Ask HN: High-level math, useful?
9 comments
Some higher-level math will be important, other parts will not be. The parts that will be more useful are more on the analysis side (real analysis, complex analysis, functional analysis, convex analysis, Fourier analysis, probability theory). These are higher math and are very applicable, or are prerequisites to understanding the applied stuff (convex optimization, dynamical systems, control, ...).
It helps to know what a topology is, but not much more, and you would learn enough "on the way" in learning analysis properly. It helps to know what groups are, because they do show up in practical things, but you don't really need to know full-up "group theory". (They show up because they capture the idea of symmetries, and it is useful in certain practical situations to talk about something being symmetric with respect to various transformations, e.g. under permutations or rotations or whatever. But in this case you don't tend to do much analysis actually using group theory beyond this.) A whole course on abstract algebra is not necessary unless you're interested. It may help in some indirect way of "helping you think better", it may not.
See, say, http://junction.stanford.edu/~lall/engr207c/ as an example of an EE course that does a fair amount of math.
(Also, above, I don't mean 'applicable' in the very indirect sense of "helping you think better" -- I mean people use it to do real stuff. Whether you want to do that stuff is another story -- there are certainly good things in EE/CS that don't require this kind of math.)
It helps to know what a topology is, but not much more, and you would learn enough "on the way" in learning analysis properly. It helps to know what groups are, because they do show up in practical things, but you don't really need to know full-up "group theory". (They show up because they capture the idea of symmetries, and it is useful in certain practical situations to talk about something being symmetric with respect to various transformations, e.g. under permutations or rotations or whatever. But in this case you don't tend to do much analysis actually using group theory beyond this.) A whole course on abstract algebra is not necessary unless you're interested. It may help in some indirect way of "helping you think better", it may not.
See, say, http://junction.stanford.edu/~lall/engr207c/ as an example of an EE course that does a fair amount of math.
(Also, above, I don't mean 'applicable' in the very indirect sense of "helping you think better" -- I mean people use it to do real stuff. Whether you want to do that stuff is another story -- there are certainly good things in EE/CS that don't require this kind of math.)
Mathematics and Statistics are fundamental tools of Science and Engineering, tools you need in your kit. Linear algebra, analysis, statistics, discrete math, complex analysis and optimization are about the minimum. Abstract algebra, topology, differential geometry, combinatorics, number theory, and so forth all may prove useful and enriching. The problem is that you never know what will be useful.
Years ago, abstract number theory was seen as pure mathematics unsullied by practical applications. And then along came cryptography.
Years ago, abstract number theory was seen as pure mathematics unsullied by practical applications. And then along came cryptography.
I'm only guessing here, not being an engineer or a mathematician, but... wouldn't that be rather dependent on what kind of engineering you're getting into?
Yes, poorly defined question. I guess electrical engineering and CS has a lot more math and then your specialization matters obv.
As an EE student nearing a bachelor's, I haven't seen all that much use for more abstract mathematical topics in my classes. I would say it is much more important to be very comfortable with the more basic, concrete topics that will show up all the time: algebra, ODEs and PDEs, (vector) calculus, Fourier/Laplace transforms, linear algebra, probability. An EE specializing in the physics side of things may wind up using abstract algebra or other higher math, but that is a pretty small percentage of engineers.
My feeling is that topology can help your abstract thinking as it's likely to be in engineering, but abstract algebra less so. Having said that, if you're not actually inclined to take these subjects, you'll probably end up struggling and not caring.
Topology, and the insights it brings, can make some of the analysis and linear algebra make more sense - there are unifying concepts and structures. Abstract algebra is more about symmetries and actions, and while also useful, possibly don't give the same sorts of insights.
Just my $0.02.
Topology, and the insights it brings, can make some of the analysis and linear algebra make more sense - there are unifying concepts and structures. Abstract algebra is more about symmetries and actions, and while also useful, possibly don't give the same sorts of insights.
Just my $0.02.
Doesn't topology past a certain basic level rely on algebra pretty heavily?
There are two flavors of tolopogy: Point/Set and Algebraic. Point/Set topology is really useful in understanding how things can go wrong, and what's true for sure in analysis. Algebraic topology is less obviously useful for things like engineering.
It depends on what particular field within electrical engineering you're going to pursue. My specialty is signal processing and the general feeling in our field is that more math is always better. I haven't used topology or abstract algebra on the job. A solid understanding of the basics will take you pretty far. I'd choose depth in those fields before I worried about breadth.
