I got my math PhD degree more than 10 years ago, and my major is algebra. I have published some papers in some journals such as Adv. Math.. I really love math, but I am still thinking I haven't got a good and complete math education, and do not have a good picture of math. Someone may argue that I can teach math to myself. The problem is that I am a teather in a university now; I have to teach a lot of classes and have other duties. Now I have an idea which sounds crazy: I want to do another math PhD. But the problem is I am almost 40 years old. So I am wondering if this is possible. Does someone have experience of such a second PhD?
The question is really strange. As a university professor you can attend other professor's courses, in addition to self-education. If the time and duties is the main problem, you can take an unpayed leave, or buy out some courses. There are many ways of reducing your duties and obtaining some extra time for self-study, by sacrificing some of your salary. Given the ratio of professor's salary to the graduate student support (at least in the US) you will be still better off than by switching to a graduate school. Let me add that in most US departments graduate students have the same teaching load as professors, sometimes more.
Speaking of the age, this is not a obstacle: I have at least one friend who enrolled to PhD after 40 (having a master degree before and working as a high school teacher), and he publishes first-class results.
I don't have any experience of exactly this, but my two cents are that it's not a crazy idea if that's what you really want to do. You can be the master of your own destiny (unless there are family issues involved, in which case it should be a joint decision).
Somewhat related, I was once contacted by a professor of surgery at my institution who wanted to do an independent study of advanced calculus. His story was basically that he had become a doctor because his family wanted him to, but he now realized that math was his true passion. Needless to say, he was surprised to discover how much he had to learn even to get to an undergraduate math degree, but he persevered. He's now a professor of finance at a well-known university.
Do you know what area you would do the second Ph.D. in? I have often fantasized about studying subjects far from my expertise, but when I get to the nitty gritty it usually ends up in me realizing that I wasn't really that interested in them after all. Maybe another idea is to find a program you might apply to, look at their course offerings, and gauge how excited you would be to take some of those classes.