User:Milogardner
From Wikipedia, the free encyclopedia
This is the user page for Milo Gardner, whose login name on Wikipedia is User:Milogardner.
Interest in ancient and medieval mathematics and economic history date to undergrad history of mathematics (Eves), Theory of Equations (Borofsky), history of number theory (Ore) and 15 units of history of economic thought classes taken in 1963 and 1964. These academic views merged into technical 1957 to 1959 military code breaking experiences after 1987. Assisting a retired electrical engineer, who wished to publish RMP 2/n table findings gleaned over 15 years, ancient math fragments began to be translated and freshly decoded. Little did I know that 20 years later my friend's Rhind Mathematical Papyrus micro views would be updated to meta views by considering a range of ancient and medieval Egyptian fraction texts. Personal studies have translated, decoded, and published two ancient Egyptian fraction texts, the Egyptian Mathematical Leather Roll(EMLR) and the Akhmim Wooden Tablet(AWT) and created several Wikipedia articles.
Previously unreported aspects of ancient arithmetic notations and methods have been pointed out, and confirmed by several methods, one being cryptanalysis. Another was Occam's Razor, the simplest method is likely the historical method. Several unifying Egyptian fraction aspects of the ancient arithmetic have connected the oldest ancient texts to medieval texts covering a period of over 3,000 years. The oldest theoretical and practical threads included two arithmetic systems that considered p and q as prime numbers. The theoretical thread was associated with astronomical topics as well the body of MK mathematical texts. Two remainder arithmetic systems reveal two unifying topics. The first arithmetic system wrote binary quotient binary, and scaled Egyptian fraction remainders. The second arithmetic system wrote integer quotient and unscaled Egyptian fraction remainders.
The ancient arithmetic systems had been extensively written out, applied, and double checked by scribes within weights and measures units and other applications. One purpose of MK weights and measures units assisted in the management and control of Pharaoh and absentee landlord inventories in rigorous ways. Scribal mathematics as a consequence included abstract (theoretical) and applied (practical) building blocks. That is, practical control units were created from theoretical models, one being a money system, in a manner that allowed double checking of inputs, outputs and wages to take place as implemented on a large farm: http://www.reshafim.org.il/ad/egypt/texts/heqanakht.htm . An accounting analysis of the absentee landlord's accounts was published in 2002 (http://www.blackwell-synergy.com/doi/abs/10.1111/1467-6281.00107) showing that the monetary/commodities system managed and controlled the estate " ... by converting the concrete into an abstract theoretical value via the use of ‘monies of account’ to attain a measure of economic and social reciprocity." It is clear that Pharaoh and absentee landlords across the Ancient Near East used versions of this system for over three thousand years.
Personal math history publications appear in journals, internet blogs, Wikipedia articles, PlanetMath articles (i.e http://planetmath.org/encyclopedia/EgyptianFraction2.html), and math history discussion groups. My writings parse scholarly aspects of Egyptian mathematical texts dating to the British Museum's 1864 receipt of the RMP and the EMLR. To review, the RMP began to be discussed by scholars after 1879 via a pirated copy. Chace proposed in 1927 that the RMP translation task was complete. However, the RMP's sibling document, an EMLR multiple method had been excluded from Chace's analysis. Apparently the EMLR was unrolled, and additively read in 1927 as an attempt to conform to additive views of Egyptian mathematics. The scope of this hobby reconnects Egypt's under reported RMP, AWT, EMLR, Reisner Papyrus, Kahun Papyrus, and other texts written within Hekat (volume) units. Ancient theoretical and practical arithmetic oversights continue to be pin-pointed, and at times, confirmed by linkages to Egyptian Middle Kingdom and medieval texts. Egyptian fraction remainder arithmetic, for example, was continuously used for 3,200 years and applied in region-wide trading weights and measures, as documented as late as the Liber Abaci in 1202 AD.
Concerning region-wide trading weights and measures systems, scholars have long known that Egyptian, Greek, Hellene, Coptic, Arab and medieval mathematicians used Egyptian fraction methods for trade and other purposes. However, the ancient arithmetic methods that once connected Near East cultures have been reported by scholars in fuzzy and fragmented details. Theoretical and practical Egyptian fraction arithmetic threads documented in fragmented ancient and medieval texts are slowing re-emerging, thread by thread.
To fairly parse Egyptian fraction threads from one medieval era, around 800 AD Arab mathematics copied Egyptian fraction arithmetic and Hellene trading units into base 10 numerals. The Liber Abaci is the best known of the European texts. A Pisa resident named Leonardo (Fibonacci), a son of Pisa merchant, had traveled to nearby Arab trading ports with his father. The Liber Abaci was Europe's arithmetic textbook for over 200 years.
