Induction as a scientific methodology has been known (since Hume) to be impossible.

I agree with Hume about just about everything. You're misreading him. Induction definitely isn't impossible. We do it all the time. Scientists do it for a living. Hume certainly didn't think it was impossible. What he thought was that there was no deductive reason for expecting that today will be like yesterday. They only justification is induction itself. Thus, any inductive argument begs the question. But his solution definitely wasn't to throw it out and wallow in extreme skepticism. He thought induction was inevitable (not even something we will, just part of psychological habit formation) and was pretty much the only way of having knowledge about anything.

Hume's position is basically my position. Though I have some sketchy arguments in my head that might let us go farther than Hume, I'm more than comfortable with that. Now it turns out that if your psychological habit formation occurs in a certain way (the Bayesian way) you'll start winning bets against those who form beliefs in different ways. It also lets us do statistical/probabilistic experimentation which would never falsify anything but can provide evidence for and against theories. It also explains why we like unfalsified theories that have been tested many, many times more than unfalsified theories that have rarely been tested.

If Deutsch has other arguments you can spell out here I'd be happy to hear them.