Department of Philosophy

318 Old Main,

University of Arkansas

Fayetteville, AR 72701

P 479-575-3551

F 479-575-2029

E-mail: phildept@uark.edu

# Philosophy and Mathematics

Philosophy and mathematics have had a long and fruitful interaction. Zeno’s paradoxes were constructed with the philosophical purpose of demonstrating that all change is illusory, but raise important and challenging questions about geometry, measure, limits, and the nature of infinity. Gottlob Frege’s interest in the nature of mathematical truth prompted his invention of modern mathematical logic, a subject of central importance in understanding the nature of proof and of computation more generally. David Hilbert’s philosophical view that mathematical truth reduces to provability prompted mathematical work of great profundity, notably Gödel’s incompleteness theorems and Alan Turing’s related work on computability. Concerns about the nature of numbers, sets, and the paradoxes of set theory, have animated the work of both philosophers and mathematicians.

Mathematics is philosophically interesting. It raises profound *metaphysical *questions: *what* are numbers or sets or mathematical objects more generally? What makes an axiom
of a mathematical theory *true*? Indeed, are mathematical claims the kinds of things that can be true of false at
all? It also raises challenging *epistemological* questions: given that they do not seem to be part of the physical world, how do
we know about numbers and other such abstract entities? Philosophy can also be beneficial
to the mathematician. Formal logic is both a subject of mathematical interest in
its own right, and a gateway to the mathematical study of provability and computation.
Moreover, understanding what genuinely follows (and does not follow) from esoteric
results such as Gödel’s first incompleteness theorem demands careful discussion and
elucidation of the relevant concepts, the kind of activity in which philosophers specialize.
For mathematicians with an interest in physics, philosophical discussions of quantum
mechanics and of space, time, spacetime and the associated geometries should be of
interest.

The study of both philosophy and mathematics is much greater than the sum of its parts, since each subject can profoundly illuminate the other. Moreover, majoring in both will provide students with an especially broad intellectual skill set, combining the mathematician’s formal sophistication with the conceptual clarity and critical acuity developed by studying philosophy.

Relevant Philosophy Courses

- PHIL 1003: Critical Reasoning: Discovery, Deduction, and Intellectual Self-Defense
- PHIL 2203. Introduction to Logic
- PHIL 3943.Philosophy and Physics: Quantum Theory and the Measurement Problem
- PHIL 3943.Philosophy and Physics: Space, Time, Time Travel, and Time Machines
- PHIL 3943. Philosophy and Physics: Philosophy of Space and Time
- PHIL/MATH 4253.Symbolic Logic I
- PHIL 4093. Gödel’s Theorems
- PHIL 4073: History of Analytic Philosophy
- PHIL 4233: Philosophy of Language
- PHIL 4203. Theory of Knowledge
- PHIL 4603. Metaphysics

If you have questions, contact Professor Barry Ward (bmward@uark.edu).