King Nebhetep-ra Mentuhotep, Dynasty XL, 2060 - 2010 BC(http://www.tulane.edu/lester/images/Ancient.World/Egypt/A57.gif)
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King Nebhetep-ra Mentuhotep, Dynasty XL, 2060 - 2010 BC
(http://www.tulane.edu/lester/images/Ancient.World/Egypt/A57.gif)
To: athena-discuss@info.harpercollins.com
From: "ERIC H. CLINE" [CLINE@xavier.xu.edu]
Subject: Objects, Ideas, and Transmission
Date: Wed, 8 May 1996 10:38:39 -0400 (EDT)
On Tuesday, 7 May, Edward Kent said:
"Writings are the stuff of elites, limited in space, time, and
perspective -- so if we find stuff diffused, why not ideas as well?"
In agreement and response, I proffer the following as food for
further thought and discussion, again taken from my book _Sailing
the Wine-Dark Sea_ (Oxford 1994):
pp. 54-55: "In sum, although the evidence for Lambrou-Phillipson's
`enclave colonies' at Boeotian Thebes and at Akrotiri on Thera is
unconvincing, such hypotheses may have been on the right track. If
the above criteria are applied to available and future archaeological
data, it may yet prove possible to identify the presence of resident
foreigners in the LBA Aegean. It is likely that individual Near
Eastern craftsmen and perhaps diplomats will be (or have already
been) identified at the larger cultural centers and citadels of the
LBA Aegean, such as at Mycenae and Knossos. However, the vast
majority of the Egyptian and Near Eastern personnel seen in the
region are more likely to have been merchants and seafarers whose
domiciles, whether permanent or temporary, were located in the
port cities such as Tiryns and Kommos (cf. also Negbi 1988:357
and Morris 1990:58, 62). It is at such harbor sites, therefore, that
we should look most carefully, and with the greatest hope of
success, for evidence of resident foreigners in the LBA Aegean.
"The idea of foreign seafarers, merchants, and craftsmen
particularly in the port cities (and harbor taverns) of the LBA
Aegean is, in fact, an attractive one, for we may well imagine these
men as the simplest means of transmission for a variety of ideas and
innovations. It is clear that there was much contact between the
Aegean, Egypt and the Near East during the LBA, and transfers of
ideas and innovations no doubt occurred. We might imagine that
such transfers of ideas took place not only at upper levels of
society, but also in the taverns and bars of the port cities in the
Aegean and the Eastern Mediterranean. Where else would a sailor
or crew member spend the long weeks waiting for the wind to shift
to the proper quarter or waiting for a diplomatic mission to
conclude its negotiations but in a tavern, swapping myths, legends
and tall tales? It is easy to envision such mundane events taking
place; resulting, occasionally, in dramatic cultural influences
reaching Greece from Egypt and from elsewhere in the ancient Near
East. Certainly, the notable parallels between certain Bronze Age
Aegean epics and myths and those found in Bronze Age Syro-Palestine,
Mesopotamia, Anatolia and Egypt (cf. Burkert 1987) may
well be readily explained via the telling of such tales in the
alehouses of the Aegean and Eastern Mediterranean. Transmission
of Semitic `loan-words' and of Syro-Palestinian gentilics would also
be readily engendered by such informal contacts. Such `low-level'
transmissions may well have complemented the `high-level' or
official contacts hypothesized by Morris (1989:42-46, 1990:58, 61-62;
see also Burkert 1984 and West 1988)."
Cheers,
-- Eric H. Cline
----------------
To: athena-discuss@info.harpercollins.com
From: "ERIC H. CLINE" [CLINE@xavier.xu.edu]
Subject: Identification of "Foreigners" in the Aegean
Date: Fri, 10 May 1996 18:01:22 -0400 (EDT)
To the List:
In a continuing (and thus far fruitless) effort to stimulate further
discussion, I would like to ask the following: if there were
indeed an Egyptian conquest of the Bronze Age Aegean (as per
Bernal), or even a partial colonization by Egyptians or people from the
Near East, or even simply some resettlement, how would one go about
identifying the presence of Egyptians and other "foreigners" resident
in the Bronze and Iron Age Aegean? As an initial point of departure,
I submit the following food for thought (taken from _Sailing the Wine-Dark
Sea_: pp. 52-54); comments and reactions on-list would be welcome and,
indeed, are desired:
"How then, acknowledging the possibility that they may have
existed, can we recognize such resident foreigners in the
archaeological record of the LBA Aegean? We should, as Branigan
and Lambrou-Phillipson have suggested, set up criteria to identify such
immigrants so that future scholars and excavators can have a secure
framework from which to operate. Perhaps we can work backwards,
from the known to the unknown, for resident immigrant craftsmen
from the Eastern Mediterranean, or products from their Aegean
workshops, have been identified at a number of 9th-7th century BC
sites in the Aegean. These include Knossos, Afrati, the Idaean Cave,
and possibly Kommos on Crete, and Athens, Lefkandi, Eleusis and
Anavysos in Attica and Euboea on the Greek Mainland (Boardman
1980:37, 56-62, with references; Higgins 1980:96-101; Themelis
1983; Shaw 1989; Morris 1992:140-141, 155-161; Coldstream
1982:261-272, 1993:99-100; Sherratt and Sherratt 1993:365; but
contra Muhly 1985c and Lembessi 1975). The criteria used to identify
these Near Eastern immigrants in the Iron Age Aegean are simple:
1) votive goods; 2) burial goods; 3) burial techniques; and/or
4) religious architecture which demonstrate unmistakable Near Eastern
origins or influences (cf. also Branigan 1981:25-27 and Schofield
1983:299 for similar criteria used to identify Minoan "Community
Colonies" elsewhere in the Aegean, and further discussion in T.R.
Smith 1987:156-158).
"Strangely enough, given the occupations of the above Near
Eastern immigrants, evidences of such resident emigres have rarely
been identified in habitation or workshop areas in the Iron Age Aegean
-- only their tombs and the objects produced in their workshops have
been identified, such as goldwork found in 8th century BC contexts at
Athens, Eleusis and Anavysos in Attica on the Greek Mainland
(Boardman 1980:58; Higgins 1980:96-101; Coldstream 1982:266-267)
and bronze shields evoking Near Eastern motifs given as votive
offerings at the Idaean Cave in Crete (Dunbabin 1957:40-43;
Boardman 1980:58-59; Shaw 1989:181; Morris 1992:152-154;
Coldstream 1993:99). Thus, we should perhaps add several criteria
to those mentioned above, so that we might identify additional
evidence for such immigrants, if any exists. These might incorporate
some of Lambrou-Phillipson's criteria (1987, 1990c), if used with the
proper caution. Branigan's original criteria (1981, 1983; and
Schofield 1983, 1984) will also be of value, if extracted from their
original 'Minoano-centric' context.
"It is essential to realize that any resident Syro-Palestinians
[NB: or Egyptians, or Hittites, or ...] in the LBA Aegean would have
been vastly outnumbered by the 'native' inhabitants, namely the
Mycenaeans and Minoans. They will have been, quite literally, a
'minority' within the LBA Aegean. As such, therefore, we might turn
to anthropology in seeking a means to identify such ethnic or cultural
'minority groups' residing in the LBA Aegean specifically as a result
of an established trade network or diaspora (cf. Curtin 1984:1-14;
T.R. Smith 1987:149-161; Sherratt and Sherratt 1991:356-357). As
Harris (1988:413) says:
`These groups...have distinctive life-styles that can be
traced to the cultural traditions of another
society...and...their members are conscious of their
existence as a group set apart from the rest of the
population.'
