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.
|
||||
The rise of Mathematical Logic: from Demonstration to Laws of Thought to Foundations for Mathematics
In this article we look at the evolution of logic from its earliest form in the demonstration of truths to the rapid development of mathematical logic in the 1800s at the start of the “golden century” of logic (1850-1950). We also look at the rise and surprising dashing of hopes for the formalist program. This is Part 3 in the Ancient Mathematics series. (To read earlier parts: Part 1: Prehistoric Origins of Mathematics, Part 2: The Mathematics of Uruk and Susa). This article explores what the people of Mesopotamia wrote about, counted and produced in the last part of the fourth millenium BCE. It does so by examining the frequency of signs in the proto-cuneiform tablets from the period c.3500-3000 BCE. For those wishing to build up experiential understanding of life in the Neolithic and early Bronze Age near east, this article provides suitable references as well as practical explorations of the economic and productive activities of the people: rope-making from grass, basket-weaving from reads, baking, weaving of cloth from linen, breaking ground, sowing, reaping, making flour, etc. The first part of the article look closely at the collection of artifacts in each period that are associated with the evolution of writing and mathematics in ancient Mesopotamia, examining noteworthy individual artifacts that showcase a key development. The study uses the CDLI database of cuneiform documents, and software I’ve written that parse the files in the CDLI database, extracting counts, parsing signs, generating frequency distributions of signs, creating a proto-cuneiform glossary, and assisting in the quantitative analysis of artifacts and semi-automated translation (see CDLI parser software library written in Ruby). This is Part 2 in the Ancient Mathematics series. (Part 1: The Prehistoric Origins of Mathematics, Part 3: Exploring Cuneiform Culture 8500-2500 BCE) 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 Continue reading this article…
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. “Everything” you need for ultra-fast desktop search1. Everything(tm) is an ultra-fast desktop search utility that can scan through hundreds of thousands of files in milliseconds using a pre-built and real-time updated index.“Everything” brings order to information growing at scale (documents, photographs, source code, spreadsheets, etc.), and tames the problem of proliferating folder trees.  Everything is a fast desktop search utility that can index 1 million files in less than 1 minute, and generate search queries in milliseconds. We’ve all been in the scenario of searching through electronic documents for a document you know you prepared three, maybe four weeks prior… maybe it was longer… and now you can’t remember where you saved it… or in what format: was it a quickly written text file, a word document, a few paragraphs within One Note, on a desktop post-it note, or did you email yourself from your phone?… After trying different Windows searches in various recently used folders and looking through Word, Excel, and PDF files, and trying to remember possible filenames to search for, at some point you prepare mentally for the moment when you will give up the search and attempt to redo the missing work, salvaging as much of it as you can remember. The general problem of wasted effort locating information we know we have, occurs more often than we’d like to admit. With “Everything“, it can be better. 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. Notepad++: a programmable, extensible, feature-rich text editorNotePad++ (NPP) is an open-source programmer’s text editor with outstanding built-in features that can be further enhanced with powerful plugins and extensively customized with your own configurations. NPPs features include syntax highlighting for a large collection of programming languages, code folding, recordable macros, cloned views, selectable shortcuts, tabbed documents, and a host of other capabilities. This article describes a few of the dozens of capabilities. It also shares a pre-configured Notepad++ package that I use (20.0MB compressed, 50.0MB uncompressed, download here), which contains the configurations and capabilities I use. The file is portable and self-contained: just unpack NPP to your drive (in a separate folder to your current running instance) and run notepad++.exe from there.1  Notepad++, by Don Ho, multi-view with syntax highlghting Continue reading this article…
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! 2nd ed. January 21, 2018; 1st ed. Feb 8th, 2010 Abstract 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). |
||||
|
© 2009-2024 Assad Ebrahim To contact me on email, use: assadebrahim2000 at gmail dot com (do the obvious) |
||||
using discrete maths techniques (recurrence relations, matrix systems) to obtain a solution polynomials whose coefficients turn out to be exactly the Bernoulli numbers 
.
. In