Not much! In place of the ideal hard science stuff, for a while it's been Stack Overflow, tenacity, and a nose for useful solutions.
I feel if I had educated myself in rigor, I could implement that paper, prove solutions, and have an assortment of physical phenomena at my disposal.
All the books I have and not read speak to the need for knowledge, and while applications abound, words can only be consumed one long tape at a time.
> Where does classics and philosophy come into it?
It's your secret weapon: it's the quote or phrase or context that comes to mind, to flower one's thoughts amid numbers and process, that inspires and renews.
> What do you typically apply that toolkit to, and how does it help you?
Programming, of course! Mostly in pattern recognition for solving problems.
Whatever you decide, I hope you end up owning "something that is yours." It's hard to describe. It could be a business, or a codebase that is maintained a long time.