|
\Maria, and group members:
Thanks again for this opportunity. google groups seem not
to allow active http links. What needs to be done to achieve
that step?
The following introduction has been updated to included
additional ... non working ... links. My apologies to everyone.
I look forward to introducing Egyptian math to your google group. The Webinar should be fun.
I'd like to keep the session low key. Of course as the basic
definitions and applications roll in most participants will likely
feel their modern math glasses getting fogged up. The morning mist will clear putting pencil to paper a few times.
There may be a small number of participants that have
experience with number theory and its ancient built blocks.
For that group the 4,000 year jump back in time will be an
easy experience.Very few pencil and paper calculations
will need to be made.
The google group posts will attempt to lay out the basic
definitions and applications of Egyptian math, including:
An Egyptian Old Kingdom multiplication operation was
consistent with the modern "Russian Peasant" operation.
Egyptian scribes in the Old Kingdom and Middle Kingdom,
the period that Egyptian math will be focused upon in the
Webinar, seemed to only double the desire multiplication
elements. This aspect of the subject will not be explored
in detail. A story or two can to told at this point.
For those, like yourself, that understand Russian peasant
multiplication, a point will be made that the ancient algorithm
was modified into a finite numeration system that included finite arithmetic operations. This will be the topic of the Webinar.
The second historical fact that needs to be introduced is
the binary definition of one (1) used in the Old Kingdom
1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...
Egyptians built a mythology, named the "Eye of Horus, around this definition. An algorithmic statement rounded off statements of any rational number to six-terms, rounding off up to 1/64 of a unit. The Egyptian Old Kingdom arithmetic metaphor was active in the entire history of Egyptian base 10 arithmetic and the entire history of Babylonian base 60 mathematics. "Book of the Dead" stories can be told at this point, citing the weighing of everyone's heart (at death) on a balance beam, by a man that wore two hats, one religious and one science (the mythical leader of Egyptian
mathematics).
--
You received this message because you are subscribed to the Google Groups "MathFuture" group.
To post to this group, send email to mathf...@googlegroups.com.
To unsubscribe from this group, send email to mathfuture+...@googlegroups.com.
For more options, visit this group at http://groups.google.com/group/mathfuture?hl=en.
Edward, Thank you for the questions: A brief *** response*** will be given to each question: |
1. What do we know that we don't know about Egyptian mathematics? |
*** We know a great about its ciphered numeration system, arithmetic, algebra, geometry, weights and measures and arithmetic progressions that were consistent with Gauss summing up 1 to 100 by matching 50 pairs of 101, matching the first and last terms in the series, to the 5050 total. The Kahun Papyrus and RMP 39, 40 and 64 discuss Egyptian arithmetic progressions, found on Wikipedia per: http://en.wikipedia.org/wiki/Kahun_Papyrus *** |
2. What don't we know that we don't know about Egyptian mathematics? |
*** Aspects of cubits and other weights and measures units and calculation styles have not been found in the written records. http://planetmath.org/encyclopedia/CubitsEgyptianGeometryAreasCalculatedIn.html The fragmentary status of the hieratic mathematical texts has caused many scholars to give up, suggesting like Otto Neugebaur did in "Science in Antiquity" (1954?) that the 2/n table was a step backward mathematically. The exact opposite was true. |
3. Why didn't the Egyptians go further? |
