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Abstract
This review paper is devoted to study the conceptual difficulties that mathematics meets when attempting to describe the complexity of living matter focusing on the challenging perspective of developing a mathematical theory for living systems including mutations and selection. The quest starts with the identification of a number of common complexity features of living systems. Then, mathematical structures are derived to include these features, while mathematical models are derived by inserting in the structures models of individual based interactions. Three applications are examined by active particles methods, i.e., models of SARS2-CoV-2 pandemics, models of idiosyncratic learning in open markets and of the dynamics of prices accounting for human behaviors. A critical study, which pervades the whole paper, shows that also economics can be viewed as a behavioral science thus accounting for specific aspects typical of living systems.
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