Is reduction always a good scientific strategy? The existence of the special sciences above physics suggests not. Previous research has shown that dimensionality reduction (macroscales) can increase the dependency between elements of a system (a phenomenon called 'causal emergence'). Here, we provide an umbrella mathematical framework for emergence based on information conversion. We show evidence that coarse-graining can convert information from one 'type' to another. We demonstrate this using the well-understood mutual information measure applied to Boolean networks. Using partial information decomposition, the mutual information can be decomposed into redundant, unique and synergistic information atoms. Then by introducing a novel measure of the synergy bias of a given decomposition, we are able to show that the synergy component of a Boolean network's mutual information can increase at macroscales. This can occur even when there is no difference in the total mutual information between a macroscale and its underlying microscale, proving information conversion. We relate this broad framework to previous work, compare it to other theories, and argue it complexifies any notion of universal reduction in the sciences, since such reduction would likely lead to a loss of synergistic information in scientific models. This article is part of the theme issue 'Emergent phenomena in complex physical and socio-technical systems: from cells to societies'.
Keywords: Boolean system; emergence; information; mutual information; partial information decomposition; synergy.