Transcript Configurations versus Equations

Configurations versus Equations:
A Notational Difference
陣式和方程式:符號的異同
程貞一
聖迭戈加州大學物理系
There are historians of mathematics and
sinologists who view the presentation
of the early Chinese works in equations
with mathematical signs such the equal
‘’, plus ‘’, minus ‘’ signs and letter
symbols such as x, y, z is equivalent to
read modern concept of equations into
old work.
有些漢學家和科技史學者認為用現今
數學符號系統:等號 ‘’,加號‘’,減
號‘’, 和字母符號: x, y, z,書寫中國
古代數學,就等于把近代“equation”
(即“方程式”)的觀念帶入到古代中
國數學中。
In this talk, I would like to suggest
misunderstandings such as these can be
easily clarified, if one generalizes one’s
concept of notation from written mode
to include mechanical mode.
在此我建議如果把數學符號觀念
由書寫符號系統闊大到包含機械
符號系統,這類誤解自然就消失了。
Ever since antiquity, there were in
mathematics, the written mode and the
mechanical mode of representing
technical and quantitative mathematical
concepts and relations, and in executing
algorithm and derivations. These two
different modes lead to the creation of
written and mechanical notational
systems, respectively.
Notation in the written mode began probably
with the use of a‘hard’pen to create impressions
on tablets such as the cuneiform writings, or on
turtle shells and ox bones such as the shell-bone
inscriptions [1] (jiăgŭwén 甲骨文), and with the
use of a ‘colored’ pen to write on papyrus,
bamboo strips (zhújiăn 竹簡), parchment, silk
fabrics or papers for expressing mathematical
concepts and for performing computations.
[[
1]. The translation “shell-bone inscriptions” is advised based on the technical terms “甲骨文”
coined by Chinese Linguistic communities. The translation “oracle-bone inscriptions” in the
West initiated by sinologists is misleading, depicting only one aspect of the inscriptions. The
turtle shells and ox bones were the favored writing media during the Shāng 商 dynasty and
their use probably begun in antiquity.
Notation in mechanical mode, on the
other hand, began with tools, other than
a pen for performing computations. In
this mode, one uses the devised objects
such as rods in chóusuàn 籌算 or axialsliding balls in zhūsuàn 珠算 to compose
configurational layouts for expressing
mathematical thoughts and for
performing computations.
Positional Numerals in Written Notation (left
column) and in Mechanical Notation Displayed
on Abacus in Zhūsuàn 珠算 and in Chóusuàn
籌算 on a Computational Board.
In many early civilizations, the written
mode was favored in their early stage
of mathematic developments both for
performing calculations and for
keeping records of calculations.
Consequently, written notations were
used exclusively
However, in the Chinese civilization
early computations were carried out
only with mechanical tools. The written
mode was used only for describing and
recording of the computations. Hence,
notation both in written and in
mechanical modes were used. Not until
the 17th century did computations in
written mode became common.
In order to study early mathematics in
Chinese civilization, especial its algorithm
and derivations, it is imperative to have
certain familiarities with chóusuàn 籌算
and the rod notational system used by the
Chinese mathematicians before the 17th
century. This is because mathematical
concepts and computations are expressed
all in mechanical notation not in written
notation.
Solving Mathematical Problems
In the ‘written mode’
One uses written symbols to
compose equations for expressing
mathematical thoughts and to
execute algorithms in performing
mathematical computations
Solving Mathematical Problems
In the ‘Mechanical mode’
One uses devised objects such as rods,
balls, rotations, or signals to compose
configurations, patterns or sequences
for expressing mathematical thoughts
and to execute algorithms in
performing mathematical computations.
A practical way to examine the
difference in notation between written
and mechanical modes, Let us consider
an actual problem from Problem 1 of
Chapter 8 ‘Fāng Chéng’〈方程〉‘Square
Array’ from the Jiŭ-Zhāng Suàn-Shù
《九章筭術》(Nine Chapters in
Mathematics) which is a problem with
three linear algebraic relations for three
unknowns
Rectangular-Array Configuration
(jŭzhènshì 矩陣式)
Linear Equations Set
3x  2y  z  39
2x  3y  z  34
x  2y  3z  26
It is seen that the basic layouts are similar;
the layout in current written notation is
horizontal but the array-configuration
layout is vertical. By rotating the written
linear equations 90 degrees clockwise, one
would then have the coefficients and
constants of the linear equations arranged
exactly as those given in the array
configuration of the linear equations.
Instead of the letter symbols x, y, and z, the three
variables are identified by their corresponding
rows in the rectangular-array configuration. The
first row is identified with the unknown x, the
second row with the unknown y and so on until
the row before the last row reserved for the
constant terms. By construction, each constant
in the last row equals the sum of all the entries
above within the same column of the constant.
The plus ‘’ and minus ‘’ signs are specified by
the coefficients and the constant occupying the
configuration.
For equations of higher degrees, let us
consider the problem of square root
extraction a problem from Problem 20,
Chapter 9 of the Nine Chapters in
Mathematics.
Columnar Configuration
(hánglièshì 行列式)
2nd-Degree Equation
In the written notation, the degrees of
the variable are denoted by the different
powers of the letter x for the variable.
