When designing a system, what should you optimize? If it is a user-interface or process, you should be minimizing clicks, or process steps. But for hardware-software systems, the answer is not obvious, and a common mistake is to fail to consider the end-to-end problem. This article explores what is involved in optimizing end-to-end in hardware-software systems. The goal here is to minimize the overall complexity of the system, i.e. of the triple hardware-software-user combination. The following remarks set the stage for our discussion:
“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”.2
“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]
“We are reaching the stage of development [in computer science] 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].
“The arc of change is long, but it bends towards simplicity”, paraphrasing Martin Luther King.3
Between complexity and simplicity, progress, and new layers of abstraction.
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.
I made these by taking still photographs of the action with my dad’s iPhone and then recording an audio soundtrack (for the second animation). My dad then assembled the stills into an animation and layered on the soundtrack (see below for how). I’m working to add music to it — check again soon!
The Basket (1m 18s long)(silent film) The Pets find a cozy basket. But amidst all the hustle and bustle, will anyone be able to take a nap?
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, computing, and applied mathematics are gateway subjects to modern technology.
For young learners, we believe that electronics provides an ideal entry point. It is practical, with manipulables. It is easy to see cause and effect. With the right equipment and approach, exploring electronics can begin for children as early as 3 years old.
There are many tangible benefits for young learners getting started in electronics:
fine motor skill development,
an intuition for how technological things work at a component level,
the integration of technology into the palette for imagination and creativity,
improved self-confidence,
strengthening a growth mindset,
building resilience,
raising the threshold of frustration,
better dexterity,
stronger focus.
A three year old wiring his first circuit and the joy at seeing the LED, which he selected, light up!
*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.
Thoughts on Teaching Mathematics in an Exploratory, Dialectical, Topical format.
(2nd ed. July 13th 2016; 1st ed. Jan 31, 2010)
Mathematics is a richly spun tapestry, threaded with interconnections from a multiplicity of endeavors, perspectives, and disciplines, both theoretical and applied. Yet contrary to this “non-linear” reality, the typical pattern of school and even university mathematics is both linear and restricted.
This article takes a look at what lies behind the way things are, and what could bring positive change.
An examination of mathematical methods and the search for mathematical meaning.
This article curates a reading list (most sources available freely online1) organized into a set of encounters that lie outside the standard mathematics curriculum. They are intended to enrich the reader’s understanding of mathematics and its place in scientific inquiry, increase her/his connection to the historical and philosophical questions behind the mathematics of the past and present, and gain greater satisfaction from further mathematical study. The reader should come away with a better understanding of the culture of mathematics: what mathematics is, mathematical method and meaning, and the relation of mathematics to the empirical world and to science.
We look at seven topics. These may be covered in any order, to suit your particular interests.
There is no core body of technical material to master in this course; the important thing is a feel for how, why, and in what context the core ideas of mathematics evolved, getting to the essence of their motivation, and understanding the fruits of these efforts. The course such as the below should appeal to all those who have an itch to scratch beneath the surface of mathematics, who find themselves asking “but why?”. It could be useful in all three tiers of education: secondary, post-secondary (undergraduate), and graduate, appropriately restructured.
Secondary school elective: to encourage bright students in mathematics, science and technology to enter the university with a broader perspective on the mathematics they will be rapidly learning there.
University elective course: offered as a writing-intensive seminar, intended primarily for students in the sciences and engineer: mathematics, physics, engineering.
Graduate level course: offered in the first year of graduate school in mathematics or applied mathematics as a supplementary seminar.
To ensure that the materials are always available for download, I am serving them from copies held on this site. If you are the author of any of these articles and would prefer to have the primary download originate from your site, please send me an email, and I will make the change. ↩
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