Donald Bruce Stewart wrote: 
sdowney:
  
i'm not naive enough to think they are the composition function, and
i've gathered it has something to do with free terms, but beyond that
i'm not sure. unless it also has something to do with fix points?
    

The wiki knows all! :)

    http://haskell.org/haskellwiki/Pointfree

    1 But pointfree has more points!

    A common misconception is that the 'points' of pointfree style are the (.)
    operator (function composition, as an ASCII symbol), which uses the same
    identifier as the decimal point. This is wrong. The term originated in
    topology, a branch of mathematics which works with spaces composed of points,
    and functions between those spaces. So a 'points-free' definition of a function
    is one which does not explicitly mention the points (values) of the space on
    which the function acts. In Haskell, our 'space' is some type, and 'points' are
    values.
Hm. I've been lurking for a while, and this might be a bit of nit-picking as my first post, especially given I'm still a bit of a n00b in Haskell. I've been programming a long time, though - coming up on three decades now and virtually all of it really programming, no management.

Anyway, as I understood it, the "points" were the terminal objects of the category in which you're working - in this case, pointed continuous partial orders (CPO), and the points are effectively values in the domain. The usage of "point" for terminal objects comes from the category of topological spaces, as you say,. and algebraic topology is where category theory found it's first big home - but that's not really what we're talking about here, is it?

Category theory got the term from topology, which got it from geometry. So you could say "point" is "position without dimension" - but that's just not the "point" we're talking about anymore.

So, the usage of "point" here refers a terminal object in the CPO category, which means a value of some datatype - in this particular case, a value in the domain of the function being defined. So when you give a definition that uses patterns for the parameters, the variables in the patterns get bound to the values in the domain of the function. If you write the function in a higher-order style, where you don't bind the values, your definition doesn't refer to the "point" at which it's being evaluated, hence "point-free". 
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