"Such groups are usually identified by anthropologists via
'ethnic boundary markers,' which serve to distinguish their members
from all other groups. These can entail differences in language,
religion, physical appearance, or particular socio-cultural traits
including clothing, architecture, personal adornment, food, technology,
and general lifestyle. Combinations of such 'markers' are commonly
used, since a single 'boundary marker' is seldom sufficient to allow
conclusive identification (Weber 1961; Barth 1969; Cohen 1978:385-387;
Reminick 1983:8-13; De Vos 1975; Schwartz 1975; Harris
1988:413; Peoples and Bailey 1991:375-376; Howard 1993:237).
Most such 'markers' will, of course, not have survived a burial of
3500 years and will not be present in the archaeological record.
Some, however, will still be extant, particularly the material remains
left from the above socio-cultural traits -- namely architecture, objects
of personal adornment, and artifacts resulting from new technology.
"Anthropologists have noted that ethnic or cultural 'minority
groups' frequently assimilate rapidly into the larger local community
(Barth 1969; De Vos 1975; Reminick 1983:18-40; Peoples and Bailey
1991:376-377). If this occurred in the LBA Aegean, it will be
extremely difficult to identify such 'enclave colonies' or 'minority
groups,' as Schofield (1983:296) and Lambrou-Phillipson (1987) have
pointed out (cf. also Harris 1988:416; Tournavitou 1990). Individual
residents may not ever reveal their presence in the archaeological
record; others may reveal themselves only in death -- via burial goods
or burial customs. This is the case with the 9th century BC Near
Eastern goldsmith and his family buried in the Teke tomb at Knossos,
who were identified by the burial of two 'crocks of gold' as a
foundation deposit just inside the entrance to the tomb -- a
predominantly Eastern practice -- and by the nature and style of the
goldwork found inside, which find their best parallels at Tell Halaf in
Mesopotamia (Boardman 1967, 1980:57; Higgins 1980:107-111;
Coldstream 1982:267-268, 1993:99-100; Shaw 1989:181; Morris
1992:157-158, 161 n. 55). We should note also the bronze bowl
inscribed with a personal name in Phoenician letters found in another
9th century BC burial at Knossos (Boardman 1980:37, figure 6; Cross
1980:15-17; Coldstream 1982:271-272; Shaw 1989:181 and n. 64;
Morris 1992:159; Sherratt and Sherratt 1993:365).
"However, an entire 'enclave' of resident aliens will more than
likely be apparent in some manner in the archaeological record, even
if it is only in the textual evidence, as in the case for the Assyrian
colony at Kultepe-Kanesh in Anatolia. Anthropologists have long
noted that an ethnic or cultural 'minority group' will frequently live
separately from, or separately within, the larger community -- either
voluntarily or involuntarily (Barth 1969; Milosz 1975; Cohen 1978;
Reminick 1983:8, 27-28; Curtin 1984:11-12; Harris 1988:416-417;
Peoples and Bailey 1991:387; also Branigan 1981:26; T.R. Smith
1987:157). This is, in fact, the observable case at LBA Ugarit in
Syro-Palestine, where most of the resident foreigners, who included
Cypriots, Hittites, Cilicians, Egyptians, Canaanites, Assyrians,
Ashdodians, and others, were relegated to one or two special districts,
including the port area at Minet el-Beidha, where they lived under the
supervision of the akil k ri "overseer of the merchant colony" (Astour
1965b:253, 1973:25, 1981:25; Rainey 1963:319; North 1973:126-127;
Courtois 1974:107-108; Linder 1981:35-36).
"Lambrou-Phillipson (1987, 1990c) is correct in suggesting that
radical changes in technology can imply the arrival of foreigners (cf.
also Boardman 1990; Knapp 1990c:124, 128). Moreover, although
foreign goods, changes in burial procedures, or changes in secular and
religious architecture by themselves are not necessarily indications of
resident aliens, a combination of more than one of these factors found
together in similar contexts at a single site is a good indication of their
presence. Three instances from Iron Age Crete serve as prime
examples. At Afrati (ancient Arkades), the early 7th century BC
cemetery yielded clay objects which imitate North Syrian objects in
other materials, as well as containing distinctive Near Eastern
cremation burials, in which lidded urns were set on flat dishes,
covered by inverted pots, and surrounded by a ring of stones -- the
closest parallels for these burials are found in the Iron Age cemetery
at Carchemish (Kurtz and Boardman 1971:174; Boardman 1980:60;
Morris 1992:160-161; Coldstream 1993:100). At Kommos, a tripillar
shrine inspired by Phoenician models was set up in Temple B during
the 8th century BC, while more than 200 fragments of Phoenician
pottery, mostly amphorae, were found in Temples A and B, in
contexts dating from the 10th-8th centuries BC (Shaw 1989; Morris
1992:155; Sherratt and Sherratt 1993:365). In the Teke tomb at
Knossos, as mentioned, the 'foundation deposit' indicates a Near
Eastern style of burial, while the gold objects found inside and
elsewhere show new gold-working techniques and methods (Boardman
1967). In all three of these situations, it is the combination of criteria
which gives rise to the suspicion that resident, or semi-resident,
foreigners were present at those Aegean sites.
"Thus, we might be able to identify the existence of resident
Syro-Palestinians, Egyptians, Italians or other Mediterranean peoples
in the LBA Aegean if we look for situations where more than one of
the following criteria are found together:
1. worked foreign goods deposited as votive offerings
2. worked foreign goods deposited as burial offerings
3. worked foreign goods (a substantial quantity) in a workshop or
crafts area
4. local materials worked in a manner previously foreign to the area
5. foreign non-luxury (i.e. day-to-day) items in habitation or burial
contexts
6. changes in burial techniques or architecture paralleled or inspired
by foreign antecedents
7. changes in religious architecture paralleled or inspired by foreign
antecedents
8. changes in secular architecture paralleled or inspired by foreign
antecedents
9. changes in technology paralleled or inspired by foreign antecedents
10. textual evidence attesting to the presence of resident foreigners
If there are any sites in the LBA Aegean where several of the above
criteria appear in combination, we might be able to hypothesize the
presence of resident foreigners in the Aegean during the second
millennium BC.
"The 38 Near Eastern cylinder seals at Boeotian Thebes are the
most famous objects which have been cited as evidence for Syro-Palestinian
colonists in the LBA Aegean. Such assertions were put
forth soon after the discovery of these seals by scholars trying, among
other things, to correlate these finds with the legends surrounding the
Phoenician Kadmos (Culican 1966:54-55; Sasson 1966:135 n. 53;
Astour 1967:387; Hemmerdinger 1967:232-240). They were
repudiated almost immediately (Muhly 1970:37-38, 41, 61; Edwards
1979:131-137; cf. Morris 1990:58-60, 1992:104). Since the presence
of these seals fulfills only one of the above criteria (no. 3: 'worked
foreign goods [a substantial quantity] in a workshop or crafts area')
and since there is no other convincing evidence for such an enclave of
foreigners, there is currently no basis to hypothesize any resident
Syro-Palestinians at Boeotian Thebes. As noted above, suggestions for the
imported cylinder seals as the raw stock of a local Mycenaean artisan
or as the prized collection of a Mycenaean noble are more convincing;
even Porada's explanation (1981) of these seals as a gift from
Tukulti-Ninurta I of Assyria to the Theban king seems more likely.
