I admire your determination to start a scientific career at age sixty! I once faced the "converse" problem---how to start doing research at age 14 or 15, before I'd found any mentor to help me---and it was one of the great revelations of my life that, at least in theoretical fields, the barriers standing in the way are more internal than external. So long as you have the time, and enough income to live off, you can start reading textbooks and arXiv papers. You can email the authors your questions. You can go to conferences, with or sometimes without registering (just don't tell anyone I said so ;-) ). You can sit in on classes at a nearby university (ask the professor; most are fine with it). You can talk to the professors. You can start working on a problem that interests you. If you get somewhere, you can write it up and submit it to a conference or journal. You can offer your services as a research assistant. The gates are open.
Of course there's a chicken-and-egg problem here, where the stronger your research record, the more busy researchers will want to talk to you, and the more they'll talk to you, the stronger you'll be able to make your record. Just like with dating or anything else, I guess, it does take persistence to break this cycle. :-)
And of course, you'll need to be considerate of people's time, and you'll need to decline to take it personally if some people are too busy to answer you. Just plow ahead and ask others.
Lastly, let me strongly suggest finding some other people who started their research careers later in life, and asking their advice. They'll surely be able to think of things I didn't!
Of course there's a chicken-and-egg problem here, where the stronger your research record, the more busy researchers will want to talk to you, and the more they'll talk to you, the stronger you'll be able to make your record. Just like with dating or anything else, I guess, it does take persistence to break this cycle. :-)
And of course, you'll need to be considerate of people's time, and you'll need to decline to take it personally if some people are too busy to answer you. Just plow ahead and ask others.
Lastly, let me strongly suggest finding some other people who started their research careers later in life, and asking their advice. They'll surely be able to think of things I didn't!