> Does anyone have any news about Seidenberg's old paper from the beginning
> of the sixties about the "origins of geometry" and the "Vedic altars"? Is
> there a debate now, or is it dead?
>
I found some information about Seidenberg in a set of Web pages from Truman
State University. I failed to note which page was which, but the following
can all be found starting at URL
http://math.truman.edu/~thammond/
********************************************************
Seidenberg, A. The ritual origin of counting. Arch. Hist. Exact Sci.
(1962b), 1-40.
It is common to argue that counting and other elementary mathematics
arose spontaneously throughout the world in response to a practical,
or perhaps psychological, need. Abraham Seidenberg argues instead
for a _diffusion theory_, that counting arose only once, and then
spread throughout the world. In fact, many common associations with
numbers suggest such a common origin. One such association that
Seidenberg is the idea that odd numbers are male and even numbers
are female; this is certainly well known from the Pythagoreans, but
turns out to be nearly universal. Seidenberg proposes that counting
in fact originally arose in a ritual context. Seidenberg draws from
a wide variety of anthropological sources for rituals and myths that
hint at what this common origin might have been. He finds that
counting "was frequently the central feature of a rite, and that
participants in ritual were numbered." He focuses more specifically
on creation rituals. He suggests that in the enaction of creation
myths, men and women may have come onto the scene alternately, easily
explaining the odd/male even/female association. He finds that his
ideas clarify "pure 2-counting, which is the oldest stratum of
counting we can detect." In pure-2 counting, there are separate words
for one and two and these are used to form all other number words.
He illustrates this with number words from diverse languages such as
the Gumulgal of Australia, the Bakairi of South America, and the
Bushmen of South Africa. He sheds additional light on his hypothesis
with discussions of the possible origin of counting taboos (and
connections with ritual sacrifice), of ancient one-one-correspondence
"tally" systems (e.g., counting people with stones), of taxation systems,
of money, and of gematria. Seidenberg also gives us some fascinating
examples of counting in world religions. These include the analogy
_The Lord_ : _His people_ = _the shepherd_ : _his sheep_, the analogy
_The shepherd_ : _his sheep_ = _the moon_ : _the stars_. These two lead
one to expect the moon to count the stars; and Seidenberg in fact finds
evidence of this in ancient Babylonia. He argues from the equation
_The Lord's people_ = _the stars of the heaven_ to
_The Lord's people_ = _the sand upon the seashore_ that one would
expect to find a ritual counting of sand. In fact, he finds the notion
of _Counter of the Sands_ both in Buddhism and among the Ancient Greeks.
The equation _The Lord_ = _The Counter_ seems to be confirmed in two
of the ninety-nine beautiful names of Allah, namely _The Counter_ and
the _Reckoner_; and there is further confirmation in Chapter's XV and
XIX of the Qu'ran. This is a fascinating article, connecting
mathematics with a wide variety of disciplines.
******************************************************
The Islamic World, and Abraham Seidenberg
(http://math.truman.edu/~thammond/history/AbrahamSeidenberg.html)
Seidenberg, A. The ritual origin of the circle and square.
Arch. Hist. Exact Sci 25 (1981), no. 4, 269-327.
(Reviewer: M. P. Closs.) SC: 01A10 (51-03), MR: 83h:01008.
Abraham Seidenberg advances a theory that the circle first arose in the
context of the ritual enactment of a creation myth. In many cases, stars
seem to play an important role in these myths. Seidenberg's research
suggests that participants in these myths generally moved in a circle
in imitation of the stars in the heavens. It is interesting that
individuals in these societies often move in the same direction as the
stars, and movement in the opposite direction is often considered
unlucky. The fact that the Aztec god Tezcatlipoca is missing is right
foot, forcing him to walk clockwise in a circle may be related.
