Is Popper's Methodology Applicable to the Natural Sciences?

You must mean protophysics, i.e. the principles of measurement &c.

On the topic, Laurence BonJour has a nice section on Quine in his In Defense of Pure Reason.

Popper was just flat out wrong about science. Long ago, it was worked out that if you followed Popper’s falsificationist logic straight through, the inevitable conclusion was that no progress is possible in scientific knowledge at all. It’s ironic that he is considered by many (particularly scientists) to be a defender of positivism, or something resembling it.

Someone recommended reading Kuhn. I don’t. Slim book (Revolutions), but still a waste of time - historical interest only. Kuhn spent most of the latter part of his career fighting againts “misreadings” of that book. In effect, he was repudiating it, since the “misreadings” he was battling were what just about everyone got from the text, because, well, it’s what the text says.

I don’t know if you’ve ever read “Popper and After: Four Modern Irrationalists”, by David Stove, but if reading the original arguments of Lakatos, Popper and Kuhn makes your heart sink, then reading Stove might give your heart a lift. He rips them (and Feyerabend) apart mercilessly, exposing the rhetorical tricks whereby they make their severely flawed arguments and improbable conclusions sound plausible.

In my opinion, the best picture of science is approximately that summarised in Susan Haack’s “Defending Science - Within Reason”.

I don’t really know what you mean by “problem of induction”. The two concepts I know of that are called that don’t claim that, using induction, you can’t know for sure. What they say is that using induction you can’t know AT ALL. In Hume’s version, he doesn’t say simply that we should be careful because induction doesn’t work all the time, he says that induction has no justification that doesn’t rely on induction. The problem of induction asks why induction is rational at all, not that it sometimes doesn’t work.

Try thinking of the problem of induction like this:

Imagine a chicken in a coup. Everyday the farmer stops by the coup with some chicken feed. This is the case no matter if it is raining or sunny, fall or summer, even or odd days of the month, etc…The chicken believes that today the farmer will bring feed but instead the famer brings a hatchet and lops off the chickens head. Why are people that rely on induction more rational than the chicken?

Cannot counter-induction be justified in the same way (that is, by relying on counter-induction)? Why not believe that because something has happened in the past it WILL NOT happen in the future? Someone who relies on induction and someone who relies on counter-induction will use all the same examples to show why they are acting rational.

BTW I use induction everyday of my life and would be dead without it. I just don’t know if it can be justified using an argument that isn’t circular.