"Possible evidence for a resident Syro-Palestinian in the LBA
Aegean may be seen in the reuse of Tomb Rho in Grave Circle B at
Mycenae, of LH IIA date. The tomb exhibits characteristics
comparable to slightly-later graves at Ugarit in North Syria and at
Enkomi in Cyprus (Mylonas 1966:106-107, 1983:56-57). Moreover,
the reworked tomb originally contained a lapis lazuli scarab of possible
Hyksos origin (152). Tomb Rho, like the above 9th century BC Teke
tomb at Knossos, thus meets two of the above criteria: no. 2 ('worked
foreign goods deposited as burial goods') and no. 6 ('changes in burial
techniques or architecture paralleled or inspired by foreign
antecedents'). However, it must be noted that the characteristics of
Tomb Rho have now been shown to be as comparable to local Aegean
tombs at Ayia Irini on Kea, Phylakopi on Melos and at Thorikos on
the Greek Mainland as they are to the tombs at Ugarit and Enkomi
(Dickinson 1977:64). Since none of these other Aegean graves contain
Orientalia, it is likely that only architectural influences, rather than
actual immigrants, from Syro-Palestine are reflected in these Aegean
burials, including Tomb Rho.
"Perhaps the most viable evidence for resident foreigners
physically present in the LBA Aegean may be seen at Phylakopi,
Tiryns, and elsewhere at Mycenae. First, Negbi (1988:341-345; cf.
also Morris 1992:110-111) has amply documented that the architecture
of the sanctuary at Phylakopi on Melos during the LH IIIC period, and
perhaps during the LH IIIA and IIIB periods as well, is related and
indebted to Levantine sacred architecture present in Canaan during the
Middle and Late Bronze Ages. Moreover, there are numerous
Orientalia found in and around the Phylakopi sanctuary, including two
Canaanite 'Smiting God' statuettes (14-15), an Egyptian stone pendant
(84), an Egyptian or Syro-Palestinian faience scarab (124), and a Near
Eastern rhyton made of ostrich eggshell (950). Thus, at Phylakopi we
meet two of the above criteria: no. 1 ('worked foreign goods deposited
as votive offerings') and no. 7 ('changes in religious architecture
paralleled or inspired by foreign antecedents'). Negbi (1988:357) is
almost certainly correct in suggesting that:
`the minor shrine of Phylakopi was reserved for foreign
cult that was presumably practiced by Canaanite
seafarers engaged in East Mediterranean trade.'
"Second, Negbi (1988; followed and enlarged by Morris
1992:108-109) has pointed out that portions of the Cult Center at
Mycenae may also have antecedents in Levantine sacred architecture.
If true, then the Orientalia found in the Cult Center (69, 97, 100, 119-120)
take on further importance, for the same two criteria for resident
foreigners at Phylakopi would be present at Mycenae: no. 1 ('worked
foreign goods deposited as votive offerings') and no. 7 ('changes in
religious architecture paralleled or inspired by foreign antecedents').
Moreover, if the faience plaque of Amenhotep III (97) found in Room
31 of the Cult Center is of local Mycenaean manufacture (Cline
1990:210), then we would fulfill a third criteria -- no. 9 ('changes in
technology paralleled or inspired by foreign antecedents') -- and would
have evidence for at least one Egyptian craftsman resident at LBA
Mycenae.
"Third, at Tiryns the relatively large numbers of Cypriot non-luxury
goods found only in LH IIIB contexts at this site -- nine
terracotta wall brackets and three ceramic vessels (399-400, 632, 788-796)
-- may imply the existence of resident foreigners. Furthermore,
Hirschfeld (1990) has determined that a number of LH III vessels
manufactured locally at Tiryns had Cypro-Minoan marks (at least 16
different signs) incised after firing but before they left the Greek
Mainland; corresponding vessels with similar marks have been found
on Cyprus itself. This may indicate a Mycenaean export trade aimed
specifically at Cyprus, and the presence of individuals at Tiryns who
were familiar with the Cypro-Minoan marking system (if not
themselves Cypriote). Two of the above criteria may thus be fulfilled
at Tiryns: no. 4 ('local materials worked in a manner previously
foreign to the area;' e.g. incising with Cypro-Minoan marks) and no.
5 ('foreign non-luxury [i.e. day-to-day] items in habitation contexts').
While these may simply be indicative of reciprocal trade -- Mycenaean
vessels for Cypriot wall brackets and pottery -- they might also be seen
as tentative evidence for resident Cypriots at Tiryns.
"The possible Syro-Palestinian personal names found in the
Linear B texts (A-ra-da-jo, Pe-ri-ta, Tu-ri-jo, and po-ni-ki-jo) also may
very well be an indication of the presence of individual Syro-Palestinians
in the LBA Aegean. They are, however, unlikely to be
evidence for entire 'enclave colonies.' Moreover, the extensive
contact and trade between the LBA Aegean, Egypt and the Near East
provide a ready explanation for the numbers of Semitic "loan-words"
found in the Linear B texts (above), without resorting to the notion of
colonies or conquest by Syro-Palestinians. These few words in the
Linear B tablets are a far cry from the extensive textual documentation
which should be present if entire 'enclave colonies' were in residence
at Akrotiri, Thebes, Tiryns, Pylos, Knossos or Mycenae, although it
is conceivable that such records have not yet been found or that they
were kept on perishable materials such as papyrus.
"In sum, although the evidence for Lambrou-Phillipson's
'enclave colonies' at Boeotian Thebes and at Akrotiri on Thera is
unconvincing, such hypotheses may have been on the right track. If
the above criteria are applied to available and future archaeological
data, it may yet prove possible to identify the presence of resident
foreigners in the LBA Aegean."
Thoughts and reactions, anyone?
Cheers,
-- Eric H. Cline
------------------
To: athena-discuss@info.harpercollins.com
From: "ERIC H. CLINE"
Subject: Egyptian Objects in Greece
Date: Tue, 7 May 1996 12:09:47 -0400 (EDT)
In reply to Chuck Grimes' posting on Sunday, 5 May regarding
Egyptian artifacts in Greece:
Chuck, I am afraid that you are incorrect to state:
"So where are the Egyptian artifacts in Greece? As far as I know
there are none."
There are, in fact, some 236 Egyptian objects found in Late Bronze
Age contexts on the Greek mainland, Crete and the Cycladic
islands. There are even more Egyptian objects found in later
contexts in the Aegean area, e.g. during and after the so-called
"Orientalizing" period during the first millennium BC.
Allow me to take this opportunity to introduce some relevant hard
data into this discussion by quoting from the chapter entitled "Egypt
and the Late Bronze Age Aegean" in my recent book (_Sailing the
Wine-Dark Sea: International Trade and the Late Bronze Age
Aegean_; Oxford: Tempus Reparatum, 1994):
p. 31-32: "Much of the archaeological evidence, namely the
Mycenaean and Minoan pottery found in Egypt, is well known.
