This from John D. Cook is a great example of how to respond to strong but highly disputable scientific claims — in this case Michael Atiyah’s claim to have proven the Riemann hypothesis:

Atiyah’s proof is probably wrong, just because proofs of big theorems are usually wrong. Andrew Wiles’ proof of Fermat’s Last Theorem had a flaw that took a year to patch. We don’t know who Atiyah has shown his work to. If he hasn’t shown it to anyone, then it is almost certainly flawed: nobody does flawless work alone. Maybe his proof has a patchable flaw. Maybe it is flawed beyond repair, but contains interesting ideas worth pursuing further.

The worst case scenario is that Atiyah’s work on the fine structure constant and Todd functions is full of holes. He has made other big claims in the last few years that didn’t work out. Some say he should quit doing mathematics because he has made big mistakes.

I’ve made big mistakes too, and I’m not quitting. I make mistakes doing far less ambitious work than trying to prove the Riemann hypothesis. I doubt I’ll ever produce anything as deep as a plausible but flawed proof of the Riemann hypothesis.