Nomenclature The Challenge of Complexity Little Science, Big Science Complexity Chapter Overviews After Thoughts Yesterday's Science Simple Linearity Complicated Linearity Nonlinear Dynamics After Thoughts Appendix Chapter 2 New Ways of Thinking Why Now? Through the Looking Glass Non-differentiablity can be Physical The size effect After Thoughts Simple Fractional Operators Random Walks Fractional Derivatives Fractional Rate Equations After Thoughts Tomorrow's Dynamics What We Think We Know; Linear Systems 126 5.2 Fractional Linear System Applications of FLE Control of Complexity Fractional Logistic Equation Fractional Leibniz Rule After Thoughts Appendix Chapter 5 Fractional Cooperation HRV and Lévy Statistics Fractional Wave Equations Turbulence Fractional Magnetization Equations Fractional Search Hypothesis The Network Effect After Thoughts Strange Statistics Fractional Hamiltonian Formalism Anomalous Transport Fractional Fokker-Planck Equation Fractional Sturm-Liouville Theory Fractional Kinetics Equation Physiology and Complexity Loss Diffusion Entropy After Thoughts Appendices Chapter 7 What have we learned? Why the fractional calculus? Conformation Bias and Scientific Truth Final Thoughts.