[SeqFan] Leibniz, I-Ching and Binary Arithmetic.
Antti Karttunen
Antti.Karttunen at iki.fi
Fri Jan 9 02:00:44 CET 2004
Here is something concerning the origins of the binary system.
Especially the section about "periodicities" (#71a) might interest
those of us that have been immersed into the experimental way
of doing mathematics. More of that in the next posting.
Excerpts from
Gottfried Wilhelm Leibniz:
Writings on China
Translated, with an Introduction, Notes and Commentaries
by Daniel J. Cook and Henry Rosemont, Jr.
Open Court Publishing Company, Chicago and La Salle, 1994.
From "Remarks on Chinese Rites and Religion" (1708).
#9 And thus, as far as I understand, I think the substance of the
ancient theology [21] of the Chinese is intact and, purged of additional
errors, can be harnessed to the great truths of the Christian
religion. Fohi, the most ancient prince and philosopher of the
Chinese, had understood the origin of things from unity and nothing,
i.e. his mysterious figures reveal something of an analogy to
Creation, containing the binary arithmetic (and yet hinting at
greater things) that I rediscovered after so many thousands of
years, where all numbers are written by only two notations, 0 and 1.
So: [22]
0 1 10 100 1000 10000 etc.
signify 0 1 2 4 8 16 etc.
The numbers are expressed . Figures .
as follows: . of Fohi .
.............................
0 | 0 . . .
1 | 1 0 . - - . 0 . 0
10 | 2 . . .
11 | 3 1 . --- . 1 . 1
100 | 4 . . .
101 | 5 .............................
110 | 6 . . .
111 | 7 00 . - - . 0 . 0
1000 | 8 . - - . .
1001 | 9 . . .
1010 | 10 01 . - - . 1 . 1
1011 | 11 . --- . .
1100 | 12 . . .
1101 | 13 10 . --- . 10 . 2
1110 | 14 . - - . .
1111 | 15 . . .
10000 | 16 11 . --- . 11 . 3
etc | etc. . --- . .
--------+------- . . .
.............................
The figures of Fohi [23] also signify two, four, eight, sixteen,
thirty-two, sixty-four, as reproduced by Kircher and others -
of which only two, four, and eight are herein inscribed [24]
- which all the Chinese until now did not understand, but which
the Reverend Father Bouvet correctly noticed corresponds to my
binary arithmetic. [25].
[21]: The word "ancient" here is crucial to Leibniz's arguments:
he believes the moderns have merely lost the true meaning(s) of
what is contained in their oldest books - in the best Hermetic
tradition.
[22]: This outline of the philosopher's system of binary arithmetic
is fleshed out in part IV of the Discourse. These tables and those
in #71 of the Discourse, are reproduced exactly as Leibniz wrote
them in the autographs. (Except that I have left out the trigrams
"000" - "111" from the above "Figures of Fohi" table, for the
lack of space -- AK.)
[23]: I.e. the trigrams of the Yi Jing.
[24]: Leibniz inscribed the yin and yang line, then paired them,
and then inscribed the eight trigrams.
[25]: In his 4 November 1701 letter to Leibniz, Widmaier (2),
pp. 147-169. Most Leibniz commentators have attributed to him
the linking of his binary notation to that of the Yi Jing.
This acknowledgement makes clear that the credit is Bouvet's.
From "Discourse on the Natural Theology of the Chinese" (1716).
[IV. Concerning the Characters which Fohi, Founder of the Chinese
Empire, Used in His Writings, and Binary Arithmetic]
#68 It is indeed apparent that if we Europeans were well
enough informed concerning Chinese Literature, then, with the
aid of logic, critical thinking, mathematics and our manner of
expressing thought - more exacting than theirs - we could uncover
in the Chinese writings of the remotest antiquity many things
unknown to modern Chinese and even to other commentators thought
to be classical. Reverend Father Bouvet and I have discovered
the meaning, apparently truest to the text, of the characters
of Fohi, founder of Empire, which consist simply of combinations
of unbroken and broken lines, and which pass for the most ancient
writing of China in its simplest form. There are 64 figures
contained in the book called Ye Kim [175], that is, the Book
of Changes. Several centuries after Fohi, the Emperor Ven Vam [176]
and his son Cheu Cum, and Confucius more than five centuries later,
have all sought therein philosophical mysteries. Others have even
wanted to extract from them a sort of Geomancy and other follies.
