3 + 4 == 7
This equation can be interpreted as expressing the thought that the
++<<<<<<< HEAD:topics/_week3_what_is_computation.mdwn
+complex expression `3 + 4` evaluates to `7`. The evaluation of the
+expression computing a sum. There is a clear sense in which the
+expression `7` is simpler than the expression `3 + 4`: `7` is
+syntactically simple, and `3 + 4` is syntactically complex.
+
+Now let's take this folk notion of computation, and put some pressure
+on it.
++=======
+ complex expression `3 + 4` evaluates to `7`. In this case, the
+ evaluation of the expression involves computing a sum. There is a
+ clear sense in which the expression `7` is simpler than the expression
+ `3 + 4`: `7` is syntactically simple, and `3 + 4` is syntactically
+ complex.
+
+ It's worth pausing a moment and wondering why we feel that replacing a
+ complex expression like `3 + 4` with a simplex expression like `7`
+ feels like we've accomplished something. If they really are
+ equivalent, why shouldn't we consider them to be equally valuable, or
+ even to prefer the longer expression? For instance, should we prefer
+ 2^9, or 512? Likewise, in the realm of logic, why shold we ever
+ prefer `B` to the conjunction of `A` with `A --> B`?
+
+ The question to ask here is whether our intuitions about what counts
+ as more evaluated always tracks simplicity of expression, or whether
+ it tracks what is more useful to us in a given larger situation.
+
+ But even deciding which expression ought to count as simpler is not
+ always so clear.
++>>>>>>> working:topics/week3_what_is_computation.mdwn
##Church arithmetic##
(y((\xx)y)) ~~> yy
etc.
++<<<<<<< HEAD:topics/_week3_what_is_computation.mdwn
+In the arithmetic example, there is only one number that corresponds
+to the sum of 3 and 4 (namely, 7). But there are many sums that add
+up to 7: 3+4, 4+3, 5+2, 2+5, 6+1, 1+6, etc.
++=======
+ Likewise, in the arithmetic example, there is only one number that
+ corresponds to the sum of 3 and 4 (namely, 7). But there are many
+ sums that add up to 7: 3+4, 4+3, 5+2, 2+5, 6+1, 1+6, etc.
++>>>>>>> working:topics/week3_what_is_computation.mdwn
So the unevaluated expression contains information that is missing
from the evaluated value: information about *how* that value was