Doctor of Philosophy, The Ohio State University, 2019, Philosophy

The primary aim of this dissertation is to discuss the epistemological fallout of Godel's Incompleteness Theorems on Hilbert's Program. In particular our focus will be on the philosophical upshot of certain proof-theoretic results in the literature. We begin by sketching the historical development up to, and including, Hilbert's mature program, discussing Hilbert's views in both their mathematical and their philosophical guises. Godel's Incompleteness Theorems are standardly taken as showing that Hilbert's Program, as intended, fails. Michael Detlefsen maintains that they do not. Detlefsen's arguments are the focus of chapter 3. The argument from the first incompleteness theorem, as presented by Detlefsen, takes the form of a dilemma to the effect that either the infinitistic theory is incomplete with respect to a certain subclass of real sentences or it is not a conservative extension over the finitistic theory. He contends that Hilbert need not be committed to either of these horns, and, as such the argument from the first incompleteness theorem does no damage to Hilbert's program. His argument against the second incompleteness theorem as refuting Hilbert's Program, what he calls the stability problem, concerns the particular formalization of the consistency statement shown unprovable by Godel's theorem, and endorses what are called Rosser systems. The success of Detlefsen's arguments critically depends upon the precise characterization of what exactly Hilbert's program is. It is our contention that despite Detlefsen's attempts, both of the arguments (from the first and second incompleteness theorems) are devastating to Hilbert. The view that Detlefsen puts forth is better understood as a modified version of Hilbert's general program cast as a particularly strict form of instrumentalism. We end by analyzing the coherence of Detlefsen's proposal, independently of the historical Hilbert. In response to Godel's Incompleteness theorems several modified or partia (open full item for complete abstract)

Committee: Neil Tennant (Advisor); Stewart Shapiro (Committee Member); Christopher Pincock (Committee Member) Subjects: Philosophy

Doctor of Philosophy (PhD), Ohio University, 2014, Curriculum and Instruction Mathematics Education (Education)

Deciding on the truth value of mathematical statements is an essential aspect of mathematical practice in which students are rarely engaged. This study explored undergraduate students' approaches to mathematical statements with unknown truth values. The research questions were 1. In what ways and to what extent do students use intuition and analysis to decide on the truth value of mathematical statements? 2. What are the connections between students' process of deciding on the truth value of mathematical statements and their ability to construct associated proofs and counterexamples? 3. What types of systematic intuitive, mathematical, and logical errors do students make during the proving process, and what is the impact of these errors on the proving process? Clinical task-based interviews utilizing the think-aloud method revealed students' reasoning processes in depth. Twelve undergraduate students each completed four mathematical tasks requiring them to decide on the truth value of a statement and prove or disprove it accordingly. Through analysis of the data, I developed a framework for distinguishing among types of reasoning based on their cognitive and mathematical properties. The framework identifies four distinct categories of reasoning – intuitive, semantic-empirical, semantic-deductive, and syntactic – each with subcategories. The students in this study used all four types of reasoning for deciding on the truth value of the statements in the tasks. Their use of semantic-deductive and syntactic reasoning mirrored mathematicians' use of these reasoning types for decision-making. With the exception of one task, the students' decision-making and construction processes were generally connected. Connections in which the construction process was based on decision-making process mostly facilitated proving. However, simultaneous decision-making and construction processes often led to overturned decisions. Regarding intuitive decision-makin (open full item for complete abstract)

Committee: Robert Klein (Committee Co-Chair); Allyson Hallman-Thrasher (Committee Co-Chair) Subjects: Mathematics Education

PHD, Kent State University, 2007, College of Education, Health, and Human Services / Department of Teaching, Leadership and Curriculum Studies

Research projects have long indicated incompetence in justification of mathematical arguments among students of high school as well as college level. This research sought to understand the mental proof schemes that students possess, and to determine if these schemes progress when the Moore method is used in teaching a mathematics course. This study is significant in two major ways: First, it confirmed the mental schemes of proofs proposed by Harel & Sowder. Second, the study examined the effect of the Moore method on students' learning and appreciation of proofs. Using a qualitative analysis design the study showed that the Moore method has positive affects on students' conceptualization of mathematical proof, their self-confidence in their abilities, their appreciation of the relevance of proofs, and their ability to think autonomously. The Moore method allowed students to experience mathematics first hand. They built a coherent body of knowledge in which they created their own proofs.

Committee: Michael Mikusa Genevieve Davis (Advisor) Subjects: