Transcript Heuristics in Ancient Arabic and Chinese Mathematics

Heuristics in Ancient Arabic
and Chinese Mathematics
and its use in textbooks
Prof. Dr. Bernd Zimmermann
from University of Jena
at University of Xi‘an
August 2002
Heuristics:
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Methods to find conjectures
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Methods to find proofs
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Methods to (re)invent mathematics
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By analysis of history one might find
methods/heuristics, which proved to be
most fruitful (“invariants”)
B. Zimmermann ICM Beijing 2002
Example 1: Analogy
Archimedes ? Kepler
R
U
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Example 2: Analysis
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“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
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Jaouiche, K.: Ibn al Haitham: Kitab at-tahlil wa-t-tarkib.
Ouvrage d’al-H,.asan ibn al al-H,.asan ibn al Haitham
sur l’analyse 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 Unpublished manuscript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, 91d:01002 01A30 01A20 11-03, S.
1822, April 1991-Issue 91d.
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Ibn al Haitham, the method of
analysis and perfect numbers
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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 Euclid’s Elements. Cambridge
University Press, Cambridge 1925. Vol. 2, p. 421)
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Modern form: If m=(1+2+22+23+…+2n)2n and
(1+2+22+23+…+2n)[=(2n+1-1)] is prime, than m is perfect.
B. Zimmermann ICM Beijing 2002
Ibn al Haitham, the method of
analysis and perfect numbers
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Starting point of analysis:
Given an(y) even perfect number. What
structure might it have?
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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
496=1+2+22+23+24+31+62+124+248
=(25-1)+31(1+2+22+23)=(25-1)(1+24-1)
= (25-1)24
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Ibn Sinan and heuristics
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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
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Example of a
problem.
“Viviani’s” 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
B. Zimmermann ICM Beijing 2002
Al Sijzi and problem fields
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“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 A’C2+B’C2 ;
A’C’2 + B’C’2 ?
C‘
A‘ A
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C
B
B‘
Al Sijzi and problem fields
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Heuristics from ancient Chinaapplied in a German textbook
volume of a sphere
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Some questions about occurrence
of heuristics in ancient China
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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?
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