Wednesday, July 20, 2011

Similar but different

Eric Temple Bell wrote "Men of Mathematics" in 1937. It is a fun way to
stroll through some of the history of mathematics. You may think that
history is bad enough and math is worse so that the history of mathematics
would have to be extra-boring. I once thought so. But in my college days,
I was interested in the mental gymnastics mathematicians have gone through
to arrive at our present-day rich array. John Kemeny (math professor and
later president of Dartmouth, inventor of the important computer language
BASIC) realized that the ordinary mathematics curriculum of arithmetic,
algebra, trigonometry, geometry and calculus was too limited. So he began a
new field of math usually called "finite
Kemeny emphasized that more mathematics has been invented and applied
the invention of calculus 300 years ago than all the math up to that time
but school kids never hear about it.

Eric Temple Bell organizes the history of mathematical thinking along two
major strands and shows that mathematics can be helpfully viewed as
alternating views of reality: continuous or discontinuous. The current
digital age is deeply rooted in the discontinuous side but motion, whether a
cannon ball or a rocket ship, has been very helpfully viewed as continuous.

Humans often have difficulty with multiples: twins, triplets, several
sources. See the movie
consider the problems that would emerge if you cloned yourself four

Statistics can be viewed as the study of variation, as in the multiple
measures of men's heights or the weight of salmon, or of anything that shows
itself in multiple forms. The old approach, and a good one, is to
substitute some sort of average for multiple observations. So, we could say
that men tend to be 5'10" or 6'. The average is always a model or a myth,
clearly not equal to many real-life examples.

Statisticians have many strategies for dealing with variation and
multiplicity but once in a while, none are needed. When the weight of the
lightest man is greater than the weight of the heaviest salmon, you can stop
right there, even if you don't really know what a man or a salmon is. With
a situation of no overlap between the two groups, you are dealing with a
statistically significant difference without even using statistics. True
that the salmon differ and so do the men, but you can use the rule that the
light examples are salmon and the heavy ones are men.
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