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# history-page2 - • Prehistoric ◦ Tallies ◦ OF-topic by...

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Numbers and Counting General Counting is arguably the most fundamental/primitive mathematical action. Modern science of combinatorics. Counting arrangements. E.g., how many ways can 67 be written as the unordered sum of positive integers? More generally, assessing the quantity of objects in a collection. In competition for most basic with assessing the magnitude of a physical quantity. · Difference is discreteness versus continuity. · Parallel to time versus space. · Which do we see first in children? Story of comparing hairs on two animals · 1-to-1 correspondence · cardinality 1-to-1 correspondence with · physical tokens (pebbles, finger-positions) · sounds · written symbols (numerals) Natural Numbers = N = { 1 , 2 , 3 , 4 , 5 , 6 , . . . } (abstract quantity, not the symbol)
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Unformatted text preview: • Prehistoric ◦ Tallies ◦ OF-topic by de±nition ◦ Claims are: ⋆ Speculative ⋆ Tentative ⋆ Unconvincing ◦ Calinger: “far more fundamental than language” ⋆ Not terribly advanced animals have ability to count into the high single digits. · Anecdote about crows being able to count to ±ve but not six. · The canonical (but not only) work is The Number Sense by Stanislas Dehaene. • ²or the historic era, we now address the question, what numeral systems did various ancient civilizations develop and when? In a bow to history over heritage, we look at: ◦ Mesopotamia ◦ Egypt ◦ Indus Valley ◦ Yellow River ◦ Greek ◦ Mesoamerican ◦ Roman ◦ Independence? Still a matter of dispute....
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