Egyptians did go farther, so far that Greek, Arabs and medieval added little to the arithmetic notation, modifying the structure of the unit fraction notations, without modifying the number theory foundations. Unit fraction arithmetic continued for 3,600 years, ending in Europe with the 1585 AD creation of our modern base 10 decimal system. Fibonacci's Liber Abaci, a 1202 AD text
http://liberabaci.blogspot.com/ is a case in point. Fibonacci took 124 pages of a 500 page treatise (that was Euope's arithmetic book for 250 years -until Ottomans over ran Byzantine culture) to describe seven rules (distinctions in Sigler's 2002 Latin to English translation). Egyptian fraction arithmetic to Fibonacci, Arabs, Greeks and Egyptians was demonstrated, often without summary algebra statements (info that I have taken the liberty to add ... from time to time). **** |
You mention some points on 1. below. For Greek math, the Antikythera mechanism demonstrates that far more went on than is documented, and thus that |
*** this was a machine that was cranked one day at a time. To my knowledge the modular arithmetic that "Archimedes" employed to create the gears has not been decoded in Greek arithmetic metaphors, writing 1/2 as a ciphered beta'. |
2. is important. Similarly for 3., Galileo used only Greek math and observations easily available to the Greeks to work out the first approximation to gravity: parabolic paths in a constant field. |
*** Here your question jumps near to our algorithmic base 10 decimal era. By the time of Napier (*early 1600s), Stevin's decimal notation was improved near to the one that we use today. *** |
Why did the Greeks never notice the parabolic shape of a water fountain, and attack this problem themselves? They would surely have solved it. The Kepler and Newton versions required better observations and stronger math. Why didn't the Greeks attack both after coming so close? |
*** You are asking questions that are not present in the historical record. As you may know, we have zero Classical Greek mathematical texts.What we have are "copies" and revisions of reported Classical Greek texts that stressed geometry ... without reporting Greek arithmetic. *** |
And why didn't the Egyptians continue on into the realms opened up by Greeks trained in Egypt? |
*** Your history is backwards. The hieratic arithmetic that will be Webinar'ed next month existed 1,000 years before Classical Greek culture. Plato and others were trained in Egyptian schools, and Greeks continued Egyptian arithmetic, algebra ... and so forth ... in Greece ... Alexandria's library burnt down ... had it not .. many of your probing questions would have been answered. Best Regards, Milo --- On Thu, 6/17/10, Edward Cherlin <eche...@gmail.com> wrote: |
Kirby, Maria, et al Reading Gadalla's book online: http://books.google.com/books?id=evYSzuCLXt4C&dq=gadalla,+egyptian&printsec=frontcover&source=bn&hl=en&ei=f1gfTPiTG8aLnQeOocnrDQ&sa=X&oi=book_result&ct=result&resnum=5&ved=0CC4Q6AEwBA#v=onepage&q&f=false discussed an Old Kingdom (hieroglyphic) list of Egyptian mythical characters placed on a valid ancient unity stage. The story-line that interests Bruce Friedman and myself considers Anibus as recorded in hieratic script. Anibus took the dual head of Egyptian math, an entity that weighed every dead Egyptian's heart to determine each person's worth in life (in the Old Kingdom and Middle Kingdom). Gadalla's book did not focus upon Anibus' math duties recorded in the Middle Kingdom. When Anibus is focused upon in hieratic script, parsed from spiritual duties, the remaining Middle Kingdom mathematical duties describe a range of solutions to the Old Kingdom "Eye of Horus" problem: http://en.wikipedia.org/wiki/Eye_of_Horus The Middle Kingdom solutions ONLY dealt with rational numbers. Before discussing the weights and measures side of writing hieratic volume units in various classes of unities (10/10, 64/64, 320/320 and so forth) the ancient love of Middle Kingdom unities topic will be covered within Gadalla's broader hieroglyphic view, I agree with the general sense of the wholeness of life, and the unity principles that many gods/characters played in the non-mathematical aspects