Actually, the 64 figures represent a Binary Arithmetic which
apparently this great legislator [Fu Xi] possessed, and which I
have rediscovered some thousands of years later.
#68a In Binary Arithmetic, there are only two signs, 0 and 1,
with which one can write all numbers [177]. When I communicated
this system to the Reverend Father Bouvet, he recognized in it
the characters of Fohi, for the numbers 0 and 1 correspond to
them exactly [178] if we put a broken line for 0 and unbroken
line for the unity, 1. This Arithmetic furnishes the simplest
way of making changes, since there are only two components,
concerning which I wrote a small essay in my early youth, which
was reprinted a long time afterwards against my will [179].
[175]: Yi Jing.
[176]: Again, King Wen; this time the spelling is Bouvet's.
"Cheu Cum" is Zhou Gong, the Duke of Zhou, and King Wu's brother.
He was one of Confucius's favorite cultural heroes.
[177]: At this point Leibniz wrote, but then crossed out,
the following: "I have since found that it further expresses the
logic of dichotomies which is of the greatest use, if one always
retains an exact opposition between the numbers of the division."
[178]: At this point, Leibniz wrote, but then crossed out
the following: "provided that one places before a number as many
zeroes as necessary so that the least of the numbers has as many
lines as the greatest."
[179]: In a letter to Rémond in July 1714, Leibniz wrote about
"a little schoolboyish essay called 'On the Art of Combinations'
('De Arte Combinatoria' -- AK), published in 1666, and later
reprinted without my permission."
Gerhardt, III, 620. See Loemker, p. 657.
So it seems that Fohi had insight into the science of combinations,
but the Arithmetic having been completely lost, later Chinese
have not taken care to think of them in this [arithmetical] way
and they have made of these characters of Fohi some kind of
symbols and Hieroglyphs, as one customarily does when one has
strayed from the true meaning (as the good Father Kirker [180] has
done with respect to the script of the Egyptian obelisks of
which he understands nothing). Now this shows also that the
ancient Chinese have surpassed the modern ones in the extreme,
not only in piety (which is the basis of the most perfect morality)
but in science as well.
#69 Since this Binary Arithmetic, although explained in the
Miscellany of Berlin [181], is still little known, and the mention
of its parallelism with the characters of Fohi is found only in
the German journal of the year 1705 of the late Mr. Tenzelius [182],
I want to explain it here - where it appears to be very appropriate
- since it concerns justification of the doctrines of the ancient
Chinese and their superiority over the moderns. I will only add
before turning to this matter that the late Mr. Andreas Müller,
native of Greiffenhagen, Provost of Berlin, a man of Europe, who
without ever having left it, had studied the Chinese characters
closely, and published with notes, what Abdalla Beidavaeus wrote
on China. This Arab author remarks that Fohi had found a
"peculiare scribendi genus, Arithmeticam, contratus et Rationaria",
"a peculiar manner of writing, of arithmetic, of contracts, and
of accounts" [183]. What he says confirms my explanation of the
characters of this ancient philosopher-king whereby they are
reduced to numbers.
[180]: Athanasius Kircher, 1601-1680.
[181]: "De periodis columnarum in series numerorum progressionis
Arithmeticae Dyadice expressorum," by P. Dangicourt, in
Miscellanea Berolinensia, I (1701), 336-376. This and the Tenzel
article cited below were both instigated by Leibniz himself.
See Zacher, p. 1.
[182]: "Erklärung der Arithmeticae binariae, ...," in Curieuse
bibliothic oder Fortsetzung der Monatlichen Unterredungen einiger
guten Freunde, ed. W.E. Tenzel (Frankfurt and Leipzig, 1705), pp.
