2nd ed. Jan 2024 (with standalone chapters), 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.
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.
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 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
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.
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)
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.
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)
Building a fully analog electronic piano using only resistors, capacitors, and transistors, is an insightful experiment in electronic sound generation from first principles. I designed and built a 13-key analog piano in early 2019 using discrete through-hole components on a breadboard powered off a 9V DC battery. The design creates 13 astable multivibrator oscillator circuits, each able to be tuned to a given note frequency in the C5 to C6 range. The outputs of the oscillators are collected (mixed) to create a polyphonic analog audio signal that is amplified and run through an 8-ohm speaker. The device fits into an 11x25cm footprint. Check out how it sounds! (To hear the explanation of how it works, start at the beginning.)
Electronics is a gateway subject to modern technology, along with computer programming and applied mathematics. Getting started in electronics is easier than one may imagine and not prohibitively expensive. With the right approach, exploring electronics can begin for children as early as 3 years old. I’ve been play-testing these ideas with my children, Adam (3 yrs & 4 months) and Jasmine (6 yrs & 10 months), and a couple of teenagers (13 and 14 yrs). Read on for the journey plan, and a photo gallery of what we’ve built so far.
Adam having wired his first circuit and seeing his selected blue LED lit!
*New!* (29 Aug 2020) – Turtle Logo v1.8 (portable) is available! Developer kit with source code included. Suitable from ages 3 years to adult. (970 lines of Forth code).
1. Inspiring the next generation of technology builders.
A challenge facing parents and teachers is how to help children develop ‘builder’ relationships with technology rather than being limited to the passive consumption of content created by others. The consensus on what’s important for older kids and adults is clear: coding. This enables children to participate in the creation of their own technological “micro-worlds” — environments rich in educational potential.[14]
This autumn, spurred by having our own young children (one aged 4 years, the other 16 months), we began an experiment, the result of which is a Turtle Logo program for Windows computers (freely downloadable) that is simple enough to be accessible for children from 3 years and older, while providing an extensible platform that can grow with the child.
The long-term goal is to enable children to express their creativity, artistry, and natural ‘builder’ impulses using coding, computer graphics, and robotics as readily as the previous generation could using paints, brushes, and building blocks.
Turtle Logo – Inspiring the next generation of technology builders.
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