In the columnar configurational
representation, the degrees of the
variable are denoted by the positional
orders in the column with respect to the
constant term of the problem at the
bottom of the column.
These comparisons reveal that
configurations and equations have the
same function in their respective
representations. There is actually a oneto-one correspondence between the two
representations. This implies that one
should be able to transform between the
two representations without changing
the content of the mathematics.
This demonstrated that once one’s
concept of notation is generalized
from written mode to include
mechanical mode. One would then
realized that the concept of equation is
not a new mathematical concept, it
appeared earlier as configurations in
mechanical notations in Chinese
mathematical work.
The presentation of the early Chinese
works in equations with mathematical
signs such the equal ‘’, plus ‘’,
minus ‘’ signs and letter symbols such
as x, y, z is merely a one-to-one transfer
of notation from mechanical mode to
the written mode.
It should also be noted mechanical notation
is an important part of mathematics because
history has shown that the development of
mathematical tools was an important part of
mathematical progresses. Attempts to
improve calculating tools began actually
very early in Chinese civilization.
Among the driving forces for improving
calculating tools are speed, repetitive
calculations, versatilities, etc. Facing the
growing needs in commerce for faster
calculations, the early Chinese invented
suànpán 算盤, a mechanical tool with
axial-sliding balls (or beats) for faster
calculations. During the Táng 唐 and Sòng
宋 dynasties, suànpán developed into an
important calculating tool primarily in
commerce.
To aid zhūsuàn, a number of algorithmic
rhymes and verses, originated from the
chóusuàn, were modified and adopted for
zhūsuàn. These rhymes and verses were
usually recited subconsciously during
calculations to facilitate the recalling of
algorithms and to speed up repetitive
calculations. Such algorithmic rhymes and
verses can be viewed as instructions for
operating mechanical devises in performing
repetitive computations.
For further improvement in handling
mechanical calculations, it would be
desirable, if such instructions as well as the
mechanical notation can be built in the
mechanical devises instead of carrying out
manually and tracking mentally as in both
zhūsuàn and chóusuàn.
The next generation of calculating tools
involved wheels and gears instead of rods
and balls. With such wheel-gear calculating
machines, the mechanical notation were
expressed in terms of patterns of rotational
sequences of the interlocking geared
wheels. The first machine capable of
addition was a water-powered armillary
clock invented in 1092 by astronomer Sū
Sòng 蘇頌 (1020-1101).
In his documents Jìn-Yíxiàng Făzhuàng
《進儀象法狀》(Report on the Armillary
Clock), Sū Sòng 蘇頌 stated, in discussing
the interlocking wheel-and-gear operating
system of the clock, that the machine
“makes measurements and perform
calculations” (qìdù suànshù “器度算數”);
and “keeps time and reporting time by
drumming drums and ringing bells” (sīchén
jígŭ yáolíng 司辰擊鼓搖鈴)”.
The first mechanical device capable of
adding numbers in Europe was also a
calculating clock invented in 1623 by
Wilhelm Shickard (1592-1635), a professor
of mathematics at the University of
Tübingen. Twenty years before the
invention of the ‘Pascaline’ adding device
in 1643 by the French mathematician Blaise
Pascal (1623-1662).
The adding function was based on the
movement of six dented wheels, geared
through a wheel which with every full turn
allowed the wheel located at the right to
rotate 1/10th of a full turn. By design, it
was able to handle up to six-digit numbers.
The adding feature was devised to help
performing multiplication with a set of
Napier's cylinders included in the upper
half of the machine. An overflow
mechanism rang a bell.
In modern electronic calculations, one
uses the binary numeral system for
implementing calculations. The “zero”
and “one” digits are designated by the
“off” and “on” signals respectively.
Notation adopted in such electronic
calculations is expressed in signalpatterns which are built in integratedcircuits in chips and in microprocessor
for calculators and computers.
By the 17th century, Europe began to
develop innovative devises for assisting
calculations. The first such a devises
was the Napier’s bones consisting of
strips of wood or bones written with
multiplication tables which was built in
1617 by the Scottish mathematician
John Napier (1550-1617) for
performing arithmetic operations.
Through this device, he also
developed logarithms to convert
multiplication into addition. Later in
1633, the slide rule was invented
based on logarithms by the English
mathematician William Oughtred
(1574-1660).
分析中西古代數學運算方式和推導思
路上的異同,顯然可見由於運算方式
上的差異,中西建立了不同符號體系
和發明了不同運算工具;由於推導思
路上的差異,中西創建了不同的推理
系統和精煉出不同的運算程序。但是
歷史証實,這些異同對世界數學的發
展曾多次出現相互推動和相互補助的
功能。
“方程式”是用書寫表達代數法總結數
學問題的方式,以便在筆算中進行開根演
算求解。在觀念上“方程式”與“陣式”
完全一樣。因為“陣式”是用算籌表達
代數法總結數學問題方式,以便在籌算中
進行開根演算求解。由此可見,陣式和方
程式之間的差異並不在數學觀念上,而在
符號系統上。陣式應用于機械符號系統
中然而方程式應用于書寫符號系統中。
把陣式改變為方程式,僅僅是一個一對一
的符號轉換,不牽涉到數學的內容。