I think studying higher maths make for a good problem solver, but you are missing out on practicality and people skills.
For instance, in my life, I went all the way up to PhD level courses in mathematics. This translated into me being able to go into my start-up and solve all sorts of problems. I was also solving problems that were already solved because I was ignorant of what was considered good programming practice.
Like source control. Because I could, I wrote my own source control. Like an IDE. Because I could, I wrote my own.
I attribute my fantastic problem solving skills to my mathematics back-ground. After all, when dealing with proofs in abstract algebra and the creative process of making up strange sets that exhibit strange behaviors... Most things seem trivial at some level.
The fundamental problem of mathematics is that you spend a lot of time solving fake or stupid problems to build up the ability to solve real problems. If you are going to go into the cutting edge of research, then you will need those skills. Otherwise, you will be very good at solving artificial problems.
I want to say that it will make you a better engineer, but I'm very biased. I feel like it has helped me compared to my peers in terms of raw engineering power.
For instance, in my life, I went all the way up to PhD level courses in mathematics. This translated into me being able to go into my start-up and solve all sorts of problems. I was also solving problems that were already solved because I was ignorant of what was considered good programming practice.
Like source control. Because I could, I wrote my own source control. Like an IDE. Because I could, I wrote my own.
I attribute my fantastic problem solving skills to my mathematics back-ground. After all, when dealing with proofs in abstract algebra and the creative process of making up strange sets that exhibit strange behaviors... Most things seem trivial at some level.
The fundamental problem of mathematics is that you spend a lot of time solving fake or stupid problems to build up the ability to solve real problems. If you are going to go into the cutting edge of research, then you will need those skills. Otherwise, you will be very good at solving artificial problems.
I want to say that it will make you a better engineer, but I'm very biased. I feel like it has helped me compared to my peers in terms of raw engineering power.
Studying higher math does not automtically mean you miss out on practicality and people skills. Perhaps in your case it did, but a good engineer will be dealing with some of that.
It's important to have a balance - there are infinitely many things that you could advise would be useful, and time is limited. When solving problems it's important not to dive in immediately, but also to ask "What of this has already been done?"
But even then, solving some of the problem first gives an appreciation of what has been done, and often makes you better understand the strengths and limitations of existing solutions. Your example of an IDE is one where re-doing it from scratch is unlikely to give a better result, but Linus re-did the source control idea, and did it better.
It's important to have a balance - there are infinitely many things that you could advise would be useful, and time is limited. When solving problems it's important not to dive in immediately, but also to ask "What of this has already been done?"
But even then, solving some of the problem first gives an appreciation of what has been done, and often makes you better understand the strengths and limitations of existing solutions. Your example of an IDE is one where re-doing it from scratch is unlikely to give a better result, but Linus re-did the source control idea, and did it better.
That's very true.
Part of the culture of learning higher math however rewards/tolerates esoteric behavior where practicality is just not valued.
I think it is all about related rates. If you are studying abstract algebra now hard-core, then you are missing out on doing some cool Kinect hacks now. Or, you are missing out on chatting up the girls over at the pub.
I found math very ... addicting, and I wish I had learned balance sooner. Instead, I thought it was a lot of fun to sit down every evening and grind on problems from "Berkeley Problems in Mathematics"
Part of the culture of learning higher math however rewards/tolerates esoteric behavior where practicality is just not valued.
I think it is all about related rates. If you are studying abstract algebra now hard-core, then you are missing out on doing some cool Kinect hacks now. Or, you are missing out on chatting up the girls over at the pub.
I found math very ... addicting, and I wish I had learned balance sooner. Instead, I thought it was a lot of fun to sit down every evening and grind on problems from "Berkeley Problems in Mathematics"
In most mathematicians and theoretical physicist I know this tolerance for "esoteric behavior" probably caused exactly opposite effect: they are probably too much sociable and cool.
Was it the actual algebra you learned that helped you code these things? Or just the mental discipline of sitting down until the problem is solved?
If you want to learn it, learn it. There is no other justification required. If you enjoy learning it, then it will benefit you.
I don't have as much experience as others here (~2 years) but I can tell you that it definitely changes the way you think, not just about engineering problems but life situations too, and the change is not something you might predict from the sub-topic you're learning about. Math is really just the art of thinking precisely (lots of inventing beautiful abstractions and interesting examples).
Things like topology and abstract algebra, does it teach you anything that is actually applicable in electrical engineering? Does it evolve your abstract thinking skills which could make you a better engineer?