81-112. See Zacher, p. 210. Leibniz omits mention of his own,
earlier endeavor concerning binary arithmetic and its
relationship to the characters of Fu Xi, written in 1703;
"Explication de l'Arithmetique Binaire qui se sert des seuls
caracters 0 et 1; avec des Remarques sur son utilitité, et
sur ce qu'elle donne le sens des anciennes figures Chinoises
de Fohi," par M. Leibniz, Histoire de l'Academie royale des
Sciences, Année 1703; avec les Memoires ... pour la meme Année,
Paris 1705 [Mem.], pp. 85-89. A more readable and available
edition of this work is found in Zacher, pp. 293-301.
[183]: There is little evidence that the hexagrams were used for
these purposes. Leibniz is quoting directly from Tenzel. See
also Zacher, p. 159. And see Remarks, #9.
#70 The ancient Romans made use of a mixed arithmetic, quinary
and denary, and one still sees reminders of it in their counters
[184]. One sees, from Archimedes' work on the counting of the sand
[185], that already in his time something approaching denary
arithmetic was understood (which has come down to us from the
Arabs and which appears to have been brought from Spain, or at
least made more known by the renowned Gerbert, later Pope, under
the name of Sylvester II [186]). This prevalence of base 10
arithmetic seems to come from the fact that we have 10 fingers,
but as this number is arbitrary, some have proposed counting by
dozens, and dozens of dozens, etc [187]. On the other hand,
the late Mr. Erdard Weigelius resorts to a lesser number predicated
on the quaternary or Tetractys like the Pythagoreans [188]; thus,
just as in the decimal progression we write all numbers using
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, he would write all numbers in his
quaternary progressions using 0, 1, 2, 3; for example 321 for him
signifies 3 x 4² + 2 x 4^1 + 1 or rather 48+16+1, that is 65
according to the ordinary system. (I get 48+8+1 = 57. -- AK).
[184]: Leibniz is talking of the Roman numerals, which except for
unity (I), are based on either five (V, L, D) or ten (X, C, M),
hence the mixed nature of their numeration or counting, but not
necessarily of their arithmetic, about which little is known.
The purpose of the Roman counters, originally pebbles (calculi),
is uncertain; it has been argued that they were used in games such
as backgammon or checkers, or even like poker chips. See Smith, II,
165-66.
[185]: "Archimedes saw the defects of the Greek number system,
and in his Sand Reckoner he suggested an elaborate scheme of
numerations, arranging the numbers in octads, or the eighth powers
of ten." Ibid., I, 113.
[186]: Gerbert, who was pope from 999 to 1003, has traditionally
been held responsible for introducing the Arabic numerals into
Christian Europe, which he probably learned while studying in
Spain. Leibniz is wrong in thinking that Gerbert introduced
the decimal or denary system, rather than simply the nine characters.
"He probably did not know of the zero, and at any rate he did not
know its real significance." Ibid., II, 74-75; see also, Ibid., I,
195-196.
[187]: Leibniz himself toyed briefly with a base 12 (and even
mentioned a base 16) number system and may have gotten the idea
from Pascal. See Zacher, pp. 17-21.
[188]: Erhard Weigel (1625-1699) was professor of Mathematics
at the University of Jena, where Leibniz followed his lectures
for one semester in 1663. Having been influenced strongly by
the Pythagorean and other mystical traditions in mathematics,
Weigel saw the number 4 as the perfect number and constructed
a base 4 number system. Although Weigel influenced Leibniz in
many areas (e.g., the need for linguistic and legal reforms in
Germany), the latter saw no need for such a number system.
Practically, a base 10 or even higher number system (12 or 16)
shortened calculations and condensed enumeration; theoretically,
a base 2 system was best, since it had the simplest and most
easily analyzable base. Couturat, pp. 473-474.
#71 This gives the opportunity to point out that all numbers
could be written by 0 and 1 in the binary or dual progressions.
Thus:
1 1
10 2 10 is equal to 2
1000 8 100 is equal to 4
10000 16 1000 is equal to 8
100000 32 etc.
