Measuring size, dimension, etc.

How do you measure the -size- of some weird thing?
[I'm stating this vaguely.]

You -compare- it to some existing object.
This shows up -all over the place-
- cardinality of sets
- measure/content (and integrals)
- dimension of vector spaces
- dimension in analysis
- dimension in algebraic geometry
...and dates back to Euduxous (sp?) and Archimedes!
(cool, eh?)

Here's how this works:
- cardinality of sets
  Given sets $A,B$, we can ask if $A \inj B$,
  or if $A \surj B$; these both induce orderings
  We use the 
  
  Indeed, you can compare a set $X$ to the ordinals,
  asking:
  ``which ordinals map -into- X?
                       -onto- X?
               does X map -into-?
                          -onto-?''
  maps can go -either- way, and you can ask if you -fit- or -cover-
  
  You can also simply ask if you're -isomorphic- to something.
  
- measure/content (and integrals)
  Here you compute the areas/volumes of shapes either by -filling-
  them with known shapes (rectangles, triangles, squares)
  or -covering- them with known shapes.
  [This is the method of exhaustion,
   and leads naturally to notions of sup,
   though that only was realized as a separate notion by Weierstrass,
   regarding whether optima in calculus of variations are -realized-!]
  
- dimension of vector spaces
  The definition of dimension of a vector space in terms of a basis is
  -exactly-:
  is there a -monic- $K^n \to V$?
    (linearly independent set)
  is there an -epic- $K^n \to V$?
    (spanning set)

  You can -dually- ask it in terms of -coordinates-
    (maps $V \to K^n$)

  For modules, you ask for the minimal number of generators,
  as finitely generated modules have ``enough projectives/frees''
  (so everything can be covered by something)
  (but you don't generally have enough -injectives-!
   think of $\bQ$ and $\bQ/\bZ$ for fun)
  Holy -shit-!
  We went from ``what is dimension'' to homological algebra in a few
  steps!!!!

- dimension in analysis
  This is trickier:
  the existence of space-filling curves means that you -cannot-
  use the naive surjection definition
    (but differentiability fixes this)
  Also, Cantor's set theory shows that -as sets-, $\bR^1$ and $\bR^2$
  are the same -- you need continuity.

  Here one generally uses ``local patches''
    ((local) comparison to a model space)

  You can also use subtler ideas, like Hausdorff dimension:
  the size of coverings -scales- like some power/dimension

- dimension in algebraic geometry
  Here the ``which space do you map onto finitely-many-to-1''
  definition was the standard for affine/projective varieties for a
  long time.
  Later on people linearized matters and asked about the dimension of
  the cotangent space, or used charts and covered with quasi-affine
  patches.
