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
Also linked, but out of scope for this paper, is the impact of institutional values in enhancing/suppressing innovation. Laws limiting exploitation by the powerful were put in place by Sargon of Akkad, Gudea and Entemena of Lagash, and Hammurapi of Babylon. The military policies of King Shulgi of Ur III stimulated massive state investment, drove institutional innovation but suppressed individual innovation. In the freedoms of the Old Babylonian period we see indiviual innovation thrive. See (Hoyrup/1991) and (Hoyrup/2009: 31-32) for a survey and further reading. ↩
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)
If you haven’t done so already, you may want to start by reading the Preface to the Computing Series: Software as a Force Multiplier, Sections 1-3.
1. Total Commander: a programmable, extensible, feature-rich two-panel orthodox file manager
Total Commander (TC) is more than just a two-panel orthodox file manager for Windows. It is a swiss army knife of computing utilities and is the first piece of software that I install on any Windows computer on which I’m working.1 Total Commander, used well, is a force multiplier.
Tenets of the TC approach:
Two-panes is the natural way to think about most file and directory operations (source panel, destination panel).
Keys beats mouse for speed and accuracy. Make the keyboard use easy. List of keyboard shortcuts and description of features
A computing platform should be fully extensibility. Adding your own tools should be easy.
Portability secures your investment: grab your totalcmd folder, copy to a new computer, and everything should work seamlessly.
I’ve been using TC continuously since 2001. Over the years, I’ve put together a Total Commander Expansion Pack (lite and full) for the TC platform that conforms to the tenets and further extends TC platform’s capabilities with tools I have found valuable. Both are pre-configured downloads to allow unpack and start using with minimum fuss (I use them when switching computers). Feel free to download and give them a try. Feedback or questions welcomed in the comments.
Total Commander Expansion Pack Winter 2024 – Toolbar View with Integrated Applications, Download from link.
“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”1
At First Glance
Forth2 is a remarkable computer language born in 1968 that exemplifies the low-fat computing ethos of its inventor Charles ‘Chuck’ Moore (b.1938), for whom simplicity and minimalism were central themes of his life’s work.3 Forth was the first in a series of revolutionary ideas in computing practice4, followed by a simple VLSI design environment written entirely in Forth (OKAD5) and which Chuck used to design Forth-in-hardware chips6 culminating in the GA-144, a 144-core low-power (0.4uW standby power, 7 picojoules per instruction), high-performance (1.4 nanoseconds per instruction, or 700 MIPS per core) parallel computer on a single chip. Continue reading this article…
OKAD was built on top of OK, which was Chuck Moore’s One Kilobyte operating system. OKAD2 was built within ColorForth and benefited from 10 years of Chuck Moore’s experience designing chips. ↩
The history is interesting: 1) Jeff Fox’s history, 2) Chuck Moore’s history↩
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!
Abstract
This three part paper explores solving the sum of powers problem using discrete maths techniques (recurrence relations, matrix systems) to obtain a solution polynomials whose coefficients turn out to be exactly the Bernoulli numbers . Part 1 (this paper) solves the problem using recurrence relations in a way which a high school student could emulate for small . In Part 2, we develop a general recursive solution that works for arbitrary , from which we can build a table of values to assist in finding the coefficients of the solution polynomial, coefficients that are precisely the Bernoulli numbers discovered in 1713. In Part 3, we show how by transforming the problem into a linear system, we may obtain a direct (non-recursive) solution which directly calculates the Bernoulli number for any power . Source code is provided for all solutions.
Readers who are interested in this topic are referred also to lovely paper by Bearden (March 1996, American Mathematical Monthly), which tells the mathematical story and fills in the history (thanks to a reader for this great reference). Continue reading this article…
Abstract
We continue the 3-part paper exploring how one might solve for themselves the general case of the sum-of-integer-powers problem for arbitrary , the coefficients of whose solution are the famous Bernoulli numbers (1716). In this paper we show to how obtain a -th order recurrence relation that can be used to iteratively obtain the closed form polynomial for for any given . Source code is given for computing these polynomials using Maxima, an open-source (free) symbolic computation platform. Continue reading this article…
Abstract
This is the last in the 3-part series of articles on finding for oneself the solution to the sum of integer power problem, and in the process discovering the Bernoulli numbers. In Part 3 (this paper), we find a direct closed-form solution, i.e. one that does not require iteration, for the general case of the finite-summation-of-integer-powers problem . Having established in Part 2 that the closed-form solution is a polynomial, the summation is here rewritten as the sum of the independent monomials (), where the are unknown coefficients. Using the recurrence relation , we obtain a linear combination of the monomials, which reduces to an easily solvable -by- triangular linear system in the unknown coefficients of the closed-form polynomial solution. Maxima and Octave/Matlab codes for directly computing the closed-form solutions are included in the Appendices.
*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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using discrete maths techniques (recurrence relations, matrix systems) to obtain a solution polynomials whose coefficients turn out to be exactly the Bernoulli numbers 
.
. In 
for arbitrary 
, the coefficients of whose solution are the famous Bernoulli numbers (1716). In this paper we show to how obtain a 
for any given 
independent monomials 
(
), where the 
are unknown coefficients. Using the recurrence relation 
, we obtain a linear combination of the monomials, which reduces to an easily solvable 
-by-