Of course, we should be perfectly aware of the danger of reading
our concepts into the minds of our forefathers. One hundred years
ago, when Cantor's transfinite numbers were the rave of the day, a
mathematician writing about Zeno's paradoxes proposed their solution
with help of transfinite ordinals; more recently other faddish
attempts were made to explain them with the notions of non-standard
analysis ! In the same vain, a title such as "Goedel-Escher-Bach"
makes me cringe.
I conclude with two stories destined to illustrate the difficulty
to determine the threshold of mathematics even in our contemporary
culture.
app. 1.
I grew up among businessmen and -women. In retailing, they had to
tell the customer the total sum of cash to pay for his several
purchases; only that was entered into the "cash register" which,
at best, was a mechanical machine. In wholesaling, they had to
check their invoices and compute discounts expressed in
percentages. If they had to determine the sum of 4 items each at
0.47 , they would not multiply, adding 1.60 plus 0.28 to 1.88 ,
but would multiply 4*0.50 = 2.00 and subtract 4*3 = 0.12 .
These people usually had left school at the age of 14 . I do not
know whether they had learned their ways of mercantile
computation at school; probably they adopted them through
imitation from their elders during their apprenticeship. But
certainly they understood what they were doing, without knowing
that they were applying a 'distributive law' in 4*(0.50 - 0.03) =
4*0.50 - 4*0.03 . Would it be justified to call their reasoning
mathematical ?
app. 2.
Thirty years ago, the price of gasoline in this country had risen
to about 0.70 uoc (units of currency) per (metric) litre. At the
gas station, the devices measuring the litres filled into a car
computed the price by multiplying them with the factor 0.yz where
the two digits yz were parameters set by the station's operator.
Twenty years ago, the price of gasoline rose above 1.00 uoc . So
for the measuring devices a problem analogous to our k2y arose,
and by and by they all were replaced by new ones which would
accept three parameters to express the new prices of x.vw uoc .
At gas stations not yet equipped with the new devices, the
operators temporarily set their two parameters yz such that
2*0.yz = 1.vw , and at the cash register they then charged the
double sum of that printed on the measuring device's control
slip.
One day in that summer of 1979 , I stopped by at a station
operated by a big supermarket chain. They still had the old
two-parameter devices, and there was only one attendant at the
cash register - a young man, probably still in his teens. He took
the control slip, which stated half the sum I had to pay, took a
pen and wrote that amount a second time below the printed one.
Then he added the two identical sums, digit by digit from the
rear, and safely arrived at the total.
W.F.