Randomness is a pervasive phenomenon in science and nature which manifests itself in patterns arising from events whose individual outcomes are unpredictable. A characteristic feature is that the unpredictability of individual events is counterbalanced by a remarkable predictabiity of properties of events when considered en masse; for example, statistical properties. A surprising difficulty lies in attempts to give a mathematical definition of the notion of randomness. A natural naive definition would decree data to be random were it to exhibit every statistically certain property. However, nothing at all satisfies this overly-strict definition. The project proposes a new definition of randomness, according to which data is declared to be random if it satisfies every statistically certain property that can be specified 'independently' of the data itself. To formulate this, the language of mathematics needs to be extended with 'information independence' as a new primitive notion. Axioms for this notion will be given and justified by consideration of a motivating model: the Random Topos. Applications to the formulation of probability theory will be investigated.