These Mycenaean and Minoan goods, primarily ceramic vessels
which will have originally contained wine, oil or perfume, are found
throughout Egypt during the New Kingdom Period. Ongoing
excavations continue to increase the number of these objects --
which now number more than 1800 vessels of various shapes and
sizes (Bell, personal communication). Although LH/LM [Late
Helladic/Late Minoan] I-II pottery is relatively scarce, such sherds
have been found at nine sites, including Abusir, Memphis, Gurob,
Kahun, Sedment, Abydos, Deir el Medina, Thebes and Armant
(Vincentelli and Tiradritti 1986:327-334; Kemp and Merrillees
1980:226-245; Manning 1990:93). A new LM IA or IB sherd,
which may prove useful for chronological studies, has recently been
excavated at Kom Rabia (Memphis) in Egypt (Warren and Hankey
1989:138-146, esp. 139 [RAT 530.1301]). LH/LM III pottery as a
whole appears to have been consistently imported throughout most
of the 14th-12th centuries BC. Such pottery has been found at
approximately thirty sites in Egypt, from Marsa Matruh on the
northwest coast to Sesebi in the far south (MAP 2). It should be
noted that the accumulating evidence clearly indicates that the
importation of Mycenaean pottery was not unique to Akhenaten, his
capital at Tell el Amarna, or the 'Amarna Period', as previously
suspected. Rather, such pottery was in use over great areas of
Egypt and was imported by a number of Pharaohs, from Amenhotep
III to Ramses II (Redford 1983:482; Cline 1987:13-16).
"The other half of the extant archaeological evidence,
namely the Egyptian imports in the LBA Aegean, include transport
amphorae, storage jars, jugs, bowls and vases in ceramic, stone,
and glass, as well as scarabs and figurines of faience, frit and
steatite. More than half of these are functional items rather than
trinkets, imported consistently over the course of the Late Bronze
Age. The perishable trade goods, including perhaps grain, textiles,
and metals sent between the two areas (e.g. Barber 1991:311-357;
Bernal 1991:482-489), must also be taken into account; although
these goods have long since disappeared, they can be seen depicted
in Egyptian tomb paintings and are occasionally mentioned in
written texts.
"The high points of contact between Egypt and the Late
Bronze Age Aegean seem to be during the reigns of Thutmose III,
Amenhotep III, Ramses II, and possibly Ramses III. In all, there
are some 236 Egyptian objects found in good LH/LM I-IIIC
contexts. Of these, 75 are on the Mainland, 120 are on Crete, 11
are on Rhodes, nine are on the Cycladic Islands, 16 are on the Ulu
Burun (Kas) wreck, and five are on the Cape Gelidonya wreck.
These do not include the objects found in contexts too broad for
assignation to a specific period (i.e. LH I-III). It is interesting to
note that the situation in the LH/LM III Aegean, with regard to the
importation of Egyptian objects, has become completely reversed as
a result of excavations in the past sixty years. For instance, in 1930
Pendlebury (1930a:84-85) listed only two Egyptian objects found in
LM III contexts on Crete; there are now 53.
"It is likely that direct commercial trade between the Aegean
and Egypt began at least as early as the emergence of the Minoan
palatial centers (cf. Phillips 1990). As mentioned, by the time of
the LH/LM I-II periods in the Aegean, Egyptian objects comprise
the vast majority of the Orientalia. There are 82 Egyptian objects
in these contexts, compared to fewer than 25 objects from any other
Near Eastern country (TABLE 3, FIGURE 2). Fully 67 of these 82
Egyptian objects (82%) are found on Crete, most in LM IB contexts
(TABLE 2, FIGURE 1). Even these numbers are misleading, for
many of the objects found outside Crete (e.g. a number in the Shaft
Graves at Mycenae) appear to have reached their final destinations
via Crete. Quite a few were reworked by the Minoans, being
altered and readapted for Aegean purposes after their importation
(Phillips 1989)."
p. 32: "Tomb paintings and literary references in Egypt during this
time provide further evidence for contacts with the Late Bronze Age
Aegean (Vercoutter 1956; Strange 1980; Sakellarakis and
Sakellarakis 1984; Wachsmann 1987; Laboury 1990). References to
the LBA Aegean and to Aegean peoples are far more common in
Egypt than in any other Near Eastern country. The term Kft(j)w,
vocalized as Keftiu, is most likely the Egyptian name for the island
of Crete and the Bronze Age Minoans (Vercoutter 1956:33-38, 116-122;
contra Strange [1980], who attempted to equate the term with
the island of Cyprus; and contra Morris 1992:102-103, who
suggested that the term is a more general reference to all Aegean
and even Levantine seafarers). The term Iww hryw-ib nw W3d-wr,
translated as "the Isles in the Midst of the Great Green," is usually
taken as a reference to the Cycladic islands of the Aegean, perhaps
including Crete (Vercoutter 1956:125-127, 149-157; contra
Vandersleyen 1985:44-46). The term Tj-n3-jj is to be read Tanaja
(possibly vocalized as a variation of *Danaoi) and is most likely a
specific reference to the land of the Mycenaeans, in the Late Bronze
Age Peloponnese (Faure 1968:145-147). These three terms occur
primarily during the 18th, 19th, and 20th Dynasties, but earlier
references to Keftiu and to the Isles do occur; and later examples of
all three terms can be found as well. A final term h3w-nbw.t (Hau-nebut),
long thought to refer to Greeks and the Aegean, is more
likely an allusion to areas in Syro-Palestine and has not been
included in these discussions (cf. Vercoutter 1947, 1949;
Vandersleyen 1971:140-174, 1988:78-80; Iversen 1987:54-59; Nibbi
1989:153-160; contra Bernal 1991:416-417, who argued for the re-establishment
of this term as a reference to the Aegean).
"In 1956 Vercoutter stated that references to Keftiu occur
most frequently in Egyptian documents and tombs dating to the 15th
century BC: a total of 16 times. He further stated (1956:114-115)
that the following centuries saw a decrease in the occurrence of the
term Keftiu: only two in the 14th century, three in the 13th, and one
in the 12th century BC. However, matters have changed somewhat
since Vercoutter's day. Some of the occurrences cited by him, in
particular unlabelled scenes in tombs ostensibly depicting "Aegean"
peoples, have been thrown out as unreliable for serious
considerations. Other examples have been noted and added to the
corpus over the past thirty years.
"Thus, we now have the following instances in Egypt:
Middle Kingdom Period: 1 Keftiu, 1 Isles, 0 Tanaja
17th-16th century: 3 Keftiu, 1 Isles, 0 Tanaja, 1 Generic
15th century: 8 Keftiu, 5 Isles, 1 Tanaja, 3 Generic
14th century: 6 Keftiu, 2 Isles, 3 Tanaja, 2 Generic
13th century: 6 Keftiu, 4 Isles, 2 Tanaja
12th century: 1 Keftiu, 9 Isles, 0 Tanaja
These findings indicate a consistent pattern of contact between
Egypt and the Aegean throughout the Late Bronze Age (see
FIGURE 7)."
p. 35: "In addition to the archaeological evidence in the Aegean, we
also have textual evidence for this international trade. There are
two textual references to Egypt and the Egyptians found in the
Linear B texts in the Aegean. These appear only in the tablets
found at Knossos: mi-sa-ra-jo = "Egyptian" and a3-ku-pi-ti-jo =
"Memphite" or "Egyptian." The former term, mi-sa-ra-jo is
interesting, as it apparently comes from the Semitic word for Egypt,
Miraim, more commonly found in Akkadian and Ugaritic
documents in Mesopotamia and Syro-Palestine. The latter term,
a3-ku-pi-ti-jo, may also be derived from a Syro-Palestinian reference to
Egypt, for an Ugaritic name for both Egypt and the city of
Memphis was ikupta, which corresponds to the ikuptah of the
Amarna letters and to t-k'-pt in Egyptian (see Virolleaud
1953:192). It is used in the Linear B tablet as the name of an
individual who was in charge of a flock of 80 sheep at the Cretan
site of su-ri-mo. The same name, again used for an individual, is
also used later by Homer (Od.II.15). As Palaima (1991:280) states:
`personal names derived from foreign toponyms also attest to
overseas contacts at some stage prior to the dates of the tablets on
which they are recorded.'"
