Ask HN: If I work with applied mathematics should I still study analysis? | Hacker News Reader

Ask HN: If I work with applied mathematics should I still study analysis?

26 pointsmeri_dian23 comments

Is studying analysis helpful for those people who aren't working on proofs every day, i.e. engineers, people in finance, etc, who still need to apply mathematical techniques?

The speculative answer if I had to guess is yes, because understanding the theoretical basis of something can't hurt, but I'd like to get some answers from people who actually work in applied quantitative fields. Has your study of analysis helped you, either directly or indirectly?

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The closest thing to feeling like I had a genuine superpower came after I studied a bunch of pure mathematics – the (richly interconnected, actually conceptually easier to manipulate after you've grown accustomed to them) building blocks of applied mathematics – and was then able to dazzle people who had only been exposed to applied mathematics with genuinely new, and often beautiful, perspectives on things they thought they already understood the whole story of. Pure mathematics, like analysis and algebra, makes it all immensely more rich and personal.

I feel the same way. Intellectually it seems like it's been all downhill since leaving grad school :).

Did you complete grad school all the way to a doctorate? I'm in grad school right now weighing the decisions to leave a year after a master's degree.

No, I left a couple of years after my MS degree, while I was starting to do work on my dissertation. While I do think that CS isn't as intellectually satisfying as math, it's definitely more lucrative, so I have zero regrets.

I can give you one data point. I have worked on computer vision (implementing, training, and running convolutional neural nets), and have also applied machine learning methods on natural language processing. I have never taken analysis and I do not feel like I am missing out. In most applied machine learning fields nowadays, the "math" of the it is used more as a gateway into intuition, not really anything rigorous. For example people in machine learning research nowadays use the word "manifold" a lot, but I would wager most do not have a rigorous background in topology. In applied machine learning, the details of your problem domain matters more than understanding the derivation of your tools, this is true even in computer vision, which used to be a lot more rigorous before convolutional neural nets. However, if you are looking to take a rigorous course in probability, I would recommend analysis since it is the foundation of probability, you cannot derive a lot of basic theorems in probability without it. I took a rigorous class in probability once without analysis, and the gaps in knowledge was Clear everyday. If you are determined to tackle theoretical course just as a confidence builder, then I would recommend linear algebra since it has far wider applications. Linear algebra is also a must if you want to understand many optimization tools used in machine learning.

As for finance, I have a no direct knowledge here, but I do know when companies like D.E. Shaw recruit here at Penn for quantitative analysts, a strong math background (including analysis) is a must. Whether they use it on a day to day basis is unclear. However, taking one analysis class will not be sufficient for such firms, they are looking for many years of formal schooling and in some cases, publications.

Subspaces, Point-Set Topology, Compactness, Matrix Spaces, Limits, Linear Dual Spaces, Hilbert Spaces, Banach Spaces, Subgradients, Gâteaux derivative, Connected Sets, Path Connected Sets, Cauchy Sequences, Power Series, Taylor's Theorem, Properties of Monotonic Functions, Linearity, Hessians, Jacobians, Stone-Weirstrass, Heine-Borel, Fixed Point Theorems, Contraction Principle, Lebesgue Measure, Measurable Functions, L2, Fourier Series....

As an applied mathematician with additional degrees in CS and Engineering, I have used them all. More importantly, the knowledge of those topics helped me understand many algorithms and they lead to a deeper understanding of probability (Lebesgue Measure).

do you mentor?

It's ridiculously easy to apply a method which is not applicable if you are not at least a little aware of the theory behind it. You will get wrong results and you will never know why. The wrong results might never be found out because the impact is so complicated that you cannot even point out the source. I have sadly seen this happen too often in various places. But because our other systems are robust enough, this issue (even though it affects things) is not really heeded to until something really catastrophic happens (like 2008 crash).

Studying analysis gives you more tools to model changing systems. The more tools you have, the easier it is to abstract things. That's really all applied mathematics is, abstracting stuff to things you can do mathematical operations on. When counting sheep, all you're really do is abstracting the sheep as units and then performing simple arithmetic.

So why should you have all these tools to abstract stuff? Because you want to do something new and innovative. If this really isn't that interesting to you then don't bother but every time I find a new way to model something and try it out (often with unsuccessful result, though) I get giddy with joy. The few times I come up with something new and people actually value what I've contributed with it feels like heaven, however small these contributions end up being.

