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Innovation is often assumed to be the work of a talented few, whose products are passed on to the masses. Here, we argue that innovations are instead an emergent property of our species' cultural learning abilities, applied within our societies and social networks. Our societies and social networks act as collective brains. We outline how many human brains, which evolved primarily for the acquisition of culture, together beget a collective brain. Within these collective brains, the three main sources of innovation are serendipity, recombination and incremental improvement. We argue that rates of innovation are heavily influenced by (i) sociality, (ii) transmission fidelity, and (iii) cultural variance. We discuss some of the forces that affect these factors. These factors can also shape each other. For example, we provide preliminary evidence that transmission efficiency is affected by sociality—languages with more speakers …