Many people might not bother to define complexity, thinking that we know it when we see it. Scientists should not afford such luxury. I will provide a compact but comprehensive overview of the different ways that systems can be complex, offering an aggregate definition. I will discuss the role of complexity measures, and why complexity cannot be captured by a single number. This work was done in collaboration with James Ladyman, published with Yale University Press in 2020. At the other end of the spectrum of complexity science is the application to real-world problems. I will present two examples from recent work. The project ‘Aiding the mitigation of and adaptation to climate change using the tools of complexity science’ was done in collaboration with the Green Climate Fund, founded by the UN members in 2014. Equally, political systems are more and more focus of computational and mathematical investigations. I will present conceptual work on the stability of democracy, a collaboration with an international and interdisciplinary group of scientists.
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