Discussants on this list might also be interested in the data originally
presented by me in:
"Of Shoes and Ships and Sealing Wax: International Trade and the
Late Bronze Age Aegean," _Expedition_ 33/3 (1991) 46-54 (with
M.J. Cline). [NB: written for a "lay" audience]
"Contact and Trade or Colonization?: Egypt and the Aegean in the
14th - 13th Centuries B.C.," _Minos_ 25/26 (1990-91) 7-36.
"Amenhotep III and the Aegean: A Reassessment of Egypto-Aegean
Relations in the 14th Century BC," _Orientalia_ 56/1 (1987) 1-36.
Cheers,
-- Eric H. Cline
-------------------
To: athena-discuss@info.harpercollins.com
From: "Staffas V. Broussard" [SVBLL@jazz.ucc.uno.edu]
Subject: Greek Algebra and Eurocentrism
Date: Mon, 20 May 1996 12:34:17 -0600 (CST)
Stirling,
Thanks for the reply. I agree with much that you say.
You asked for clarification on "genetic relationship" and
"Eurocentrism."
I borrowed the term "genetic relationship" from linguistics.
There, a genetic relationship is established between two
languages when they are shown to have the same phonetic
correspondences according to "sound laws" within a particular
language family. This is contrasted with a typological
relationship which is based on resemblances between languages due
to borrowing and contaminations from contact and other cultural
influences. I didn't intend to draw an analogy here between
biological evolution and cultural development, but only used the
term metonymically in the sense that language is a part of
culture.
However, that a genetic relationship ( genetic in analogy
with linguistics or evolution) as opposed to a relationship
dependent on contact and borrowing has been asserted between the
two cultures can not be denied. Read any middle or high school
history text, browse through any public library, examine the
journals. Or, for example, listen to Heidegger, one of the
greatest European philosophers of this century, go on and on
about the intimate relationship between the German and Greek
languages, the only languages suitable for philosophy, a
Eurocentric notion indeed. It is this assertion that I
questioned.
************
Why Eurocentrism Should Be Included In This Debate
************
In reference to Eurocentrism and Eurocentrics, you wrote on
19 May 1996:
I am interested in what they say only in the
negative sense. The declaration that some one else
is a "eurocentric" seems to me to be primarily a code
word for "you are a nazi racist" -implying somehow that
the target of the accusation supports the aryan state
models. Further in almost every case when I read that
some group of people "are unable to uncover their basic
assumptions" I find the subtext is "and those basic
asusmptions are that their culture is one big
conspiracy against people like me".
From your post, I know that you would agree that there is
a need to tell the true story of mathematics. On 19 May 1996,
you wrote :
What irks me about teaching of Algebra in America
is that while Geometry is taught with reference to
names and people, and so is calculus and analytic
geometry, Algebra, which was not developed by
Europeans in any significant way, one gets simply
formulas and proofs. No mention of the long arguments
which finally lead to the full acceptance of negative
solutions to quadratics or the bionomial theorum.
Silence.
However, the phenomena of which you speak has a cause,
namely, the almost universal belief in the 19th and early 20th
century, a belief that is alive and well today, that non-European
cultures were incapable of making genuine mathematical and
scientific discoveries. This is part of what I am referring to
when I speak of Eurocentrism. Nazis did not invent the science
of race, or scientific racism, neither did they invent
Eurocentrism, which has been a general feature of European
civilization since the late 18th century. Much of it was
invented by European intellectuals. But, this is part of the
story also, which must be told. Since any corrective which seeks
to give more accurate and complete presentations of the history
of mathematics, will also need to address historiographic issues
in the history of mathematics. And it must be told, because my
people need to recover from the humiliation and degradation of
European slander against their humanity promulgated through
European cultural hegemony for at least the last two hundred
years.
In my reading of European historians of mathematics and
science, I have found two Eurocentric features almost always
present: 1) the privileging of Greek mathematics over the
mathematics of other ancient cultures and 2) the characterization
of mathematics in such a way that it becomes a Greek creation.
These two features partially constitute the Eurocentrism of which
I speak. Moreover, the Eurocentric search for identity through
history clearly has a history and we can identify some of its
themes: 1) Common values are shared by modern European and
ancient Greek civilization. Rationality and scientific method
are examples. Differences between the two civilizations are
excluded in order to construct an isomorphism between them; 2) by
ignoring singularities along the path of European development, a
continuous zigzag is drawn from ancient Greece to modern Europe.
Selection and combination, metaphor and metonymy are tools
of historical writing. Here, selection includes omission. It is
evident that cultural beliefs influence selection and combination
in historical writing. However, for Eurocentric historians, what
seems to be missing is an awareness that the beliefs described
above influence what they select and combine in writing histories
of science and mathematics and, how in their selection and
combination, they reinforce those same beliefs.
I offer the following examples of Eurocentric writing. The
first two dealing with Indian mathematics and the others with
Greek and Egyptian mathematics.
Indian Mathematics and Proof
Consider the opinions of the following two historians
concerning Indian Mathematics. Let's begin with a late opinion
from Lloyd (1990)[[1]].
It would appear that before, in, and after the
Sulbasutra [the earliest know evidence of mathematics
from India], right down to the modern representatives
of that tradition, we are dealing with men who tolerate,
on ocassion, rough and ready techniques. They are
interested in practical results and show no direct
concern with proof procedures as such at all.
(Lloyd, 1990, p. 104)
And from a text found in every university library
in the world written by Kline (1972)[[2]]:
There is much good procedure and technical facility,
but no evidence that they (i.e., the Indians) consider
proof at all. They had rules, but apparently no
logical scruples. Moreover, no general methods or new
viewpoints were arrived at in any area of mathematics.
It is fairly certain that the Hindus (i.e., the Indians)
did not appreciate the significance of their own
contributions. The few good ideas they had, such as
separate symbols for the numbers, were introduced
causally with no realization that they were valuable
innovations. They were not sensitive to mathematical
values. (Kline, 1972, p.190)
An early Indian Proof:
Now consider the following example from the Apastamba
Sulvastura (which can be dated anywhere from 1000 B.C to 200
A.D.) The translation is taken from Seidenberg [[3]]. An altar is
described in the form of an isosceles trapezium with its eastern
base 24 units, its western 30, its width (from M to L) 36. The
text says that the area is 972 square units. The priest proceeds
to prove the propositions. I have omitted certain phrases in the
proof. The abridged proof follows:
E
F A M D
_____************************
| * |*
| * | *
| * | *
|********************************
B L E C
W
One draws a line from D toward C to the point E which is 12
(padas from the point L). Thereupon one turns the piece cut off
(i.e., the triangle DEC) around and carries it to the other side
(i.e. to the north). Thus, the vedi obtains the form of a
rectangle. In this form (FBED) one computes its area.
