If you haven’t done so already, you may want to start by reading the Preface to Knowledge Engineering & Emerging Technologies.
January 31st, 2024 (4th ed)
The aim of this article is to encourage you to take an end-to-end perspective in your designs, seeking to minimize the overall complexity of your system, of the hardware-software-user combination. To achieve this, it is helpful to understand how computing, and within that, how the notions of the sacred and the profane have evolved over the past 60 or so years.
The following remarks set out a ‘true north’ perspective for this conversation:
- “We are reaching the stage of development where each new generation of participants is unaware both of their overall technological ancestry and the history of the development of their speciality, and have no past to build upon.” – J.A.N. Lee, [Lee, 1996, p.54].
- “Any [one] can make things bigger, more complex. It takes a touch of genius, and a lot of courage, to move in the opposite direction.” – Ernst F. Schumacher, 1973, from “Small is Beautiful: A Study of Economics As If People Mattered”.
- “The goal [is] simple: to minimize the complexity of the hardware-software combination. [Apart from] some lip service perhaps, no-one is trying to minimize the complexity of anything and that is of great concern to me.” – Chuck Moore, [Moore, 1999] (For a succinct introduction to Chuck Moore’s minimalism, see Less is Moore by Sam Gentle, [Gentle, 2015]
- “The arc of change is long, but it bends towards simplicity”, paraphrasing Martin Luther King.
The discussion requires a familiarity with lower-level computing, i.e. computing that is close to the underlying hardware. If you already have some familiarity with this, you can jump straight in to section 2. For all backgrounds, the discussions in the Interlude (section 4) make for especially enlightening reading. Whether you find yourself in violent agreement or disagreement, your perspective is welcomed in the comments!
 Between complexity and simplicity, progress, and new layers of abstraction. Continue reading this article…
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 (9,033 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 (2,806 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)
Continue reading this article…
Abstract This brief note explores the use of fuzzy classifiers, with membership functions chosen using a statistical heuristic (quantile statistics), to monitor time-series metrics. The time series can arise from environmental measurements, industrial process control data, or sensor system outputs. We demonstrate implementation using the R language on an example dataset (ozone levels in New York City). Click here to skip straight to the coded solution), or read on for the discussion.
 Fuzzy classification into 5 classes using p10 and p90 levels to achieve an 80-20 rule in the outermost classes and graded class membership in the inner three classes. Comparison with crisp classifier using the same 80-20 rule is shown in the bottom panel of the figure. Continue reading this article…
Short Articles on Data Science
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This article looks at Propositional Logic, also called Statement Calculus, from a combinatorial and algebraic point of view (Sections 3-6), its implementation in software (Section 7), and its application to digital electronics (Section 10). Historical sections cover the shift in viewpoint from classical logic based on Aristotle’s syllogism to modern symbolic logic (Section 2) and the axiomatization of logic (Section 9). (See logic sourcebook for the original source papers (1830-1881) that drove this shift.)
In Section 7, we implement the grammar of the statement calculus in the Symbolic Logic Simulator (SLS), a program written in 28 lines of Forth code, that allows computer-aided verification of any theorem in Propositional Logic (see Appendix 1 for source code). The program makes it straight-forward to explore non-obvious logical identities, and verify any propositional logic theorem or conjecture, in particular see Appendix 2 for key identities in the statement calculus (duality, algebraic, and canonical identities).
The concept of linguistic adequacy is developed in Section 8 and the NAND Adequacy Theorem is proved showing that NAND can generate all logical operations. A corollary is that any digital logic circuit can be built up entirely using NAND gates, illustrated using the free Digital Works software.
Continue reading this article…
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