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Except from the goal of finding some new truths, proving some theorems, or new ideas. What is the career path for someone pursuing a PhD in pure mathematics?
Continuing in research and trying to land a professor's position somewhere ? Then a PhD student should also worry about landing a post-doc before finishing his PhD ?
Why do I keep hearing that : "the competitiveness in pure math is rather bad these days" and "an academic position in pure math nowadays is a tough choice" ?
For most of us who followed that path, if you have to ask the question then it may not be right for you and you should explore all your options before you jump.
There is a saying that "You don't choose mathematics. Mathematics chooses you."
For myself and many others we never thought of any other options.
Yes, at the moment the academic job market is terrible. Most pure math doctorates will need to deal with it, though there are a few other options, though mostly in applications.
Yes, post docs seem to be a necessary step. Not entirely pleasant but not entirely unpleasant if you meet new people with new ideas.
But a lot of academics in other fields are also "idea driven". It is a great life if you can manage to get in the door. People pay you to think. Not much could be better than that.
And, It is hard to predict what your career path might be several years in the future. Things change. When I started grad study the future looked extremely bright for PhD mathematicians. But the successful moon landing ended all of that and universities discovered they had already hired too many people and the faucet immediately closed. Overnight, essentially.
But the opposite can also happen. Some new challenge can open doors that nobody saw before. In CS, AI and data science has had something of that effect recently.
If you really love something, try to do that and try to avoid substitutes even if you need to compromise for a while as you build a reputation.
Let me add that I'm not implying that some people are "fit" for math and others are not. That isn't true, at least in my experience. It is a matter of desire and training as well as hard work. To repeat a comment made to another answer here (and elsewhere on this site): My teachers wanted to hold me back a year, primarily for poor performance in math at about age 10 (My mom saved me from that one). My first truly positive educational experience in any area was a high school geometry class. I really liked solving those problems and bought problem books to have more to solve. I finally learned my multiplication tables after earning a doctorate in math. There were other struggles along the way, but each defeat caused a renewed effort to succeed. Luckily I had good mentors.
Let me try to provide a somewhat different though not entirely distinct perspective, namely that this is a totally reasonable series of questions to ask oneself and the mere act of considering them seriously isn't some type of heretical disqualification from pursuing a PhD. But you should think very carefully about your motives. The top line takeaway is that the opportunity cost of doing practically any PhD is massive.
Except from the goal of finding some new truths, proving some theorems, or new ideas. What is the career path for someone pursuing a PhD in pure mathematics? Continuing in research and trying to land a professor's position somewhere?
The major work one produces when doing a PhD in math is finding some new truths, proving some theorems, and refining some interesting new ideas. Suppose you do not think these will be satisfying for you or you are generally have different priorities than being "intrinsically satisfied" by your work. Then the pay, working conditions/job benefits, and future opportunities provided by the job are roundly beaten by a whole host of alternative vocations which require similar types and levels of training to enter.
There are vanishingly few jobs other than being a professor for which a PhD in mathematics is an essential qualification. For those where a PhD is helpful but not strictly necessary, there are usually other preparatory jobs you can take which will also qualify you and compensate more (except for the manner of work) compared to doing a PhD.
Why do I keep hearing that : "the competitiveness in pure math is rather bad these days" and "an academic position in pure math nowadays is a tough choice" ?
I've never been presented with a convincing framework that uses hard data and discusses comparative questions over time like this. By anecdote everything fluctuates wildly by sub-field and depends heavily on which two points in time you're comparing. If you are talking to someone more than 20 years older than you about the ease of getting jobs in their sub-field, it is almost a default that professorships are harder to get in it than when they got theirs because they obtained a job in the past and are fulfilling that potential demand in the present.
Is you question WHY or SHOULD?
from the content of your question, it sounds like "should", but the title is too obvious. If you're not really asking, I will provide a lemma at the end.
Anyhow, welcome to the club of:
Actually, since STEM includes Mathematics, it reminded me of the joke that "everybody thinks he's better than average." Oh wait... it's not even a joke!
But here are some good parts too: Every mathematician here decries the statement: "Mathematics chooses you". Even the ones who wrote it!! :) But some people do believe it: otherwise it didn't exist! Assume you're gullible enough to believe this stereotype: "Mathematics chooses you". Then, following the logic of it, you will not pursue Math in the first place. What does that tell you about the "average IQ" of Math vs NOT Math set of students? Therefore: it's a self reinforcing idea! QED.
Corollary, the proof I leave for the reader as it's too trivial: Lemma: If you're a mathematician then you think that if someone doesn't choose to pursue a PhD in math then he/she has a low IQ.
Jokes aside, if you have no better alternatives, doing a degree in Applied Math or just Math is worth it! (Example: being from Moldova and no money). Later, you can easily switch from Math to any other field: Banking, Insurance, Telecom, Bio-Chemistry, Software Engineering etc.. Mobility: It provided many people I know a legal ticket to US/EU/other and an exit from abject poverty to a rather decent life.
While you could be born to have a mathematical mind and work as a mathematician as Buffy writes, or like someone with synesthesia has cognitive advantages in arts/music, remember a PhD in math from starting your studies to graduating with a PhD does often take 10 years alone. As a postdoc there will be maybe another 10 productive years, but at some point your job will become routine and even hinder you from inventive progress climbing up the ladder of responsibility. (It's really interesting from that perspective to read the Perelman case, who apparently had to leave Harvard to be able to focus on mathematical research or Einstein and when and where he made his biggest discoveries).
My strong advise to all my students is to try out as much and different opportunities as possible during their study/PhD career. If your primal goal is to get a PhD in mathematics and a follow-up academic position, then you wouldn't have asked this question here.
One would think in studies like pure math or astrophysics, why don't end mainly Perelman, Einstein, Wolfram like prodigies on professor positions, and nowadays maybe even less than a century ago? Partly and funnily because a PhD in such scientific branches is pretty much needless if you have theoretically groundbreaking ideas and can realize them (without a need of big funding sources). PhD and professort titles are side-effects in the careers of such researchers, they rather think in scientific projects before earning any degree while on academia.se one can read hundreds of questions on the ideal PhD and how to get it. (For me the best sign, academica offers to many PhD positions, but that is a side note)
If you are already reasoning before starting this path how much financial or job security it guarantees you, your path is already doomed. I had a similar decision during my physics studies to make, either pursuing astrophysics for PhD/professorship as the latter is the only way to work in astrophysics apart from becoming a ESA/NASA engineer or joining a branch of physics that gives more financial security, life planning but still offers many options to do fundamental and intellectually challenging research as a physicist.
And there are many and it's actually the majority of physicists that started physics for particle or astrophysics, graduated with a PhD in such, but ended doing in job life completely different things due to becoming bored by the necessary strong focusing after 10 years of studying or not hitting a job in academia.
Ofcourse pursuing PhD in pure mathematics and applying all the innovative, creative and analytical power solving unsolved problems is very difficult task. One should not choose mathematics just for profesionality but one should choose mathematics with both affection towards it plus targetting a job. But then looking at the competitiveness market in academia might demotivate one morally. However competition is everywhere and most of all succesful researchers (PhD+Post docs) get academia job in time. The world need mathematicians more and more in every academic institutions because the language of pure mathematics as well as applied mathematics enlarge the idea of advanced world and natural sciencies are devloping continuously based on mathematical language developped by pure mathematicians. I am myself final year PhD in number theory and I am optimistic because I can see most of the seniors got academia job. So don't lose hope if you really love pure mathematics with heart not in terms of professionalism only.
