The Paradigm of Early Greek Mathematics

Applying Kuhn (Chapters 1–3) to Early Greek Mathematics

In what ways was the compass-and-straightedge tradition of early Greek mathematics a paradigm in Kuhn’s sense? How did it shape the research questions, methods, and standards of proof for generations of mathematicians?

Kuhn’s Structure

In Chapters 1–3 of The Structure of Scientific Revolutions, Thomas Kuhn establishes foundational concepts regarding the sociology and epistemology of scientific developmena.t

  • Pre-Paradigm Phase: Disorganized fact-gathering, competing foundational schools, lack of shared standards, and constant debates over elementary principles.
  • Paradigm Emergence: A transformative achievement that attracts an enduring group of adherents while leaving open-ended problems for the redefined group of practitioners to resolve.
  • Normal Science as “Mopping Up”: Research firmly based upon one or more past scientific achievements. It focuses on three primary classes of factual problems:
    1. Determination of significant fact: Increasing the precision and scope of facts the paradigm demonstrates are particularly revealing.
    2. Matching facts with theory: Demonstrating direct agreement between paradigm predictions and nature.
    3. Articulation of theory: Resolving ambiguities, determining physical constants, and reformulating laws to expand the scope of the paradigm.
  • Normal Science as Puzzle-Solving: Research guided by strict rules, shared instrumentation, theoretical criteria, and an assured outcome, where failure to achieve a solution reflects on the investigator rather than the paradigm.

Assignment Instructions

Using Dunham’s text as a starting point, answer the following questions in a short essay (aim for less than 1000 words). Be sure to support your claims with specific examples from the text and any additional sources you consult. Be concise, and to the point.

  • What was the pre-paradigm phase of mathematics before the Greek tradition?
  • Identify a transformative achievement that helped to establish the compass-and-straightedge tradition (Greek mathematics) as a paradigm? Explain why it was transformative.
  • How did the compass-and-straightedge tradition shape the research questions?
  • How did it shape the methods and standards of proof for generations of mathematicians?
  • Briefly discuss two problems that were outside of the scope of the paradigm of early Greek mathematics.
  • Due: September 3