From 2e064948c0cb722be400441476fd13b2add6d636 Mon Sep 17 00:00:00 2001 From: Chris Barker Date: Sun, 26 Sep 2010 22:44:34 -0400 Subject: [PATCH] edits --- week3.mdwn | 14 +++++++++++--- 1 file changed, 11 insertions(+), 3 deletions(-) diff --git a/week3.mdwn b/week3.mdwn index a55659cb..88b1c4f9 100644 --- a/week3.mdwn +++ b/week3.mdwn @@ -585,9 +585,10 @@ rather than recursive functions. 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 +601,8 @@ 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 +618,12 @@ 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.zas.gwz-berlin.de/mitarb/homepage/sauerland/jacobson99.pdf) +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. -- 2.11.0