Yes, controlled experiments with appropriate conditions in place can produce proteins
No. Controlled experiments with conditions similar to those that, according to our scientific understanding, prevailed at the time. That's quite important. Because it shows that it
can happen.
but the mathematical likelihood that even a single protein was derived out of the cosmic chaos defies Borel's Law by leaps and bounds. Regardless of the expanse of time the probablility of such an event is mathematically impossible. Correct me if I'm wrong here, but this is what I've understood.
That it happened in a lab, in conditions similar to what prevailed at the time on Earth shows that it's not all that improbable and certainly not "mathematically impossible." But even if I were to grant you that the chances of it happening are vanishingly small (say, 1/(2
128, which is approximately equal to 3×10
-39. That number is
vanishingly small. But it's not impossible. Consider also that, to the best of our understanding and knowledge, the age of the Universe is 13 billion years give or take a few hundred million... Given that much time, and perhaps concurrency, all of a sudden, it doesn't seem all that impossible, does it?
Now, about Borel's Law. I've read about one guy who twisted Borel's Law and used it to wave his hands and make grandiose claims. I believe he stated that "mathematicians agree events with odds beyond 10
50 don't happen" (I'm paraphrasing his exact quote; I don't remember the original phrasing) which is, to put it bluntly, complete and utter bullshit. After you stop laughing at the whole 10
50 bit (which by itself shows that the man was clueless about statistics and probability theory), which mathematicians agree with that statement? Certainly not all - I have a degree in mathematics and I don't agree with that phrasing. Besides, even if they all agreed, what does the consent of mathematicians mean vis-à-vis unlikely events actually occuring? Why not quote the consent of astrologers? Or poker champions? Or New York-based chefs?
The reality is that he most likely misunderstood and misinterpreted what he was reading. Borel claimed that, even in the cosmic scale, an event with a probability of 10
-50 has a
negligible probability of occurring, which is certainly true. But a negligible probability is not the same as "it's impossible." It's exceedingly improbable to flip a coin 512 times and get 512 heads -- specifically it's 1/(2
512) which is approximately equal to 7.5 × 10
-155. That number is
vanishingly small. But it's not impossible and exceedingly unlikely events do happen.