Study maths as second degree?

Good morning.

TL; DR: For a long time I convinced myself that computing science (and in particular AI/data science) was the way, until I realised I dislike how everything is monopolised by the industry, and that CS has become solving very 1st world problems (in a bad way). Would it make sense to try taking another degree/doing a PhD in mathematics?

Here is some (long) background about me to explain how did I arrive at such point:

I am a computing science student who has been dealing with identity crises for a long time. I enrolled in computer science because I thought it was an easy way to find a job and make decent salaries (which is partially true) but I came to realise coding is more and more off-putting.

Back in the days of high school, I used to take part to mathematics competitions and studied some advanced stuff like maths for winter camps in preparation to IMO, despite never reaching that level. During my last year of high school I considered whether to continue computing science (Italian high schools have fixed subjects you can take according to the type of your school, so mine had a lot of cs/swe classes) - for which I started to lose my initial fascination after doing cs-ish stuff for 5 years -, mathematics because I like problem solving a lot, and physics. After some considerations I opted for CS as I had clear ideas about what I could do and learn.

At first I was sceptical about joint degrees, thus I did not apply for a joint CS+Maths degree, and I am regretting it now. I guess it's a cultural thing: in my country we always believed that you should study one thing and do it well, so I could not initially accept the Anglosaxon idea of joint degrees. I thought that, despite interesting, one would have ended up having incomplete education about both.

Fast forward to last year: I went down the rabbit hole of data science and for a while I was quite happy of my decision. Until... well, until I realised data science is clearly applied science. And with that a number of factors come in, for example that much of the stack is proprietary (i.e. runs on non-free software), is being monopolised by a small number of companies and that you cannot really do actual data science without both data and computing resources. Data-wise, you need to perform massive (and unethical) data mining on users to stay on top of the concurrence. Computing-wise, the university has some resources, yes, but they can be nowhere as near as the ones that - say - Google, Nvidia, OpenAI etc. have when they train language models like BERT and GPT-2/3 from scratch. Which in practise means your research will not be truly free until you work for a company that wants to make profit out of your work.

I have worked with people coming from one of the aforementioned companies and it sounds like my worst fears were true: extreme pressure to get things done, intellectually dead applied research (e.g. writing a lot of parsers just to get the Google Assistant to 'understand' the intent of your utterance), actually no freedom of self-defining your work until you become a senior researcher etc.

And, last but not least, the actual 'usefulness'. At first I thought "I am going to do CS to solve real-world problems" like treatments for cancer, optimizing aqueduct planning in developing countries etc. I thought of things like combinatorial optimizations, bioinformatics etc.; however, I ended up doing NLP - the 'easiest' field of ML one can get into at the moment - and realised how little it actually "help" end users. Yes, you can make good information retrieval systems but at the end of the day it mainly serves the purpose of making better and better advertisements. Even if you put DS aside, you quickly realise much of software engineering is mainly about making internal tools, dashboards, websites for the nth corporate etc... Why not other classical CS? Because:

1. AI is clearly "the future". For example, I considered doing formal verification for a while. FV is a tool humans use to proof check their code by - to simply put it - "convert it" into a theorem and run a proof assistant on it, but if an AI can program (yes, current AI models can program, even if at a toy level) and debug itself then what's the point? It would end up being a technique for Good Old-Fashioned AI (GOFAI) - which, to plainly put it, fails to be AI.

2. Other fields of CS that are ML-free are at the moment mainly in combinatorics (e.g. stringology and graph theory) or numerical optimization or constraint programming. Now, I can't get to like the second one. About the first one... many cool problems in combinatorics are unfortunately very NP, so basically untreatable. Approximated algos are nice but I don't feel like they are 'challenging' enough. You are not cracking a problem but simply finetuning algorithms to better cover edge cases.

Yes, I am aware I can do Data science on medical data (which is something I am seriously trying to do at the moment). But the problems about availability of data, computing resources and, lastly, actual interests of your employer remain.

So, given that much of CS is about solving very 1st world problems... why not just agreeing on solving very artificial puzzles? But this time, nice problems like the ones I find in mathematics? The small issue here is that my knowledge of maths is a bit rusty: yes I know some bits of real analysis, linear algebra, general topology, number theory, category theory, ring theory etc. but I never fully really wondered what I would like to study if I were to do a PhD. Which is why I think it's unlikely I would apply for a PhD in maths even if I were eligible, as I do not have enough preparation to clearly define a research goal.

But then I question myself: would I be up for doing 3/4 more years of studies at a university for Mathematics? I am 21 now and I am starting to feel like I am wasting my youth. Will I do research as usual? How would I sustain myself during this period? What are the policies for second-degrees students (I am European and study in Scotland, but will likely go back to the mainland because of Brexit - thus some financial advice would be great too).

Is there anyone here who has a similar background, in particular wanted to migrate from the "cool" CS to mathematics? What made you do the switch?

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