X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2Fweek1_advanced_notes.mdwn;h=c5e589f23c678f7f54173ec964d7d641ae4aafdf;hp=8f0fb85114b40b94cb3987aec0a4ae153eb67574;hb=a4c6be4f4d84375904d19a74b7a781f7b204e7f2;hpb=ade558370a8eae453217df959d56f56de5df5c30

diff --git a/topics/week1_advanced_notes.mdwn b/topics/week1_advanced_notes.mdwn
index 8f0fb851..c5e589f2 100644
--- a/topics/week1_advanced_notes.mdwn
+++ b/topics/week1_advanced_notes.mdwn
@@ -190,16 +190,25 @@ or like this:
 
     lambda (x, y). x + y
 
-They permit you to appreviate the first λ-expression as simply `(10 - )`. We know there's an argument missing, because the infix operator `-` demands two arguments, but we've only supplied one. So `(10 - )` expresses a function that takes an argument `x` and evaluates to `10 - x`. In other words, it expresses λ`x. 10 - x`.Similarly, `( & ys)` expresses a function that takes an argument `x` and evaluates to `x & ys`. And --- can you guess what the last one will be? --- `( + )` expresses a function that takes two arguments `(x, y)` and evaluates to `x + y`.
+They permit you to appreviate the first λ-expression as simply `(10 - )`. We know there's an argument missing, because the infix operator `-` demands two arguments, but we've only supplied one. So `(10 - )` expresses a function that takes an argument `x` and evaluates to `10 - x`. In other words, it expresses λ`x. 10 - x`. Similarly, `( & ys)` expresses a function that takes an argument `x` and evaluates to `x & ys`.
 
-Wait a second, you might say. Isn't that last operation exactly what `+` does *already*? Why am I making a distinction between `+` and `(+)`? The difference is that bare `+` without any parentheses is an *infix* operator that comes between its arguments. Whereas when we wrap it with parentheses, it loses its special infix syntax and then just behaves like a plain variable denoting a function, like `swap`. Thus whereas we write:
+All of this only works with infix operators like `-`, `&` and `+`. You can't write `(1 swap)` or `(swap 1)` to mean λ`x. swap (1, x)`.
+
+Can you guess what our shortcut for the last function will be? It's `( + )`. That
+expresses a function that takes two arguments `(x, y)` and evaluates to `x + y`.
+
+Wait a second, you say. Isn't that just what `+` does *already*? Why am I making a distinction between `+` and `( + )`? The difference is that bare `+` without any parentheses is an *infix* operator that comes between its arguments. Whereas when we wrap it with parentheses, it loses its special infix syntax and then just behaves like a plain variable denoting a function, like `swap`. Thus whereas we write:
 
     x + y
 
-if we want to instead use `( + )`, we have to instead write:
+if we want to use `( + )`, we have to instead write:
+
+    ( + ) (x, y)
+
+It may not be obvious now why this would ever be useful, but sometimes it will be.
 
-    (+) (x, y)
+All of these shorthands `(10 - )`, `( & ys)` and `( + )` are called "sections". I don't know exactly why.
 
-Confession: actually, what I described here diverges a *tiny* bit from what OCaml and Haskell do. They wouldn't really write `(+) (x, y)` like I just did. Instead they'd write `(+) x y`. We will look at the difference between these next week.
+Confession: actually, what I described here diverges *a bit* from how OCaml and Haskell treat `( + )`. They wouldn't really write `( + ) (x, y)` like I did. Instead they'd write `( + ) x y`. We will look at the difference between these next week.