🚀 go-pugleaf

In article <1992Nov5.213739.4045@galois.mit.edu> jbaez@riesz.mit.edu
(John C. Baez) writes:

>For example: "a group is a subset of the set of permutations of a set
>that is closed under taking inverses and products".  Yuck!
>These things should be theorems, not definitions.  

This is true, but only with hindsight, I think.  As Gerald Edgar has noted,
some think that we should proceed from the abstract to the specific, but
this approach often suffers from the problem of motivation.  Objects such as
groups and manifolds were not defined intrinsically at first, because they
arose in some specific context.  Only afterwards did it become clear what
properties to abstract and how this would help.  Frequently the original,
clumsier definition is closer to intuition and easier to motivate.

Groups are an excellent example.  Usually the axioms are just thrown at the
student without any particular motivation.  The student isn't told that the
motivation for the associativity axiom is that groups usually feel most at
home when acting on a set, and functional composition is "automatically"
associative.

This isn't to say, as I read in Spivak once, that ontogeny must recapitulate
phylogeny in pedagogy.  Still, for the purposes of motivation it may be
useful to look at the history of mathematical concepts.
-- 
Tim Chow     tycchow@math.mit.edu
Where a calculator on the ENIAC is equipped with 18,000 vacuum tubes and weighs
30 tons, computers in the future may have only 1,000 vacuum tubes and weigh
only 1 1/2 tons.                               ---Popular Mechanics, March 1949