In 1454 the Ottoman Empire ended Byzantine and Venetian access routes to the Silk Road. New European trade routes to the Orient were needed. Portugal provided access to the India and China around the horn of Africa, Spain found New World trade routes connected to the Philippines and China rather than to Japan within an economic system that led to mercantilism. As one consequence Fibonacci's Egyptian fraction trading units were no longer needed. With the Moors removed from Spain in 1492, and Renaissance Europeans generally disapproving of Islamic political power and Islamic trading weights and measures systems, Europeans were motivated to create a replacement decimal system and replacement trading units. The older medieval theoretical arithmetic including 1-9 Hindu-Islamic numerals slowly added algorithms after 1492 creating a new decimal thread by 1600. The new base 10 decimal system contained new trading units. By 1900 AD and the scholarly effort to fairly read the RMP and its 2/n table were severely hindered. Few hints of Egyptian fraction's oldest and newest theoretical threads were available to scholars during the majority of the 20th century.
Today, after 130 years of RMP research, scholars are connecting ancient and medieval theoretical Egyptian fraction arithmetic patterns. Jens Hoyrup recently reported http://akira.ruc.dk/~jensh/Publications/Egyptian_math.pdf stressing algorithms was used by scribes. Large gaps in the practical and theoretical Egyptian fraction record have slowed scholarship efforts. Yet, the oldest Egyptian fraction fragments are revealing several theoretical and practical secrets. The early Egyptian fraction texts provide documentations of arithmetic methods that defined Egyptian fraction as a unified body of knowledge. The late medieval portion of the record contains theoretical threads that are connected to the RMP, Ahmes, the EMLR student scribe's writings and other texts.
Several early scholars had worked on the oldest Egyptian fraction records incorrectly suggested that Ahmes' Egyptian fraction methods had been marked by intellectual decline, by only writing fractions in additive arithmetic. Since 1945, and the work of E.M. Bruins and others, a major upgrade in the intellectual decline view began to be formalized. Scholars began to publish evidence that the oldest Egyptian fraction notations were attested to have been advanced compared to the Old Kingdom's Eye of Horus arithmetic. By 2002 the upgrade level of scholarship began to publish theoretical arithmetic topics, one defining p and q as prime numbers within the oldest Egyptian fraction texts.
A glimpse of the oldest Egyptian fraction texts is reported by: http://egyptianmath.blogspot.com. Three 2002 papers marked a major change in theoretical Egyptian fraction arithmetic and theoretical and practical weights and measure applications. One EMLR paper details the 1900 BCE 26 line text translating, and decoding the data character by character, line by line, in a journal paper published in India. The paper exposes algebraic aspects of a single multiple method, with hints of theoretical arithmetic. The second and third 2002 papers, written by Pemmerening and Vymazalova, opened doors to reading closely related theoretical and practical weights and measures topics.
A non-optimal EMLR multiple method became optimal in the Rhind Mathematical Papyrus 2/n table defined a single multiple method. Optimal RMP 2/n table multiples were created from an unclear red auxiliary LCM technique. The RMP multiple method appears in the 1202 AD Liber Abaci within seven rational number conversion methods. The Liber Abaci's author Fibonacci details four of the seven methods in a way that created RMP 2/n table members. A journal paper has been authored on the 1900 BCE Akhmim Wooden Tablet(AWT) connecting a volume unit partitioning method to the RMP 2/n table and Liber Abaci Egyptian fraction topic. The AWT generally partitions a volume unit written as (64/64)) by n into binary quotients and scaled 5R/n 1/320 unit remainders.
On a broad level, the Liber Abaci documents 3,200 years of Egyptian fractions trading units used by Arabs, Hellenes, Greeks and Egyptians with all four cultures employing varying levels of theoretical arithmetic. Cudo's to L.E. Sigler for fully translating the Latin Liber Abaci into English in time for the book's 800th anniversary. The translation allows hard to read Latin to be available to English readers. Almost any English reader can jump back 800 to 4,000 years and ponder simple and complex Egyptian fraction arithmetic, sometimes theoretically based, and most often practically based. Had Sigler's translation been available 50 to 100 years ago the 2008 confusion that confounds Egyptologists and math historians alike related to the use of theoretical Egyptian fraction arithmetic, may have been resolved.
That is, 1920's scholarly views of Egyptian fraction math are expected to merge Ahmes and Fibonacci arithmetic, covering a period of over 2,850 years, into one form of Egyptian fraction arithmetic.Occam's Razor has pointed out a multiple method used by an EMLR scribe, Ahmes, and Fibonacci, and others. The multiple method generally converts rational numbers into concise, optimal or elegant Egyptian fraction series. Journals will be filling gaps in the Egyptian fraction multiple story line. It is expected the Egyptian fraction story line will be restoring large chunks of MK arithmetic, algebra, weights and measures, and economic considerations within a context of Ahmes arithmetic's validated in medieval Egyptian fraction arithmetic.
