Date: Sat, 18 May 1996 17:10:47 +0100
To: uunet!athena-discuss%info.harpercollins.com@uunet.uu.net
Subject: Egyptian Science, the Greeks, and Mathematical PROOF
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,
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