15 Days of Black History Month
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Beginnings of Counting and Numbers Tallies and Tokens
Beginnings of Counting and Numbers Tallies and Tokens Picture Link (http://www.flickr.com/photos/quadrofonic/834667550/) Bone Tallies • The Lebombo Bone is a portion of • The radius bone of a wolf, a baboon fibula, discovered in the discovered in Moravia, Border Cave in the Lebombo Czechoslovakia in 1937, and mountains of Swaziland. It dates dated to 30,000 years ago, has to about 35,000 years ago, and fifty‐five deep notches carved has 29 distinct notches. It is into it. Twenty‐five notches of assumed that it tallied the days of similar length, arranged in‐groups a lunar month. of five, followed by a single notch twice as long which appears to • terminate the series. Then Picture Link starting from the next notch, also (http://www.historyforkids.org/learn/africa/science/numbers.htm) twice as long, a new set of notches runs up to thirty. • Picture link (http://books.google.com/books?id=C0Wcb9c6c18C&pg=PA41&lpg=PA41 &dq=wolf+bone+moravia&source=bl&ots=1z5XhaJchP&sig=q_8WROQ1Gz l4‐6PYJ9uaaNHLhOM&hl=en&ei=J8D‐ TZSgIuPTiALxn4CEBQ&sa=X&oi=book_result&ct=result&resnum=4&ved=0 CCsQ6AEwAw) Ishango Bone • Ishango Bone, discovered in 1961 in central Africa. About 20,000 years old. Ishango Bone Patterns • Prime 11 13 17 19 numbers? • Doubling? 11 21 19 9 • Multiplication? • Who knows? 3 6 4 8 10 5 5 7 Lartet Bone • Discovered in Dodogne, France. About 30,000 years old. It has various markings that are neither decorative nor random (different sets are made with different tools, techniques, and stroke directions). -
Writing the History of Mathematics: Interpretations of the Mathematics of the Past and Its Relation to the Mathematics of Today
Writing the History of Mathematics: Interpretations of the Mathematics of the Past and Its Relation to the Mathematics of Today Johanna Pejlare and Kajsa Bråting Contents Introduction.................................................................. 2 Traces of Mathematics of the First Humans........................................ 3 History of Ancient Mathematics: The First Written Sources........................... 6 History of Mathematics or Heritage of Mathematics?................................. 9 Further Views of the Past and Its Relation to the Present.............................. 15 Can History Be Recapitulated or Does Culture Matter?............................... 19 Concluding Remarks........................................................... 24 Cross-References.............................................................. 24 References................................................................... 24 Abstract In the present chapter, interpretations of the mathematics of the past are problematized, based on examples such as archeological artifacts, as well as written sources from the ancient Egyptian, Babylonian, and Greek civilizations. The distinction between history and heritage is considered in relation to Euler’s function concept, Cauchy’s sum theorem, and the Unguru debate. Also, the distinction between the historical past and the practical past,aswellasthe distinction between the historical and the nonhistorical relations to the past, are made concrete based on Torricelli’s result on an infinitely long solid from -
The Maban Languages and Their Place Within Nilo-Saharan
The Maban languages and their place within Nilo-Saharan DRAFT CIRCUALTED FOR DISCUSSION NOT TO BE QUOTED WITHOUT PERMISSION Roger Blench McDonald Institute for Archaeological Research University of Cambridge Department of History, University of Jos Kay Williamson Educational Foundation 8, Guest Road Cambridge CB1 2AL United Kingdom Voice/ Ans (00-44)-(0)1223-560687 Mobile worldwide (00-44)-(0)7847-495590 E-mail [email protected] http://www.rogerblench.info/RBOP.htm This version: Cambridge, 10 January, 2021 The Maban languages Roger Blench Draft for comment TABLE OF CONTENTS TABLE OF CONTENTS.........................................................................................................................................i ACRONYMS AND CONVENTIONS...................................................................................................................ii 1. Introduction.........................................................................................................................................................3 2. The Maban languages .........................................................................................................................................3 2.1 Documented languages................................................................................................................................3 2.2 Locations .....................................................................................................................................................5 2.3 Existing literature -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
