Andrew Arana
University of Notre Dame

"Methodological purity as a mathematical ideal"

ABSTRACT: The principle of methodological purity, roughly speaking, says that we should prove propositions, or solve problems, of one kind, using only propositions or methods of that same kind. This principle has been held as an ideal by many thinkers (philosophers and mathematicians) going back to Aristotle. We explain why one might hold such a view, focusing on an account rooted in Aristotelian concerns about knowledge. We then present two case studies from the history of mathematics that appear to violate this principle and explain their consequences. We conclude by pointing out several potential resolutions of these consequences.

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