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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 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.
At First GlanceForth2 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.
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:
*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).
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. Updated: 2014-11-23 Pencil & Notebook are hard to beat when diagrams, equations, and words are desired in roughly equal measure, which is common when working as an engineer, mathematician, or product designer. To be sure, there are good digital tools for subsets of these (example LaTeX), but not when all three are combined. But now there’s a new suite of digital freehand tools that are almost good enough to set aside the notebook and pencil permanently. The core of this toolkit is a vector-based graphics canvas (software) and an ultra-precise stylus (hardware).  Example of Freehand Digital Pencil, using Adobe Ideas. If you use Apple iOS, you may already be using Adobe Ideas, which does the job brilliantly and is free (see this excellent article by Simon Raper). [1] But Adobe has no plans to release this to either Windows RT or Android, so for the rest of us, that’s a show-stopper, until now, with MetaMoji… Duelling with pistols. If you were the one issuing the challenge, your dilemma was that custom dictated that your adversary be allowed to shoot first. Only then, if you were still able to shoot, would you be permitted to seek “satisfaction”. How much of an advantage does the first shooter really have? In this article, we build a simple probability model, and implement a numerical model in a few lines of R code.  Two gentleman face off in the snow. Convention dictates the challenged shoots first. |
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© 2009-2026 Assad Ebrahim To contact me on email, use: assadebrahim2000 at gmail dot com (do the obvious) |
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