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---
[email protected]
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Peter Henrici 1973 (Algorithmic Mathematics) (Dialectic
Mathematics)
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Peter Henrici, Computational complex analysis, in The Influence
of Computing on Mathematical Research and Education, (ed.) J.P.
LaSalle, Proceedings of Symposia in Applied Mathematics, 20 (1974),
79-86.
Dialectic mathematics is a rigorously logical science, where
statements are either true or false, and where objects with
specified properties either do or do not exist.
Algorithmic mathematics is a tool for solving problems. Here we
are concerned not only with the existence of a mathematical object,
but also with the credentials of its existence.
Dialectic mathematics invites contemplation.
Algorithmic mathematics invites action.
Dialectic mathematics generates insight.
Algorithmic mathematics generates results.
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Jean-Luc Chabert, Evelyne Barbin, et al,
A History of Algorithms : From the
Pebbles to the Microchip (English
translation of Histoire dalgorithmes : Du
caillou la puce, 1994; new edition
2010), 1999.
http://sharebooks21.com/a-history-of-algorithms-from-the-pebble-to-the-microchip/
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Robert McNaughton, Elementary Computability, Formal Languages,
and Automata, 1982.
1. The algorithm must be
capable of being written in a
certain language: a language is
a set of words written using a
defined alphabet.
2. The question that is posed
is determined by some given
data, called enter, for which the
algorithm will be executed.
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3. The algorithm is a procedure
which is carried out step by
step.
4. The action at each step is
strictly determined by the
algorithm, the entry data and
the results obtained at previous
steps.
5. The answer, call exit, is
clearly specified.
6. Whatever the entry data, the
execution of the algorithm will
terminate after a finite number
of steps.
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Ren Descartes
(1596-1650)
Descartes three dreams on
November 10, 1619
Quod vitae sectabor iter (What path shall I take in life? )
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Unification of all sciences
by reason
Method:
(a) accept only what is so clear in ones mind as to exclude any
doubt ;
(b) divide difficulties into smaller ones ;
(c) reason from simple to complex ;
(d) check that nothing is omitted .
CARTESIANISM
Cogito, ergo sum
(I think, therefore I am)
WORLD MATHEMATIZATION
http://en.wikipedia.org/wiki/File:Descartes_Discours_de_la_Methode.jpg
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Ren Descartes(1596-1650)
When I was younger, I had
studied a little logic in
philosophy, and geometrical
analysis and algebra in
mathematics, three arts or
sciences which would appear
apt to contribute something
towards my plan..
Ren Descartes, Discours de la
mthode pour bien conduire sa
raison, et chercher la vrit dans les
sciences (1637)
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Ren Descartes(1596-1650)
But on examining them, I
saw that, regarding logic, its
syllogisms and most of its
other precepts serve more to
explain to others what one
already knows, or even, like
the art of Lully, to speak
without judgement of those
things one does not know,
than to learn anything new..
Ren Descartes, Discours de la
mthode pour bien conduire sa
raison, et chercher la vrit dans les
sciences (1637)
-
Ren Descartes(1596-1650)
Then, as for the geometrical
analysis of the ancients and
the algebra of the moderns,
besides the fact that they
extend only to very abstract
matters which seem to be of
no practical use, the former is
always so tied to the
inspection of figures that it
cannot exercise the
understanding without
greatly tiring the imagination,
[while]..
Ren Descartes, Discours de la
mthode pour bien conduire sa
raison, et chercher la vrit dans les
sciences (1637)
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[imagination,] while, in the
latter, one is so subjected to
certain rules and numbers
that it has become a
confused and obscure art
which oppresses the mind
instead of being a science
which cultivates it. This was
why I thought I must seek
some other method which,
while continuing the
advantage of these three, was
free from their defects.
i
Ren Descartes(1596-1650)
Ren Descartes, Discours de la
mthode pour bien conduire sa
raison, et chercher la vrit dans les
sciences (1637)
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Yale Babylonian Collection
7289, c.1700B.C.
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1
(n) (1) + (n + 1) (1) = 1 (n, n + 1) = 1 [n, n + 1] = n (n +
1)
2
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(4 5 LCM )
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( Euclidean algorithm)
. ),(
],[ BA
ABBA
.] ,[
] ,[ ,
. ] ,[
] ,[ ,
2
211212
1
111
r
rrBrBrrkB
r
rBABArBkA
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Fermats Little Theorem (1640)
We will examine different
proofs by L. Euler, A-M.
Legendre, C.F. Gauss, J.
Tannery.
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Proof 1 (Tannery, 1894)
[ Essentially the crux is the Pigeonhole
Principle.]
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[This can be phrased as a proof by
mathematical induction, if you like.]
Proof 2 (Euler, 1736/1741; Legendre, 1798)
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Proof 3 (Euler, 1755/1761; Gauss, 1801)
[This idea is same as that Lagrange used in 1770
to prove his theorem on cosets of a subgroup.
Basically the crux is the Pigeonhole Principle.
Once we know the result, we can streamline a
proof by using the Euclidean algorithm.]
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RSA algorithm for
public-key cryptography
(1977)
Ronald L. Rivest, Adi Shamir,
Leonard M. Adelman
Example:
Encryption
Decryption
The algorithm hinges on the Fermat-Euler
Theorem.
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ABCD
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https://imp3.webmail.hku.hk/horde/imp/download/?mime=5e7935083b4950dcecfb5e184f869bdb&actionID=112&id=4&index=28673&thismailbox=mail%2Fsent-mail&fn=/Screenshot_15_05_2013_21_59.png
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Datum 43 (2005-2006) 1 4
ABCD ABD = 50 DBC = 80 ADB = 70 BDC = 40 ACB ACD BAC DAC
x + y = 60
x + w = 50
y + z = 70
w + z = 60
x = z 10
y = 70 z
w = 60 z
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A : z = 40 B: z = 30 C : z = 35
(a) A
(b) B
(c) C
(d)
A
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, )( sin)sin(sin
sindbLdcb
z
)sin()sin(
)cos(sinsin2
)sin(
sin
)sin(
sin22
2
dbca
dcba
db
b
ca
aL
[]
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A procedural approach helps us to prepare more solid ground on
which we build up conceptual understanding. Conversely, better
conceptual understanding enables us to handle the algorithm with
more facility.
procedural vs conceptual knowledge
process vs object in learning theory
computer vs no-computer in learning environment
symbolic vs geometric emphasis
in learning / teaching
"Eastern" vs "Western" learner
algorithmic mathematics
and dialectic mathematics
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Algorithmic Mathematics and
Dialectic Mathematics --- The Yin
and Yang in Mathematics Education
SIU Man Keung
Plenary Lecture given at the ICTM2,
Crete, July 2002.
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([email protected])
GeoGebra