Yea - you don't have to go immensely deep into Analysis, but having some experience in it will give you an understanding of what makes the algorithms "tick" and when they break-down.

This is especially true for things that involve numerical analysis - approximation techniques/diff eq's/etc.

And just for clarity - while any class on Analysis is mostly proofs, it's not really about the study of proofing techniques. It will use proofs as a vehicle to discuss _analytic_ properties of functions (how limits behave at infinity, etc).

tl;dr yes.

A lot of applied mathematics (physics, quantitative finance, some machine learning) rely a lot on analysis. In a more general light, I would advice against skipping bricks of something that big and that close to your direct field because "it is useless".

I can probably add the perspective of a physicist training where analysis is an integral part of the mathematics education (typically taking ~1/3 of the time of a bachelor degree (bachelor~=3yrs)). In addition, I (partially) work (study) in the applied mathematics nowadays (CFD/PDEs, numerical methods) and without doubt, analysis tools are still everyday ingredients to understand what's going on. So definetly: Yes

Hard to say, we don't know what your calc, diff EQ, LA, probability/stats etc classes were like. You could self study Pugh, Abbott or Spivak. At a bare minimum you can read Lara Alcock/s little red book on analysis. Or you might unfortunately be in the position that you don't quite have the support group, focus or prep for Pugh or Abbott's books.

The other question is, what are you giving up to study real analysis? If you don't have great software engineering skills, along with related skills (linux/network admin, database design/tuning) there's a good argument that you would be better off developing those skills instead.

It's helpful if you are reading literature actively. If you are interested in some new technique and want to understand the theoretical aspects of it, then knowing analysis will be useful in deciphering the presented proofs. It may also be helpful when implementing said methods as you will better be able to formulate sanity checks and such for your implementation.

If you want to go deep on stochastic processes / probability theory - useful in finance - then you'll need it for sure. Otherwise probably not. Whether "needing" to know it is a good reason for not learning it is another question. If you're asking the question then you should probably just learn at least a little for general interest.

Worth taking statistical analysis at the least. Depends on what you want to do. "applied quantitative fields" is a very wide brush to use.

Can anyone recommend analysis textbooks suitable for self study?

Calculus, by Spivak (an analysis textbook despite the name). I found this book clear, with good exercises, and it starts at the very beginning. I think the most difficult exercises are in the early chapters, so stick with it!

I wouldn't recommend Spivak for self-study and a first exposure to analysis. It's known to be notoriously difficult even for good students.

I learned from this Dover book[1]; I think it's pretty good. From there you might move up to Baby Rudin[2], but it might have a lot of typos, or big gaps in the exposition that are taken to be obvious but require several steps to fill in, since that was certainly the case for Papa Rudin[3].

[1]: https://www.amazon.com/Introduction-Analysis-Dover-Books-Mat... [2]: https://www.amazon.com/Principles-Mathematical-Analysis-Inte... [3]: https://www.amazon.com/Real-Complex-Analysis-Higher-Mathemat...

Without knowing your background, Pugh's and Abbott's are usually good choices as are Schramm's Dover, and while not really analysis texts, don't forget the classic fundamentals/foundations books by Courant/Robbins What is Mathematics, Ian Stewart/D Tall Foundations of Math, Felix Klein etc

Here's lots of reviews: https://www.maa.org/tags/real-analysis

https://math.stackexchange.com/questions/tagged/real-analysi...

https://www.reddit.com/r/math/wiki/faq#wiki_what_are_some_go...

Try Terence Tao's Analysis I and II.

It really depends. For learning and using ML/DL it hasn't helped me much, the bottleneck is stats and linear algebra for that field. Even if it helps, is it the most valuable and broadly useful tool you can add to your current toolkit as of right now? Likely not. Learning to write proofs helped in other ways, the content wasn't actually terribly important.

(1) I don't think this is necessarily the right place to ask this. HN is not particularly known for mathematics, it's better to ask at a math focused site like https://math.stackexchange.com/

(2) My generic answer to the question is - NO. You can be a good quant, engineer, etc. without having the knowledge of how to prove the theorems in a real analysis undergraduate course. However, knowing the proofs may be helpful if you decide you want to do write theoretical research papers in a particular area, even just for the sake of "mathematical maturity".

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