TheMathGuy

I recall a friend telling me that there did exist deterministic formulations of quantum mechanics. Perhaps I should ask him where he saw this. Just because we can't determine it from observations doesn't mean it isn't determined.

Part of the difficulty with the discussion of whether or not randomness exists is the problem of defining just what, exactly, randomness is. It's not as simple as it looks. Let's say you have a binary sequence. Perhaps you'd like to define this sequence as random if there does not exist a mathematical formula for determining the next bit in the sequence given the previous bits, but every sequence encountered in real life is always finite. And given a finite sequence, it is no problem to construct a formula that describes every bit in the sequence. Just make your formula sufficiently large, and presto! And if you think you have a formula that works, and it gets the next bit wrong, that doesn't necessarily prove the sequence is random either. There may always be some other formula that would have gotten it right. The probabilistic model is simply the last mathematical model that we have to resort to when we can find no better model (or modeling it deterministically would be infeasible) and observations lead us to have a reasonable degree of confidence that the distribution is uniform. But even the randomness tests themselves are subject to question. Just because these tests don't find a pattern doesn't mean there doesn't exist a test that will. We have in our heads this intuitive idea of "randomness" as being without pattern. But what kind of "pattern" are we to allow here? Could not the data itself be thought of always as a "pattern", just perhaps a highly complex one? "True randomness" is unfortunately not even a well-defined concept, which we would need to have if we can ever hope to have a meaningful discussion on whether or not "true randomness" actually exists.

Yes. (at least when it comes to methods that only involve a compass and unmarked straightedge) ...in exactly the same way as we should give up looking for two integers whose ratio is exactly the square root of 2. A mathematical proof is very different than a scientific fact, which has been established by empirical observation and could always potentially be disproven by further empirical observation (however unlikely). A mathematical proof, once it has been confirmed to be valid, can never and will never change. Two plus two will always equal four, never five. That's not to say there won't be constructions with an unmarked straight-edge and compass that will come very close to trisecting an angle, but there can never be such a construction that gets it exactly right for every angle. In particular, an angle of 60 degrees cannot be trisected in such a way, since this would imply that the cosine of 20 degrees is the root of some quadratic polynomial with rational coefficients, and one can prove using the laws of trigonometry and algebra (particularly field theory) that this is not the case (see http://en.wikipedia.org/wiki/Angle_trisection). However, notice that the Wikipedia article also points out that there are other methods of trisecting an angle. it's not like the problem is impossible by any means.