
Ask HN: If I work with applied mathematics should I still study analysis? - meri_dian
Is studying analysis helpful for those people who aren&#x27;t working on proofs every day, i.e. engineers, people in finance, etc, who still need to apply mathematical techniques?<p>The speculative answer if I had to guess is yes, because understanding the theoretical basis of something can&#x27;t hurt, but I&#x27;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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tw1010
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.

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

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aisync
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.

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monaghanboy
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.

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xiaolingxiao
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.

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irchans
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).

~~~
hulkisdumb
do you mentor?

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proginthebox
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).

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pimmen
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.

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mattxxx
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).

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frantzmiccoli
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".

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ktpsns
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

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gtani
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.

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tnecniv
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.

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nicodjimenez
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.

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grigjd3
Worth taking statistical analysis at the least. Depends on what you want to
do. "applied quantitative fields" is a very wide brush to use.

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bronxbomber92
Can anyone recommend analysis textbooks suitable for self study?

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gtani
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://www.maa.org/tags/real-analysis)

[https://math.stackexchange.com/questions/tagged/real-
analysi...](https://math.stackexchange.com/questions/tagged/real-
analysis+book-recommendation?sort=votes&pageSize=15)

[https://www.reddit.com/r/math/wiki/faq#wiki_what_are_some_go...](https://www.reddit.com/r/math/wiki/faq#wiki_what_are_some_good_books_on_topic_x.3F)

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myaso
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.

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rweba
(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/](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".

