Hi Jesse and Bruno:
To consolidate my response:
Are you going to respond to my most recent post? It made points that you did not really address in your response to Bruno.
The All contains all information [is this controversial?] but that must add up to no net information content if my total system is to have no information. The small amount of external information necessary to define the All is balanced to zero net information by the other components of the system.
It is controversial that mathematics contains any information in the first place--by the most commonly-accepted definition of information in information theory, I don't think it would, simply because there is no room for multiple possible answers to a given question.