🚀 go-pugleaf

In article <1992Nov6.023524.6664@galois.mit.edu> tycchow@phragmen.mit.edu (Timothy Y. Chow) writes:
>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.

Well, I think it should go like this.  First various examples.  Then the
definition as a distillation of the common features of the examples.
Then perhaps some theorems saying "every group is isomorphic to a
permutation group" or "every finite group is isomorphic to a matrix
group" to show that, in a certain sense, the examples they know are all
there really is.  In other words, I don't think one should give
watered-down definitions, but one *certainly* shouldn't throw
definitions at someone who is not prepared to enjoy them.  (If you know
a bunch of examples it's positively fun to see the definition that
distills their essence.)