[jdonaldson]

Eigenvectors and Eigenvalues show up everywhere, although sometimes it's in the form of an iterative estimate (PageRank is basically the power method estimation of the first eigenvector of a connected graph of web pages).

They're in the same class as logarithms and Fourier transforms IMHO. You won't need to calculate them by hand, but you should know what they do and why they're important.

[madengr]

I use logarithms and FFTs daily, but got a C in Linear Algebra, after getting A in all other EE/Math/Phys courses. It kicked my ass once we got to proofs.

[xevb3k]

Interesting, perhaps that’s it... I’ve used FFTs a few times, but even FFTs have never been make or break in terms of getting a pipeline or analysis working well.

[cowboysauce]

The Fourier transform is actually linked with linear algebra. You can think of it as taking a vector in an infinite dimensional Hilbert space (your signal) and decomposing it into its components (the amplitudes of the frequencies).

[stephencanon]

The Fourier transform as used in practice in signal processing is ~always discrete, hence finite-dimensional. It's literally "just" a rotation with an especially nice decomposition that allows efficient multiplication.

[danharaj]

A particular fft in practice may be discrete but the relation between fft's of different resolutions of the "same" signal hints at the infinite structure which bundles up all the finite subspaces into one conceptual object.

[WhompingWindows]

Why should you know what they do and why they're important? How does that practically change my R code?

[angersock]

Because, say, knowing about Fourier transforms can help you write more efficient filtering or open up new ways to view your data--perhaps there's a really interesting behavior in the frequency domain you'd miss otherwise.

If you just want to be a statistical script kiddie you do you. :)

[ItsMe000001]

When working with real world data almost everything is more important than being able to use the most abstract methods "to extract the last bit of data". It's often extremely fuzzy to begin with, the collection process to what it represents, for me, while I love math and see it as the "magical language" in a magical world, I find common sense and a certain kind of work ethics go soooo much further than any math Ph.D. I (with a CS degree) wanted to start another study (over a decade later, thirsting for new theory and new knowledge) and chose math - but that was around the same time edX and Coursera etc. got up and running with loads of courses, and I ended up ditching the additional math degree for loads and loads of courses in fields like medicine and biology, deciding to go for breadth instead. Of course, that's purely personal and even as an anecdote not worth much to anyone. It's just that I too got disillusioned with (higher and higher) math as an helpful tool in practical life. Of course I still see the benefits in many fields, but I think a surprisingly low number of very good specialists may be all we need. The rest of us can just ask them when we actually do need something.

Right now I'm taking a "math. modeling" course. Still, the only use case I ever found was... other courses! I already modeled a little bit in a biology course. Sure, in real life I could model this or that, but the truth is that a very rough estimate guided by experience and "feeling" has always been enough. There are too many variables that cannot be accurately measured, so going for a nice model is kind of useless.

For example, I was just asked today about the performance of the crypto-hash-connected data storage and exchange library I wrote. Now that sounds like something I could model! Only experience tells me that's useless. The only worthwhile answer is to set up a concrete scenario, with a concrete app using it, concrete network and concrete systems, and test it. Could be anything from smartphones to well-connected servers. Sure I could create a sophisticated model and simulation - and it would be useless.

Maybe I'm just a bit, or more than just a bit, disappointed that all the considerable amount of math I learned in my life didn't seem to be of nearly as much use as I would have hoped. I'm also frustrated each time such a topic comes up and everyone is so excited about how great it is, and I always feel like I'm missing something despite trying hard, like the color blind guy looking at paintings. I mean the usefulness to me, not understanding it.

[pphysch]

As a younger person (finishing up a Math BS) this resonates with my perspective.

IMHO, it comes down to individual beliefs about mathematical realism. Is there anything inherently real about math, or is it just a man-made, arbitrary set of cognitive tools? Is it valid to presume the existence of a Grand Mathematical Framework that can solve any problem a priori? Or, is every problem unique and independent of mathematical developments?

From the little I've read about Math history, it seems pretty clear that the Problems came first, and the Mathematics followed. Infintesimal calculus, game theory, etc. were mathematical ideas developed primarily to solve real problems. Then 20th century formalism came along and rebranded much of mathematics under a "clean" framework, while giving little attention to the human environment in which much of it was developed.

To me, it is a great shame that abstract mathematical concepts are made further abstract (e.g. in math education) by distancing them from their human roots. Instead of forcing oneself to understand this mathematical "new testament", I think it's far more productive to adopt this sort of irreverent attitude towards math as you describe.

Einstein:

>"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."

[hyperpallium]

Reminds me of economic rationalism: model makes sense, and conclusions seem logical and compelling - but you can't tell if the model represents reality or even if crucial factors have been omitted.

[aje403]

Perhaps colleges can start adding an "applied" math major with a focus on subjects that more directly involve the human environment more directly to alleviate the problem you're describing?

[akvadrako]

Applied math is definitely a major. Do you mean something else?

[hyperpallium]

I'm doing some fluid simulation (CFD), and the actual code for finite differences is simple. But the analysis (of stability etc) is more mathy, and I don't feel confident reading other needed papers, because they are couched in math.

I mean, I can and have coded it, but can't be sure how it how it will behave in all situations.

So I can understand these papers, I'm going back to study math properly. I'm not fully convinced it's really needed (though how could I tell?), but I'm fully convinced it's needed to understand the papers.

Math is the latin of CFD.

[bigger_cheese]

I work in Engineering and feel somewhat similar about CFD. There's always some element of doubt lurking in the back of my mind "is it really correct in all circumstances".

The most useful info I learnt at university were a couple of equations: Bernoulli's (for general observation about expected pressure drop), Ergun's (for flow through packed beds) and the general laws of thermodynamics.

Those are mostly enough to be able to sketch out an intuitive 'guess' about expected behavior in a large range of systems and the underlying math is not particularly demanding.

[hyperpallium]

Have you seen the Method of Manufactured Solutions (MMS)?

You guess ("manufacture") a solution to the PDE. Then you plug it into the PDE. It won't be correct, so you just add source terms to make it correct. You now have an analytical solution for this PDE with those source terms. You can now run your simulation (with those source terms) and compare it with the "correct" solution.

Unfortunately, you'll introduce discretization error, because your delta x and delta t aren't infinitessimal. But using the order of accuracy of your discretization method, you know how the error should change with delta x and delta t. So you graph error against delta x, on log-log, and see if the curve matches the order of accuracy. Apparently, it shows up even minor bugs in your code really well.

This approach scares me a little, because how do I know my math is correct? I need better appreciation of how the order of accuracy interacts over time as well as space; the behaviour of error summaries; and how log-log plotting works.

[xevb3k]

Really interesting, thanks! Yes, it’s certainly been my experience that adding more domain knowledge has helped me a huge amount.

[Jack000]

well you don't need this stuff until you do - some things still need definite, analytical performance guarantees.

I'd be pretty nervous riding an airplane that didn't use modern control theory, or going over a bridge that didn't use FEA - or an self-driving car that ran on a raspberry pi instead of a RTOS...

[ItsMe000001]

Why do you think in absolutes? I'm tempted to cite a certain Dilbert... but that would make me appear incredibly rude, which I really don't want to be. Of course it is easy to find examples where it's needed, and it is easy to see so too. So? Did I propose at any point "Nobody needs higher math for anything" or something similar? I don't see a need to argue about an argument never made.