What to do after a Master's Degree in Pure Mathematics? This is not exactly a math question but I figured this would be the best place to ask - I suppose mathematicians are best qualified to answer it.
This summer I will finish my thesis in pure mathematics and therefore the master's degree itself. My thesis is in Mathematical Logic (concerning functional interpretations); I have a strong background in algebra and a bit of geometry too. 
My plan is to proceed to a PhD, which is actually what I would love to do and is my first option. But I wanted to have a plan B (and C, D, ...) in case things don't go as expected. So the question is: what kind of job outside academia would allow me, simultaneous, to


*

*explore what I've learned so far and use (some of) it

*eventually allow me to keep developing my knowledge on mathematics


I mean, a lot of people suggest working at banks, etc - but that's finance, statistics, not pure mathematics. Is there anything out there that meets my needs or am I asking for too much?
Edit: I have a minimal programming background (this basically means I learned the basics of the language C), I very much like programming (specially the logic behind it) and would definitely be interested in learning more.
 A: It is very difficult to find a non-academic job for pure mathematicans, who want to work on pure mathematic like algebra, logic ore something like that.. But there are still some rare jobs in private business.
My suggestions for you are the following fields:


*

*Artifical intelligence / Machine Learning (optimal websearch, intelligent robots, etc.)$\;\;\rightarrow$ logic

*Computer graphics (rendering, animation, etc.)$\;\;\rightarrow$ parts of algebra

*Teaching (but it might be difficult to keep developing your knowledge without doing it privately)
Not long ago I had the same problem like you and decided to make my PhD. Until now it seems to be the right decision. :-)
A: I believe with a background in pure mathematics, you can get a job in cryptography and possibly work for GCHQ and other similar security agencies. The work can be very exciting and you will certainly be working with bright minds and learn in the process. As a general rule, any job in industry will require programming skills so I'd encourage you to brush up on your coding. Although you've already said you don't want to work in finance, I would still encourage you to look into becoming a quantitative analyst. The work requires a high level of mathematics and they are still able to work in other industries apart from banking and finance.
However, if you can find any other practical applications of your research to any industry, with a bit of convincing, there's no reason why they can't hire you. If your line of research has some applied mathematics, there are many more career options available to you. 
A: It seems to me that expecting to do work in pure mathematics outside of academia is perhaps,
as you put it,
asking for too much.
After all,
when doing pure mathematics,
one studies abstract concepts irrespective of whether or not they have any useful physical (or otherwise useful from a practical applications point of view) interpretations. Therefore,
if you are working for a company
(i.e., help the company gain a profit)
or a government entity
(i.e., work for the people),
it is to be expected that your work will be focused on applying mathematics to solve practical problems.
As Danny mentioned in his answer,
many domains in industry make use of notions from algebra,
logic,
as well as other fields of mathematics in that are considered pure mathematics.
However,
it must be said that this work still more closely resembles that of an engineer or of an applied mathematician than that of a pure mathematician in that,
the focus of the work is to apply algebra/logic etc. to solve applied problems
rather than to study algebra/logic for the sake of furthering our understanding of mathematics.
If applying pure mathematics to solve practical problems satisfies your desire to explore and use what you've learned and learn new things (which you would most certainly do a lot of even if you work as an applied mathematician),
then this is probably a good option for you.
If you work hard on you programming skills,
that together with your technical degree should make you a fairly competitive candidate.
As for specific examples if you're interested,
other than what Danny mentioned,
I can think of the following:


*

*National intelligence agencies usually hire mathematicians to work on cryptography related problems. See the NSA in the US, for example.

*Companies like IBM, Microsoft and Google usually have research divisions that hire mathematicians (although such positions are sometimes considered as part of academia and may require a PhD).

