1. DMS220 Machines Codes Cultures Fall 2017
There was a time before number.
DMS220 Machines Codes Cultures Fall 2017

2. DMS220 Machines Codes Cultures Fall 2017

3. DMS220 Machines Codes Cultures Fall 2017
But the first man who noticed the analogy between a group of seven fishes and a group of seven days made a notable advance in the history of thought…
Alfred Whitehead. Mathematics as an Element in the History of Thought. 1957
DMS220 Machines Codes Cultures Fall 2017

4. DMS220 Machines Codes Cultures Fall 2017
Counting ability in chimps
DMS220 Machines Codes Cultures Fall 2017

5. DMS220 Machines Codes Cultures Fall 2017
Crows as the ultimate problem solvers
DMS220 Machines Codes Cultures Fall 2017

6. DMS220 Machines Codes Cultures Fall 2017
DMS220 Machines Codes Cultures Fall 2017

7. DMS220 Machines Codes Cultures Fall 2017
limits of perception – 4 objects
DMS220 Machines Codes Cultures Fall 2017

8. DMS220 Machines Codes Cultures Fall 2017
one to one correspondence:
… every element in set A maps to distinctly one element ins set B and visa versa. Ifrah. p 10
DMS220 Machines Codes Cultures Fall 2017

9. DMS220 Machines Codes Cultures Fall 2017
one, two, and many.
DMS220 Machines Codes Cultures Fall 2017

10. DMS220 Machines Codes Cultures Fall 2017
Ifrah, The Universal History of Numbers, p10
Applications of one to one correspondence
DMS220 Machines Codes Cultures Fall 2017

11. DMS220 Machines Codes Cultures Fall 2017
Counting requires:
- The ability to assign a ‘rank-order’ to each element in a procession
- The ability to insert into each unit of the procession the memory of all those that have gone past before.
- The ability to convert such a sequence into a stationary vision
DMS220 Machines Codes Cultures Fall 2017

12. DMS220 Machines Codes Cultures Fall 2017
Ifrah, The Universal History of Numbers, p16 / 22
Anthropomorphic origin of counting systems
DMS220 Machines Codes Cultures Fall 2017

13. DMS220 Machines Codes Cultures Fall 2017
Ifrah, The Universal History of Numbers, p7
DMS220 Machines Codes Cultures Fall 2017

14. DMS220 Machines Codes Cultures Fall 2017
First stage
Only the lowest numbers are within human grasp. Numerical ability remains restricted to what can be evaluated in a single glance. "Number" is indissociable from the concrete reality of the objects evaluated. In order to cope with quantities above four, a number of concrete procedures are developed. These include finger-counting and other body-counting systems, all based on one-for-one correspondence, and leading to the development of simple, widely-available ready-made mappings.
Second stage
By force of repetition and habit, the list of the names of the body-parts in their numerative order imperceptibly acquire abstract connotations, especially the first five. They slowly lose their power to suggest the actual parts of the body, becoming progressively more attached to the corresponding
number, and may now be applied to any set of objects.
Third stage
A fundamental tool emerges: numerical nomenclature, or the names of the numbers.
DMS220 Machines Codes Cultures Fall 2017

15. DMS220 Machines Codes Cultures Fall 2017
Mayan Number system
DMS220 Machines Codes Cultures Fall 2017

16. DMS220 Machines Codes Cultures Fall 2017
Cardinal numbers:
Natural numbers, 1 2 3
Cardinal numbers answer the question: how many?
Ordinal numbers:
Place numbers, first, second, third
Ordinal numbers answer the question: in which sequence?
Nominal numbers:
A number as a name: “the famous 10” (on the back of a athlete)
DMS220 Machines Codes Cultures Fall 2017

17. DMS220 Machines Codes Cultures Fall 2017
The origin of zero l
DMS220 Machines Codes Cultures Fall 2017

18. DMS220 Machines Codes Cultures Fall 2017
Three types of numbering systems
DMS220 Machines Codes Cultures Fall 2017

19. DMS220 Machines Codes Cultures Fall 2017
The Day We Learned to Think
DMS220 Machines Codes Cultures Fall 2017