Seidenberg suggests that the creation myth is the origin for the dance
around the may pole, which is for example observed near the summer
solstice in northern Scandinavia today. Analogous rituals may play (or
have played) a role in a wide variety of other cultures as well; examples
are found in the Aztecs, ancient Indians, American Indians, and Greeks
(Spinning tops may have a ritual significance as well.) Special support
is given to Seidenberg's these through the fact that in some cases, a
pole may have been set up at an angle so as to point towards the pole
star. Seidenberg notes that the moon might have motivated the circle
rather than the stars, but the sun is unlikely to. His investigations
tend to confirm this, and also suggest that lunar culture is older than
solar culture. Seidenberg believes that the square arose from the circle,
through the process of dividing a group into a dual organization, where
for example members of one group marry someone in the other group and
also (as he notes) play complementary roles in ritual. If a society
divides a second time, one can think of it dividing the tribal circle
into four parts. He finds some evidence of this as well. The four
parts naturally define a square. His theory therefore implies that the
circle arose first and that the square arose as a dual form of the
circle; there is some other evidence (e.g., architectural) that may
tend to confirm this. Seidenberg mentions several interesting dualities
involving the circle and the square. The Altar of Heaven in Peking, for
example, exhibits the equations _Heaven_ : _Earth_ = _circle_ : _square_
= _three_ : _two_ = _South_ _North_ = _White_ : _Yellow_. In Sinhalese
art he finds the equation _circle_ : _square_ = _standing_ : _sitting_.
In the Omaha tribe he finds the equations that
_Sky_ : _Earth_ = _superior_ : _inferior_ = _one_ : _two_. He also
notes the equations _Heaven_ : _Earth_ = _Male_ : _Female_ and
_Male_ : _Female_ = _one_ : _two_. The former is well known, and the
latter is extensively discussed in Seidenberg, A., The ritual origin of
counting (http://math.truman.edu/~thammond/history/Myth.html#roc). The
ancient Egyptians appear to be an exception as they associated the square
with the earth and the circle with the sky. A fascinating paper.
********************************************************
Seidenberg, A. and Casey, J. The ritual origin of the balance.
Arch. Hist. Exact Sci. 23 (1980/81), no. 3, 179--226.
(Reviewer: M. P. Closs.) SC: 01A10, MR: 82j:01008.
The author's trace the beginnings of the balance back to a rituals
where principals contended against each other on a kind of see-saw
(somewhat similar sports are of course known from medieval times).
The grain-crusher and water-lifter are similar, and perhaps derived
from, the see-saw; the fact that one stands on these suggested to
the authors that the contestants may have been standing on the see-saw.
The authors note that in ancient Egypt, one's heart was believed to
be weighed against a feather in order to decide whether one would be
able to enter the afterlife. Other parts of the body, such as hair,
can be used to represent an individual, and in other instances these
may have been weighed instead; the authors give examples of rites
where hair is weighed. An interesting use of the balance in Greece
is from the _Iliad_ where Zeus weighs Achilles and Hector on pans of
a balance. "That of Hector sinks toward Hades and Hector falls,
slain by Achilles." An even more interesting weighing ritual was
once common in the far east, where a ruler was balanced against a
quantity of a precious substance such as gold, and gave that substance
(and thereby symbolically himself) to his people. The authors found
many other interesting examples in a wide variety of cultures and world
religions. The authors believe that only items of ritual significance
were weighed at first, and that widespread commercial use came much
later. Although the authors don't focus greatly on this, they also
briefly discuss the different kinds of balances (and the balance-like
instrument used to carry loads on the shoulders) and the weight multiples
that were used on balances.
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The following quote also gives some information about van der Waerden's
theses.
For an analysis of pre-Greek and Greek mathematics there are many shelves
of books to refer to, for example ((BOYER 1968), (HEATH 1921), (NEUGEBAUER
1957), (van der WAERDEN 1963)). Now we just try to underline some aspects
we believe noteworthy for our concerns.
