regarding sand and multiplication:-
another crazy analogy i was toying with. i imagine Mr Sand-man
piling up sand (+) and taking away sand (-). i imagine his
counter-part Mr Hole-man living upside down in the ground, who
treats holes as "real stuff" and sand as absence of holes. When
Mr Sand-man digs a hole say 6 feet below ground (-6), he's piling
up something "real" for Mr Hole-man. If Mr Hole-man living
upside-down below ground uses his magic spade to dig away one
feet of hole "-(-3)", he's in fact filling up one feet of sand
(+3). If Mr Hole-man does this twice, then it is (-2 × -3), same
as filling up the hole (2 × 3).
it seems we got to tweak our ontologies to get this to work.
regarding discs:-
i was part of a Singapore team that did the AlgeTools software
that included AlgeDisc, an electronic version of "number discs".
You can do this just as well with coins (heads vs tails) or
coloured discs. There are 2 types of discs e.g. blue (positive)
and red (negative). a blue and a red disc forms a zero-pair, so
a blue and a red disc annihilate each other. you can also
put in pairs of blue and red discs at will with impunity.
so put in say 6 such zero-pairs. -2 × -3 means two times
_removing_ sets of 3 red discs. that leaves the 6 blue discs
i.e. the operation has the same effect as adding two sets of
blue discs (2 × 3).
regarding structure and agency, arbitrariness:-
i think now it's time for me to pop this question that has been
bugging me for some time. i am convinced that mathematics is
socially constructured, but i am not so convinced that mathematics
is _merely_ socially constructured. if we vary across cultures
and different human activities, we might find different ways
in which patterns and structure can be expressed and yet we might
find commonalities / analogies. the question i am asking is:
is maths just a ball game determined by some group of nerds who
happen to be in power and dominate the discourse, or is there some
invariant, something deeper in maths that can transcend and unite
language, culture, activity .... ?
Foo Keong,
NIE, Singapore
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