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The conclusion I've come to is this: The kind of person who is useful in engineering is that person who gets to the bottom of things. They not only find a way

The conclusion I've come to is this: The kind of person who is useful in engineering is that person who gets to the bottom of things. They not only find a way to understand the problem, they also find a robust solution. You come across this kind of thing on HN often, like that article about the guy who opens safes with a device hidden in a marker pen. For 99% of people, that level of effort is incomprehensibly deep. I mean you really have to know a heck of a lot about the subject, and you have to be willing to hurdle quite a lot of issues in the solution as well. Math is the ultimate get-to-the-bottom-of-things subject. That kid who asks you to explain Euler's identity, who wants to know why only 23 people are needed for half of such groups to share a common birthday, and wonders whether there's something deeper than complex numbers when looking at polynomials. The kid who diligently expands his binomials just to see that the formula is actually right. 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. So yes, you don't need advanced math to do a lot of these things, and maybe doing advanced math doesn't even help you do these things. But math is a sort of flag that suggests you can get to the bottom of whatever problem you're faced with.

epigramx 4y ago
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.
trashtester 4y ago
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.
- lordnacho 4y ago
  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.
  - zasdffaa 4y ago
    > 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.
  - trashtester 4y ago
    > 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. )
EdwardDiego 4y ago
> 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?
- dr_dshiv 4y ago
  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.
  - kritiko 4y ago
    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.
    - dr_dshiv 4y ago
      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.
edanm 4y ago
> 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."
- lordnacho 4y ago
  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.
- trashtester 4y ago
  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.
raxxorraxor 4y ago
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.
HeroOfAges 4y ago
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!
- 988747 4y ago
  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.

epigramx 4y ago

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.

trashtester 4y ago

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.

lordnacho 4y ago
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.
- zasdffaa 4y ago
  > 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.
- trashtester 4y ago
  > 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. )

EdwardDiego 4y ago

> 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?

dr_dshiv 4y ago
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.
- kritiko 4y ago
  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.
  - dr_dshiv 4y ago
    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.

edanm 4y ago

> 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."

lordnacho 4y ago
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.
trashtester 4y ago
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.

raxxorraxor 4y ago

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.

HeroOfAges 4y ago

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!

988747 4y ago
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.