Temporal Topos Methods for the Philosophy of Natural Sciences

This book is a first attempt to unify and explain, through the language of pure mathematics called categories and sheaves, the mechanism of mental activities. Humanities and philosophy meet physics and mathematics at both levels of the microcosm and macrocosm.   The purpose of this book is to provide systematic and unifying methods for the physical and the cognitive aspects of a conscious entity. The philosophical thoughts of the West and the East will be formulated and presented in terms of categories and sheaves using pure mathematics.  A definition-based rigorous approach and method using categorical sheaf theory is a completely new attempt to formulate the ontological philosophies, which must be consistent with quantum physics. For example, it is the author's intent to capture reality as a pair of the cognitive realm and the physical realm, for which the author uses the notion of (temporal) topos theory, as a pair of an initial object and a terminal object of temporal topos. This book allows philosophers and natural scientists to interact to discuss and observe the physical world and the cognitive world.  That is, a brain existing in the physical world is simultaneously functioning in the cognitive world. The author provides detailed descriptions for the process from the small to the large, which is a part of descent-sheaf theory in mathematics by gluing the local data to obtain a global object. A description of a process of "understanding" (and realizing-recognizing-discovering) is also formulated in terms of temporal topos language.  The author provides all the needed mathematical background so that readers whose main interests are in humanities can enjoy and appreciate the rigorousness and the harmony of humanities and natural sciences.