I wasn't meant to imply that in general, only for the relation between physics and philosophy that Hawking had in mind. You don't have to "keep up" for doing ph

I wasn't meant to imply that in general, only for the relation between physics and philosophy that Hawking had in mind. You don't have to "keep up" for doing philosophy in general (that would be impossible, cf. see below). But since I'm working in formal ethics, I can answer your last question. Many problems of value theory and practical reasoning can be solved with formal methods and have been studied by decision theorists since the 50s. For example: parity, value incommensurability, non-additive value aggregation, claims about the intransitivity of betterness, etc. There are mathematical models for all of these. You also cannot make informed claims about, say, practical reasoning and how reasons combine to a from decision or intention to act without knowing at least Arrow's theorem, the various ways in which it can be circumvented, the limitations of qualitative value aggregation, and some fundamental results from measurement theory. Similar things can be said about normative systems (laws, moral codes, deontic rules, etc.), which are also studied by computer scientists and AI. To give another example, social choice is an important part of politics and I don't think you could do proper political philosophy without knowledge of social choice and related methods from economics. Problems of distributive justice and and fairness have been studied extensively in welfare economics and they are not all based on "egoistic agents" and utility maximization. Mathematical method is needed to show that philosophical claims are consistent and to perhaps make them more precise. That certainly doesn't require keeping pace with the latest developments, though, it only requires awareness and some (limited) understanding of old results in applied mathematics, in fields like logic and model theory, order theory and measurement theory, or statistics. Obviously, all of this is just my personal opinion and you can find plenty of math-haters in philosophy.