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Explore the complex and contested nature of algebra as a mathematical discipline through this conference talk that examines two pivotal historical case studies. Delve into how both historical mathematicians and modern historians have grappled with defining algebra, often making choices based on tacit assumptions about its purpose, usage, language, and objects. Investigate Isaac Newton's approach to algebra by examining the specific context, tasks, and values that shaped his algebraic practice, which differ significantly from how 17th-century mathematics is typically narrated today. Analyze the Abel-Ruffini theorem as a second case study, understanding how this proof of the impossibility of solving general polynomial equations of degree five or higher not only represented a mathematical breakthrough but fundamentally reconfigured algebra's disciplinary boundaries and objects, with particular attention to the role of commutativity in this transformation. Discover how these case studies reveal strikingly different conceptions of algebra across different historical periods and contexts. Engage with the methodological challenge of contextualizing algebra as a discipline, considering both the perspectives of historical actors and present-day historians and mathematicians who interpret and edit past mathematical texts. Examine how the interplay between historical and contemporary categories creates opportunities for dialogue between historians and mathematicians, ultimately exposing the problematic and contingent assumptions underlying various definitions of algebra throughout history.
