- Organic space and time: regularity and predictability of natural phenomena
- Patterns need naming: start of farming, cluster numbers enough
- Called reckoning: facilitating predictability; the moon as a marker of time
- 4236 BCE: marking days into thirties: why 60 and 360 are important
- Numbering in thirties would have generated a need for a symbol system
- How reckoning becomes numbering
- Early reckoning was object specific; universal numbering was not understood: that is, descriptive of the objects
- Early languages did not have universal numbers
- Different names for the same numbering according to application
- Advanced to become descriptive of classifying objects (crowd of people; sheep in a flock)
- Later model groupings with size constancy against which other groups can be compared (two wings, three leafed clover, four legged animals)
- Later a need to divide large clusters into stages
- Quantities split and joined to other quantities, and across types
- Problem how to count (reckon) when there is no logical object connection: a dynamic for a homogeneous system
- Yet had to remain in touch with their object environment: needed: body and sensory experience (finger counting; the body; felt, understood and close)
- Larger numbers: trade
- Numbering limited to symbolic transfers of objects: here is the foot - became a foot in length
- Need for strict ordering before an abstracted numerical system
- A crucial movement from the object attachment to acting in order around the object
- Act became universal: homogeneous
- Convention needed
- Even then the object world was still a comfort!
- Abstraction was taking place in imposing order (we do it: how to efficiently get up from bed to breakfast and to go out: attachments and memory devices)
- Abstraction facilitates theory: potential for mathematics
- Symbols must be agreed
- Systematic handling of number
- Use of powers in the radix or base (example: twelve, known as the duodecimal scale)
- The moon still important!
- Inconsistency of symbolism [and base] least a problem with little counting
- Awkwardness: still increasingly high levels of computation
- Systems used the principle of repetition: higher value symbols to reduce length
- NOT the same as a system of position: Integers varied in one position but the same choice of them in different base positions
- Value from integer rising and the position
- Zero crucial
- Hindus' zero system works because it is visual: symbols do not need counting along
- Mathematicians divided: 1500s.
- Abacists of old and the new style algorists of the Hindu-Arabic system
- Superior system won: England adopted tens in 1600s
- Facilitated science and the ages of European civilisation
- Moderns seduced by number
- Behaviour rarely is purely rational (ancestors linked to objects)
- Extremely large/ extremely small: maths begins to warp
- Predictability becomes probability
- Virtual number iterations produce patterns (in the natural world?)