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

DMS220 Machines Codes Cultures Fall 2017

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 http://www.unz.org/Pub/NewmanJames-1957v01-00402 DMS220 Machines Codes Cultures Fall 2017

DMS220 Machines Codes Cultures Fall 2017 Counting ability in chimps https://www.youtube.com/watch?v=VCy7M6ueSpE DMS220 Machines Codes Cultures Fall 2017

DMS220 Machines Codes Cultures Fall 2017 Crows as the ultimate problem solvers https://www.youtube.com/watch?v=AVaITA7eBZE DMS220 Machines Codes Cultures Fall 2017

DMS220 Machines Codes Cultures Fall 2017 https://www.youtube.com/watch?v=TRF27F2bn-A DMS220 Machines Codes Cultures Fall 2017

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

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

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

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

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

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

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

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

DMS220 Machines Codes Cultures Fall 2017 Mayan Number system https://www.youtube.com/watch?v=Ybvb7oy_WV0 DMS220 Machines Codes Cultures Fall 2017

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

DMS220 Machines Codes Cultures Fall 2017 The origin of zero https://www.youtube.com/watch?v=LQEuywkWa2U http://www.slate.com/articles/video/video/2016/06/an_animated_history_of_the_number_zero_video.htm l DMS220 Machines Codes Cultures Fall 2017

DMS220 Machines Codes Cultures Fall 2017 0123456789 Three types of numbering systems https://www.youtube.com/watch?v=L6gstFGC7u4 DMS220 Machines Codes Cultures Fall 2017

DMS220 Machines Codes Cultures Fall 2017 The Day We Learned to Think https://www.youtube.com/watch?v=mqG-I4VfJ-g DMS220 Machines Codes Cultures Fall 2017