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This book explains the quantitative reasoning required by the fractional calculus applied to complex physical, social, and biological phenomena. Fractional calculus is inextricably linked to complexity and provides a way to think systematically about complex phenomena without restricting it to a particular discipline. The book presents the case for science moving away from the fragmented disciplinary constructs devised to make the understanding of complex phenomena intellectually manageable, thus regaining its integrated nature as "natural philosophy" and underscoring how fractional calculus is the vehicle for accomplishing this task.