X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week3.mdwn;h=e8a86e90bbc180f69d98d3af58680b40e2edf143;hp=a55659cb97b6e0f12616770115fc8680eb2ed821;hb=c794ad08ee1ab7a1297fcc1bf1ec3f9b81b4a612;hpb=cd20a0a226f35177c21ef48bcabfc59316e3e489
diff --git a/week3.mdwn b/week3.mdwn
index a55659cb..e8a86e90 100644
--- a/week3.mdwn
+++ b/week3.mdwn
@@ -583,11 +583,14 @@ truth and circularity](http://tinyurl.com/2db62bk) for an approach
that is similar, but expressed in terms of non-well-founded sets
rather than recursive functions.
-HOWEVER, you should be cautious about feeling too comfortable with
+##However...##
+
+You should be cautious about feeling too comfortable with
these results. Thinking again of the truth-teller paradox, yes,
-ω
is *a* fixed point for `I`, and perhaps it has
+Ω
is *a* fixed point for `I`, and perhaps it has
some a privileged status among all the fixed points for `I`, being the
-one delivered by Y and all.
+one delivered by Y and all (though it is not obvious why Y should have
+any special status).
But one could ask: look, literally every formula is a fixed point for
`I`, since
@@ -600,7 +603,8 @@ So the Y combinator is only guaranteed to give us one fixed point out
of infinitely many---and not always the intuitively most useful
one. (For instance, the squaring function has zero as a fixed point,
since 0 * 0 = 0, and 1 as a fixed point, since 1 * 1 = 1, but `Y
-(\x. mul x x)` doesn't give us 0 or 1.) So why in the reasoning we've
+(\x. mul x x)` doesn't give us 0 or 1.) So with respect to the
+truth-teller paradox, why in the reasoning we've
just gone through should we be reaching for just this fixed point at
just this juncture?
@@ -616,6 +620,12 @@ fixed point for this referential function: if this pen cap is the
referent of *this noun phrase*, then it is the referent of (3), and so
for any object.
+The chameleon nature of (3), by the way (a description that is equally
+good at describing any object), makes it particularly well suited as a
+gloss on pronouns such as *it*. In the system of
+[Jacobson 1999](http://www.springerlink.com/content/j706674r4w217jj5/),
+pronouns denote (you guessed it!) identity functions...
+
Ultimately, in the context of this course, these paradoxes are more
useful as a way of gaining leverage on the concepts of fixed points
and recursion, rather than the other way around.