Hacker News Reader

> Math is the ultimate get-to-the-bottom-of-things subject.

I love math and am decently good at it.

But I don't agree with this statement. I don't think math is the ultimate get-to-the-bottom-of-things subject, I think it's an ultimate get-to-the-bottom-of-things subject. And I think that makes a big difference.

It's easy to set up a filter which only very intelligent/whatever people pass, and say "yep, this is the correct filter, everyone passing this filter is intelligent". But we might be missing a lot of people we'd get otherwise! What if someone is totally uninterested in math, but she happens to love biology, and spent her whole life "getting-to-the-bottom-of" how bodies work? Or substitute in chemistry, exercise science, psychology, or any other field. People can be totally uninterested in and bad at math, but still geniuses of their respective fields.

Note: I'm not sure you disagree with what I wrote, you yourself wrote " math is a sort of flag that suggests you can get to the bottom of whatever problem you're faced with."

I agree with what you say, you can be smart and be into other subjects, definitely.

The thing about math is you can plumb the depths without much resource. You don't need to check your ideas against the physical world, something that costs time and money that most kids will not have access to.

So it's more like if someone is good at math, that is a good sign that they are the kind of person were talking about, but if they're not that doesn't mean they aren't.

Pure math for its own purpose is for a select few. For most people, math is a set of tools that enables them to better understand certain aspects of reality (or of non-reality, perhaps, in some cases).

In some of those cases, you only require basic math. Some require intermediate math and some relatively advanced math.

But if you want to be a "genius" if either biology, chemistry or psychology, you most likely will need more math than most people learn in high school. Almost every kind of scientific experimentation requires a firm grasp of statistics (way too many papers are written and even published with sub-par stats in many fields). And in most STEM fields, before you can get "genius" level specialization, you will need a relatively broad mastery of the field, and both Biology and Chemisty, there are plenty of things to learn that require at least some entry level collage math.

> Pure math for its own purpose is for a select few.

There is a vast swath of research effort directed to modern geometrical foundations of physics that look pretty much like pure math.

The difference between physics and pure math is not in how it "looks". A lot of physics already "looks" like math, but whether it leads to some testable conclusions.

I would say that String Theory has been more similar to Math than Physics for a while, in that it hasn't really led to many interesting experiments.

As somebody who's cursorily interested in what people would call "Foundation of Mathematics", I don't even think the "ultimate" part is justified.

Why is 1 + 1 = 2? This isn't really a mathematics problem (I know about axiomatic systems, they were invented relatively late in the timeline of mathematical history, invented as a fiction to avoid getting to the bottom of the messier things, i.e. the social context of how mathematics was actually invented and is used). Sidenote: I have written an article (not in English) about how numbers were a linear to log(N) time complexity reduction trick for recording information, and that 1+1=2 presumes a reductionist and somewhat capitalist view of the world...

Anyway, to me, mathematics is a subject that invents abstract concepts and rewards those who are capable of understanding and believing in them. That kind of thinking is probably useful in a lot of areas that have business value. The make-believe part is, I would humbly suggest, quite an important feature too. It really fits nicely into our capitalist framework playing the money game where they'd want you to believe $1+$1=$2 is an axiom, and it's best not to reach out beyond the closed, axiomatic system and ask where those $ comes from or the messy social implications of the "math".

> 1+1=2 presumes a reductionist and somewhat capitalist view of the world

Hopefully, not all Chinese engineers think that...

> mathematics is a subject that invents abstract concepts and rewards

Not according to much of the history of math. Mathematics was borne of practical needs, and it has been extremely useful in science and engineering. (Sure, you can think of the number 2, say, as an abstract concept, but to most people it is a very concrete thing - as is the definite integral, etc.)

Yes, a philosophical journey can be a journey down a rabbit hole, so one should be careful...

> Hopefully, not all Chinese engineers think that...

:) The main point of divergence between "Western Capitalism" and "Socialism with Chinese characteristics" isn't whether it's ok to accumulate capital, but who controls it :) I think both agree that capital is a great idea for building strong nations.

> Not according to much of the history of math. Mathematics was borne of practical needs, and it has been extremely useful in science and engineering. (Sure, you can think of the number 2, say, as an abstract concept, but to most people it is a very concrete thing - as is the definite integral, etc.)

I agree, and... that's kind of my point. I intended to say modern mathematics is a subject that invents abstract concepts... Historically numbers were invented to solve a practical need, and then some hobbyists came and had too much fun with it, ending up with funny things like Peano and ZFC. I've seen too many people trained in mathematics that forgot the social-historical context came first, and jump straight to the newly invented axiomatic systems to explain things as basic as 1+1=2, which to me doesn't fit well with the "ultimate get-to-the-bottom-of-things" description.

