Origins of number

MDPT

Culture and Nature

• How are numbers used and conceptualised by people?– What differences and similarities are there

between peoples from various communities?

• In what ways do numbers use people?– Is it the case that humans• cannot not count? • cannot not notice ‘more’ versus ‘less’?

Main theme

• As humans we are predisposed to use number for/as properties.

• In human societies, this predisposition has flourished in many different ways.

• This short presentation will refer to– historical snapshots,– biological results,– cultural positions.

Historical snapshots

• An ancient artefact from Africa• Representations of number from– Middle East– South America

• A method of calculating from China

Mathematical artefacts: engraved bones

There are now several archaeological findings of bones with tally-mark engravings.

A famous example is the Ishango bone, discovered in 1960 on the border of modern-day Congo & Uganda, it is ~20,000 years old.

What does this artefact tell us about the mathematics of that community?

There are different interpretations of the marks on the bone.

Historians of mathematics (who interpret the marks) use them as evidence to make claims about the cultural practices of the people that produced the marks.

These include the practices of :• doubling/halving;• adding and subtracting 1 from

10 or from 20;• lunar/periodic tracking.

Babylonian civilisation from ~2000-600BC

Babylonian number system

• a positional system with a base of 60 rather than the system with base 10 in widespread use today.

• Babylonians constructed tables to aid calculation: – Two tablets found at Senkerah on the Euphrates in 1854

date from 2000 BC give squares of the numbers up to 59 and cubes of the numbers up to 32.

– The table gives 82 = 1,4 which stands for

1*60 + 4 = 64 – and 592 = 58, 1 (= 58*60 +1 = 3481).

Why base 60?• The Babylonians divided the day into 24 hours, each

hour into 60 minutes, each minute into 60 seconds. • This has survived for 4000 years!

– Notations for sexagesimal numbers, e.g., 5 hours, 25 minutes, 30 seconds include • 5h 25' 30"

• the ‘sexagesimal fraction’ 5 25/60 30/3600 • 5; 25, 30.

– the number 5; 25, 30 - in sexagesimal form - can be expressed as a base 10 fraction: 5 4/10 2/100 5/1000

– i.e. 5.425 in decimal notation.

Mayan civilisation 2000BC –900AD

Mayan positional number system

Chinese calculating rods

• These are calculating devices that are pre-cursors of the modern Chinese abacus.– Evidence of use dates from the first century BCE.

• The procedure for calculating and the procedure for representing are very close.– This is unlike the current world-wide standard of

representing number using ‘1’, ‘2’, ‘3’ etc. and calculating with number using algorithms like ‘column arithmetic’ for addition.

Counting rods in vertical and horizontal layouts

Place value represented by layout style

Place value is read left to right (as in ‘HTU’)• ‘Units’ (a.k.a. ‘Ones’) are represented VERTICALLY• ‘Tens’ are represented HORIZONTALLY• ‘Hundreds’ are represented VERTICALLY• ‘Thousands’ are represented HORIZONTALLY• and so on- gaps left for zero in that order of magnitudeFor example:

Use matchsticks for rods

There is difference in opinion as to whether they worked from the highest order of magnitude down to the units or from the units up as we do

Adding practice1. 567 and 3122. 567 and 3523. 567 and 3584. 567 and 3385. Your choice.

Neuroscience and mathematics

• This is a developing area of research with new results being published regularly

• Brain imaging, gives information about location of brain activity when a person is performing certain tasks– techniques include fMRI (functional Magnetic

Resonance Imaging), PET scans (Positron Emission Tomography).

• Eye movement tracking is used to attribute attention.

We can’t help seeing quantities!

Although material objects are not directly associated with a unique number– e.g., a single apple is not just ‘one’ but can be

associated with many molecules, several pips and a range of colours,

babies (just a few days old) can distinguish between visual presentations of one, two or three dots– perceptual grasp of number is pre-linguistic;– humans and some animals have this number

perception ability;– ‘subitizing’ is the term used for seeing a small number

at a glance.

Exact numbers and estimates

• Examples of research into how people recognise, represent and process number:– Recognising small numbers activates a different part

of the brain from estimating two quantities.• This ‘functional independence’, which is found in other

aspect of mathematics, suggests that there is no such thing as being ‘globally bad at mathematics’ (Ann Dowker)

– Language skills areas of the brain and mathematical skills areas overlap but are not the same• You can be good at mathematics without high overall

intelligence.

Mathematics common to all cultures

• Counting• Locating• Designing• Playing• Explaining• MeasuringFrom: Alan Bishop ‘Mathematical Enculturation’ 1988

A few reference links

• History– http://www-history.mcs.st-andrews.ac.uk/history/• Roman numerals

– http://www.novaroma.org/via_romana/numbers.html

• Neuroscience– http://faculty.washington.edu/chudler/image.html – http://www.educ.cam.ac.uk/centres/neuroscience/