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 May 1st, 2023 (7,540 views) |
Topic: Education, Technology
2nd ed. Jan 2023 (before the ChatGPT/LLM AI release), 1st. ed. 2005
Overview
In the intersection between Mathematics, Modern Statistics, Machine Learning & Data Science, Electrical Engineering & Sensors, Computer Science, and Software Engineering, is a rapidly accelerating area of activity concerned with the real-time acquisition of rich data, its near real-time analysis and interpretation, and subsequent use in high quality decision-making with automatic adjustment and intelligent response. These advances are enabled by the development of small, energy efficient microprocessors coupled with low-cost off-the-shelf sensors, many with integrated wireless communication and geo-positional awareness, communicating with massive high-speed databases. For teams able to bridge the disciplines involved, the potential for economically productive application is limitless.
 Traditional science and technology disciplines are in the outermost ring, often isolated from each other. The result of their integration is driving the areas out of which a large portion of technology in the coming decades is likely to appear. Continue reading this article…
By Assad Ebrahim, on March 14th, 2023 (10,646 views) |
Topic: Maths--General Interest, Technology
Updated May 2024 (added non-banking financial sector), Updated March 2023 (added latest bank collapses in US/EU). First published July 5, 2010 (two years after the financial collapse triggered Great Recession).
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.)
Continue reading this article…
TinyPhoto is a small rotating photobook embedded graphics project that uses the low-power ATtiny85 microcontroller (3mA) and a 128×64 pixel OLED display (c.5-10mA typical, 15mA max). This combination can deliver at least 20 hrs of continuous play on a 3V coin cell battery (225mAh capacity). TinyPhoto can be readily built from a handful of through-hole electronic components (12 parts, £5) organized to fit onto a 3cm x 7cm single-sided prototype PCB. The embedded software is c.150 lines of C code and uses less than 1,300 bytes of on-chip memory. TinyPhoto rotates through five user-selectable images using a total of 4,900 bytes (yes, bytes!) stored in the on-chip flash RAM. The setup produces crisp photos on the OLED display with a real-time display rate that is instantaneous to the human eye with the Tiny85 boosted to run at 8MHz. A custom device driver (200 bytes) sets up the OLED screen and enables pixel-by-pixel display. Custom Forth code converts a 0-1 color depth image into a byte-stream that can be written to the onboard flash for rapid display. It is a reminder of what can be accomplished with low-fat computing…
The magic, of course, is in the software. This article describes how this was done, and the software that enables it. Checkout the TinyPhoto review on Hackaday!
 Tiny Photo – 3cm x 7cm photo viewer powered by ATTiny85 8-bit microcontroller sending pixel level image data to OLED display (128×64 pixels), powered by 3V coin cell battery. Cycles through 5 images stored in 5kB of on-chip Flash RAM. (Note, this is 1 million times less memory than on a Windows PC with 8GB RAM). The magic is in the software. Continue reading this article…
This article explains how to use the Arduino toolchain to program microcontrollers from the Arduino IDE using their bootloaders, and also burning bootloaders directly onto bare microcontroller chips. It covers developing interactively with Forth (rapid prototyping), and moving your creations from a development board (Nano, Uno) to a standalone, low-cost, low-power, small footprint chip such as the ATMega328P or ATTiny85 or ATTiny84. Each of these microcontrollers is powerful, inexpensive, and allows using 3V batteries directly without the need to boost voltage to 5V. Additionally, we describe how to build an inexpensive (under £5), standalone 3-chip Atmel AVR universal bootloading programmer that you can use to program all of the chips above.
Continue reading this article…
Voice controlled hardware requires four capabilities: (1) vocal response to trigger events (sensors/calculations-to-brain), (2) speech generation (brain-to-mouth), (3) speech recognition (ear-to-brain), and (4) speech understanding (brain-to-database, aka learning). These capabilities can increasingly be implemented using off-the-shelf modules, due to progress in advanced low-cost silicon capable of digital signal processing (DSP) and statistical learning/machine learning/AI.
In this article we look at the value chain involved in building voice control into hardware. We cover highlights in the history of artificial speech. And we show how to convert an ordinary sensor into a talking sensor for less than £5. We demonstrate this by building a Talking Passive Infra-Red (PIR) motion sensor deployed as part of an April Fool’s Day prank (jump to the design video and demonstration video).
The same design pattern can be used to create any talking sensor, with applications abounding around home, school, work, shop, factory, industrial site, mass-transit, public space, or interactive art/engineering/museum display.
 Bringing Junk Model Robots to life with Talking Motion Sensors (April Fools Prank, 2021) Continue reading this article…
By Assad Ebrahim, on April 12th, 2021 (179 views) | Enter your password to view comments.
Topic: Building Technology, Education, Electronics, Personal File, Technology
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