I mean, to cite a specific example, while the exercise is probably entertaining to some, anyone seriously believing 2+2=4 requires thousands of lines of "proof" is probably not very good at getting to the bottom of things in a general sense. (ref: http://us.metamath.org/mpegif/mmset.html#trivia )

I'm... not sure I'm following most of what you're saying (in this and your other comment).

> Why is 1 + 1 = 2? This isn't really a mathematics problem

Umm, so first of all, depending on what you mean, "why is 1+1=2" is definitely a maths problem. It's exactly in the foundations of math, as you said. Or rather, it makes more sense to think of it as "what is our system for deriving true statements, that lets us do things like 1+1=2?".

> (I know about axiomatic systems, they were invented relatively late in the timeline of mathematical history, invented as a fiction to avoid getting to the bottom of the messier things, i.e. the social context of how mathematics was actually invented and is used)

That's not really why axiomatic systems were developed. It was part of trying to give better foundations to math, because mathematicians ran into a lot of actual problems, mostly beginning with calculus. They managed to reach paradoxes, weird conclusions, etc, and wanted to start being more rigorous in how they treated everything in math, which has led to lots of flourishing in maths, so it was probably a good idea.

> [...] 1+1=2 presumes a reductionist and somewhat capitalist view of the world...

This I don't understand at all. What does this have to do with capitalism? The rest of your comment is equally weird to me. Mathematicians like to invent concepts and play with them, sure, I'm fine with that description, but how does this have anything to do with the capitalist system?

> [From your other comment] I mean, to cite a specific example, while the exercise is probably entertaining to some, anyone seriously believing 2+2=4 requires thousands of lines of "proof" is probably not very good at getting to the bottom of things in a general sense. (ref: http://us.metamath.org/mpegif/mmset.html#trivia )

I think you're confusing two things here. For the most part, most mathematicians don't need to bother with "foundations" questions, or reach a point where they're proving 1+1=2. It can be helpful to "peak under the hood" of various definitions to understand how they are defined rigorously, and more importantly, it can be fun. But most mathematicians use numbers as a black box, never needing the detailed definition of how numbers work.

As for taking 2000 lines to prove this, that's probably only when you want to rigorously use a computer proof or something from first first principles. Given a set of axioms (e.g. Peano), I'm pretty sure it's easy to prove 1+1=2 in only a few steps. And again, this is mostly of interest to people who actually want to work on this kind of foundational thing, not the average mathematician I wouldn't think.

Yep. But "math" at various levels of depth is also helpful in guiding and inspiring thought.

For instance, without stats, decisioning processes quickly become "gut feelings". But a little stat can be misleading, as seen in problems like p-hacking. To use statistical thinking generically, not by following some recipe, you need to understand calculus.

Algerbra (linear- and abstract) similarly help structuring logical problems. Linear algebra builds an intuition for organizing problems into dimensions, and and can be useful for semmingly unrelated efforts, such as organizing work (by finding activities that are orthogonal or linearly dependent), and more related effort, such as vectorizing code.

Abstract algebra is useful for thinking about symmetries and for transforming concepts into each other. It creates bridges between areas of mathmatic, like complex numbers and matrices. (Things like Euler's identity becomes easy to grasp if considering the unit circly on the omplex plane as a group, and this in turn remove the "magic" from complex numbers.)

In computer science, algebra can provide intuitions for seemingly unrelated tasks, such as refactoring. Basically, tranform the "algebraic structure" of your code to a simpler one.

Precisely how much of this kind of benefit people will get from knowing math probably depends on their talent for logical thinking. At the lower end, anything beyond basic arithmetic may not provide much value.

But the close someone come to the gifted side of the spectrum, the more secondary benefits may be gained from knowing more math. A gifted person with the same education as an average person may be twice as productive, buth add some math, and that becomes 4x.

Now, if the hypothesis above is true, math would turn a linear curve into a parabolic curve. Math/stats could both be used to test that, but even just coming up with the hypothesis requires some inclination to think mathmatically, beyond mere arithmetic.

And, obviously, if the different between average and good math education is the same as the difference between the area under a parabolic vs linear curve, someone who learned calculus would instantly know how massive the economic benefit might be.

For sheer beauty, study math.

But economically, I think it's much like any advanced level STEM course. In such courses you come across a huge range of things, but the main benefit is that you now know that X is a thing that people have investigated. If you know that X is a field of some depth, that completely changes your own course of action.

The thing is I'm not sure the actual meat is necessary. Yes it sounds criminal, but what you just indexed everything and came back to learn it properly when needed? When I look at HN type people that seems to be the actual way things happen, rather than learn X -> apply X. It's more like people kinda sorta remember there was something that sounded like it might be useful in this other area, then they circled back, read up on the details, and applied it.

> For sheer beauty, study math.

I wish. I wonder what proportion of people reading your post don't see or feel the beauty in it. I think you need a special and probably quite rare 'sixth sense' to perceive it.

