This book offers a systemic perspective on cognition and the development of mathematics. First the yoyo model is discussed: a recent development of systems research which has yielded revolutionary applications. 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 then turns 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.

Dr. Jeffrey Yi-Lin Forrest, also known as Yi Lin, holds all his educational degrees (BS, MS, and PhD) in pure mathematics from Northwestern University (China) and Auburn University (USA) and had one year of postdoctoral experience in statistics at Carnegie Mellon University (USA). Currently, he is a guest and specially appointed professor in economics, finance, systems science, and mathematics at several major universities in China, including Huazhong University of Science and Technology, National University of Defense Technology, Nanjing University of Aeronautics and Astronautics, and a tenured professor of mathematics at the Pennsylvania State System of Higher Education (Slippery Rock campus). Since 1993, he has been serving as the president of the International Institute for General Systems Studies, Inc. Along with various professional endeavors he organized, Dr. Forrest has had the honor to mobilize scholars from over 80 countries representing more than 50 different scientific disciplines.
Over the years, he has served on the editorial boards of 11 professional journals, including Kybernetes: The International Journal of Cybernetics, Systems and Management Sciences, Journal of Systems Science and Complexity, International Journal of General Systems, and Advances in Systems Science and Applications. And, he is the editor of the book series entitled Systems Evaluation, Prediction and Decision-Making, and the editor of the book series "Communications in Cybernetics, Systems Science and Engineering", both published by Taylor and Francis with the former since 2008 and the latter since 2011.
Some of Dr. Forrest's research was funded by the United Nations, the State of Pennsylvania, the National Science Foundation of China, and the German National Research Center for Information Architecture and Software Technology.
Professor Jeffrey Forrest's professional career started in 1984 when his first paper was published. His research interests are mainly in the area of systems research and applications in a wide-ranging number of disciplines of the traditional science, such as mathematical modeling, foundations of mathematics, data analysis, theory and methods of predictions of disastrous natural events, economics and finance, management science, philosophy of science, etc. As of the end of the summer of 2013, he had published over 300 research papers and over 40 monographs and edited special topic volumes by such prestigious publishers as Academic Press, Elsevier, Kluwer Academic, Springer, Taylor and Francis, Wiley, World Scientific, and others. Throughout his career, Dr. Jeffrey Forrest's scientific achievements have been recognized by various professional organizations and academic publishers. In 2001, he was inducted into the honorary fellowship of the World Organization of Systems and Cybernetics.

Part 1 Basics 1. Where everything starts 2. Elementary properties of systemic yoyos Part 2 The mind 3. Human body as a system 4. The four human endowments 5. Character and thought Part 3 Mathematics seen as a systemic flow: A case study 6. A brief history of mathematics 7. Actual and potential infinities 8. Are actual and potential infinity the same? 9. Inconsistencies of modern mathematics Part 4 Next stage of mathematics as a systemic field of thought 10. Calculus without limit 11. A new look at some historically important paradoxes 12. An attempt to rebuild the system of mathematics Afterword A systemic yoyo model prediction Part 5 Appendices: A close look at the systemic yoyo model Appendix A Theoretical foundation of the yoyo model Appendix B Empirical evidence of the yoyo model