From the Yellow Nile to the Blue Nile. the Quest for Water and the Diffusion of Northern East Sudanic Languages from the Fourth to the First Millenia BCE"
This lecture was delivered in ECAS 2009 (3rd European Conference on African Studies, Panel 142: African waters - water in Africa, barriers, paths, and resources: their impact on language, literature and history of people) in Leipzig, 4 to 7 June 2009. "From the Yellow Nile to the Blue Nile. The quest for water and the diffusion of Northern East Sudanic languages from the fourth to the first millenia BCE". Dr. Claude Rilly (CNRS-LLACAN, Paris) The quest for water and hence, for food supply, is a key issue in the appearance and diffusion of languages in the Sahelian regions of Africa. Climate changes, as occurred from the end of Neolithic period down to the second millenium BCE, played a major role in the redistribution of populations along the Nile river and its tributaries and can explain the appearance of a recently defined linguistic family, namely Northern East Sudanic (NES). This paper must be considered as a synthesis of several recent publications I wrote on this subject, so that I shall have to refer the reader, more often than not, to these earlier studies. Detailed demonstration of all these points would require much more time than is allotted to me. The Northern East Sudanic language group In his seminal study published in 1963, J. H. Greenberg divided the languages of Africa into four major phyla or superfamilies, namely Afroasiatic, Niger-Congo, Khoisan and Nilo-Saharan. If the three first phyla were more or less obvious, Nilo-Saharan was not so easily constituted, requiring from Greenberg a long work to merge twelve different families into one phylum. -
Nrr,O- S¡,Hanax Rnvrsrrno L
Nordic Journøl of,lfrican Studies 8(2): 108-1 j8 (1999) N íl o- S aharan Revis it e d Nrr,o- S¡,Hanax Rnvrsrrno l. SupponrrNc EvIDENcE PERTTI MIKKOLA University of Helsinki, Finland 1.1 Srnlcr CzuteRIe., GRmNeBRc's D¿,t¡, The total length of the test list2 was 93 items; altogether 68 glosses involving ca. 103 observed to occur in Greenberg's Nilo-Saharan etymology lists. Inthe INIRonucttoN etymologies were statistical approach only one of these (#225r in the combined list) 'who?' was significantly above the expected by random coincidence (at the 95 per cent level), In Nilo-Saharan is the mostproblematic case in Greenberg's (196311966) classification of addition, nearly significant cases were etymologies #209 'tooth' (in Eastem Sudanic) and African languages. Previously the role of chance in mass comparison has #224'white 3'. Other important etymologies within the sample found more often than the ry'm-ary (Mikkola been investigated with approach 1998). The test comprised a expected included notably: 'meît?'(one of the common African roots), 'dog'(which is languages in Greenberg's Nilo-Saharan word lists. standard sample from l8 Consonants a Wanderwort), 'mouth 2' (Eastern Sudanic),'man l'(mostly Eastern Sudanic) and 'to were grouped into I types corresponding to natural classes, any vowels were accepted. kill2lto die 2' (rwo roots; also found in Niger-Congo). The probabilities of every different word-initial Cvc-sequence type were investigated, Though the quantþ of strict similarities between Nilo-Saharan lineages is usually and the expected dishibutions were compared with the observed scores. Around a dozen insignificant, we ought not to forget that some languages had incomplete data. -
Kanuri and Its Neighbours: When Saharan and Chadic Languages Meet
3 KANURI AND ITS NEIGHBOURS: WHEN SAHARAN AND CHADIC LANGUAGES MEET Norbert Cyffer 1. Introduction' Relations between languages are determined by their degree of similarity or difference. When languages share a great amount of lexical or grammatical similarity, we assume, that these languages are either genetically related or else they have been in close contact for a long time. In addition to genetic aspects, we also have to consider phenomena which may lead to common structural features in languages of different genetic affiliation. We are aware, e.g., through oral traditions, that aspects of social, cultural or language change are not only a phenomenon of our present period, we should also keep in mind that our knowledge about the local history in many parts of Africa is still scanty. The dynamic processes of social, cultural and linguistic change have been an ongoing development. In our area of