3rd ed. Aug 2023 (expanded appendices). 2nd ed. Nov 2019 (revised to include advances in linguistics, genomics, interpretive theory, and Mesopotamian mathematics); 1st ed. (Dec 29, 2009)
Part 1 in Ancient Mathematics series. (Part 2: The Mathematics of Uruk and Susa 3500-3000 BCE, Part 3: Exploring Cuneiform Culture 8500-2500 BCE)
Abstract
How far back in time can we trace mathematical understanding and mathematical practice? When did humans acquire the neurological circuitry for the cognitive and linguistic capabilities on which mathematics depends? Advances in multiple disciplines over the past 30 years have fundamentally changed what we know about our past and about the biological capacity for, and cultural impulses behind, cognitive precision (language, number sense, cultural transmission). Exploring these questions will take us on a journey across archaeology, Assyriology, artifact analysis (close reading theory), anthropology, genomics, linguistics, neurobiology, and animal cognition.
 The Anthropology and Archaeology of Conceptual Thought leading to the Birth of Mathematics Continue reading this article…
By Assad Ebrahim, on March 14th, 2023 (9,037 views) |
Topic: Maths--General Interest, Technology
Updated March 21, 2023, following two bank collapses in the US and the collapse of Credit Suisse in Europe. First published July 5, 2010, two years after the start of the Great Recession.
Mathematical Finance is an area of applied mathematics that has seen explosive growth over the past 30 years as the U.S. financial markets became deregulated during the late 1980s and 1990s. From a technical point of view, Mathematical Finance uses a broad range of sophisticated mathematics for its financial models: from the partial differential equations of mathematical physics, to stochastic calculus, probabilistic modeling, mathematical optimization, statistics, and numerical methods. The practical implementation of trading strategies based upon these mathematical models requires designing efficient algorithms as well as exploiting the state-of-the-art in software engineering (real-time and embedded development, low latency network programming) and in computing hardware (FPGAs, GPUs, and parallel and distributed processing). Taken together, the technical aspects of mathematical finance and the software/hardware aspect of financial engineering lie at the intersection of business, economics, mathematics, computer science, physics, and electrical engineering. For the technologically inclined, there are ample opportunities to contribute.
The ideas of financial mathematics are at the heart of the global free market capitalist system that is in place across most of the world today and affects not only economics but also politics and society. It is worthwhile to understand the essential mechanics of the modern financial world and how it has arisen, regardless of whether we agree with its principles or with the impact of the financial system on social structures. In this article, I’ll motivate the origins of financial mathematics through a simplified account of the rise of the modern financial marketplace. (Update: 2012. A highly recommended graphic novel account is Economix, by Michael Goodwin, which was published in 2012. His website – linked above – has some great context explaining developments since 2012 in a similar highly accessible style!)
Separate from the technical content, there is a kernel of core financial ideas that every literate citizen should understand. In 1999, Clinton and Congress removed the last remaining restraints on the financial industry that were put in place by Roosevelt in the famous Glass-Steagall Act of the 1930s specifically to address the root causes of the banking crisis of 1929 and the subsequent Great Depression. Ten years after Glass-Steagall was repealed, the Bear Stearns/Lehmann Brothers collapse brought on the financial crisis of 2008 and the Great Recession whose consequences were a decade of stagnation in western economies, the acceleration of income inequality, and a resurgences of nationalism across UK, Europe, and the US. The severity of the crisis led to a limited package of restraints being placed on the US financial system in 2010 through the Dodd-Frank act, but this was repealed by Trump in 2018, leaving the financial market again largely deregulated.
Update 2023: Five years on from this, we see in March 2023 the start of US bank failures beginning with the Silicon Valley Bank (SVB), the 16th largest bank in the U.S., and the collapse in just one week of two US banks that together hold over $300 billion in total assets which is more than the largest single US bank failure history in 2008 (Washington Mutual). The contagion is spreading to Europe with Credit Suisse impacted, the 2nd largest bank in Switzerland and one of the top 8 global investment banks of US/UK/EU. Housing market decline, bank failures, and increasing unemployment mark the three signs of a coming recession – the bank failures have completed the set. (See Recommended Reading).
Continue reading this article…
This is Part 2 in the Ancient Mathematics series. (Part 1: The Prehistoric Origins of Mathematics, Part 3: Exploring Cuneiform Culture 8500-2500 BCE)
Summary The written mathematics of ancient Iraq and Iran (Mesopotamia, Khuzistan) developed out of an administrative/bureaucratic program to control the surplus raw and manufactured goods of the settled societies of the late neolithic/early bronze age: grains & grain products, sheep & other herded animals, jugs of dairy fats & beer, rope & textiles. It evolved through a sequence of literary and mathematical innovations, each making more efficient the ability to record quantitative/metrological information and use it for planning and control. Initially, impressed tokens and pictographs were used whose meaning was clear by association. Subsequently, this repertoire was written signs was expanded in a consious effort to provide a standard, all-encompassing collection of signs/symbols (ideographs/logograms) that could represent all aspects of importance in early thought (professions, animals, foods, containers, textiles, etc.). The standard sign lists were spread through scribal schools to produce the scribes that administered the temple economies of the early city-states.
Uruk was the hegemonic centre of this innovation in mathematics and writing, starting from 3500 BCE. The increased administrative control generated economic efficiencies accelerating Uruk’s growth and which supported greater military effectiveness and the ability to dominate neighboring polities and support longer distance trading missions [Adams/2005], [Algaze/2013]. The success of Uruk’s structures had the effect of radiating the new inventions outward throughout the Greater Mesopotamian region (evidence in Aratta/Susa adoption of writing/adminstrative control), even reaching Anatolia (Turkey) in the far north (Uruk expansion phenomenon).
