I think this is not about randomness vs objectiveness but rather randomness vs nothing being actually random. Objectivity has it's own opposite: relativity. Because of relativity we have big problems determining what's what, therefore objectivity is recommended specially when dealing with advanced physics, universal laws and such things.
1+1 will always equal 2 only if the correct rules are applied, just as the author said.
Otherwise, 1+1 = 2 is not true. As a rule-based example: 1+1 = 10 if the base rule is that we deal with a binary system.
To discover if there is any true randomness somewhere in the Universe may not be that easy. For example anything natural is related to Golden Ratio and/or Fibonacci, so there's nothing random in living organisms or plants (not even at DNA level). Going from DNA level and atoms to a very large scale like galaxies, we will notice that even the stars have a non-random arrangement in galaxies.
So from an Universal point of view, the vote goes against randomness (note: not to be confused with chaos).
That being said, what's left to do is try to generate our own synthetic randomness, just to prove that it can be done. But in this situation, we may find that this isn't easy either. In the case of computers, it's well known that numbers generated using them aren't genuinely random. Any piece of software needs some kind of unpredictable physical input to be able to generate from there. But what about humans ? Well, most of the choices they make are based on something quite pre-determined so we don't get much randomness from there. but as we trick the computer giving it physical input, we could find a way to bypass the human pre-determined concepts so we get something random out of it.
Let's design a situation. We have the drawing of a cube. Any sane mature human should understand that those lines drawn in 2D are the representation of a 3D cube. But the orientation of the cube cannot be determined from the 2D drawing, so each human would have to pick a way he thinks the cube orientation is.
By choosing the orientation the humans think the cube may have, we can say that we generated a random sequence of 0s and 1s. But is it truly so ? If we do this test to thousands or more, we may find that a selection is way more preferred compared to the other. That means some other pre-determined factor (like spatial orientation capabilities, geometry knowledge, the distance between the eyes and object) influences the choice. So in order to actually have the random result we must consider all these factors when making the test. So basically, if we are able to account for everything involved, we may find a trick to generate something truly random. Otherwise...slim chances.