Heuristics in Ancient Arabic and Chinese Mathematics and its use in textbooks

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Heuristics in Ancient Arabic and Chinese
Mathematics and its use in textbooks
Prof. Dr. Bernd Zimmermann from University of
Jena at University of Xian August
2002
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Heuristics

• Methods to find conjectures
• Methods to find proofs
• Methods to (re)invent mathematics
• By analysis of history one might find
  methods/heuristics, which proved to be most
  fruitful (invariants)

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Example 1 Analogy
Archimedes
? Kepler
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Example 2 Analysis

• Now, analysis is the path from what one is
  seeking, as if it were established, by way of its
  consequences, to something that is established by
  synthesis.
• That is to say, in analysis we assume what is
  sought as if it has been achieved, and look for
  the thing from which it follows, and again what
  comes before that, until by regressing in this
  way we come upon some one of the things that are
  already known, or that occupy the rank of a first
  principle. We call this kind of method
  'analysis', as if to say anapalin lysis
  (reduction backward).
• In synthesis, by reversal, we assume what was
  obtained last in the analysis to have been
  achieved already, and, setting now in natural
  order, as precedents, what before were following,
  and fitting them to each other, we attain the end
  of the construction of what was sought. This is
  what we call 'synthesis'.
• (Pappos in Jones A. (ed. . transl.) Pappus of
  Alexandria. Book 7 of the Collection. Part 1.
  Springer, New York 1986. 1986, p. 82)

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Ibn al Haitham, the method of analysis and
perfect numbers

• Jaouiche, K. Ibn al Haitham Kitab at-tahlil
  wa-t-tarkib. Ouvrage dal-H,.asan ibn al
  al-H,.asan ibn al Haitham sur lanalyse et la
  synthèse. Unpublished manuscript Paris 1991.
• Rashed, R. Ibn al-Haytham et les nombres
  parfaits. In Historia Mathematica 16 (1989),
  343-352.
• Hogendijk, J. P. Review of Rashed 1989,
  Mathematical Reviews Sections, 91d01002 01A30
  01A20 11-03, S. 1822, April 1991-Issue 91d.

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Ibn al Haitham, the method of analysis and
perfect numbers

• Euclid Prop. 36 If as many numbers as we please
  beginning from a unit be set out continuously in
  double proportion, until the sum of all becomes
  prime, and if the sum multiplied into the last
  make some number, than the number is perfect.
  (Heath T. L. The Thirteen Books of Euclids
  Elements. Cambridge University Press, Cambridge
  1925. Vol. 2, p. 421)
• Modern form If m(1222232n)2n and
  (1222232n)(2n1-1) is prime, than m is
  perfect.

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Ibn al Haitham, the method of analysis and
perfect numbers

• Starting point of analysis
• Given an(y) even perfect number. What structure
  might it have?
• A. H.s goal was not the conversion of the
  theorem of Euclid, but its heuristic foundation!
  
• A. H. tries to generalize the experience of the
  analysis of the example 4961222232431621
  24248 (25-1)31(122223)(25-1)(124-1)
  (25-1)24

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Ibn Sinan and heuristics

• Bellosta, H. Ibrahim ibn Sinan On Analysis and
  Synthesis. In Arabic Sciences and Philosophy,
  vol. I (1991), pp. 211 - 232
• Content Classification of problems analysis and
  its role in the determination of the class of
  each problem synthesis reaction to criticism

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Ibn Sinan and heuristics

• Example of a problem.
• Vivianis theorem
• In any equilateral triangle the sum of the
  distances from a point P within the triangle from
  all three sides is always the same.

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Al Sijzi and problem fields
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Al Sijzi and problem fields

• Move A and B in such a way out of or into the
  Thales-circle, that these points are symmetric to
  the center of this circle.
• Move C on the old Thales-circle. What is
  AC2BC2 AC2 BC2 ?

C
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Al Sijzi and problem fields
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Al Sijzi and beyond

• Given two fixed points A and B in the plane. What
  is the locus of all points C in the plane, so
  that AC2BC2const.?
• Given two fixed points A and B in the plane. What
  is the locus of all points C in the plane, so
  that ACnBCnconst.? Relations to Fermat-curves?

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Al Sijzi and beyond
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Heuristics from ancient China- applied in a
German textbook
volume of a sphere
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Some questions about occurrence of heuristics in
ancient China

• What about other testimonies concerning use of
  heuristic methods in ancient China?
• In which way the results from the Nine Chapters
  of Mathematical Technique or other famous
  ancient books were created?