Martin Heidegger's Mathematical Dialectic: Uncovering the Structure of Modernity

Martin Heidegger is generally regarded as one of the most significant—if also the most controversial—philosophers of the 20th century. Most scholarly engagement with Heidegger’s thought on Modernity approaches his work with a special focus on either his critique of technology, or on his more general critique of subjectivity. This dissertation project attempts to elucidate Martin Heidegger’s diagnosis of modernity, and, by extension, his thought as a whole, from the neglected standpoint of his understanding of mathematics, which he explicitly identifies as the essence of modernity.

Accordingly, our project attempts to work through the development of Modernity, as Heidegger understands it, on the basis of what we call a “mathematical dialectic.“ The basis of our analysis is that Heidegger’s understanding of Modernity, both on its own terms and in the context of his theory of history [Seinsgeschichte], is best understood in terms of the interaction between two essential, “mathematical” characteristics, namely, self-grounding and homogeneity. This project first investigates the mathematical qualities of these components of Modernity individually, and then attempts to trace the historical and philosophical development of Modernity on the basis of the interaction between these two components—an interaction that is, we argue, itself regulated by the structure of the mathematical, according to Heidegger’s understanding of the term.

The project undertaken here intends not only to serve as an interpretive, scholarly function of elucidating Heidegger’s understanding of Modernity, but also to advance the larger aim of defending the prescience, structural coherence, and relevance of Heidegger’s diagnosis of Modernity as such.

Ground and Electronic Excited States from Pairing Matrix Fluctuation and Particle-Particle Random Phase Approximation

The accurate description of ground and electronic excited states is an important and challenging topic in quantum chemistry. The pairing matrix fluctuation, as a counterpart of the density fluctuation, is applied to this topic. From the pairing matrix fluctuation, the exact electron correlation energy as well as two electron addition/removal energies can be extracted. Therefore, both ground state and excited states energies can be obtained and they are in principle exact with a complete knowledge of the pairing matrix fluctuation. In practice, considering the exact pairing matrix fluctuation is unknown, we adopt its simple approximation --- the particle-particle random phase approximation (pp-RPA) --- for ground and excited states calculations. The algorithms for accelerating the pp-RPA calculation, including spin separation, spin adaptation, as well as an iterative Davidson method, are developed. For ground states correlation descriptions, the results obtained from pp-RPA are usually comparable to and can be more accurate than those from traditional particle-hole random phase approximation (ph-RPA). For excited states, the pp-RPA is able to describe double, Rydberg, and charge transfer excitations, which are challenging for conventional time-dependent density functional theory (TDDFT). Although the pp-RPA intrinsically cannot describe those excitations excited from the orbitals below the highest occupied molecular orbital (HOMO), its performances on those single excitations that can be captured are comparable to TDDFT. The pp-RPA for excitation calculation is further applied to challenging diradical problems and is used to unveil the nature of the ground and electronic excited states of higher acenes. The pp-RPA and the corresponding Tamm-Dancoff approximation (pp-TDA) are also applied to conical intersections, an important concept in nonadiabatic dynamics. Their good description of the double-cone feature of conical intersections is in sharp contrast to the failure of TDDFT. All in all, the pairing matrix fluctuation opens up new channel of thinking for quantum chemistry, and the pp-RPA is a promising method in describing ground and electronic excited states.