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…
4th ed. Jan 2024; 3rd ed. May 2023; 2nd ed. Dec 2009; 1st ed. Sep 2004
“It is not philosophy but active experience in mathematics itself that alone can answer the question: `What is Mathematics?'” – Richard Courant & Herbert Robbins, 1941, What is Mathematics?, Oxford University Press)
“An adequate presentation of any science cannot consist of detailed information alone, however extensive. It must also provide a proper view of the essential nature of the science as a whole.” – Aleksandrov, 1956, Mathematics: Its Content, Methods, and Meaning
‘What is mathematics?’ Much ink has been spilled over this question, as can be seen from the selection of ten respected responses provided in the footnote, with seven book-length answers, and three written in the current millenium. One might well ask, is there anything new that can be said, that should be said? We’ll start by clarifying what a good answer should look like, and then explore the answer proposed.
The rest of the paper follows the structure below:
1. Criteria for a Good Definition of Mathematics
2. Definition 1: covering mathematics up to the end of the 18th century (1790s)
3. Two Perspectives
Mathematics as Dialectic (Lakatos)
Mathematics shaped by its Anthropology (Hoyrup)
4. Definition 2: covering all mathematics, including contemporary mathematics
5. The emergence of contemporary mathematical practice from 1800s onward
6. Three Facets of Mathematics
1. Mathematics as an Empirical Science
2. Mathematics as a Modeling Art
3. Mathematics as an Axiomatic Arrangement of Knowledge
7. Mathematics "from the inside": Mathematicians writing about Mathematics
8. Continue Reading
9. References
Continue reading this article…
2nd ed. June 2023; 1st ed. April 2010
The term “mathematical maturity” is sometimes used as short-hand to refer to a blend of elements that distinguish students likely to be successful in mathematics. It is a mixture of mathematical interest, curiousity, creativity, persistence, adventurousness, intuition, confidence, and useful knowledge.[1],[2],[3]
With advances in machine learning, computer science, robotics, nano-materials, and many other quantitative, fascinating subjects, students today have increasingly more choice in technical studies besides mathematics. To attract and retain mathematics students, it is important that mathematics instruction be experienced as both intellectually and culturally rewarding in addition to being technically empowering. Losing students from mathematics who are otherwise capable, engaged and hard-working is tragic when it could have been avoided.
In this article, building on observations gained over the years teaching and coaching students in mathematics, we consider how enriched mathematics instruction (inquiry-based/discovery learning, historiography, great ideas/survey approaches, and philosophical/humanist) can help (1) develop mathematical maturity in students from at-risk backgrounds and prevent their untimely departure from quantitative studies, (2) strengthen the understanding of those that are already mathematically inclined, (3) expand mathematical and scientific literacy in the wider population.
Continue reading this article…
By Assad Ebrahim, on March 14th, 2023 (9,039 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…
By Assad Ebrahim, on December 25th, 2020 (2,808 views) |
Topic: Maths--General Interest
This is a collection of short articles and reflections on topics of current interest. For older short posts, see here: #1-199 (Feb 2014-Oct 2019)
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In this article we look at the ideas of George Peacock whose 700-page opus A Treatise on Algebra (1830) transformed classical algebra into its modern form as an abstract symbolic science, free from the physical interpretation of quantity that had previously restricted it.
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By Assad Ebrahim, on March 18th, 2020 (4,098 views) |
Topic: Maths--General Interest
Revised Nov 2022, Jan 2023
In this article we look at the evolution of logic from its earliest form in the demonstration of truths to the rapid development of mathematical logic in the 1800s at the start of the “golden century” of logic (1850-1950). We also look at the rise and surprising dashing of hopes for the formalist program.
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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For under £10, you can put together a microcontroller development platform, ready to program directly from your PC over USB using free Arduino software. Once programmed, your microcontroller will run autonomously, untethered from your PC, powered by as small a battery power supply as a single 1.5V AAA or 3V CR2032 coin cell. You can have it interact with its environment using dozens of low-cost sensors and motors. Everything you need to explore the exciting world of embedded systems is available to you, typically for less than a day pass on the London underground.
 A homebrew Arduino Nano microcontroller development kit for under £12 (including optional OLED display) Continue reading this article…
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