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Part of this article is bullshit, there is nothing "fundamentally random" about physical processes. It is just that you can't measure it without disturbing it. so you must use probability to help you.
I'm not sure I understand.

Imagine something like radioactive decay. We know how probable decay is, but we cannot know when an atom is going to decay.

We cannot predict when an individual atom is going to decay, thus that process is fundamentally random?

You have to make a distinction between what happens on a microscopic (the atom) and macroscopic (decay of an amount of atoms) level. On a microscopic level you need a stochastic model to describe the decay of an atom. With a lot of atoms (I don't remember thr tipping point) you can use a deterministic model, which is the mean of of the distribution of the stochastic model.
The core of quantum physics is that everything is "fundamentally random". From http://en.wikipedia.org/wiki/Heisenberg%27s_uncertainty_prin... "...the uncertainty principle actually states a fundamental property of quantum systems, and is not a statement about the observational success of current technology."
"In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known. In layman's terms, the more precisely one property is measured, the less precisely the other can be controlled, determined, or known."

It doesn't tells us that physical process is fundamentaly random (as in that randomness is objective reality)

Randomness doesn't come from the Heisenburg uncertainty principle, it comes from the Born rule.

(and by the way, all this talk of randomness mainly applies to the copenhagen interpretation. Other interpretations, like many-worlds, do not really have 'randomness', though the meaning of the born rule in such an interpretation is unclear)

My view is that of Copenhagen interpretation. where this randomness is also not the part of objective reality.

Many-worlds interpretation is even more extreme and less proven than view that reality is by nature random

I thought the success of Bell's Theorem suggested that certain processes (radioactive decay) are "fundamentally random" in that there isn't a latent state variable which is causal on the process.
The authors of this article write in their abstract: "Are there fundamentally random processes in nature? Theoretical predictions, confirmed experimentally, such as the violation of Bell inequalities, point to an affirmative answer. However, these results are based on the assumption that measurement settings can be chosen freely at random, ..." [1]

From its inception, quantum mechanics has used the language of probability to make calculations (this is the "Born rule" that another poster refers to), but this does not mean that the universe behaves randomly. Perhaps, as you suggest, this language of probability conceals human ignorance. To test this, we define a physical property called "realism" that says, roughly, "every experiment has a result that was knowable beforehand". If realism does not hold, then the results of some physical experiments are somehow randomly determined at the moment that the experiments are performed. A set of famous experiments (the "Bell's inequality experiments" [2]) have shown that, with certain assumptions, "local realism" does not hold, i.e. locality and realism do not simultaneously hold. Locality is the idea that it is possible to perform two experiments in sufficiently separated regions of the universe so that they cannot possibly influence each other. If general relativity were a complete theory, for instance, then we would have locality (because no signal could travel faster than light).

Almost all non-specialist physicists (physicists who use quantum mechanics but do not spend time trying to assess its foundations) accept locality and accept that the Bell experiments rule-out local realism. This means that they give-up realism, and these physicists say that nature is "fundamentally random". Some physicists try to give-up locality instead, and you will hear them talk about "non-local hidden variable theories". Other physicists, such as the authors of the linked-to "Nature" article, play carefully with the assumptions behind using the Bell experiments to rule-out local realism. For instance, they point-out the assumption that "measurement settings can be chosen freely at random". If this assumption does not hold, neither does Bell's proof.

All-in-all you are in excellent company if you say that physical processes are fundamentally random, but there are many specialist physicists who grapple with this question in their daily work.

[1] http://arxiv.org/abs/1105.3195 [2] http://en.wikipedia.org/wiki/Bell_test_experiments