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Numbers, numerals, operators 3
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Recovery/review I: basis of cardinal & ordinal numbers
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in the beginning, humans (and some animals) have a raw number sense
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(can tell when number has changed—only works for small numbers)
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this gets extended to a notion of cardinality by establishing a correspondence (bijection; isomorphism)
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this leads to tallies (“unary notation”; from “cut”) and pebbles (“calculate” from stone)
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in this context we can recover at least the sum & product operations of SPF
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merging groups of pebbles; substitution of groups for individuals (perhaps of different “type”)
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early uses of numbers in this sense are often “typed” according to what they count
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this gets extended later to a more abstract notion of number independent of the objects
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eventually, we get an ordinal sense of number from counting
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(these might be “nonsense words”, but seem to be based on body parts (e.g., fingers or digits) used for correspondence)
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Peano formalizes natural numbers in the late 1800s (as formal logic & set theory are rising)
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an abstract “linguistic” or term-structured basis, plus axioms to characterize behavior
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in Haskell, these fall out of the algebraic definition & the notions of constructor & pattern-matching
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the SPF functions are easily defined by iteration over each other, in sequence, from successor
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iteration is roughly primitive recursion, the computational view of logical induction provided via Peano’s axioms)
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we can recover integers (with negatives) and rationals (with fractions) using a general construction
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any monoid (algebraic structure with associative operation and unit) can be extended to a group (with inverses)
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… but not responsible for this construction this semester!
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Numerals
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numbers: the abstract entities; numerals: the linguistics, written (string) forms
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an obvious (but poor) choice for numerals is to use tallies
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(even then, bundles soon arise: see diagonal mark to make "5")
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a sense of order amongst symbols of an alphabet can be extended to pairs and lists
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for pairs, we simply identify a dominant and subordinate role, then arrange a la pets or cards
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we can extend the lexicographic ordering technique to lists, which are really just iterated products
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when counting a common type of thing in larger numbers, natural to make & count “bundles”
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this is what will lead to the notion of a base (‘b’ for bundle, ‘b’ for base)
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the operation which “counts” lex-ordered products is: sumProd b j k = b * j + k
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(relative to base b, curried over j and k: j bundles via product, plus k extra “units”
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the “inverse” operation (relative to base b) is divMod (from the Haskell Prelude)
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but note that this only works when restricted so that k < b (as we would have in a symbolic/alphabetic/digital context)
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we can extend these operations to lists in an natural way using folds & unfolds
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very concise, revealing definitions
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traditional mathematical definition of numeral “meaning” (or evaluation) uses indices
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positional notation is just the coefficients of a polynomial in the base
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relative to the same restriction as above (symbolic/alphabetic/digital context)
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what about algorithms to compute number operations via their numerals?
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remind of standard addition and multiplication algorithms, as usually presented (with very stylized layout)
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historically, there was a time when such computations were considered college-level topics
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these algorithms are accessible to elementary students only because of intense pedagogical development
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for polynomials, these algorithms (very usefully) extend the “FOIL” mnemonic taught in schools
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Alternative numerals/notations
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tallies
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use (e.g.) List of Unit—in Haskell, [()]
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addition is just appending—in Haskell, (++)
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multiplication is just substitution—in Haskell, (concatMap . const)
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dyadic/bijective numerals a la logician Raymond Smullyan
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uses the same “formula” as conventional positional notation
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… but no digit 0: instead, use a digit b (where b is the base)
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no leading 0s, so a 1-1 correspondence with numbers!
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(but 0 is represented by the empty string)
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hereditary prime decomposition as a basis for numerals
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the choice of a base is particularly arbitrary
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a much more “universal” choice would be to use prime decomposition
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we can pull back to the exponents, just as we normally pull back to the “coefficients”
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… but then we should recursively express those exponents in the same notation!
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