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Chris Barker
<barker@kappa.(none)>
Mon, 27 Sep 2010 02:44:34 +0000
(22:44 -0400)
committer
Chris Barker
<barker@kappa.(none)>
Mon, 27 Sep 2010 02:44:34 +0000
(22:44 -0400)
week3.mdwn
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b/week3.mdwn
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week3.mdwn
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week3.mdwn
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-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,
HOWEVER, you should be cautious about feeling too comfortable with
these results. Thinking again of the truth-teller paradox, yes,
-<code>&
o
mega;</code> is *a* fixed point for `I`, and perhaps it has
+<code>&
O
mega;</code> is *a* fixed point for `I`, and perhaps it has
some a privileged status among all the fixed points for `I`, being the
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
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
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?
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.
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.
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.