investigation we can confirm this from the 11 th century. Here, the linguistic landscape kept changing throughout time. The wider Lake Chad area provides a good example for these developments. For example, Hausa, which is today the dominant language in northern Nigeria, played a lesser role as a language of wider communication (L WC) in the past. This becomes obvious when we assess the degree of lexical borrowing in the languages that are situated between Hausa and Kanuri. However, during the past decades, we observed a decrease of Kanuri influence and an increase of Hausa . • Research on linguistic contact and conceptualization in the wider Lake Chad area was carried out in the project Linguistic Innovation and Conceptual Change in West Africa. -
Ethnolinguistic Structuring of Sorghum Genetic Diversity in Africa and the Role of Local Seed Systems
Ethnolinguistic structuring of sorghum genetic diversity in Africa and the role of local seed systems Ola T. Westengena,b, Mark Atam Okongoc, Leo Onekd, Trygve Berge, Hari Upadhyayaf,g,h, Siri Birkelandb, Siri Dharma Kaur Khalsab, Kristoffer H. Ringa, Nils C. Stensethb,1, and Anne K. Brystingb aCentre for Development and the Environment, University of Oslo, NO-0317 Oslo, Norway; bCentre for Ecological and Evolutionary Synthesis, Department of Biosciences, University of Oslo, NO-0316 Oslo, Norway; cMinistry of Agriculture, Eastern Equatoria State, Torit, South Sudan; dCollege of Applied and Industrial Sciences, Juba University, Juba, South Sudan; eNoragric, Norwegian University of Life Sciences, NO-1432 Aas, Norway; fInternational Crops Research Institute for the Semi-Arid Tropics, Patancheru 502 324, India; gDepartment of Agronomy, Kansas State University, Manhattan, KS 66506; and hUWA Institute of Agriculture, University of Western Australia, Crawley, WA 6009, Australia Edited* by Hans R. Herren, Millennium Institute, Arlington, VA, and approved August 18, 2014 (received for review January 27, 2014) Sorghum is a drought-tolerant crop with a vital role in the live- recombination of these types in different environments (2, 14, 15). lihoods of millions of people in marginal areas. We examined On the basis of morphology, Harlan and de Wet (16) classified genetic structure in this diverse crop in Africa. On the continent- sorghum into five basic and 10 intermediary botanical races (16). wide scale, we identified three major sorghum populations (Central, The race “bicolor” has small elongated grains, and, because of the Southern, and Northern) that are associated with the distribution “primitive” morphology, it is considered the progenitor of more of ethnolinguistic groups on the continent. -
10 Nigeria: Ethno-Linguistic Competition in the Giant of Africa
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by SOAS Research Online 10-Simpson-c10 OUP172-Simpson (Typeset by spi publisher services, Delhi) 172 of 198 August 31, 2007 15:49 10 Nigeria: Ethno-linguistic Competition in the Giant of Africa Andrew Simpson and B. Akíntúndé Oyètádé 10.1 Introduction Nigeria is a country with an immense population of over 140 million, the largest in Africa, and several hundred languages and ethnic groups (over 400 in some estimates, 510 according to Ethnologue 2005), though with no single group being a majority, and the three largest ethnic groups together constituting only approximately half of the country’s total population. Having been formed as a united territory by British colonial forces in 1914, with artificially created borders arbitrarily including certain ethnic groups while dividing others with neighbouring states, Nigeria and its complex ethno- linguistic situation in many ways is a prime representation of the classic set of problems faced by many newly developing states in Africa when decisions of national language policy and planning have to be made, and the potential role of language in nation- building has to be determined. When independence came to Nigeria in 1960, it was agreed that English would be the country’s single official language, and there was little serious support for the possible attempted promotion of any of Nigeria’s indigenous languages into the role of national official language. This chapter considers the socio- political and historical background to the establishment of English as Nigeria’s official language, and the development of the country over the subsequent post-independence era, and asks the following question. -
E-Thesis Vol 2
