Type of Document Dissertation Author Hammontree, John Stephen Author's Email Address steve.hammontree@vanderbilt.edu URN etd-04172014-220702 Title Hypercomplex-Transfinite Numbers: A Theory of Chaos, Totality and Modality Degree PhD Department Philosophy Advisory Committee

Advisor Name Title Jeffrey S. Tlumak Committee Chair John Lachs Committee Member Michael Hodges Committee Member Steven T. Tschantz Committee Member Keywords

- Philosophy of Mathematics
- Neoplatonism
- Modality
Date of Defense 2014-04-27 Availability unrestricted AbstractThe present discussion considers how chaos theory might become positioned to subsume modal categories of necessity, determinism, free will and contingency. The strategy is to: (1) propose a theory of transfinite probability as a science of contingency, (2) propose a theory of hypercomplex-transfinite (HT) numbers as a science of necessity, (3) demonstrate that these two theories are isomorphic to one another, and (4) produce an algebraic synthesis of these theories by means of HT number theory and HT modality, thereby producing a synthesis of necessity and contingency. Because the logical structure of the Mandelbrot fractal is the same as the logical structure of HT number theory, chaos theory has the same logical power as HT theory, and can thus be interpreted as deeply involved in the synthesis accomplished through HT modality.Chapters 1–3 present the background and context for the philosophical concerns motivating the present study. Chapter 1 describes the classical dichotomy of Being vs. Nonbeing, and then describes how Neoplatonists of the via negativa envision a synthesis of these modal opposites. Chapter 2 describes the twentieth century crisis in mathematical foundations, the field of study most widely regarded as a locus classicus of logical necessity. Chapter 3 describes modal problems in broad outline, giving special emphasis to questions of contingency. Chapter 4 proposes a theory of transfinite probability, contrasting it with the standard mathematical theory of classical probability. Chapter 5 proposes a unified theory of hypercomplex-transfinite (HT) numbers. This system gives expression to an HT ramified zigzag theory. Chapter 6 shows that, not only does HT ramified zigzag solve the set theoretic paradoxes and Gödel’s incompleteness theorem, but it effects a synthesis of HT numbers and transfinite probability within the rubric of HT modality, thus accomplishing the fundamental objective of the present study. Chapter 7 reflects on how the material proposed in chapters 4–6 relates to the issues discussed in chapters 1–3, and reorients questions of modality in terms of the philosophical-mathematical program proposed here. Interestingly, HT theory implies that P = NP.

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