9.3 Medieval universities and scholastic mathematics

Medieval universities played a crucial role in advancing mathematics during the Middle Ages. They provided a structured environment for learning and research, combining classical knowledge with new ideas. The curriculum centered on the liberal arts, including mathematical subjects in the quadrivium.

The Oxford Calculators, a group of scholars at Merton College, made significant contributions to mathematics and physics. They developed new approaches to quantifying motion and change, laying the groundwork for future advancements in calculus and analytical geometry.

Medieval University Curriculum

Liberal Arts Foundation: Trivium and Quadrivium

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• Trivium formed the foundation of medieval university education
  • Grammar focused on Latin language and literature
  • Logic taught formal reasoning and argumentation techniques
  • Rhetoric developed persuasive speaking and writing skills
• Quadrivium built upon the Trivium with mathematical subjects
  • Arithmetic covered number theory and basic calculations
  • Geometry explored Euclidean principles and spatial relationships
  • Astronomy studied celestial bodies and their movements
  • Music examined mathematical relationships in harmonics and rhythm
• Liberal arts curriculum prepared students for advanced studies in theology, law, or medicine

Scholasticism: Method and Philosophy

• Scholasticism emerged as the dominant intellectual approach in medieval universities
• Emphasized logical reasoning and critical analysis of authoritative texts
• Utilized dialectical method to reconcile contradictions in philosophical and theological writings
• Key figures in scholastic tradition included Thomas Aquinas and Peter Abelard
• Quaestio disputata served as a primary teaching tool
  • Involved formal debates on specific topics
  • Students practiced defending and refuting arguments
• Scholastic approach influenced mathematical studies
  • Encouraged precise definitions and logical proofs
  • Led to advancements in logic and early foundations of mathematical rigor

Oxford Calculators

Merton College and the Oxford School

• Merton College at Oxford University became a center for mathematical innovation in the 14th century
• Group of scholars known as the Oxford Calculators or Merton School emerged
  • Focused on applying mathematical reasoning to physical problems
  • Developed new approaches to quantifying motion and change
• Notable members included Thomas Bradwardine, William Heytesbury, and Richard Swineshead
• Merton College provided institutional support for mathematical research
  • Well-stocked library with mathematical and scientific texts
  • Collaborative environment fostered intellectual exchange

Key Figures and Their Contributions

• Nicole Oresme made significant advancements in mathematics and natural philosophy
  • Developed early concepts of analytical geometry
  • Proposed graphical representations of variable quantities
  • Explored infinite series and the nature of continuity
• Thomas Bradwardine applied mathematics to theology and physics
  • Authored "Tractatus de Proportionibus" examining relationships between velocity, force, and resistance
  • Contributed to the development of logarithmic concepts
• Richard Swineshead, known as "Calculator," wrote influential works on physics and mathematics
  • "Liber Calculationum" addressed problems of motion, intensity, and infinity
  • Explored concepts of acceleration and deceleration

Mean Speed Theorem and Its Implications

• Mean Speed Theorem, also known as the Merton Rule, represented a major breakthrough in kinematics
• Stated that a uniformly accelerating body travels the same distance as a body moving at a constant velocity equal to the mean of the initial and final velocities
• Formulated mathematically as: $s = \frac{1}{2}(v_0 + v_f)t$ Where s is distance, v₀ is initial velocity, vf is final velocity, and t is time
• Theorem laid the groundwork for later developments in calculus and physics
  • Provided a method for calculating distance traveled under acceleration
  • Influenced Galileo's work on falling bodies in the 16th century
• Application of the theorem extended beyond physics
  • Used in economic calculations and financial modeling
  • Contributed to the development of graphical representations of motion and change