What If We Change Our Axioms? A Feminist Inquiry into the Foundations of Mathematics

Introduction

Modern mathematics is based on the axiomatic method. We choose axioms and a deductive system—rules for deducing theorems from the axioms. This methodology is designed to guarantee that we can proceed from “obviously” true premises to true conclusions, via inferences which are “obviously” truth-preserving. The study of this method is itself a branch of mathematics called Foundations, an appropriate metaphor, as the mathematicians who initiated this field were attempting to reconstruct mathematics on a secure basis. This quest for certainty, ¹ the desire for an ultimate guarantor of Truth unsituated in time, space or history, the dream of order, regularity, repeatability, predictability, ² and the ideal of a “pure” disembodied reason that is the hallmark of western mathematics, was bequeathed to us by the early Greeks. This mind-independent ³ knowledge is usually called “objective.” The salient properties of objective knowledge are its impersonality, verfiability, (en)durability, and transcendence. According to one mathematician, RH (on a recent e-mail discussion list):

Anything that doesn’t have these attributes is not knowledge. I don’t think of math as “objective”; I think of it as true. I don’t see that the “subject” enters into the subject at all. We represent in an innocent pure context-free way as far as we can, because that is the way to the truth. In possessing the truth there is power. If we cease to represent things in an innocent, context-free way (for example, if we start special pleading for political causes) we lose access to the truth and access to power.

This is the vision of a mathematics that “sets the standard of objective truth for all intellectual endeavors.” ⁴

Critics ⁵ have challenged this fantasy of “self-empowerment through purity and control.” ⁶ They have contested the belief that it is possible (or even desirable) to arrive at any knowledge that is completely context-free, independent of who knows what and when. New and interesting questions arise if we give up as myth the claim that our theorizing can ever be separated out from the complex dynamic of interwoven social/political/historical/cultural forces that shape our experiences and views. Considering mathematics as a set of stories produced according to strict rules, ⁷ one can read these stories for what they tell us about the very real human desires, ambitions, and values of the authors (who understands) and listen to the authors as spokespersons for their cultures (where and when). This paper is the self-reflective and self-conscious attempt of a mathematician to re-tell a story of mathematics that attends to the relationships between who we are and what we know.

The Greek Story

My retelling begins with Parmenides, a Greek philosopher of the sixth century B.C. ⁸ His main extant work is a poem entitled “On Truth.” He believed that Reality, the One, true Being, must be changeless, consistent, homogeneous, “autonomous and explainable in its own terms, a perfect unified self subsistent whole.” ⁹ In his poem, Parmenides tells the story of his journey to his heart’s desire, logical truth. The quest culminates in a meeting and conversation with the goddess of truth, who reveals to him the sacred teachings of “what is.” Philosophers have long decried as “unfortunate” ¹⁰ the choice of poetry to describe a search for logical truth. Logic, created as a language to transcend natural language, is supposed to be perfectly transparent, simply revealing that thought which exists apart from any individual. The deficiencies of natural language that are the very basis of poetry—its ambiguity, metaphoricity, shifting multiplicities of meaning, object descriptions that are not delineated sharply (pinned down)—have to be excised ¹¹ in order to “capture” meaning. Logical truth (like objective knowledge) must be independent of both its origins (genesis) and the person (man) who speaks it. In philosophy, committing either the genetic fallacy or the ad hominem fallacy is an error in reasoning tantamount to sinning, and invalidates whatever point (argument) one is making.

Yet this is precisely what makes Parmenides’s...