> But economically, I think it's much like any advanced level STEM course.

I agree. Almost all of those would be some kind of applied mathematics, though.

I was not arguing above for the importance of studying pure math exclusively. For most people, math is a force multiplier when combined with some other knowledge/skill.

> It's more like people kinda sorta remember there was something that sounded like it might be useful in this other area, then they circled back, read up on the details, and applied it.

Absolutely. By having a good and broad generic understanding of a field, you can circle back to it. Understanding something in a more abstract way makes that easier, as it promotes a deeper understanding rather than just memorizing facts.

For instance, if you think of euler's identity in terms of a rotation group it becomes very easy to prove it (at least for someone with a mostly visual understanding of math an physics). But just as important, the rotation group gives a much better understanding of many cases where complex numbers are useful, ranging from physics to electronics to electrical engineering. Quantum wave functions, capacitors/inductors and electrical motors all have this kind of rotational symmetry in common.

So while some colleges may teach quantum mechanics, electronics or electrical engineering with a shallower math curriculum as the base, some abstract algebra can help gain deeper understanding of each, compared to memorizing the formulaes without fully understanding them.

Also, even if many with a STEM degree often end up in a different field than what they studied in collage, the ability to apply math and statistics can often be useful way outside of the field studied initially.

(At least, that's my experience as a former physicist. While I rarely need quantum mechanics, the ability to manipulate probability density function, find and describe symmetries, and have developed a quantitative intuition that spans a 50 orders of magnitude remains useful, both professionally and even as a citizen. )

I feel like that's my kind of approach, but I guess you also need another kind of person to just keep drilling away at some weird little math problem that has absolutely no practical use for decades until it suddenly does.

You might be describing forms of OCD too. Sometimes the "engineering-smart" thing to do is to not look into it. There's only so much time.

I hadn't heard of the "Birthday Paradox" before you mentioned it your comment, but your description of it isn't quite right. It's not that half the people in a group of 23 will share a birthday, it's that there is a 50-50 chance that two people in the group will have the same birthday. Still, very interesting and I learned something today!

That's how i understood GP's "half of such groups" statement - as a 50-50 chance (half of all groups, not half of the people in the group). But maybe I was biased because I heard of "Birthday Paradox" before.

True, but higher level math also isn't higher level math the same way that you don't want to get heart surgery by your ophthalmologist. Engineering behaves the same too. There are luminaries for everything, but the vast majority of engineering feats were developed by specialists.

> That kid is also the one who will figure out why your rocket is malfunctioning, how your car will drive itself, how to fix carbon dioxide, and every other problem that we call a technical problem.

That kid is also the one who will enter academia and spend the rest of their life taking part in intradepartmental warfare as they desperately strive for tenure.

Equally possible, right?

Hahaha, oh wow, that’s so funny. <depressed af>

Actually, I think it is a different personality type. The kid who does well at math because rules are provided clearly and success is guaranteed somehow is different from the kid who wants to get to the bottom of it from a curiosity perspective. Both are driven partially by ego but one is more fear driven and the other more joy driven.

Of course, years of academic training might drive the joy out of anyone, but many end up in academics because they just wanted to keep getting the positive validation through clearish rules. These people struggle in entrepreneurship but can do great with corporate advancement.

I think it's a mistake to sort people into such clearly delineated personas, especially if you're going to make judgments about them being driven by "fear" or "joy".

Somebody could be very intrinsically driven by the pursuit of truth, but still appreciate the importance of politics in increasing earnings and status and thus outwardly appear to your 2nd type.

Those same people might "struggle in entrepreneurship" with regards to inventing wholly new ideas, but be good at fundraising or copying. (Let's make it an NFT...) Just something to consider.

Well, I agree. What is the use? Well, inaccurately simple models can sometimes help make complex phenomena manageable. If we roughly assume that there are real variations in people’s motivation (eg variations in curiosity, status-seeking, fear, joy) we can see whether certain variations predict certain kinds of success. If so, we can think about how to support these educationally. If we find that curiosity-based motivation in math is highly predictive of certain kinds of success, we might want to see how to cultivate that motivation through pedagogy.

But there's so few of those academic jobs. Plenty of kids have the curiosity but won't want to take part in the tenure race.

Emmanuel Derman wrote about the mass of PhDs in the 1980s who went into finance. This will just keep happening.

Plus you don't need to get a PhD at all to be one of those guys who solves problems.

>Equally possible, right?

Not equally, since more PhDs end up in industry than academia.

Also, those in academia routinely consult and solve these problems for industry, they make startups, they solve problems that further advance industry. Many bounce between careers on both sides.

It's silly to think those in academia don't work on and solve real world problems and only spend time fighting politics. Those research grants are for producing research, and those corporate grants are for producing items useful to those companies.