of life. Both were present in the Old Kingdom and Middle Kingdom. The story-line that Bruce and I will be telling begins with the algorithmic version of one, recorded in the Old Kingdom per 1 = 1/2 + 1/4 + 18 + 116 + 1/32 + 1/64 + ..., a cursive notation that generally threw away upto a 1/64 unit. All algorithmic notations, in any era, MUST do the same thing, rounding off by set standards. Babylonians did the same thing in base 60. Scribe wrote inverse tables of fractions as small as 1/91 as 1/90, throwing away the difference in an algorithmic cursive notation. Modern binary computers, use only zero and, record numbers and mathematical statements within algorithms, rounding off by standard based rules.. The IEEE organization sets up moder 'rounding off standards for analogy and digital computer's use of fixed and floating point notations, and other situations, per: http://en.wikipedia.org/wiki/IEEE_Standards_Association No algorithmic notation, from any era, in any culture, exactly records all numbers and mathematical statements. For example, irrational and higher order numbers and algorithmic statements were not allowed in ancient Egypt as proofs. Duplation statements were often seen in the RMP recorded by Ahmes in 1650 BCE. However, proof of the accuracy of implicit 'algorithmic' statements were explicitly proven by finite arithmetic methods (one of the glossary entries that are under development). Jumping to the modern era, and considering the highest use of algorithmic calculations,as supercomputers came online, Gregory and Albert Chudnovsky of Columbia U., created an algorithm notation based on polar coordinates and Little Fermat. ;Little Fermat is a 2^n type number, setting a standard when the next parallel computer is asked to come online. Working with the IBM Watson think tank, a short-term project turned into a five year project, ending about 20 years ago. When completed the "Little Fermat" operating system worked, speeded up the run time of supercomputers 1,000 times, discussed by: http://www.thefreelibrary.com/Little+Fermat:+a+scheme+for+speeding+up+multiplication+leads+to+a...-a08986294 Jumping back to the the time of the Egyptian Middle Kingdom, scribes had tired of the "Eye of Horus" 'algorithmic round-off problem. Scribes limited arithmetic calculations to rational numbers, thereby eliminating algorithmic calculations a core idea. When a higher order number like pi was needed to solve an area of a cylinder, in the definition of a hekat, one value was set for the number, 256/81. This 256/81 value for pi over-estimated the volume of 1/320 of a heat, named ro by using 3/16 rather than 3.14159. Ahmes in RMP 38 multiplied one hekat, written as 320 ro, by 7/22, an inverse of our modern estimate for pi (22/7) and found 101 + 9/11 ro with 9/11 equal to unit fraction series (2/3 + 1/3 + 1/6 + 1/11 + 1/22 + 1/66)/10) Taking a RMP 38 DISCUSSION from: http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html "In RMP 38 two rational numbers, (35/11)/10 = 35/110 = 7/22, were multiplied 320, by a doubling ' method citing: 1. Initial calculation (320 ro)*(35/11) = (320 ro)*(2/3 + 1/3 + 1/6 + 1/11 + 1/22 + 1/66)/10 = 101 + 9/11 ro 2. Proof (101 9/11 ro) was multiplied by 22/7, and returned one hekat , 320 ro. This class of hekat calculation infers that the traditional Old Kingdom pi value of 256/81 was corrected by considering : " ... that (7/22) and (22/7) were shown and proved to be inverses, and that the AE scribes were skilled and aware of the natural inverse operations of multiplication and division. In effect, the AE were adept at finding reciprocals" (Bruce Friedman)! " Inverse calculations of this type were spread across the RMPs 87 problems. Friedman uses the term 'natural; inverse operations of multiplication and division' Literally, Ahmes defined division as an inverse to multiplication ,and defined multiplication as inverse to division, a set of two modern rules that modern children learning both arithmetic must memorize, and learn to apply. This paired set of