Our approach stresses the role of signs in the architecture of knowledge.
by Nesselmann (NESSELMANN 1842) between three different types of algebra:
- "rhetorical" algebra is only the mathematical usage of natural
language, with its syntax and lexicon; in addition there are specific
geometrical
terms, analogically stemmed from anatomy or practical activities, and
numbers, used just to record quantitative information. A typical example
is the Babylonian mathematics.
- "syncopated" algebra has some more algebraic notations (unknown
quantities, equality, powers, etc.), as in Diophantus.
- "symbolic" algebra is the modern notation, introduced by Viete and
Descartes, in which lexicon and syntax are specifically designed, and,
most of all, there are rules of symbolic manipulation to rewrite the
expressions. Such rules in the earliest versions were simply 'written'
and 'argumented' and the idea of proof was 'rigorous' but not
'algorithmically defined'.
A crucial point in this report is the relation between arithmetic/algebra
and geometry. To this aim, we can distinguish, following van der Waerden
((van der WAERDEN 1983), 74), three kinds of algebra:
- a "mixed" algebra of the Babylonian type, in which line segments are
added together and equated to numbers,
- a "numerical" algebra in which only rational numbers are admitted as
coefficients and solutions of equations, as in Diophantus,
- a "geometric" algebra, like the algebra of Omar Khayyam and Viete, in
which line segments, area and volumes are strictly kept apart.
Babylonian and Egyptian algebra was rhetoric and substantially mixed,
linked to the 'palace'-centred economy, without a dominant role played
by trade and money, aimed instead to record and compute economic and
administrative data, managed by a class of priests, the 'scribes',
called by Herodotus arpedonaptai (rope-stretchers). They had the monopoly
of the intellectual functions, mathematics and writing as well.
In Mesopotamian and Egyptian science it is impossible to distinguish
between religious, theoretical and practical aspects. Babylonian
mathematics was fully integrated in an astrological religion with clear
arithmetic features.The writings of Seidenberg ((SEIDENBERG 1962),
(SEIDENBERG 1963), (SEIDENBERG 1978)), have shown sufficient elements
to speak, at least, of the 'ritual past' of mathematics.
The numerical system was not completely unified: in Babylon the
sexagesimal system was used for theoretical and astronomical problems,
mixed systems were used for practical computations, e.g. concerning
dates, weights, areas, etc. And we can find clear decimal traces in
the sexagesimal system. There was no distinction between exact and
approximate computations and there was some confusion between
heterogeneous magnitudes, e.g. segments and areas could be added or
subtracted from each other. The same numbers could have different
names when referred to different kinds of objects. Babylonian
mathematical system was, since the beginning some four thousand
years ago, positional, and, in ellenic age, a special sign was
introduced as a placeholder where a numeral was missing. It was the
ancestor of the 'zero'. We underline as alco Cinese mathematics
employed a positional system, whose base was substantially decimal
and that employed a 'zero' sign to represent missing 'ciphers'.
Babylonian clay tablets show tables of Pythagorean triples and there
are good reasons to believe that they were found not empirically, but
computed by rules linked to a thorough understanding of the so called
"Pythagoras' theorem" (van der WAERDEN 1983). We can be quite sure
that this theorem was well known before Greek mathematics in many
cultures, from China to Egypt, and we may suppose it was proved by a
reasoning on some figure similar to fig.2. Such kind of ostensive
proof can be explicitly found in Indian mathematics (VdW), and maybe
to such example Wittgenstein thought when he referred in his
Philosophical Investigations, 144, to the "Look this!" of the Indian
mathematicians.
This kind of constructive outline of the theorem has been proposed
by Bretschneider and accepted as "the most tempting" by Heath in
its edition of Euclid's Elements (HEATH 1956), with a reservation we
will consider in the following. The result is apparent from the figures
and it needs just few words of comment, addressed to acknowledge
the presence in the two equal squares of four equal triangles. Their
suppression leaves in the first figure the square on the hypotenuse,
and in the second the two squares traced on the other sides of the
triangle.
