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How do I get rid of the destructive mindset that Applied and Statistics are "lesser" than Pure Math

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Might i ask whether you've actually taking classes in (mathematical) statistics? Apart from the 'introductory statistics' class i feel like you quickly see how much mathematical rigour statistics has to offer.

I have also often heard this stereotype of statistics, but once i actually took some statistics classes i have seen that there is a good bit of math involved, usually functional analysis, measure theory, topology.
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The same thing happens with experimental and theoretical physics. I myself looked down on experimental physics, and I'm not proud of it... it's a very ignorant mindset.

What changed things for me was developing a better understanding of the field, and actually working in the field. I now see experimental and theoretical physics as two sides of the same coin, with very unique skill sets. It is also very helpful to find a research advisor or some sort of mentor who invokes a deep sense of respect from you. That is what happened to me.

If you have this feeling in the back of your mind that things are "lesser", it may be helpful to understand and admire some of the innovative work that is being done on the forefront of the field. It may invoke that humbling feeling of "Jeez, I dont know if I would have ever thought of that". This happened to me in the context of learning about a VERY creative measurement method that utilized spin sensitive materials to perform selective measurement.
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I've found myself having the opposite problem; the theory and proofs came easily, but actually doing things that are useful is pretty hard. I don't mean this in the self-deprecating way, but rather that it can actually be really hard to actually think of ways to apply knowledge to solve useful and interesting problems.
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I actually have this same problem, any time I learn math for computer science I always feel like it's inferior and I get insecure that I don't know math
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Do some science. Try to model some dynamical systems and you'll quickly gain an appreciation for numerical methods for solving differential equations. Try and fit some models to some data, and you'll quickly appreciate all the statistical tools that allow you to unravel meaningful insights.

Transitioning from pure mathematics to astrophysics was a real struggle at first. But once I realised I could just leave the strict mathematical rigour behind and start flying by the seat of my pants, I started to really appreciate applied maths and statistics.
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As someone who has done lots of both pure and applied math, it just comes with learning about what people actually care about and what makes an impact in society.

Hardly anyone actually cares about pure math, and nobody cares about the obscure theorems that you know.

The notion that pure math is more important than applied math just comes from the toxic aspects of academia and is ultimately rooted in insecurity.
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One of the things which breaks that mindset for me, is thinking about and trying to really appreciate the 'multiple constructibility' of mathematics. In other words, that our ordinary arithmetic of naturals, and countless other nonsense-to-us-but-internally-consistent arithmetics besides, and our set theories, ZFC and ZF~C and all the others, all of them... they all have unfathomably many equivalent, alternative constructions, in all kinds of symbolic and computational languages. And insofar as they can all construct each other, these equivalencies are infinitely nested.

That is to say, Von Neumann numbers are no more or less real than Peano numbers, are no more or less real than Peano numbers constructed *out of* Godel-numbering of Von Neumann numbers. All three of these systems of positive integers behave in the same way and have an equivalency between them, and the equivalence is shared by uncountably many others. But out of all those possibilities, we happen to have a theory of 'numbers', rather than, say, a theory of 'schmumbers', which behave exactly like numbers except for that they differ in some totally-indescribable way.

Somehow, this all gives me a feeling of "arbitrariness" about the foundational mathematics that we humans happen to have based our everything on. If we encountered extraterrestrial intelligence it's possible that all their Pure Maths would be formulated in a completely bizarre, unfamiliar way to ours. And so, pure mathematicians are really just like all the other hyper specific niche researchers. It's *all* stamp collecting.
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During grad school, I studied elliptic curves and modular forms -- pure math by any standard. After getting my PhD, my first job was at Microsoft, and my first assigned task was to create cryptosystems based on elliptic curve isogenies. Eight years later (after having left Microsoft in the meantime), I succeeded in doing just that. I not only witnessed the transformation of a topic from pure math to applied math, I helped bring that transformation about. To me, the whole idea of dividing math into pure and applied areas is antiquated. So-called pure math is just future applied math by another name.
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There is no quantitative discipline that is objectively lesser than any other such discipline. There are different reasons to pursue various areas of Mathematics, Statistics or Computer Science.

We happen to live at a time when ideas in Computer Science and Statistics are changing the world, often in tandem. I’m sure you have already observed many Computer Graphics and Finance -related applications, given your interests. Whether your motivation is fortune or impact (or both), you are in a great place to be. There are so many challenging areas in Machine Learning, Virtual Reality, Internet of Things, Network Theory, Information Retrieval, etc (which themselves have applications in your fields of interest) that depend on Statistics and Computer Science. Maybe you feel that these areas don’t demand the same level of rigor, but that does not inherently make them lesser or even less useful. It’s more a matter of what’s required to be effective.

Keep challenging yourself and study increasingly more difficult topics. If your passionate about application, look for cool projects or research directions to pursue. If you get a chance, speak to experts and faculty. Eventually you will understand that the field has formidable open problems and unbridled potential for applications. When you appreciate these fields and find yourself in awe of what they can accomplish, you will no longer feel this way. Additionally, what others think about your field is next to irrelevant, if you are excited and challenged by it. Best of Luck.
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I decided to study math because it seemed like it was the "purest" and most "solid" major to get, I believed that I had to first be really good at pure math to be able to do something else next. I also finished my major (about a year and a half ago). I've got the following unorganized  ideas:

1) Everything in my major happened quite fast, I have forgotten many of the proofs ( markov chain lemmas, separation axioms, definition of Sobolev spaces, proofs in functional analysis etc).

2) I think I'd be able to digest everything better if I re learned it at age 30.

3) Some of the researchers seem to have forgotten a lot of stuff as well, and are more concentrated on the specific stuff that they need to publish.

4) The "punch" that results have has seemed to go way down the harder everything gets. Quadratic reciprocity was a banger, Bass's p theorem did not feel as incredible.

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