facts are not often recognized by Peet, Chace, Gillings and the 1920s translators of Middle Kingdom mathmatical texts. Ahmes recorded In RMP 35-38, plus RMP 66, http://planetmath.org/?op=getobj&from=objects&id=11838 and worked with 320 ro problems rather than one hekat units to solve several classes of problems by adding back once missing rounded off calculaitons, by writing exact finite statements. But what was Ahmes doing in a manner that exactly translates ancient arithmetic statements into our modern base 10 decimal algorithm? Peet, Chace, Gillings, et al, guessed at Ahmes' division operation, improperly suggesting that 'single false position' a medieval root finding operation may have been used in the Middle Kingdom. Ahmes only added back missing "Eye of Horus" fragments that would have been present in the Old Kingdom's binary numeration system. A new Middle Kingdom finite numeration system was created that avoided the 'blind eye' of any algorithmic arithmetic notation. In the Middle Kingdom scribes created non-algorithmic weights and measures, geometry, algebra, arithmetic progressions and arithmetic statements that will be introduced in the webinar, and translated into modern arithmetic statements that validates that scribal division was only inverse to scribal multiplication, inverting the rational number divisor, and multipling, without using 'single false position' or an algorithm. A glossary of terms will be posted to allow novice students of Egyptian mathematics to parse mythological terms from the abstract number theory terms used by Ahmes and other Middle Kingdom scribes. The glossary will contain two sections, a small set of Old Kingdom algorithmic rules, with a correcting set of Middle Kingdom finite arithmetic rules. The second section will include modern arithmetic terms (usually number theory based) that describe the Middle Kingdom correcting rules. For example, the Old Kingdom definition of one: 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ... added back 1 dja, and 5 ro in weights and measures calculations, and defined a hekat unity as 64/64 (as Hana Vymazalova proved in 2002) when the answers to n = 3, 7, 10, 11 and 13 division problems: (64/64)/n = Q/64 + (5R/n)ro (with Q a quotient, R a remainder, scaled to (5/5) to allow 1/320 units named ro to appear with divisor n was in the range 1/64 < n < 64). were multiplied by the initial divisor n, returning (64/64) in the Akhmim Wooden Tablet http://en.wikipedia.org/wiki/Akhmim_wooden_tablet example, the AWT scribe divided one hekat by 3 by completing the following steps: a.. (64/64)/3 = 21/64 + 1/192 = b. (16 + 4 + 1)/64 + (5/3)ro = c. (1/4 + 1/16 + 1/64)hekat + (1 + 2/.3)ro steps that Hana Vymazalova did not decode. Proof: steps that Hana Vymazalova properly decoded in 2002 a. ((1/4 + 1/16 + 1/64)hekat + (1 + 2/3)(ro] times 3 = b. [(21/64 x 3) = 63/64 + (5/3*(1/320)]x 3 = 5/320] c. 63/64 + 1/64 = 64/64 Again, my apologies for the slowness of creating the needed glossary. A draft should be available in one week. Best Regards, Milo Gardner http://en.wikipedia.org/wiki/User:Milogardner --- On Thu, 6/17/10, kirby urner <kirby...@gmail.com> wrote: |
|
Kirby, Maria, et al
Reading Gadalla's book online:
http://books.google.com/books?id=evYSzuCLXt4C&dq=gadalla,+egyptian&printsec=frontcover&source=bn&hl=en&ei=f1gfTPiTG8aLnQeOocnrDQ&sa=X&oi=book_result&ct=result&resnum=5&ved=0CC4Q6AEwBA#v=onepage&q&f=false
discussed an Old Kingdom (hieroglyphic) list of Egyptian mythical characters placed on a valid ancient unity stage.
The story-line that Bruce and I will be telling begins with the algorithmic version of one, recorded in the Old Kingdom per
1 = 1/2 + 1/4 + 18 + 116 + 1/32 + 1/64 + ...,
a cursive notation that generally threw away upto a 1/64 unit.
All algorithmic notations, in any era, MUST do the same thing, rounding off by set standards.
Babylonians did the same thing in base 60. Scribe wrote inverse tables of fractions as small as 1/91 as 1/90, throwing away the difference in an algorithmic cursive notation.