In general, at least in Babylon, there was an almost complete know-how
in mastering the resolution of quadratic equations. In the clay tablets
we find the text of problems and the steps of their solution, and in
a following section we give two examples of such procedure. Hence, we
can guess the implicit knowledge they needed . So, we can be quite sure
that Babylonian scribes knew a lot of properties which now we call
'algebraic', though they had no algebra.
It is likely that such properties were found by the geometrical
reasoning we will describe in more details furtherly dealing with the
II book of Euclid's Elements.(HEATH 1956). We have also to remind that
in Babylonian, as well as Greek, mathematics the computations were
accomplished by pebbles on abacus-like boards, and then the geometric
representation of numbers, as witnessed also by the Pythagorean
'arithmetic' philosophy, was quite natural. For example, in fig.3 we
show the hypothetical 'proofs' of the formulas (here and in the
following ^2 means the square)
(a+b)^2=a^2+2ab+b^2 and (a+b)(a-b)=a^2-b^2
It is straightforward to recognise the similarity between the figure
proving Pythagoras' theorem and the above figure showing the 'square
of a binomial' and 'product of sum and difference' relations. Of these
kinds of 'proofs' there is no trace in Egyptian or Babylonian texts.
However, it is possible to account for this lack, considering the oral
and formulaic character of the cultural 'reproduction' in those ancient
civilisations, based on gestural and oral techniques and for which
writing had, beyond its ritual role, just a 'working memory' function.
Before the Greek 'school', for the scribes the religious-intellectual
learning had an apprenticeship structure, in which there were neither
'formulas' nor 'theorems', and geometrical constructions by drawings
and ropes (remind that the Greek name of the "scribes" was 'arpedonaptai',
that is 'rope-stretchers') were both the proof and the memorisation
tool of the geometrical 'property' (see also (van der WAERDEN 1983)).
Thus there is no need of distinguishing, as van der Waerden suggests,
two different Neolithic traditions of mathematic teaching: one based
on problems with numerical solutions, and another based on geometric
constructions and proofs. Actually, the latter was the proof theory of
the former.
This old geometric proof theory, where the words of the priest-teacher
were the time-scheduling of the construction in the space of the
theorem-figure, will deliver, in the making of Greek mathematics, the
inner structure of our "syntactic proof theory", where the formal proof
in the book is the process of construction of the theorem-proposition.
The reader of Euclid can directly realise as his proofs were most of
all comments to the construction of the figures.
In the Indian texts 'Sulvasutras', a proof of the procedure to compute
the area of a trapezium can be found explicitly written (see (SEIDENBERG
1962)), but, in my opinion, this accounts more for a better developed
role of writing in the Indian culture of that time, than for a substantial
difference in mathematics. By the way, such proof is exactly the
descriptionof the construction by which an isosceles trapezium can be
transformed in a rectangle. For a more detailed criticism of Seidenberg's
analysis, see Lloyd.
The Babylonian numerical system was very advanced, the equations practice
very skillful, the geometry comprehended at least complex properties of the
right-angled triangle. Nevertheless there was a "lack of explicit
statements of rules" and the "absence of clear cut distinctions between
exact and approximate results". Never they seem to raise "questions about
the solvability or unsolvability of a problem" nor "was there any
investigation into the nature of proof." (Boyer, (BOYER 1968), 44)
No proofs or theoretical questions: mathematical Babylonian texts contain
either problems or numerical tables.(NEUGEBAUER 1935) (NEUGEBAUER 1945) It
is worth noticing that the algebraic notation was of geometric origin ,
though in Babylonian (differently from Egyptian) mathematics the
geometrical problems played no special role (see (NEUGEBAUER 1957)).
Note: The unknown quantities are often called 'length', 'width' and
their product 'area'. The product of a number for itself is called
'square', the product of different numbers is called an 'oblong number'
(see (van der WAERDEN 1983)). Moreover the most direct interpretation
for the normal form of the quadratic equations is the geometric one
(see Neugebauer (NEUGEBAUER 1957)).