Mathematics, cognition, and you!

In what follows I argue for an epistemic bridge principle that allows us to move from real mathematics to ideal mathematics (and back again) without losing anything that is characteristic of either methodological class. Mathematics is a collection of actions performed in the pursuit of mathematical understanding. The actions are the processes of proving claims that come in such forms as lemmas, theorems, or conjectures. These proofs can be accomplished through a variety of means including (though not limited to) logical deduction, geometric intuition, diagramming, or computer assistance. What is common to each of these is the characteristic of being convincing to a sound mathematical mind. Even though what in particular makes each of these methods of proof convincing differs, that they are convincing is enough to usher forth mathematical understanding. The chapters of this dissertation explore (i) a new naturalistic metaphysics for under- standing “where mathematics comes from”, (ii) recent psychological findings in the the nature of mathematical reasoning, (iii) the concatenation of two historical forms of reasoning (real and ideal), (iv) the possibility of an epistemic bridge between real and ideal methods, and (v) the implication of this new bridge principle for the long-standing concern that Go ̈del’s Incompleteness Theorems shake the foundation of modern mathematics.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Philosophy

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2017

Author dc:creator

• Bohall, Jordan Michael

Contributors dc:contributor

• Carr, Jennifer
• Sanders, Kirk
• Waskan, Jon
• Hummel, John

• Philosophy
• Mathematics

Statement dc:rights

• Copyright 2017 Jordan Bohall

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/97532

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/97532