Richard Hamming in his talk about research says that scientists have to put up with stress, and he talks about his ulcers (and he was amongst other things a mathematician). I thought mathematical research should be mostly thinking/ living a life of the mind and thus while involved shouldn't be anxiety-producing... Is mathematical research typically stressful? Are there good researchers that don't have stress?
I'm not a mathematician, but being an academic in a tenure-track position may be stressful itself if you're on a publish-or-perish program. I.e., either you generate (mathematical) results year to year in six or seven years or you get fired. The stress may not be in the work itself but in the built up angst thinking about how achieving each new result quickly is, and it can be so bad that sitting down to work on a new problem or even thinking about it can be stressful.
(I write from a U.S. perspective.) The stresses of the job of academic mathematician include many that have little to do with mathematics itself, but are due to various commodified forms of mathematics. I'll get back to the stresses of research itself at the end.
For example, there're the issues of getting a job after the PhD, getting a tenure-track job after that, and getting tenure after that. These are not research issues per se. Similarly, "getting grants" certainly helps travel to conferences to see other work and promote one's own (though this may be less critical nowadays with Internet), but/and is sometimes used as an "objective external" evaluation of one's research program, so that getting tenure in upper-tier places may be impossible if one does not have an external grant.
The game of "getting grants" is much as what a skeptic would imagine, apart from various degenerative "streamlined" aspects as time goes on. That is, one must propose a thing that one can approximately do, but it is not considered entirely ok to propose to do things one already secretly knows one can do, etc. And "do" has to mean "in a year or two, at most", or it won't help the final report on the grant, nor on the re-application. Also, nowadays, apparently NSF wants its "panels" to favor certain general directions, certain subjects, etc, rather than attempting to seriously evaluate all proposals, thus "steering" research insofar as funding affects it. In particular, it's harder to get grants if you're not doing an "approved" thing.
"Publication in peer-refereed journals" is somewhat less "coerced" than federal funding, "status" is very important for getting tenure, and journals will reject papers perceived as insufficiently high-status to match them. Status is the chief commodity journals possess, and squandering it would be a mistake. But, of course, not all topics of research have equal status. Trying to establish status for a particular topic, or spend one's personal status-currency to do so, is a game in itself.
The long history of mathematics, and the fact that (mostly) things don't "become wrong" in mathematics, mean that there is a looooong backstory, and much low-hanging fruit is gone. (I can see the substantial changes in the 40+ years I've been observing...) Lots of things have been done, and many of the ideas that pop into the heads of people have popped into others' heads before. Stress: how to think of a new thing, that is worthwhile, and, once observed, other people would wish they'd thought of it themselves... but they hadn't??? Sounds like there'd be a scarcity issue, and, in many ways, there is. People have to find "ecological" niches that will generate sufficient status in dept heads' and deans' worlds to get jobs, get tenure.
And, yes, for many of us there is stress in doing the work itself, but there is perverse pleasure in it, or at least no viable alternative. I might claim that it would be silly to try to make a living as an academic mathematician unless one did care about the details so much that confusion or frustration did provide significant stress. Maybe there are better and worse stresses... But, yes, I'm unhappy and bored if I'm not confused or provoked by some mathematical thing, and if I'm confused or provoked I don't sleep well, etc. Just great. :)
But, seriously, apart from the job/business-aspect stresses (which are often hard to overlook), I would indeed claim that there is genuine stress (mostly of a less venal sort) in doing serious mathematical research. Significant scholarship is a prerequisite for not reinventing wheels, and for not thinking that trivial exercises are "research".
Probably the conclusion is that unless the "positive" piquant stress of the confusion of research is sufficiently fun/gratifying, the ugly/bad stress of the business aspect is possibly expensive. But the job/business stresses are universal, apparently, simply manifest in different ways.
First of all, I think it is helpful to separate the stress and anxiety which arises as part of being a mathematician, and that which arises due to being an academic, the latter referring to all the stress surrounding the job, such as publishing, finding a permanent position, grant applications, dealing with students and supervisors and relationships and so on, none of which are unique to being a mathematician.
So I assume you're more interested in the former. You state that
I thought mathematical research should be mostly thinking/ living a life of the mind and thus while involved shouldn't be anxiety-producing...
But surely living a life of the mind, as you put it, opens the doorway to a unique kind of anxiety?
For me the notion of a low stress job is one where I have some concrete task or goal which I am able to carry out successfully each day, one that doesn't involve a great deal of effort, after which I am able go home in the evening and switch off.
Mathematics is anything but this. At a given point in time a researcher will probably have a set of vague goals, problems they want to solve or areas they want to get to know better. They might be in the process of writing up a new result, tidying up the details and watching it develop into something publishable. This is can be quite satisfying. But for the most part progress in mathematics is extremely painful, involving long periods of frustration where you might be going nowhere. This in turn often leads you to doubt not only your methods and approach but sometimes even your own ability and value as a mathematician. Moreover it is rarely easy to switch off. I often find myself 'relaxing' with family or friends, when in reality the whole time I'm unable to ignore the research problem which is persistently gnawing away at me.
Don't get me wrong, I love my profession and I wouldn't change it for the world. But like any creative pursuit it sucks you in, and while there are brief moments where you see something and feel elated, there is a lot of time where you're banging your head against a wall day after day.
Of course, anyone working in a vaguely scientific subject might have broadly similar experiences. But I think that these experiences are particularly acute the more towards the theoretical end of the spectrum you get, and maths is right there at the apex. When you're stuck there's typically little input from the real world to help you, no experiments you can run to test your approach, and once you're past graduate level often only a handful of people in the world who even understand your work.
As a result, it can feel like you're living alone in a big dark room with no idea where the light switch is, all the time worrying that actually there might not be a light switch after all, and that maybe you walked into the wrong room in the first place! So we have our fair share of stress and anxiety in mathematics. But to avoid slipping into self-pity as well I think I'll stop here!
All people, everywhere, need food, clothing, and shelter. 99% of us or more gain those things by working. Work is guaranteed to cause stress at some point in your career, regardless of the career.
I've spent my entire life trying to avoid stress, without success. It's a stupid endeavor considering the fact that I'm an engineer that works in a production environment. Bean-counters are ALWAYS watching my productivity. It brings back memories of when I used to think that the practical application of science and math would be a utopian existence. LOL! I personally don't know of any engineer who would choose to do it again.
Here is a list of supposedly low-stress, high-paying jobs. I believe very little of it.