Modern binary computers, use only zero and, record numbers and mathematical statements within algorithms, rounding off by standard based rules.. The IEEE organization sets up moder 'rounding off standards for analogy and digital computer's use of fixed and floating point notations, and other situations, per:
http://en.wikipedia.org/wiki/IEEE_Standards_Association
No algorithmic notation, from any era, in any culture, exactly records all numbers and mathematical statements. For example, irrational and higher order numbers and algorithmic statements were not allowed in ancient Egypt as proofs. Duplation statements were often seen in the RMP recorded by Ahmes in 1650 BCE. However, proof of the accuracy of implicit 'algorithmic' statements were explicitly proven by finite arithmetic methods (one of the glossary entries that are under development).
Jumping to the modern era, and considering the highest use of algorithmic calculations,as supercomputers came online, Gregory and Albert Chudnovsky of Columbia U., created an algorithm notation based on polar coordinates and Little Fermat. ;Little Fermat is a 2^n type number, setting a standard when the next parallel computer is asked to come online. Working with the IBM Watson think tank, a short-term project turned into a five year project, ending about 20 years ago.
When completed the "Little Fermat" operating system worked, speeded up the run time of supercomputers 1,000 times, discussed by:
http://www.thefreelibrary.com/Little+Fermat:+a+scheme+for+speeding+up+multiplication+leads+to+a...-a08986294
Jumping back to the the time of the Egyptian Middle Kingdom, scribes had tired of the "Eye of Horus" 'algorithmic round-off problem. Scribes limited arithmetic calculations to rational numbers, thereby eliminating algorithmic calculations a core idea.
When a higher order number like pi was needed to solve an area of a cylinder, in the definition of a hekat, one value was set for the number, 256/81.
This 256/81 value for pi over-estimated the volume of 1/320 of a heat, named ro by using 3/16 rather than 3.14159. Ahmes in RMP 38 multiplied one hekat, written as 320 ro, by 7/22, an inverse of our modern estimate for pi (22/7) and found
101 + 9/11 ro with 9/11 equal to unit fraction series (2/3 + 1/3 + 1/6 + 1/11 + 1/22 + 1/66)/10)
Taking a RMP 38 DISCUSSION from:"In RMP 38 two rational numbers, (35/11)/10 = 35/110 = 7/22, were multiplied 320, by a doubling ' method citing:
1. Initial calculation
(320 ro)*(35/11) = (320 ro)*(2/3 + 1/3 + 1/6 + 1/11 + 1/22 + 1/66)/10 = 101 + 9/11 ro
2. Proof
(101 9/11 ro) was multiplied by 22/7, and returned one hekat , 320 ro. This class of hekat calculation infers that the traditional Old Kingdom pi value of 256/81 was corrected by considering : " ... that (7/22) and (22/7) were shown and proved to be inverses, and that the AE scribes were skilled and aware of the natural inverse operations of multiplication and division. In effect, the AE were adept at finding reciprocals" (Bruce Friedman)! "
Inverse calculations of this type were spread across the RMPs 87 problems. Friedman uses the term 'natural; inverse operations of multiplication and division' Literally, Ahmes defined division as an inverse to multiplication ,and defined multiplication as inverse to division, a set of two modern rules that modern children learning both arithmetic must memorize, and learn to apply. This paired set of facts are not often recognized by Peet, Chace, Gillings and the 1920s translators of Middle Kingdom mathmatical texts.
Ahmes recorded In RMP 35-38, plus RMP 66,
http://planetmath.org/?op=getobj&from=objects&id=11838
and worked with 320 ro problems rather than one hekat units to solve several classes of problems by adding back once missing rounded off calculaitons, by writing exact finite statements.
But what was Ahmes doing in a manner that exactly translates ancient arithmetic statements into our modern base 10 decimal algorithm? Peet, Chace, Gillings, et al, guessed at Ahmes' division operation, improperly suggesting that 'single false position' a medieval root finding operation may have been used in the Middle Kingdom.