1000000 64
And accordingly, numbers are expressed as follows:
0 0
These terms correspond with the hypothesis; 1 1
for example: 10 2
111 = 100 + 10 + 1 = 4 + 2 + 1 = 7 11 3
11001 = 10000 + 1000 + 1 = 16 + 4 + 1 = 25 [189] 100 4
101 5
They can also be found by continual addition of 110 6
unity, for example: 111 7
1 1000 8
.1 1001 9
---- 1010 10
10 1011 11
1 1100 12
11 1101 13
---- 1110 14
.1 1111 15
---- 10000 16
100 10001 17
1 10010 18
---- 10011 19
101 10100 20
.1 10101 21
---- 10110 22
The points denote 110 10111 23
unity which is kept 1 11000 24
in mind in ordinary ---- 11001 25
calculation. [190] 111 11010 26
---- 11011 27
.1 11100 28
---- 11101 29
1000 11110 30
11111 31
100000 32
etc. etc.
[189]: The mistake is Leibniz's; 4 should be 8. This section
should be read together with Remarks #9. (I.e. the tables are
reproduced exactly as Leibniz wrote them in the autographs.)
[190]: See #72 for explanation.
#71a But, to continue, if one wishes to make a table expressing
terms for all the natural numbers in order, one need not calculate,
since it is sufficient to note that each column is periodic, the
same periodicity recurring ad infinitum: the first column runs
0, 1, 0, 1, 0, 1, etc.; the second 0, 0, 1, 1, 0, 0, 1, 1, etc.;
the third 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, etc.;
the fourth 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1,
0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, etc. And so on
with further columns, assuming that the empty places above each
column are filled with zeroes. Thus one can write these columns
at once and accordingly make up the table of natural numbers
without any calculations. This is what one can call enumeration.
#72 As for addition, it is simply done by counting and making
periods when there are numbers to add together, adding up each
column as usual, which will be done as follows: count the unities
of the column; for example, for 29, look how this number is written
in the table, to wit, by 11101; thus you write 1 under the column
and put periods under the second, third and fourth column
thereafter. These periods denote that it is necessary to count
out one unity further in the column following.
#73 Subtraction is just as easy. Multiplication is reduced to
simple additions and has no need of the Pythagorean table, it
sufficing to know that 0 times 0 is 0, that 0 times 1 is 0,
that 1 times 0 is 0, and that 1 times 1 is 1.
#74 In division there is no need to tally as in ordinary
calculation. One must only see if the divisor is greater or less
than the preceding remainder.
#75 These are simplifications that have been proposed by a clever
man since the introduction of this Arithmetic into certain
calculations [191]. But the principal utility of this binary
system is that it can do much to perfect the science of numbers
because all calculations are made according to periodicity. It is
some achievement that the numerical powers of the same order,
made by raising the ordered natural numbers, however high
the order, never have a greater number of periods than the
natural numbers themselves which are their roots ...
[Text breaks off at this point.]
[191] Leibniz is referring to Arithmeticus perfectus, Qui Tria
numerare nescit, seu Arithmetic dualis (Prague, 1712) by
W. J. Pelican. Pelican apparently showed how one can use the
binary system for other calculations as well, i.e., fractional
arithmetic, roots, and proportions. See Zacher, p. 211.
From Introduction, II. Sources of Leibniz's Knowledge of China,
Among the contemporaries of Longobardi and Sainte-Marie who
were more sympathetic to Ricci was Martino Martini, S.J. (1614-1661).
After studying in the Collegium Romanum - Kircher was his tutor
in mathematics - he went to China in 1643, the year before the
final collapse of the Ming Dynasty. ...
Martini wrote several works on China, the most significant
of which was Sinicae historiae decas prima. Res a gentis origine
ad Christum natum in extrema Asia, published in 1658. Largely
a work on Chinese history, it contained a fairly accurate
chronology of early imperial reigns as the Chinese had established
them. Some of the earliest reigns were seen by some Chinese
(and virtually all modern scholars) as legendary, but Martini
accepted them as fact. The chronology begins with Fuxi in 2952 BCE,
which was troubling to many readers of Martini's work, because James
Ussher's Biblical chronology had been published only a few years
before and had persuaded many that creation had taken place
in 4004 BCE, and the Noachian flood in 2349 BCE. But if Fuxi had
truly reigned over six hundred years earlier, and there were
no breaks of succession to the throne, then Noah could not
be the universal patriarch.
The Sinicae historiae also provides what was probably the
first depiction in a European book of the sixty-four hexagrams
of the Yi Jing, or Book of Changes, with which Leibniz was
intrigued throughout his mature study of things sinological.
