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This book offers a systemic perspective on cognition and the historical development of mathematics. First the yoyo model is discussed: a recent development of systems research, which has brought forward revolutionary applications of systems research in various fields of traditional disciplines. After revealing the systemic structure of thought, mathematics is analyzed by using the age-old concepts of actual and potential infinities. A new look is cast at some important paradoxes, each of which played a crucial role in the development of mathematics. Attention is then turned to constructing the logical foundation of two different systems of mathematics, one assuming that actual infinity is different than potential infinity, and the other that these infinities are the same.This volume will be of interest to academic researchers, students and professionals in the areas of systems science, mathematics, philosophy of mathematics, and philosophy of science.