The Map of Mathematics

the Myth of Race

Human Origins

Historiography

Historiography is the study of the methodology of historians in developing history as an academic discipline, and by extension is any body of historical work on a particular subject. The historiography of a specific topic covers how historians have studied that topic using particular sources, techniques, and theoretical approaches. Scholars discuss historiography by topic – such as the “Historiography of the United Kingdom“, the “Historiography of Canada“, “Historiography of the British Empire“, the “historiography of early Islam“, the “historiography of China – and different approaches and genres, such as political history and social history. Beginning in the nineteenth century, with the ascent of academic history, there developed a body of historiographic literature. The extent to which historians are influenced by their own groups and loyalties – such as to their nation state – is a debated question.^[1]

The research interests of historians change over time, and there has been a shift away from traditional diplomatic, economic, and political history toward newer approaches, especially social and cultural studies. From 1975 to 1995, the proportion of professors of history in American universities identifying with social history increased from 31 to 41 percent, while the proportion of political historians decreased from 40 to 30 percent.^[2] In 2007, of 5,723 faculty in the departments of history at British universities, 1,644 (29%) identified themselves with social history and 1,425 (25%) identified themselves with political history.^[3]

Science under Islam

the documentary “Science and Islam”

science not religion is the universal language

intelligent design

La historia de π (pi)

“π is wrong!” by Bob Palais

appeared in

The Mathematical Intelligencer Springer-Verlag New York Volume 23, Number 3, 2001, pp. 7-8.

Bob Palais gratefully acknowledges Dr. Chandler Davis, for his encouragement and editorial input.

(See also the Wikipedia entry on Dr. Davis.) The most amusing letter to the editor in response stated:

“I agree with Bob Palais’ pi-ous article, but it may be 2-pi-ous.”

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The Pi Manifesto

 

Last updated July 4th, 2011

– See more at: http://www.thepimanifesto.com/#sthash.R4ubVZpw.dpuf

Taxi cab numbers

Fermat’s Last Theorem near misses?

In mathematics, the nth taxicab number, typically denoted Ta(n) or Taxicab(n), is defined as the smallest number that can be expressed as a sum of two positive algebraic cubesin n distinct ways. The concept was first mentioned in 1657 by Bernard Frénicle de Bessy, and was made famous in the early 20th century by a story involving Srinivasa Ramanujan. In 1938, G. H. Hardy and E. M. Wright proved that such numbers exist for all positive integers n, and their proof is easily converted into a program to generate such numbers. However, the proof makes no claims at all about whether the thus-generated numbers are the smallest possible and thus it cannot be used to find the actual value of Ta(n).

The restriction of the summands to positive numbers is necessary, because allowing negative numbers allows for more (and smaller) instances of numbers that can be expressed as sums of cubes in n distinct ways. The concept of a cabtaxi number has been introduced to allow for alternative, less restrictive definitions of this nature. In a sense, the specification of two summands and powers of three is also restrictive; a generalized taxicab number allows for these values to be other than two and three, respectively.

Ta(2), also known as the Hardy–Ramanujan number, was first published by Bernard Frénicle de Bessy in 1657 and later immortalized by an incident involving mathematicians G. H. Hardy and Srinivasa Ramanujan. As told by Hardy [1]:

“

I remember once going to see him when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. “No”, he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two [positive] cubes in two different ways.”

”

The subsequent taxicab numbers were found with the help of supercomputers. John Leech obtained Ta(3) in 1957. E. Rosenstiel, J. A. Dardis and C. R. Rosenstiel found Ta(4) in 1991. J. A. Dardis found Ta(5) in 1994 and it was confirmed by David W. Wilson in 1999.^[1][2] Ta(6) was announced by Uwe Hollerbach on the NMBRTHRY mailing list on March 9, 2008,^[3] following a 2003 paper by Calude et al. that gave a 99% probability that the number was actually Ta(6).^[4] Upper bounds for Ta(7) to Ta(12) were found by Christian Boyer in 2006.^[5]

Laurette Séjourné

Laurette Séjourné (October 19, 1911 – May 25, 2003) was a Mexican archeologist and ethnologist best known for her study of the civilizations of Teotihuacan and the Aztecs and her theories concerning the Mesoamerican culture hero, Quetzalcoatl.

Laurette Séjourné was born in Perugia, Italy, as Laura Valentini Corsa,^[1] although one also finds her mentioned as Laura Bianchi.^[2] Little is known about her early years; even her precise birth date is rarely mentioned.^[3] In her prime youth, she appears to have moved to France, perhaps in connection with the fascist take-over of 1922; in later life, she still wrote in French. She married a Frenchman, Séjourné, and participated in cultural life and the world of the cinema, meeting such figures as André Breton and Jean Cocteau. Strongly politicized like many others at the times, she divorced her husband,^[4] and left occupied France in exile for Mexico, in 1941. There, she became a naturalized Mexican citizen and married another exile, the Russian novelist and revolutionary known as Victor Serge (Viktor Kibalchich or Kibaltchitch, 1890-1947).^[5] Soon after his death, she joined the Mexican Communist Party.^[6] Later, she remarried with Arnaldo Orfila, director of the Fondo de Cultura Económica and founder of Siglo XXI Editores.

Séjourné’s militant spirit can be captured from a passage like the following one:^[7]

[In] spite of extreme demographic density and the lack of machinery and work animals, the members of Precolumbian societies enjoyed physical health, individual independence, security, some leisure, which implies a distribution of resources and an integration to the collectivity that in our days would seem a utopia. From all of this follows that if we refuse to analyze the invasion that destroyed a civilized world and laid the seed of a system in which hunger, humiliation, and bloody repression constitute the only form of survivorship, contemporary underdevelopment should be a result of congenital incapacity, of the irremediable racial inferiority that justified extermination and vassalage.

Jevons paradox

In economics, the Jevons paradox (/ˈdʒɛvənz/; sometimes Jevons effect) occurs when technological progress increases the efficiency with which a resource is used (reducing the amount necessary for any one use), but the rate of consumption of that resource rises because of increasing demand.^[1] The Jevons paradox is perhaps the most widely known paradox in ecological economics.^[2] However, governments and environmentalists generally assume that efficiency gains will lower resource consumption and are an effective policy for sustainability, ignoring the possibility of the paradox arising.^[3]

In 1865, the English economist William Stanley Jevons observed that technological improvements that increased the efficiency of coal-use led to the increased consumption of coal in a wide range of industries. He argued that, contrary to common intuition, technological progress could not be relied upon to reduce fuel consumption.^[4]

The issue has been re-examined by modern economists studying consumption rebound effects from improved energy efficiency. In addition to reducing the amount needed for a given use, improved efficiency lowers the relative cost of using a resource, which tends to increase the quantity of the resource demanded, potentially counteracting any savings from increased efficiency. Additionally, increased efficiency accelerates economic growth, further increasing the demand for resources. The Jevons paradox occurs when the effect from increased demand predominates, causing resource use to increase.^[4]

Considerable debate exists about the size of the rebound in energy efficiency and the relevance of Jevons paradox to energy conservation. Some dismiss the paradox, while others worry that it may be self-defeating to pursue sustainability by increasing energy efficiency. However, conservation policies such as green taxes, cap and trade, and emissions standards do not display the paradox, and can be used to control the rebound effect.^[5] Environmental economists have proposed that efficiency gains be coupled with conservation policies that keep the cost of use the same (or higher) to avoid the Jevons paradox.^[