Working in an isolated environment
I was in a pretty similar situation, so I can sympathize. Although, personally I found it less critical for me to have these interactions. Don't get me wrong - it was not easy to work this way, but I saw it as an opportunity/challenge to develop my independence as a researcher.
However, I was able to get some ideas and interactions in the following ways:
- Presenting my work at various forums - any size, from group/lab meetings to seminars and conferences.
- By establishing a collaboration/semi-collaboration with relevant people at a different institution.
- In terms of ideas - sometimes just reading a lot of literature helped (but this can be field-dependent).
- By interacting with the "mathematically unskilled" people at my lab - might sound surprising, but sometimes you just need someone to interact with and ask basic questions even if they are not knowledgeable.
Personally I also found that I don't need these interactions to be very frequent, sometimes it is enough to interact, get a seed of an idea, and then go work on it for a month on my own.
Good luck, for me it worked out very well in the end and I can say I developed significantly as a scientist, and both sides benefited.
One more piece of advice - while you are dealing with the math, don't forget to keep in close contact with the experimentalists so you don't drift away trying to solve some problem which is not relevant.
Similar problems have been discussed before in the following Stack Exchange sites.
Academia.SE: My advisor does not provide any solutions and wants me to work alone
Workplace.SE: How to be more effective when working “alone” and with no prompt feedback
Productivity.SE: What methods exist to keep yourself motivated when working alone?
These posts ought to be able to provide you with suggestions that would assist you to help yourself through this situation. But none of which could be a black and white solution for the problem. It depends on the individual.
From your description, you share qualities of both an introvert and that of an extrovert. It seems like you could be more of an ambivert (as I am). I was in a similar condition before during my first semester of research -- research in total isolation, being the only human in the room, but never complained. In such situations, you ought to understand to what extent you could control your environment and make it suitable for your purpose. Whenever a little extroversion was needed, I just meet my peers and then get back to work. It did take a little practice. My supervisor was also of great support as I frequently report my progress too. I'm in a more of an extrovert's environment now and still making progress at the same comfortable rate.
Sometimes the constraints that you are put through could make you better. Take the change of environment positively and make do with what you have.
Hope this helps, all the best.
You mentioned one thing about part of the social process involved that particularly stood out:
Then I noticed that sometimes it is sufficient to just think about how to explain a question to someone, and the solution comes to me before even calling them.
Among software developers, that's often called rubber duck debugging:
Many programmers have had the experience of explaining a programming problem to someone else, possibly even to someone who knows nothing about programming, and then hitting upon the solution in the process of explaining the problem.
Rubber duck debugging doesn't address all the issues that you've described. I've solved plenty of problems with rubber duck debugging, but I haven't had the same kind of inspiration and collaboration with a rubber duck on my desk that I've had with colleagues at a whiteboard, so this isn't a complete solution, but it's definitely part of the "working remotely" or "working in isolation" picture.