Semester

Spring

Date of Graduation

2024

Document Type

Dissertation

Degree Type

PhD

College

College of Applied Human Sciences

Committee Chair

Sharon Hayes

Committee Co-Chair

David Miller

Committee Member

Matthew Campbell

Committee Member

Scott Davidson

Abstract

In this dissertation, I utilize the post-structural philosophy of Gilles Deleuze and Félix Guattari as a lens for investigating the proof process. Deleuze and Guattari were both post- structural philosophers who, like many in this tradition, troubled traditional notions related to stable identities, meaning, language, and mathematics. For Deleuze, sense and meaning is not the result of a sterile, transcendent effect or condition that is associated with propositions and states of affairs. Rather, it is the result of a material production that emerges from the world and that has independence from language and the mind. I apply this framework to the notion of proof, situating the work of the mathematician not in terms of the problem-solving subject, cognition, and mental processes but rather as something that emerges from constructed material assemblages.

I demonstrate this theoretical approach via a story written by Dr. David Neel from the 2019 book Living Proof: Stories of Resilience Along the Mathematical Journey published by the American Mathematical Society. Neel’s story is an autobiographical account that takes place near the end of his dissertation work in which both he and his supervisor go hiking while working out the details regarding a challenging result that is related to his research. This story illustrates that, while proof involves logical reasoning and mental processes, these aspects are only part of the construction. The axiomatic proof which they try to obtain becomes entangled with the hike and, at the level of the assemblage, what they are constructing is a “proof-hike” that serves as the genesis for the production of an actualized proof solution. Not only does this “proof-hike” eventually lead to a mathematical resolution but it also leads to a sort of ethical disposition in which Neel (2019) sees the axiomatic aspect of mathematics as only a part of an entire universe that opens up to him. The result is an intensity, an “affect,” that is both compassionate as well as embracing of the “cosmic” wonder of proof at its material genesis on the plane of immanence. It is a becoming-proof.

Drawing on the literature, I end the dissertation by formulating potential possibilities for how this approach might affect our thinking about the teaching and learning of proof. For example, I suggest approaching proof from a problematic standpoint, as opposed to an axiomatic one, where the role of exploration and wonder is foregrounded rather than solutions that lead to theorems. These approaches, I argue, have the potential to deterritorialize proof and proof instruction so that those who may not feel connected to the axiomatic side of proof can still desire it immanently.

Download

Share

COinS