I was wondering what philosophy of science people here hold, and why.

In particular, what do you think about Hume's problem of induction: that there is no basis for thinking that if certain things are repeatedly seen to act in certain ways, then they will continue to act in certain ways? I think it's clear that science and induction can never be proof, but what is our justification for even suspecting that, for instance, all swans are white, or if you drop an object it will fall to Earth? Does science even imply that we should suspect these things? If not, what use is science?

It's worth noting that 'it has tended to work in the past' is itself an inductive argument.