1 BEGINNINGS OF ART: 100,000 – 28,000 BP A NEURAL APPROACH Volume 2 of 2 Helen Anderson B.A. (University of East Anglia) M.A. (University of East Anglia) Submitted for the qualification of PhD University Of East Anglia School of World Art Studies September 2009 © “This copy of the thesis has been supplied on condition that anyone who consults it is understood to recognise that its copyright rests with the author and that no quotation from the thesis, nor any information derived therefrom, may be published without the author’s prior, written consent”. 2 VOLUME TWO MAPS Africa 1 India 2 Papua New Guinea/Australia 3 Levant 4 Europe 5 CATALOGUE 1. Skhul Cave 6 2. Qafzeh Cave 9 3. Grotte des Pigeons 13 4. Oued Djebanna 16 5. Blombos Cave 18 6. Wonderwerk Cave 21 7a. Blombos Cave 23 7b. Blombos Cave 26 8. Klein Kliphuis 29 9-12. Diepkloof 32 13. Boomplaas 39 14. Enkapune Ya Moto 41 15. Border Cave 43 16. Kisese II 47 17. Mumba 48 18. Apollo 11 Cave 50 19. Patne 53 20. Bacho Kiro 56 21. Istallosko 59 22. Üça ğızlı Cave 61 23. Kostenki 65 24. Abri Castanet 69 25. Abri de la Souquette 72 26. Grotte d’Isturitz 74 27. Grotte des Hyènes 77 28a. Chauvet Cave 80 28b. Chauvet Cave 83 28c. Chauvet Cave 86 29. Fumane Cave 89 30. Höhlenstein-Stadel 94 31a. Vogelherd 97 31b. Vogelherd 100 31c. Vogelherd 103 31d. Vogelherd 106 3 31e. Vogelherd 109 31f. Vogelherd 112 31g. Vogelherd 115 31h. Vogelherd 118 31i. -
Notes for CMTHU201 History of Mathematics in This Course We Will
Notes For CMTHU201 History of Mathematics In this course we will study five early centers of civilization and track the development of mathematics in those five areas. There are other ancient centers, but the records for these is not as complete. Our five centers are the Tigris-Euphrates region, the Nile River Valley, the Aegean Sea area, the Yellow River Valley in China and the Indus River in what is now Pakistan and the culture of India that grew from this. We will then go on to study the first civilization that benefited from the mathematical and scientific discoveries of all five of these centers, the combined Persian and Islamic societies. I Prehistory The origins of counting in a broad sense are earlier in the evolutionary process than the emergence of man. Many animal species can tell the difference between the presence of a predator and the absence of the same. This carries forward to the difference between 1 and 2 objects. Wolves and lions know well enough not to attack 2 or 3 of their kind when they are alone. Experiments have shown that crows and jackdaws can keep track of up to six items. (In the Company of Crows and Ravens, Marzluff and Angell) “Pigeons can, under some circumstances, estimate the number of times they have pecked at a target and can discriminate, for instance, between 45 and 50 pecks.” (Stanislas Dehaene, The Number Sense) Human infants can discriminate between 2 and 3 objects at the age of 2 or 3 days from birth. This hard-wired and primitive sort of counting is known as digitizing. -
Language Contact, Areality, and History: the Songhay Question Revisited 1
Language Contact, Areality, and History: the 1 Songhay Question Revisited Robert Nicolaï Institut Universitaire de France - Université de Nice 1. Historical background The proposal that Songhay be included in Nilo-Saharan (Greenberg 1963, see appendix 1 below) was questioned by Lacroix (1969) and debated by Nicolaï (1990). Large-scale studies of the internal subclassification of Nilo-Saharan (NS) conducted by Bender (1995) and Ehret (2001) nevertheless supported it. Both the latter scholars base their work on the genealogical tree model and share (in particular Ehret) the same assumption of the existence of a NS language family. Like Greenberg before them, they include the Songhay group of languages in the NS family, although each places it differently on his own genealogical tree. Their working methods are also different: Bender uses mass comparison and isoglosses while Ehret uses correspondence-based reconstruction. One might therefore assume that the question has been definitively settled, since major work by two independent scholars confirms the inclusion of Songhay in the NS family/phylum, even though subclassification varies. This is not however the case. The latest study (Nicolaï 2003) provides a detailed evaluation of the empirical proof adduced in support of the classification of Songhay as NS, and finds both Ehret's and Bender's results wanting. The analysis proceeds in three stages. • Impugning the NS hypothesis : it can be shown that the data and reasoning on which the classification of Songhay as NS is based do not withstand critical examination and must be rejected in view of their unconvincing empirical validity and their theoretical insufficiency. • Unexpected developments in lexical research involving the Afroasiatic languages : a Berber-Semitic origin can be postulated for a large number of terms invoked by both Bender and Ehret as proof of NS filiation and for many others as well.