The gains in economic power and increased resilience to subsistence unpredictability conferred by the new planning and control capabilities, set in motion the development of a bureaucratic administrative culture in the southern Mesopotamian city states that, over the next 1000 years would reach its hypertrophic apex in the ambitious Ur III program under King Shulgi to plan, manage, and control all economic/productive assets in his vast empire through mathematics (c.2050 BCE). This required an army of scribes which in turn led to the standardization and systematization of the scribal school institution responsible for producing them.
Two examples of mathematical innovation are from the cattle redistribution center Puzrish-Dagan outside Nippur during the Ur III empire. One shows perfection of the form of tabular accounting (world’s earliest normalized two-dimensional table with rows and columns and sums in both dimensions) [Robson/2003]. The other shows the population growth modeling of a cattle-herd over 10 years with projected economic yields in dairy and cheese, solving, in modern terms, population difference equations in table form (see illustrated explanation of cuneiform tablet TCL 2, no.5499, [Nissen/1993: 97-102])
In this paper, we will look in more detail at mathematical development during the archaic period of writing (3500-3000 BCE) which gave rise to a new literate and quantitative layer in society in the main urban centres of Mesopotamia. Our thesis (which we have seen play out already in Part 1) is that technology (in this case mathematics/writing) and culture (in this case the impulse to plan/control) are inextricably linked. Their development influences the trajectory of the surrounding societies.
 Ur III mathematical model projecting annual dairy/cheese yields from a herd of 4 cows and a bull with assumptions on calving rates Download article (PDF)
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By Assad Ebrahim, on April 15th, 2010 (12,288 views) |
Topic: Education, Maths--Philosophy
An examination of mathematical methods and the search for mathematical meaning.
This article curates a reading list (most sources available freely online) organized into a set of encounters that lie outside the standard mathematics curriculum. They are intended to enrich the reader’s understanding of mathematics and its place in scientific inquiry, increase her/his connection to the historical and philosophical questions behind the mathematics of the past and present, and gain greater satisfaction from further mathematical study. The reader should come away with a better understanding of the culture of mathematics: what mathematics is, mathematical method and meaning, and the relation of mathematics to the empirical world and to science.
We look at seven topics. These may be covered in any order, to suit your particular interests.
- What is Mathematics? (Its Nature and Characteristics)
- Reality, Truth, and the Nature of Mathematical Knowledge
- What is Proof? and the Problem of Certainty
- Some Readings in the History of Mathematics and the Evolution of Its Ideas
- The Search for Foundations in Mathematics
- Mathematics and Science
- Thoughts on Mathematical Practice and Mathematical Style
There is no core body of technical material to master in this course; the important thing is a feel for how, why, and in what context the core ideas of mathematics evolved, getting to the essence of their motivation, and understanding the fruits of these efforts. The course such as the below should appeal to all those who have an itch to scratch beneath the surface of mathematics, who find themselves asking “but why?”. It could be useful in all three tiers of education: secondary, post-secondary (undergraduate), and graduate, appropriately restructured.
- Secondary school elective: to encourage bright students in mathematics, science and technology to enter the university with a broader perspective on the mathematics they will be rapidly learning there.
- University elective course: offered as a writing-intensive seminar, intended primarily for students in the sciences and engineer: mathematics, physics, engineering.
- Graduate level course: offered in the first year of graduate school in mathematics or applied mathematics as a supplementary seminar.
Continue reading this article…
By Assad Ebrahim, on March 5th, 2010 (16,232 views) |
Topic: Technology
“Smart dust”, tiny leaf sensors, wearable computing — these and a host of other sensors that make measurements and communicate without requiring human intervention can now be readily integrated into dispersed systems to provide ambient intelligence, situational awareness, and the capability for adaptive behaviors or intelligent process automation.
Whether the sensor’s output is used to control the opening and closing of relays or thermostats, or to automatically raise alerts — the integration of sensors into systems is at the heart of the promise of ubiquitous computing. With the ability to place hundreds of embedded sensors within a given coverage area, each wirelessly streaming information, the possibility of self-organizing sensor networks is increasingly becoming a reality.
This article takes a look at the sensor layer of a basic ubiquitous computing stack.
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By Assad Ebrahim, on January 15th, 2010 (12,951 views) |
Topic: Maths--History, Maths--Philosophy
The development of mathematics has had many encouraging forces: societal, technological, cultural. These have served to accelerate mathematics and have been accelerated in turn, in many cases the pair becoming locked into a mutually beneficial resonance that has dramatically energized both.
In this article, I look at some of the significant catalysts, from the rise of the leisured class in ancient times to the impact of computing in modern times.
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By Assad Ebrahim, on January 3rd, 2010 (152,043 views) |
Topic: Maths--History, Maths--Philosophy
The development of mathematics is intimately interwoven with society and culture, influencing the course of history through its applications to science and technology.
But mathematics itself has changed much over its history. Even the mathematics of the early 1800s can now seem quite strange, so great have been the changes in just the past 150 years as it has been reworked in the modern abstract approach. Though advanced mathematics may now appear arcane from the outside looking in, the present state of mathematics is the result of a natural evolution of the subject. And there is much excitement promised ahead with the rise of new mathematics and application areas in subatomic and quantum physics, in the the field of statistical learning (also called artificial intelligence or machine learning), and in numerical computing and simulation.
What follows is the story of mathematics, in a nutshell.
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