Is this not a proof? There are many more instances in the
Sulvasutras and other Indian works. One can only ask how closely
did they bother to examine these works. Of course, the
Sulvasutras are religious works and given the predilection on the
part of Eurocentric historians to consider myth, ritual and
religion as unscientific, they may not have read them closely at
all. This predilection is part of their rationalist disposition,
a shadow of Greek rationalism. Yet, even, if we allow that they
did read these great works, the question arises would they have
recognized the proofs as proof. For Kline and Lloyd would have
looked for mathematical discussions within a formal deductive
system. After all, if they could find no derivations from
formally stated axioms in these works, then no "real mathematics"
could possibly inhabit these vibrant works. This is precisely
the characterization of mathematics that makes it a uniquely
Greek creation and excludes non-European mathematics from the
mathematical table.
The Transmission of Geometry to Greece
Most of this is in a previous post, but I will assume
you missed it. However, I believe the following account
illustrates how European writing and rewriting of the history of
science can reflect how one eventually invents Europe and
Eurocentric culture.
In his classic work on ancient mathematics and astronomy,
"Science Awakening" [[4]], published in 1961, B.L. Van Der
Waerden began his chapter on Egyptian mathematics by assembling
testimonies from Aristotle, Herodotus, and Democritus that praise
the mathematical abilities of the Egyptians. However, his aim
was to discredit this testimony. He wrote, "We are going to show
that Egyptian geometry is not a science in the Greek sense of the
word, but merely applied arithmetic.."
It might appear that his arguments were based on the extant
Egyptian mathematical texts; after all, he used the analytic
tools of modern scholarship and followed the tradition of another
great european scholar of ancient science, Otto Neugebauer, much
of whose work he incorporates. Neugebauer would find little to
disagree with in Van Der Waerden's work. However, Van Der
Waerden narrowly interpreted the texts, dismissing and arguing
away any evidence that might support an "hypothesis concerning a
lost Egyptian higher mathematics", and offering explanations of
the texts that in the words of Neugebauer are "preferable."
"What could the Greeks have learned from the Egyptians?" he
asked. Nothing, he concluded. After all, he found nothing in
their extant texts and the evidence from Babylonian mathematics
supplies a basis for Greek mathematics. So, "we do not need to
set up hypotheses concerning a lost Egyptian higher mathematics."
In an article published the next year, [[3]], Abraham
Seidenberg argued that the very evidence that Van Der Waerden
brings forward proves the opposite. According to Seidenberg,
from this same evidence, we must conclude that the Egyptians knew
that the ratio of the circumference of the circle to its diameter
was 4 times the ratio of the area of any circle to the square on
its diameter. "And how did they come to such a realization?"
Seidenberg asked. "Surely by using their intelligence," he
concluded. "If this was done through a geometric analysis--and
we see no other way--then this analysis, no matter how crude,
shows that Egyptian geometrical knowledge was not merely
arithmetical."
In "Geometry and Algebra in Ancient Civilizations",
published in 1983 [[5]], Van Der Waerden yielded to Seidenberg's
arguments: "Modern authors, including myself, have sometimes
adopted a skeptical view towards the idea that the mathematical
sciences were transmitted from Egypt to Greece....At present I
believe this skepticism is not quite justified and that there may
be a considerable truth in the statements of Herodotus,
Democritus, and Aristotle." However, he didn't retreated
entirely from the Eurocentric position. In his own words, "On
the other hand, we need not adopt Aristotle's opinion that the
mathematical sciences originated in Egypt. It seems much more
probable that they originated in Neolithic Europe, and they were
subsequently transmitted to China, India, Babylonia, Egypt and
Greece."
Reasonable arguments do not dampen the Eurocentric desire to
be at the origin of things. While Van Der Waerden has been
willing to abandon the Greek outpost of European civilization,
albeit, for the interior of Europe Major, many Eurocentric
scholars continue to resist. As an example in the 1990s of the
continued privileging of Greek mathematics over the mathematics
of other ancient cultures, we can read Jens Hoyrop discussing
scientific (Greek) and subscientific (Babylonia and Egyptian)
mathematics, using Aristotle to develop his ideas. At the same
time, Hoyrop [[6]] can boldly quote from Aristotle's
"Metaphysics" on the origins of science but omit from the same
quote the statement: "Thus, the mathematical sciences originated
in a neighborhood of Egypt."
The story of the European writing and rewrting of the
history of science and mathemtics will be told. It is an
important part of our history. It will be told and retold.
I hope I have clarified some things for you and thanks for
reading this far.
[[1]] Lloyd, G.E.R. (1990) _Demystifying Mentalities_, Cambridge,
Cambridge University Press.
[[2]] Kline,M. (1972) _Mathematical Thought from Ancient to
Modern Times_, New York, Oxford University Press.
[[3]] Seidenberg, Abraham (1963), "The Ritual Origins of
Geometry", _Archive for History of the Exact Sciences_,
1, pp.487-527.
[[4]] Van Der Waerden, B.L. (1961), _Science Awakening_, New
York, Oxford University Press.
[[5]] Van Der Waerden, B.L. (1983),_Geometry and Algebra in
Ancient Civilizations_, New York, Springer-Verlag
[[6]] Hoyrup, Jens (1994), _In Measure, Number, and Weight_,
New York, State University of New York Press.
Staffas
------------------
To: uunet!athena-discuss%info.harpercollins.com@uunet.uu.net
From: "S.F. Thomas"
Subject: Egyptian Science, the Greeks, and Mathematical PROOF
Date: Sat, 18 May 1996 17:10:47 +0100
paul manansala wrote:
(( cuts ))
>
> Also, concerning Egyptian math, the Afrocentric
> side would be having one difficult time if not
> for the discovery of the Rhind, Moscow, Kahun, Berlin
> and other papyri, despite the historical evidence
> and the colossal engineering works of the Egyptians.
So true...
I've been lurking here for some time, and thought finally I
should weigh in, and lend Paul a hand, not that he needs it,
he has been doing a great job here. (Hi Paul, remember me
from afojs? Glad to see you again.)
First, let me state my credentials and get that out of the
way. I have none. I am not an Egyptologist, nor am I a
mathematician. But I do have a Ph.D in a mathematical
field, and I have spent considerable time thinking about
notions of PROOF, and have written a book,
_Fuzziness_and_Probability_, which bears (tangentially to be
sure) on the subject. As to any bias I might bring to the
subject, let me identify myself as of the African diaspora,
born and raised in the Caribbean, schooled in the Western
tradition.
My bias is therefore with the Afrocentrists, for having read
Diop, James, Bernal and others, it has become clear to me
that more than a threshold showing has been made that
Western scholarship has been Eurocentric at the expense of
truth. Malcolm X said, "we have been hoodwinked, we have
been brainwashed, and we have been bamboozled," (or
something such) in referring to the lies that have been
taught us in the history books about black people and their
origins. Bernal, more scholarly, says that the Ancient
Model was supplanted in the 18th and 19th centuries by the
Aryan Model, a milder way of saying the same thing, while
adverting to the racist motives that lay beneath the
historical distortions that constitute the received
(Eurocentric) history to this day. Diop puts the matter
more plainly, as does James. James spoke of a "stolen
legacy", while Diop has accused the ancient Greeks of
nothing less than plagiarism.
