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JohnH comments on The benefits of madness: A positive account of arationality - Less Wrong
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How do we choose axioms that are good to accept as valid except for our experience with reality? This is a philosophical question so isn't often considered within mathematics. If one reads some of Eliezer's posts where he doubts the axiom of infinity (unless I am misunderstanding what he is doing) then it becomes clear that his argument for doubting the axiom is that it doesn't correspond with reality.
This is generally why choice is doubted; as it is thought that non-measurable sets do not play any role in reality they are ignored in standard statistics and calculus. If non-measurable sets do play a role within the real world then some very odd things can happen that so far no one has observed happen.
I suppose if mathematical truth is changed to be whatever is provable given all possible combinations of non-contradictory axioms then reality does not play any role in math.
I think when trying to study information and its relation to mathematical truth, we must start off practical and should be talking about provability in formal systems of logic. I don't actually know of any rigorous connections between the two notions, but I can think of an argument that "mathematical truths contain zero information" might be false based on indirect connections between existing work on proof theory and information theory. But I don't want to give that interpretation yet because I would like to first ask Skatche if he wanted to elaborate on his statement a little better or point to some references for us to read.
The philosophical question is interesting too, and I would agree that a set theory without the axiom of infinity seems pretty adequate for describing our experiences of reality. I'm not sure if its the most harmonious, however...