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Henk Bos (Aarhus, Utrecht) Luminy, 17/04/07

Arithmetic , Geometry , Algebra and Analysis as historiographical categories for early modern mathematics. Henk Bos (Aarhus, Utrecht) Luminy, 17/04/07. Historiographical Categories. Why? Redefining Geometrical Exactness 1590 – 1650: “Merging of geometry and algebra”

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Henk Bos (Aarhus, Utrecht) Luminy, 17/04/07

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  1. Arithmetic, Geometry, Algebraand Analysisas historiographical categoriesfor early modern mathematics Henk Bos (Aarhus, Utrecht) Luminy, 17/04/07

  2. Historiographical Categories Why? • Redefining Geometrical Exactness • 1590 – 1650: “Merging of geometry and algebra” • Why did it take so long? • To understand the obstacles against the merging • A methodological exercise

  3. Terminology Objects • Number(s)Rational! • Geometrical Magnitude(s)Line segments, not numbers! (no unit) • Abstract Magnitude(s)Algebra Speciosa (Viète, van Roomen) • Ratio(s)Not fractions!

  4. Terminology Operations • Arithmetical operations (on numbers) • Algebraic operations (on objects for which they are explicitly defined) • Geometrical constructions (on geometrical magnitudes) • Symbolic operations (on abstract magnitudes) • Ratio operations (on ratios)

  5. Terminology Modes of objects within arguments: • Given: supposed to be available in such a way that operations and comparisons can be performed on them;The modern term would be indeterminate. • Unknown: required to be determined by construction or (numerical or algebraic) computation.

  6. Category: Arithmetic The mathematical theory and practices that deal with numbers.

  7. Category: Geometry The mathematical theory and practices that deal with geometrical magnitudes.

  8. Category: Algebra Mathematical theories and practices that • Deal either with numbers or with (geometrical or abstract) magnitudes • Involve unknowns and/or indeterminates • Employ the algebraic operations • Involve equations

  9. Category: Analysis Mathematical methods • For finding the solutions (in geometry: the constructions) of problems • Or (but rarely) for finding proofs of theorems • Doing so by introducing unknowns. • In geometry:- If algebra is used: “Analysis nova”- If not: “Analysis veterum”, based on Euclid’s Data

  10. Categorization Comments • Result: Understanding the obstacles against merging algebra and geometry • The case of Analysis • Mathematics as activity

  11. Merging algebra and geometry Obstacles: • Irrational numbers • Multiplication, dimension and the unit • Numerical and geometrical exactness • Numerical and geometrical indeterminates (new!)

  12. Merging algebra and geometry Obstacles: • Irrational numbersAbout objects • Multiplication, dimension and the unitAbout operations • Numerical and geometrical exactnessAbout operations • Numerical and geometrical indeterminates About modes

  13. Merging algebra and geometry Merging objects: • Numbers are not Magnitudes Merging Operations: • No merging, Algebraic operations took over and wiped out the geometrical ones • Algebraic exactness difficult to define Dealing with Modes • Problematic in numerical algebra but natural in geometry

  14. Merging algebra and geometry Obstacle 3: Numerical and geometrical exactness • Arithmetical operations: exact • Algebraic operations:- In Arithmetic: not exact (approximations)- In Geometry: Quadratic operations exact, others may be • Geometrical operations: Exact if acceptable • Symbolic operations: Exactness undefined • Ratio operations: PM

  15. Merging algebra and geometry Obstacle 4: Numerical and geometrical unknowns and indeterminates The case of Analysis

  16. Mathematics as activity Modes: unknowns, indeterminates • They refer to the activity of the mathematician, not to the mathematical object • Inconsistent with a naïve Platonic view of mathematics • Yet essential for Algebra and Analysis • I.e. for a major part of mathematics

  17. Speculation 18th – 20th centuries: • The concept of “given” has lost its function“Unknowns” have become rootsEquations have become objectsProblems have become theorems • Analysis has been reduced to Platonic object status • The activity is no longer reflected inm the concepts

  18. Discussion

  19. RESERVE

  20. The case of Analysis • The classical geometrical analysis argument Given Required (unknown) • Suppose it done (counterfactual!) • Mark the givens and the unknowns esp. the required unknowns • Argue – with Euclid’s Data: • “If Objects A, B, C are given than Objects P, Q, R are given too • Argue such that modes of unknowns change to given Until the modes of the required unknowns have changed to given

  21. The case of Analysis • they are strange • these modes change during the argument • Example: the classical analysis argument • Given • Required (unknown) • Suppose it done (counterfactual!) • Mark the givens and the unknowns esp. the required unknowns • Argue – with Euclid’s Data: • “If Objects A, B, C are given than Objects P, Q, R are given too • Argue such that modes of unknowns change to given Until the modes of the required unknowns have changed to given

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