First, this:

http://www.latimes.com/news/opinion/commentary/la-oe-wertheim16may16,0,252151.story

The article seems to me to confuse some of the issues rather wilfully
(no, an assertion of the statistical likelihood of a large number's
prime-ness does not constitute proof that it is a prime. There are an
infinite number of primes. If you permit any degree of uncertainty in
your determination of whether a number is prime or not, you then will
have to live with an infinite number of false positives. If they're
sparse, and computationally very expensive to differentiate from real
primes, then they might still be useful in cryptography. But
usefulness to cryptographers is not the defining property of a prime
number, and there is no scope for "definitional drift" in the matter).
Some of the issues are real nonetheless.

The question of whether proofs performed or assisted by computer are
mathematically acceptable is interesting, because an answer in the
negative would tend to rest on the assumption that conscious grasp of
a proof by a human being is required for it to be valid. This is an
odd metaphysical phantom, which yokes consciousness to truth and
places anything that falls or strays outside of the immediate ambit of
consciousness - writing, say - under suspicion. Who knows where a
computer-generated proof has been? Better get it checked out by a
meat-head, just to be sure...

Mathematics qua "that thing that mathematicians do" is, I don't doubt,
as "postmodern" and "socially constructed" as anyone who wished for
that sort of thing might wish. But mathematics qua the object of
mathematicians' enquiry I think is not. Committees may very well meet
to confer on the validity of a candidate for proof of some conjecture,
but the validity of the candidate does not in fact depend on their
consensus. They might all agree that it's a very fine proof, only for
a flaw in its reasoning to be discovered some months later. The flaw
was always there, and will always have been there, and there is no way
that anyone can make it go away just by changing their minds about it.

Dominic