[schneems]

A concept I learned about in Knowledge Based AI (gatech ONSCS) is called “version spaces” where instead of starting at specific examples and moving to be more general or the other way around you do both as a kind of knowledge bidirectional search. I feel humans work that way too. We need both specific examples and generic models to help us converge to a deeper understanding of topics.

[xg15]

True. I feel, this is why sometimes, the notation

  a_0 + a_1 + ... + a_n

is easier to understand than the shorter

  sum(i = 0 to n) a_i

.

But on the other hand, 2 or 3 elements of the sum are usually enough, i.e. you probably wouldn't improve understanding by writing out the first 10 elements or so.

[gowld]

Neither of those forms is sufficient information to solve a problem, except for a "guess what I'm thinking" puzzle.

You need:

   sum(i = 0 to n) *f*(i)

and you'd like:

   Example: f(0) = *a_0*, f(1) = *a_1*, f(2) = *a_2*

where the stuff in *italics* is given as a concrete substitution.