That is a serious charge, and Diop succeeds, brilliantly in
my opinion, in making a prima facie case in support of the
charge. I am not an Egyptologist, as I said, therefore I
cannot offer an independent opinion of the factual claims on
which his argument rests. But the charge appears
compelling. For those who have not read it, I commend his
_Civilisation_or_Barbarism_, specifically the discussion in
Chapter 16, "Africa's Contribution: Sciences". Here is the
charge of plagiarism: First as to Archimedes:
Now, a sphere inscribed in a right cylinder of a height
equal to the diameter of the sphere is the same figure
that Archimedes chose as his epitaph, considering that
this is his best discovery (fig. 41). Thus, Archimedes
did not even have the excuse of an honest scholar who
would rediscover an established theorem, without
knowing that it had been discovered two thousand years
before him by his Egyptian predecessors [from papyrus
evidence previously elaborated in the chapter]. The
other "borrowings" in which he indulged himself during
and after his trip to Egypt, without ever citing the
sources of his inspiration, show clearly that he was
perfectly conscious of his sin, and that thereby he was
being faithful to a Greek tradition of plagiarism that
went back to Thales, Pythagoras, Plato, Eudoxus,
Oenopides, Aristotle, etc., which the testimonies of
Herodotus and Diodorus of Sicily reveal to us in
part... The epitaph of Archimedes, rediscovered by
Cicero at Syracuse, proves that this is not a myth
propagated by tradition.
Second as to Thales:
The theorem attributed to Thales is illustrated by the
figure of problem 53 of the Rhind Papyrus, written
thirteen hundred years before the birth of Thales...
The anecdote claiming that Thales discovered "his"
theorem by making the end of the shadow cast by a
stick, planted vertically, meet exactly the end of the
shadow cast by the Great Pyramid, in order to have a
figure materialize identically to that of problem 53,
would only prove that Thales actually spent time in
Egypt, that he was truly a pupil of Egyptian priests
and that he could not be the inventor of the theorem
attributed to him.
Third as to Pythagoras:
Herodotus calls Pythagoras a simple plagiarist of the
Egyptians; Jamblichus, biographer of Pythagoras, writes
that all the theorems of lines (geometry) come from
Egypt...
An Egyptian priest told Diodorus of Sicily that all the
so-called discoveries that made Greek scholars famous
were things that had been taught to them in Egypt and
which they called their own, once they went back to
their country...
Fourth as to Plato:
Plato, in the Phaedrus, has Socrates say that he
learned that the god Thoth was the inventor of
arithmetic, calculus, geometry, and astronomy
(Phaedrus, 274 C)...
Fifth as to Aristotle and Democritus:
Aristotle... acknowledges the essentially theoretical
and speculative character of the Egyptian science and
tries to explain its emergence not by land surveying,
but by the fact that the Egyptian priests were free
from material preoccupations and had all the time
necessary to deepen theoretical thought. According to
Herodotus, the Egyptians are the exclusive inventors of
geometry, which they taught to the Greeks. Democritus
boasted that he equalled the Egyptians in geometry.
And Diop's conclusion:
Therefore, no trace is found anywhere, in the texts of
antiquity, of a so-called duality of theoretical Greek
science, as opposed to Egyptian empiricism... The idea
of an empirical Egyptian science is an invention of
modern ideologues, those same ones who are looking for
ways to erase from the memory of humanity the influence
of Negro Egypt on Greece.
In the interest of brevity, I have of course left out much
detail, but I believe Diop has succeeded in making the
threshold showing of deliberate plagiarism, as opposed to
innocent borrowing, or independent rediscovery of previously
known results. If that is in fact the historical truth,
then the fact of plagiarism renders moot the question of
Greek theory vs. Egyptian empiricism, for it will never be
known where the Egyptian contribution ends, and the Greek
begins.
Still, though, it is contended by some on the list that what
the papyri do not show is mathematical PROOF, in the sense
that has come down to us in, for example, Euclid's
"Elements". If one chooses to argue *only* from the
surviving papyri, there may be a point to this contention.
But one would then have to ignore the totality of the
evidence, including the assertions attributed to Herodotus
and Diodorus, not to mention Plato and Aristotle. In any
case, even if argument is confined to the evidence solely of
the surviving papyri, the point would still be a weak one, given
the nature of mathematical PROOF itself, which I now
address.
The axiomatic method is indeed very powerful. That is what
mathematical PROOF ultimately boils down to: stating
premises which are hopefully self-evident and therefore not
themselves requiring of PROOF, then applying syllogistic
reasoning based on the premises to obtain the result
(theorem) that is sought. But it is a mistake to suppose
that PROOF has not been established unless a minimal set of
axioms has first been laid down. Theorems in one system may
be axioms in another, and vice versa, and the choice is
essentially arbitrary. Therefore, if the Egyptians had a
result, as evidenced by the algorithms in the papyri
applying those results, it seems a fair inference that they
had found some way beforehand to establish the result then
applied. That their axiomatization has not survived is not
proof that there was not one; rather, the converse seems
more reasonable, namely that if they could implement a
result, they must have got to that point by some syllogistic
reasoning process. *How* they got there may not be known,
but axiomatizations are essentially arbitrary, sometimes
explicit, sometimes only implicit, but if there is a
*result*, some axiomatization, whether efficient or
otherwise, minimal or otherwise, systematic or otherwise,
must be assumed to have preceded the result.
We sometimes forget that only a very meager theory of
meaning is required to apply the axiomatic method. Rules of
logical deduction have been elaborated which follow entirely
from logical form, for example:
All rich men are happy (Premise 1)
John is rich (Premise 2)
Therefore, John is happy (Conclusion)
Leave aside the factual truth or otherwise of the premises,
and leave aside the semantic uncertainty associated with the
fuzzy terms "rich" and "happy", the conclusion follows as a
matter of form. The two premises entail, logically, the
conclusion. Raised to perfect abstraction, we have the
tautological rule known as modus ponendo ponens, under
which, from propositions A, and A->B, we may reliably infer
B, which in the standard logic notation emerges as
[A & (A->B) ] -> B
where A and B may remain uninterpreted until we have a
special case such as
[rich & rich->happy ] -> happy
which may help us "prove" that, say, O. J. Simpson is a
happy man. Of course, in any particular case, we haven't
proved anything except that IF we assume certain premises,
THEN certain conclusions would be semantically consistent
with those premises. Mathematical proof is of the same
sort. It is ultimately empty, because it cannot by itself
establish for us the premises which we hope correspond in
some way with reality. Therefore, I am far more impressed
with a concrete result, eg. the pyramids, than I am with
some mathematical proof. When in addition the papyri
clearly indicate knowledge of various *formulae* (with the
full generality implicit in that word) clearly used in
pyramid building -- areas, volumes, trigonometric relations,
geometric series, arithmetic series, etc. -- I find it
absurd to question whether the builders understood abstract
mathematical PROOF.