Ahmes only added back missing "Eye of Horus" fragments that would have been present in the Old Kingdom's binary numeration system. A new Middle Kingdom finite numeration system was created that avoided the 'blind eye' of any algorithmic arithmetic notation. In the Middle Kingdom scribes created non-algorithmic weights and measures, geometry, algebra, arithmetic progressions and arithmetic statements that will be introduced in the webinar, and translated into modern arithmetic statements that validates that scribal division was only inverse to scribal multiplication, inverting the rational number divisor, and multipling, without using 'single false position' or an algorithm.
A glossary of terms will be posted to allow novice students of Egyptian mathematics to parse mythological terms from the abstract number theory terms used by Ahmes and other Middle Kingdom scribes.
The glossary will contain two sections, a small set of Old Kingdom algorithmic rules, with a correcting set of Middle Kingdom finite arithmetic rules. The second section will include modern arithmetic terms (usually number theory based) that describe the Middle Kingdom correcting rules.
For example, the Old Kingdom definition of one: 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...
added back 1 dja, and 5 ro in weights and measures calculations, and defined a hekat unity as 64/64 (as Hana Vymazalova proved in 2002) when the answers to n = 3, 7, 10, 11 and 13 division problems:
(64/64)/n = Q/64 + (5R/n)ro
(with Q a quotient, R a remainder, scaled to (5/5) to allow 1/320 units named ro to appear with divisor n was in the range 1/64 < n < 64).
were multiplied by the initial divisor n, returning (64/64) in the Akhmim Wooden Tabletexample, the AWT scribe divided one hekat by 3 by completing the following steps:
a.. (64/64)/3 = 21/64 + 1/192 =
b. (16 + 4 + 1)/64 + (5/3)ro =
c. (1/4 + 1/16 + 1/64)hekat + (1 + 2/.3)ro
steps that Hana Vymazalova did not decode.
Proof: steps that Hana Vymazalova properly decoded in 2002
a. ((1/4 + 1/16 + 1/64)hekat + (1 + 2/3)(ro] times 3 =
b. [(21/64 x 3) = 63/64 + (5/3*(1/320)]x 3 = 5/320]
c. 63/64 + 1/64 = 64/64
Again, my apologies for the slowness of creating the needed glossary. A draft should be available in one week.
Best Regards,
Milo Gardner
http://en.wikipedia.org/wiki/User:Milogardner
Kirby, and Math 2.0 members: Thank you Kirby for your thoughtful comments. Our post-WW I classrooms are very good at introducing the algorithmic side of number, and the various classes of mathematics, including calculus, that relies on "the limit theorem" and other classes of algorithms. Prior to WWI per: http://books.google.com/books?id=-oQXAAAAYAAJ&pg=PT155&lpg=PT155&dq=college+arithmetic,+1900&source=bl&ots=mCk68C4lz0&sig=h9hbpRyh3vaD_dEh4JPoC1o2yxs&hl=en&ei=os0gTP7CC5rrnQfzroFy&sa=X&oi=book_result&ct=result&resnum=2&ved=0CBUQ6AEwAQ#v=onepage&q=college%20arithmetic%2C%201900&f=false college arithmetic began with finite arithmetic. In 1910 an international conference was chaired by a Gottingen University math person: http://en.wikipedia.org/wiki/University_of_G%C3%B6ttingen and concluded that K -12 math education should also begin with finite arithmetic. Sadly, after WW I, President Woodrow Wilson and others acted to remove all things German from USA, British and other European K- 12 and college classrooms, by always beginning with algorithms. The modern classrooms in the USA does not include topics that were German connected prior to WW I. How many on this forum are aware of those 1920s actions? In 1963 I ran across the pre-WW I body of knowledge in "Number Theory and its History" by Oystein Ore, a classic book still in publication (thanks to Dover). Problems were written and solved within finite math, the largest body of knowledge, in my view, was the Chinese Remainder Theorem, and congruences (modular arithmetic). Gauss' 1801 treatise, "Discussions on Arithmetic" provided the first rigorous set of arithmetic foundations presented in the Western Tradition, another of my views. NCTM also sells the 1910 International conference notes that that extolled the virtues of college arithmetic and finite arithmetic. Jumping back 4,000 years, Egyptians began with finite arithmetic. The older "Russian Peasant" type multiplication http://www.google.com/search?hl=&q=russian+peasant+multiplication&sourceid=navclient-ff&rlz=1B3GGGL_en___US345&ie=UTF-8&aq=0&oq=russian+peas