Will current encryption always remain secure? Is there mathematical proof?
"Will such security always remain secure?"

No. $\:$ In particular, quantum computers will break RSA and elliptic curves.

Has it been mathematically proved that these algorithms can not be cracked, or is it a possibility?

It is a possibility, since a very-practical algorithm for the Boolean satisfiability problem
would be enough to break essentially all complexity-based cryptography.

(There might still be a secure complexity-based proof-of-work system,

although it would need a super-constant number of rounds of interaction.)

If so, how likely is it that cracking the RSA via a

mathematical weakness will be possible in the near future?

Does Shor's algorithm constitute "a mathematical weakness" in RSA?