I found your essay to be one of the hardest to rate. This is partly because it is an essay and thus too short to do justice to the topic. And thus, your essay was hard to rate because it is a breath of fresh air that I completely agree with, however, I do not support logical positivism. But I can agree with this:
"Logical positivism ... I believe that it should not have died, and that it is superior to the philosophies that replaced it."
I cannot support logical positivism because the logical positivists (of the past) often tended toward being quite "reductionist" and I also do not completely support the emphasis on logic, which is often not sufficiently defined.
You wrote: "The XX century can be seen as an era when limits to human knowledge were discovered. Goedel showed that not all truths could be proved. "
I will never disparage Gödel's beautiful proof, but I also do not unequivocally support the conclusion that he proved a (conclusive) limit. Something I hinted at in my essay is, what does a contradiction actually mean? Can we really take a contradiction to be an absolute falsehood? (Note, I do not go as far as Graham Priest and his paraconsistent ilk and call contradictions true.)
I think Gödel's proof can only be interpreted within the context of the Principia Mathematica and (David) Hilbert's program, which his proof was a direct response to. In the same way that Gödel's proof is taken to be a refutation of Hilbert's program, the Principia Mathematica is often considered to be the end of what could be called the logicist program, which Gödel also helped refute.
You wrote: "Did previous scientists really believe that someday a computer could be programmed to determine all mathematical truths and predict all physical phenomena? I doubt it. That would require a belief in an extreme form of determinism, and a depressing view of humanity."
Yeah, I actually think they did and they still do. Some people (appear) to actually think that computers might wake up and become sentient... When science became something to believe in it became Scientism; an intellectually bankrupt dogma.
I apologize for rambling; I will close with this:
You wrote: "Continuum hypothesis not meaningful"
Of course it is meaningful. Once you have Transfinite Set Theory (Hilbert's paradise), this question immediately emerges and beckons a solution. I would argue that the entire theory of the Transfinte is not meaningful. Solomon Feferman, which I have the utmost respect for, also argued that the Continuum hypothesis was not meaningful. However, I think his excellent book In the Light of Logic made my point better than it made his.
I wish you well in the contest.