Parenthetically, much of modern mathematics (the Hilbert
program) has been wasted attempting to make of it a robotic
exercise in uninterpreted symbol manipulation, except,
perhaps, very minimally, at the axiomatic outset. It is
perhaps not surprising the result of Godel that not all
theorems (of arithmetic) are decidable within such a
framework. Some results would appear to require a reversion
to a fuller theory of meaning, where deduction derives from
semantic *content* rather than merely semantic *form*.
(Which, btw, is where fuzziness comes in, so far as my book
is concerned, for fuzzy sets can be bearers or explicators
of semantic content, from which rules of deduction based on
content rather than merely of form, may be derived.) It is
the old limitation of traditional logic: one cannot use
modus ponendo ponens or other rules of logical form to
deduce from the description "bachelor" that one is an
"unmarried man". The logical robot would have to be
programmed some other way.
But man is not robot. Syllogistic reasoning, which is the
chief stock-in-trade, though not the only one, of
mathematical science, is in any case not the supreme
intellectual accomplishment of which we humans are capable.
Far from it. And there is evidence that the Egyptians put
it fairly low in the totem pole of human accomplishment.
They saw man as being a spiritual being, with a spirit
having seven divisions, in each of which the consciousness
performed different types of functions, as follows:
1 - Ba, the ability to experience omnipresence
based on the existence of the universal spirit
2 - Khu, ability to intuit the truth of a
logical premise, the oracular faculty of
prophets
3 - Shekem, ability to affect nature through
the use of spiritual power
4 - Ab,
+ the ability to see the interdependence
between all things, to love
+ the ability to analyze, to see the
abstract difference between things
+ the ability to think circumspectly, ie.
to coordinate the activity of all the
faculties of the spirit, to reason.
5 - Sahu,
+ imagination and congregative thinking--
aesthetics
__________________________________________
| + syllogistic, logical and segregative |
| thinking |
------------------------------------------
+ memory and imitative faculty, learning
6 - Khaibit, the animal soul, emotions, sense
perception, the sensual, physical movement
7 - Khab, the physical body which gives us the
illusion that we are separate beings
I do not claim to know what all of this means (see "An
Afrocentric Guide to a Spiritual Union", by Ra Un Nefer Amen
for further elaboration) but if we accept the essential
claim that this hierarchical structuring of the spirit is
due to the ancients, then it reveals a clear understanding
of syllogistic reasoning which is the foundation of the
axiomatic method attributed to the Greeks. It also puts it
rather low in the totem pole of human activity. He who can
do more can also do less, as Diop is fond of saying.
Therefore, the axiomatic method of syllogistic reasoning
appears to fall well within the wisdom systems developed by
the Egyptians, and to suggest that they fell short in that
area therefore seems unreasonable. It also suggests that we
of the 20th century who fall far short of levels 1, 2 and 3
above in accomplishment, may not yet have the ability to
appreciate fully the achievements of the ancients.
Finally, Diop tells us that the earliest date in history
known with certainty is 4236 BC, because there is evidence
that the Egyptians developed at that time the sidereal
calendar, a fact which suggests rather more than empirical
knowledge on their part:
They invented the 365-day year, breaking it down as
follows: 12 months of 30 days = 360 days, plus the 5
intercalated days .... The Egyptians knew that
this calendar year was too short, that it was lacking a
quarter of a day in order for it to correspond to a
complete sidereal revolution... [In] 4236 BC they
invented a second astronomical calendar founded
precisely on this time lag ... in the 365-day calendar
as compared to the sidereal, or astronomical, calendar.
The time lag thus accumulated at the end of four years
is equal to one day. Instead of adding 1 day every 4
years and thus instituting a leap year, the Egyptians
preferred the masterful solution that consists of
following this time lag for 1,460 years... the
Egyptians preferred to "rectify" every 1,460 years
instead of every 4 years; he who can do more can also
do less, therefore contrary to popular opinion, they
knew the leap year very well. But what is still more
amazing is that the Egyptians had equally (observed?)
calculated that this period of 1,460 years of the
sidereal calendar is the lapse of time that separates
two helical risings of Sirius, the most brilliant fixed
star in the heavens located in the constellation Canis
Major... Thus, the heliacal rising of Sirius, which
takes place every 1,460 years, coinciding with the
first day of the year in both calendars, is the
absolute chronological reference point that is the
basis of the Egyptian astronomical calendar. One gets
lost in conjectures in order to figure out *how* the
Egyptians were able to arrive at such a result from
protohistory, because it is known with certainty that
the sidereal calendar was in use from 4236 BC onward.
Supposing that a celestial phenomenon as fleeting as a
heliacal rising of Sirius had accidentally caught the
attention of the Egyptians from the fourth millenium
onward, how could they have guessed at, and verified,
within a few minutes, its rigorous periodicity, in a
time span of 1,460 years, and thus invented a calendar
on this basis? Did they arrive at this result through
empirical or theoretical means? Assuredly, the
disparagers of Egyptian civilization have their work
cut out for them!
Frankly, I see no other conclusion but that the Egyptians
had more than empirical mathematical results at their
disposal if they established the sidereal calendar. If you
add the tangible evidence of the pyramids, it seems
inconceivable to me that they had not mastered both geometry
and trigonometry in their full abstraction. If in addition,
they established the heliacal risings of Sirius to its
1,460-year periodicity, either they also mastered the
equivalent of calculus and Newtonian mechanics, or in the
alternative, the part of their civilization lost in proto-
history must extend back at least 3,000 years prior to 4236
BC for them to have been able to record the observations
necessary to establish the empirical basis for the sidereal
calendar. Or both.
The latter is consistent with recent theories (West and
others) suggesting that the age of the Sphinx is much
greater than supposed by Egyptologists, and may have been
built/carved in about 10,000 BC to herald the rising of the
astrological age of Leo. That in itself would require
knowledge of a periodicity of even higher order. The
imagination *is* transfixed. I won't even go into the Dogon
people of West Africa, and their intimate knowledge of the
orbit of Sirius B, the invisible (to the naked eye)
companion star to Sirius, and what their connections might
be to ancient Egypt.
In the light of what the Egyptians for a fact were known to
have accomplished (the pyramids, and the sidereal calendar),
the question whether the Egyptians invented mathematical
PROOF seems tangential and picayune. They knew the required
theorems (arithmetic, algebraic, geometric, and
trigonometric). If arrived at by mathematical intuition
alone, this would be even more remarkable than if arrived at
by the imperfect axiomatic method to which we are heirs
today, and for which we credit the Greeks. I can think of
an analogy. Suppose the world were destroyed (nuclear war,
asteroid collision, or whatever), and no books or libraries
remained to show what mankind of the 20th century had
accomplished. Yet some future sojourners on earth were able
to see signs of our being here, and all that was left were
some film clip showing the accomplishment of men landing on
the moon. Would it be doubted that those people of the 20th
century had to have mastered the mathematics necessary to
have pulled off the accomplishment? In its full theoretical
abstraction?
Yes, Paul, you are right. But for the few papyri that
survived the destruction of the invader, the ancient
accounts of Herodotus and Diodorus, and fortuitous pieces of
evidence such as the inscription on Archimedes tomb, the
Egyptian claim to what has popularly, and it would appear
wrongly, been attributed to the Greeks would be lost to us.
> Paul Kekai Manansala
Regards,
S. F. Thomas
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