

That doing you are referring to, how is it related to advancing the state of the art? Does it only happen on the forefront or is rediscovery just fine? If it's the former, then everything up to that point must be a cruel grind to compress millennia of discovery into less than a lifetime. Curriculums usually follow the historical development of their sub-field, just skipping the dead ends. So in theory, at any point in time you could remove the books with the spoilers, lock up the classroom and let the students rediscover on their own. That might confer the excitement, but it would be glacial. So you add a teacher, to nudge the students into the direction of the fastest path. That quickly turns into full-on spoilers and students are left with watching someone reenact discovery at break-neck speed. Is there any place in the world that teaches math in a different way? The idea is sound, faithful execution impossible.

That being said, the author was studying applied mathematics. It's kind of their job to not get excited and instead ask the boring question of "how will I ever use this in real life".

I've found some of the feeling of doing original math without actually discovering new things by 'skipping a level'. That is, either taking an advanced course or trying to understand an article that is 'above your level'. This will force you to study material at your level, but now with a purpose. When you try to understand a proof, it is because you really want to get the intuition of the proof to apply it somewhere else.

>That doing you are referring to, how is it related to advancing the state of the art?

It's related in the following way: if you are advancing the state of the art, you are really doing mathematics. The converse is not true, there are many ways to do math without advancing the state of the art.

>So in theory, at any point in time you could remove the books with the spoilers, lock up the classroom and let the students rediscover on their own. That might confer the excitement, but it would be glacial.

Essentially, yes! What you described is called Inquiry-Based Learning [1][2], and is a very well established (and woefully underutilized) educational practice.

You are correct that on the surface, it appears that IBL is slow. However, I would argue that students learn more in the same timespan.

Why? Simply put, you don't measure a house by the number of bricks. A shed you build on your own is a house, but a pile of bricks accumulated over 10 years is not.

Perhaps a student who understands the Fundamental Theorem of Calculus and the notion of limits has learned more Calculus than the one who has learned 100 ad-hoc symbolic integration techniques. It's a travesty that today's education emphasizes the latter.

>So you add a teacher, to nudge the students into the direction of the fastest path. That quickly turns into full-on spoilers...

...and it really doesn't have to be this way.

>Is there any place in the world that teaches math in a different way?

Yes, the US. In some places. Personally, my spouse designed and taught an IBL university-level math course; it's something that exists and is (very slowly) spreading.

>That being said, the author was studying applied mathematics. It's kind of their job to not get excited and instead ask the boring question of "how will I ever use this in real life".

True, but the applied mathematicians I've seen still have a lot of fun with it. That's my complaint - that the author didn't get to enjoy the kind of math he was doing. I don't think that it's because his math was applied.

[1] https://en.wikipedia.org/wiki/Inquiry-based_learning

[2] https://www.edutopia.org/blog/what-heck-inquiry-based-learni...

I agree. Let me add another item to the author's list:

7. Most problems have no solution. Doing undergraduate mathematics you can always be sure that every question is well posed and has an answer somewhere. Once you start doing research, this is not the case at all. Problems without solutions are so prevalent, that asking the right question becomes the interesting thing, not the finding of the answer.