[RightWingAtheist]

Can Science prove the existence of science?

Yes. Read The Cognitive Basis of Science (ed. Brian Carruthers; Oxford Press, 2002)

Why do we listen to scientist when we know that one of the dirty little secrets of science is that the whole scientific method (inductive reasoning) is logically dubious?

Science is much more than just "inductive reasoning." Again, read the book.

[Busywhiskers]

"Yes. Read The Cognitive Basis of Science (ed. Brian Carruthers; Oxford Press, 2002)"

Thank you. I will.