On what basis can we expect the future to resemble the past?
Per Hume, the most frequently associated Western philosopher: “there can be no demonstrative arguments to prove, that those instances, of which we have had no experience, resemble those, of which we have had experience.”
Popper claimed to have a solution: you can't prove that induction will hold, but it's the simplest (most easily falsifiable, requiring fewest assumptions) theory available. However, Popper didn't believe anything at all could be proven per se, so this is maybe less a solution to the problem of induction than an argument that the existence of the problem of induction supports his eliminativist position.