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Revised & Expanded May 2023. First published November 1998. This article provides a selection of quotes, written mostly by mathematicians, that convey especially clearly essential aspects of mathematics and its culture. Comments are collected in the endnotes. Contents
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  The Anthropology and Archaeology of Conceptual Thought leading to the Birth of Mathematics 4th ed. Jan 2024; 3rd ed. May 2023; 2nd ed. Dec 2009; 1st ed. Sep 2004
‘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 footnote1, 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.
Mathematical Finance is an area of applied mathematics that has developed rapidly during the late 80s and 90s after the deregulation of U.S. financial markets, and accelerated further in the 2000s concurrently with the rise of data science/’big data’ and computational platforms able to run complex models in close to real-time. For its financial models for risk and pricing, Mathematical Finance draws upon the partial differential equations of mathematical physics, stochastic calculus, probabilistic modeling, mathematical optimization, statistics, and numerical methods. The implementation of these often complex numerical mathematical models requires efficient algorithms and exploiting the state-of-the-art in software engineering (real-time and embedded development, low latency network programming) and 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. But the relevance goes beyond mathematics. There is a kernel of core financial ideas that are at the heart of the global free market capitalist system that is in place across most of the world today. These ideas affect not only economics but also politics and society. Ideally, every citizen in a democracy should 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. This article presents a simplified account of the rise of the modern financial marketplace including some history, and contemporary financial context. Update (2012): A highly recommended graphic novel Economix, by Michael Goodwin has just been published that presents a panoramic yet highly accessible narrative.) 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) 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. The rise of Mathematical Logic: from Demonstration to Laws of Thought to Foundations for Mathematics
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. 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.1  Ur III mathematical model projecting annual dairy/cheese yields from a herd of 4 cows and a bull with assumptions on calving rates Continue reading this article…
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© 2009-2024 Assad Ebrahim To contact me on email, use: assadebrahim2000 at gmail dot com (do the obvious) |
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