was an Old Kingdom algorithmic fragment. Its numbers and arithmetic statements were written in, an awkward 6-term rounded off notation (yes, "Eye of Horus" type numbers). By 2050 BCE, a 6-term or less finite notation scaled rational numbers to solvable unit fraction series by looking for a least common multiple m, and a set of red numbers, to write out optimized unit fraction series, a topic that will be discussed in detail in the Webinar. Egyptian finite math slowly adapted to create our modern multiplication operation, first used within mental arithmetic and mental algebra applications, two topics that will be introduced in the Webinar by citing Ahmes 87 problems, and ways to fairly decode the raw hieratic data. Greeks followed Egyptian ciphered numeration and finite math statements. Eudoxus has been given credit for developing the 1/4 geometric series, an algorithmic statement for finding areas. Archimedes used it to find the area of a parabola, first writing 4A/3 = A + A/4 + A/16 + A/64 + ... and proving the hypothesis by taking one phase of the Horus-Eye series by writing out a finite Egyptian fraction series (as Heiberg saw in 1906, and Dijksterhuis reported in "Archimedes", in 1987, consider pre-WW I views on arithmetic) per 4A/3 = A + A/4 + A/12 Note that Archimedes' proof, reported by Heiberg, did not use the "Method of Exhaustion" an algorithm that we all learned in college calculus classes. At this level, it can be said that Egyptians and Greeks both learned to begin with algorithmic infinite series statements, and proved that an exact unit series could be found that solved 'traditional' round off problems. This is the power of finite mathematics, converting most infinite series into a finite series. Sadly, Stanford's project on reading Archimedes' calculus only mentioned the post WW I ":method of exhaustion" side of the topic --on Nova, its web site, and other internet sites. Happily, mathematics is an abstract subject that speaks for itself, calling out the foundational issues to those that read the ancient texts. Over the last few years, Bruce Friedman has taken the language side of hieratic symbols.This is one topic that the US Army's Language School in Monterrey, California (which I qualified to enter -- but chose cryptanalytics) teaches, by offering the oral and written languages. Cyrillic script and Russian are phonetic, hence easily taught. Chinese Mandarin and Arabic are the toughest, the first due to its symbols and syntax and the second due to its regional idioms. Bruce and I tend to confirm the work of Peet,Chace, Gillings, Clagett and others that reported complete transliterations of Middle Kingdom mathematical texts -- WITHOUT fully translating the ancient data into modern arithmetic. More on this topic in the Webinar. Code breaking of ancient mathematical texts requires at least two team members, a linguist and a person that studies the language patterns, a mathematician that also tracks the numerical patterns in the text. The second person (my role) should be skilled in chess and others games of strategy. Ancient texts offer many surprises. For those that have read about Linear B, a Minoan script, it was broken by Ventris and Chadwick, after WW II working in a math -linguist team http://en.wikipedia.org/wiki/Linear_B Chadwick was the cryptanalyst on the team. There are insufficient fragments of Linear A to decode its language side,though the Linear A does include hieroglyphic number symbols, likely used in trade. Continuing with ancient Egypt, due to the poor 20th century translations, wider interdisciplinary teams are needed to be established. In 2009 a mini-conference was chaired by two German PhDs in Egyptology, Tanja Pemmerening and Annette Imhausen per: http://mathforum.org/kb/thread.jspa?threadID=1916972&tstart=15 The usual suspects were invited to the conference, inviting no military code breakers, or serious math historians. It is hoped that future conferences will bring together the broader range of specialists with open invitations. At this point in time, the language side of Egyptology does not formally speak to the numeration and the mathematical side of hieratic mathematical texts. Tanja Pemmerening and I have informally discussed our common interests, the definition of the 1/64 dja unit, recorded with hekat weights and measures (the AWT). Tanja's work clearly shows the 'healing'' side of the "Eye of Horus" round-off problem in hieroglyphic symbols. At this time we do not agree on the two sides of Egyptian math (theoretical and practical statements that scribes used to state and prove every problem). Tanja concludes that our two bodies of work are consistent with one another, but goes no further. Annette Imhausen takes an algorithmic view of hieratic texts, follows the 1990's work of Jim Ritter. To date this group has published no new discoveries that open up previously unread sections of the hieratic math texts. For a reason I do not understand this group parses the literary symbols and subtle meanings of the mythical texts (Book of the Dead +, looking for clues to decode the hieratic texts. Annette and I have shared a small number of emails, but have not moderated our differences, and may not for several years. There is more to this story, info that can be discussed in the Webinar, for those that are interested. Thanks again for this opportunity to lay out a few historical threads. It is hoped that interesting aspects of ancient Egyptian math are being exposed. Reasons why the ancient mathematical styles are not often discussed as ancient scribes recorded their mathematics offers other interesting background info, Hopefully times are changing, thanks to Maria, Kirby, and others. Best Regards, Milo Gardner . |
Kirby, Thank you again for your very thoughtful comments: |
"This is fascinating material Milo. The 1997 Oregon Math Summit I was mentioning (not 1995 -- my mistake) also featured Ralph Abraham, who used the occasion to stump for a time-line-based approach to K-12 math learning,
actually studying the mathematics of a civilization while learning about it in historical terms, proceeding chronologically, grade by grade. Why not try it? Not every school needs to switch
tracks. Pioneering is OK (as we say in Oregon). This chrono-sequence wouldn't have to be too ploddingly adhered to (could have flash-forwards and flash-backs), would be a lot
more like the game 'Civilization' which many students have already tasted and enjoyed (another current thread with the Devlin group |
is a need for more simulations in math class)." |
The Portland and Oregon approaches to teaching children mathematics are enlightened compared to California's sterile approach. In 1990 the State of California was required to publish a K-12 Math Framework. The draft copy was written up by a kindergarten teacher from Northridge, LA County. Basic math concepts like function were muddled, as were the social learning sides of math history story lines. All of the ethnic story-lines, once traditional math framework topics, had been removed. State Board of Education, Sacramento, testimony included five to six ethnic community groups. My story line cited Gauss' child story of adding 1-100 by matching the first and last numbers in the series: 101, 50 times, that mentally were summed to 5050 by an algebraic concept (that Ahmes discussed in RMP 39. 40 and 64 following a 200 year older document, the Kahun Papyrus: http://en.wikipedia.org/wiki/Kahun_Papyrus ) Many numerate children, over the years, seem to have independently developed this basic arithmetic progression knowledge. Sadly, the California State Board of Education did not agree with our testimonies. A professional writer was hired to fix the muddled math concepts, a process that took an additional two years. In the end, California formally excised ancient and modern ethnic math story lines from its federally mandated K-12 Mathematical Framework. Luckily, knowledgeable college math professors were present (one from UC Davis, Henry Alder author of "Probability and Statistics" a text that I happily studied as an undergrad) shook their collective heads on the socially sterile final report. College math professors and average high school math teachers are aware that arithmetic remains as the main bridge to algebra. There are many classes of historical story lines that can be taken from many cultures. Reporting the human sides of how and why children can be introduced to the power of numbers, and the fundamental concepts of arithmetic and algebra, is fun. Thank you all, for not following California's 1990 lead, and showing the fun side of the long and rich history of mathematics to our children. Best Regards, Milo Gardner |