> Beyond scarcely stretching the boundaries of obscure mathematical knowledge, what tangible difference has a PhD made to my life?
The same thing a bachelors degree does for everyone else. You've proven that you can start, stick with, and complete a task that takes multiple years and a complicated set of steps.
Which feels like a 'life is suffering' weirdness. I did it and never needed my diplomas for anything. Maybe I benefited somehow but I was a coder and entrepreneur before that; I had a software company before uni. It was over 20 years ago and in hindsight I find it pretty pointless and a waste of time. Maybe I became a better problem solver on some level but unless you are going into research or are not a self starter I would not recommend it.
I went into taking my MSc explicitly to be able to add the letters to my resume - I started not long after the dot-com bubble burst, as a precaution. I don't regret it; I learned some interesting things during my thesis (the rest of it was regurgitating stuff I already knew; but it was distance learning so I didn't have to put in much effort), but similar experience - I don't really use it much. If I'd done it full time, it would have been a tremendous waste of time, though.
But a lot of the reason for this experience for me at least was that I started uni after having spent about 15 years learning to program already, and by the time I picked up again and did my masters, I'd had another 10 years of commercial software development experience.
These things are not really geared for people like us that came to them with a lot of pre-existing knowledge, but at people like my class-mates first time around that had hardly touched a computer before, and that did need a lot of hard work to come out with a good understanding of the problems.
As a hiring manager this is why I rarely care whether someone has a degree or not if they can demonstrate experience. And on the other side of the table, I only took that degree because in the UK there are still sufficiently many employers that have an obsession with degrees regardless of experience...
It also supplies the degree holder with a social signal which says: I can buy into the establishment. I say this with my newly minted BSc. in hand. In the process of obtaining it I realised that at a minimum, all you have to do to get a degree is satisfy the course requirements. I took some shitty courses that I knew were largely wastes of time [0]. Doing them anyway equips me with proof (degree) that I can submit to a system I disagree with if I have to. This excessively cynical attitude is the product of being a cog in a massive degree making machine whose graduates are on average, I'd say, mediocre [1].
[0] I'm looking at you, three years of 'numerical methods', which reduced to memorising algorithms to perform by hand.
[1] I count myself amongst the absolutely useless 'applied mathematicians' from my class.
The same thing a bachelors degree does for everyone else. You've proven that you can start, stick with, and complete a task that takes multiple years and a complicated set of steps.