Small Worlds, Mathematics, and Humanities Computing by Ryan Chartier A thesis submitted in partial fulfillment of the requirements for the degree of Master of Arts Humanities Computing University of Alberta © Ryan Chartier, 2014 Chartier ii Abstract Primarily the conversation surrounding humanities computing has been mainly focused on defining the relationship between humanities computing and conventional humanities, while the relationship humanities computing has to computers, and by extension mathematics, has been mainly ignored. The subtle effect computers have on humanist research has not been ignored, but the humanities general illiteracy surrounding computers and technology acts as a barrier that prevents a deeper understanding on these effects. This goal of this thesis is to begin a conversation about the ideas, epistemologies, and philosophies surround computers, mathematics, and computation in order to translate these ideas into their humanist counterparts. This thesis explores mathematical incompleteness, mathematical infinity, and mathematical computation in order to draw parallels between these concepts and similar concepts in the humanities: post-modernism, the romantic sublime and human experience. By drawing these parallels this thesis both provides a general overview of the ideas in mathematics relevant to humanities computing in order to assist digital humanists in correctly translating or interpreting the effects of computers on their own work and a counter argument to the commonly accepted notion that the concepts developed by mathematics are mutually exclusive to those developed in the humanities. Chartier iii Contents Abstract .......................................................................................................................................................................................... ii List of Figures ............................................................................................................................................................................. iv Chapter 1: Introduction ........................................................................................................................................................... 1 Chapter 2: Seeing the Mathematical Perspective ...................................................................................................... 12 Pure Mathematics ................................................................................................................................. 16 Applied Mathematics ............................................................................................................................. 28 Fundamentalism .................................................................................................................................... 36 Conclusion ............................................................................................................................................. 43 Chapter 3: A Brief History of Incompleteness ............................................................................................................ 45 Euclidian Geometry ............................................................................................................................... 48 Riemannian Geometry ........................................................................................................................... 59 Mathematical Consistency .................................................................................................................... 61 Mathematical Completeness ................................................................................................................. 64 Conclusion ............................................................................................................................................. 69 Chapter 4: The Mathematics of Limitations ................................................................................................................ 74 Sublime .................................................................................................................................................. 77 Limits ..................................................................................................................................................... 78 The Problem of Irrationality .................................................................................................................. 85 Set Theory .............................................................................................................................................. 87 True Infinity ........................................................................................................................................... 95 Conclusion ............................................................................................................................................. 99 Chapter 5: Virtual Reality ................................................................................................................................................. 102 Classical Computation ......................................................................................................................... 106 Quantum Computation ........................................................................................................................ 111 Mathematical Computation ................................................................................................................ 116 Human Computation ........................................................................................................................... 124 Chapter 6: Conclusion ........................................................................................................................................................ 134 Works Cited ............................................................................................................................................................................ 140 Chartier iv List of Figures Figure 1: Calvin and Hobbes ............................................................................................................................... 19 Figure 2: Constructing an equilateral triangle. .......................................................................................... 50 Figure 3: Three points on a surface. ................................................................................................................ 54 Figure 4: Graphing a function. ........................................................................................................................... 79 Chartier 1 Chapter 1: Introduction I have adopted ‘humanities computing’ in particular for certain suggestive qualities in the name: a potential still to be taken as an oxymoron, thus raising the question of what the two activities it identifies have to do with each other; the primacy it gives to the ‘humanities’, preserved as a noun in first position while functioning as an adjective, hence subordinated; and its terse, Anglo-Saxon yoking of Latinate words. I read it first as a challenge to what we think we are doing, then as its name. (McCarty 2) I was attracted to ‘humanities computing’ because of the word ‘computing’; however, to say that humanities computing is not as I expected it to be is a complete understatement, and even looking back I still do not fully understand what it was that I was expecting. I completed my undergraduate in Mathematics and Physics, and yet I found my interests pulling me towards computers and writing. To the science student, arts credits are sometimes a pleasant distraction, sometimes a university mandated-detour, and always secondary in importance to science credits. As an undergraduate, I was well aware of the division between science and art. There existed two different universes on campus, the science students and the arts students, both of whom had their own culture, their own buildings, their own dress code, and their own views on the world. Rarely, if ever, do these two pockets of reality intersect. My choice to pursue a Master of Arts was a choice of rebellion. As I progressed through my undergraduate career, certain cracks formed in the way I understood the concepts of pure and holistic science. I found the vocal rationalist movement, which was well represented in my class, to be hypocritical in their blanket dismissal of so-called ‘irrational’ subjects like religion and literature. I had reached a point where I wanted to study something new: to see the world from a different perspective. Chartier 2 I am a scientist and a mathematician; rational thinking is what I am, and it will always be what I am best at. Likewise, I have always been interested in models of the universe: explaining and understanding phenomenon as individual parts of a common whole. As a science student, I had, and to some extend still have, only a limited understanding of what it means to work in the humanities, which is why I chose to pursue a Master of Arts. To me the first step to understanding humanity as different parts of a single whole required me to understand both sides of the science / humanities rift. I like computers, I understand computers, and I have spent the better part of my life studying the philosophy behind them. Here was an opportunity to take that knowledge of computers and use it in a way different from how I was taught, offer that skill to another group who needed