Depends what you mean by "prove" and "random"... I certainly don't think the universe is
random, but that doesn't imply a
teleological purpose. The so-called "random" numbers produced by the typical computer program aren't
actually random, but they don't
mean anything either. And
rigorous "proof" is impossible outside of mathematics and formal logic - and even then, it's limited by
Gödel's incompleteness theorem. I can't even really "prove" that the universe exists at all. However, since it
appears to, I'm prepared to make some working assumptions so that I can get on with living in it.
As I see it, the
null hypothesis is that the world is exactly what it appears to be, no more, no less, and with no special magic going on in the background. If you want me to reject that null hypothesis, then the
burden of proof is on the claimant. If you want me to accept some kind of supernatural explanation, you're going to have to give me a pretty good explanation of (a) why it is
necessary, and (b) why
your supernatural explanation is better than all the other supernatural explanations other people have proposed throughout human history. So far, nobody has been able to do
either to my satisfaction.
I am an
empiricist and a
philosophical naturalist. Given that no-one has convincingly demolished either of those positions, despite a great deal of effort over several centuries by some of the finest thinkers humanity has ever produced, I doubt you're going to be able to convince me that they're wrong in this thread.
However, I am prepared to respect your right to your own beliefs, however wrong I may think they are. Are you prepared to reciprocate?