Martini credits Fuxi with the bringing of the hexagrams to
the Chinese people, and with the creation of the Chinese script
based on them.
The last of the writings on China which exerted a strong
influence on Leibniz were those authored by the French Jesuit
missionary Joachim Bouvet (1656-1730). Like Verbiest, Bouvet
had access to the throne, being a tutor of the Kang Xi Emperor's
children. ...
Bouvet thought on a grand scale, much more a metaphysician
than historian or philologist. He conveyed many original (and
sometimes far-fetched) ideas about Chinese history, language
and religion to Leibniz, based on the Figurist orientation
of some Jesuits, which attempted to tease Christian figures
out of non-European (i.e., pagan) writings. One of these ideas
was that the legendary early Chinese ruler Fuxi was not really
Chinese, but a manifestation of the "Lawgiver", akin to
Hermes Trismegistus in the West; indeed, on the basis of some
dubious etymologies and arguments Bouvet even tried to show
that Fuxi and Hermes Trismegistus were one and the same, thus
circumventing the creation chronology problem raised earlier
by Martini's work.
Bouvet further believed that by producing the eight basic
trigrams of the Book of Changes - which, doubled, are the
sixty-four hexagrams - Fuxi had provided a notation for
experiment and observation in all of the sciences. These
trigrams are made up of combinations of solid and broken
lines, but their mathematical and scientific significance
had been, according to Bouvet, lost on later Chinese, who
simply read the trigrams and hexagrams as part of a system
of prognostication. More significant, after learning of
Leibniz's work on binary arithmetic, Bouvet established
a correlation between Leibniz's notations (0 and 1) and
the broken and solid lines of Fuxi's trigrams.
...
A second insight into Leibniz's views contained in this
quote comes from the parenthetical remark: "and yet hinting
at greater things". Earlier in 1697 he had written a letter
to Duke Rudolph August, which included the design of
a medallion he wished to be struck. The obverse is merely
a portrait of the Duke, but the reverse gives binary equivalents
for several Arabic numerals, under which he inscribed
Imago Creationis. He also portrays rays of light shining down
into the watery deep, and equates the latter with Zero and
the former with "the almighty One".
Leibniz never sent this letter, but he does seem to be
sincere in believing that his binary arithmetic was theologically
as well as scientifically important, capable of unlocking secrets
both of nature and the Book of Genesis, and generating all the
natural numbers using only 0 and 1, symbolizing in a most
dramatic way the creatio ex nihilo by the one God. In another
letter to Bouvet (18 May 1703), responding to the Jesuit's
linking of the eight trigrams with the numbers 0 to 7, Leibniz
says:
... the last is the most perfect and the Sabbath, for in it
everything has been made and fulfilled; thus 7 is written 111
without 0 ...
(This last citation translated in Walker, p. 223. The original may
be found in Widmaier (2), p. 187.)
References:
Couturat, Louis. La Logique de Leibniz, 1901. Reprint; Hildesheim: Olms,
1961.
Smith, D.E. History of Mathematics. 2 vols. Boston: Ginn and Co.,
1925.
Walker, D.P. The Ancient Theology. Ithaca, NY: Cornell University
Press, 1972.
Widmaier, R. (1) Die Rolle der Chinesischen Schrift in Leibniz'
Zeichentheorie. Wiesbaden: Franz Steinder Verlag, 1983.
Widmaier, R. (2) ed. Leibniz korrespondiert mit China. Der
Briefwechsel mit den Jesuitenmissionaren (1689-1714). Frankfurt:
V. Klostermann, 1990.
Zacher, H.J. Die Hauptschriften zur Dyadik von G.W. Leibniz,
Ein Beitrag zur Geschichte des binären Zahlensystem. Frankfurt:
V. Klostermann, 1973.
---------------------------------------------------------------------------
For more information on Kircher & Leibniz with regards to
Chinese characters, see:
http://www.stanford.edu/dept/HPS/writingscience/Cryptography.html
(The beginning of article, written in overtly semiotic style, puts
me off a bit, but the actual history is still interesting.
It seems that the Chinese language and Han-zi (Kanji) characters
created quite an intellectual stir among certain Europeans. -- AK).
More information about the SeqFan
mailing list