Conceptually, my proposed materialized phenomenological view of mathematics as a performance practice eliminates three problems in philosophy and in critical studies of symbolic systems: (1) how do we represent continuity and smoothness using discrete symbolic systems? (2) how do we represent infinite things or processes? (3) why do symbolic systems have explanatory or generative power? These conundrums have occupied philosophers of mathematics, and have driven some to propose strict finitist or radically fictionalist theories of mathematics. Concretely, this study provides a design for future technologies of mathematical writing that will better augment the working practices of differential geometers. This investigation promises both pragmatic as well as conceptual results. Focusing on differential geometry through the lens of practice, I examine a-linguistic semiotic usage, and through the lens of phenomenological investigation, I examine how geometers work with continuous, smooth, or infinite structures and processes. The critical part of this project is informed by insights from literary and performance studies as well as the mathematical sciences. (PhD Dissertation, Stanford Mathematics 2001) What sort of geometric creation and performance can or cannot be supported in a writing technology that spans freehand sketching, manipulable graphics, text, symbolic computation and simulation? How do we work with and gain intuitions about supposedly abstract things like a Riemannian manifold or an infinite-dimensional function space by manipulating material marks using chalk, pencil or computer? How do we exteriorize thought using the technologies of writing and sketching when it is about processes or objects that could not be embedded in a naive space-time? I analyze in detail what differential geometric work can or cannot be performed easily in a hybrid medium provided by a generalized multi-modal writing and sketching technology.

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