INF/SUP/MIN/MAX

[Guy Steele <gls@think.com>]

   Date: Sun, 30 Jul 89 17:36:25 BST
   From: Jeff Dalton <jeff%aiai.edinburgh.ac.uk@nsfnet-relay.ac.uk>

   >From: Guy Steele <gls@think.com>
   >    Date: Wed, 26 Jul 89 15:49:33 edt
   >    From: cph@zurich.ai.mit.edu (Chris Hanson)
   > 
   >    [quotes Steele's arguments about Quux-head pennies, etc.]
   >    I don't buy these arguments: many of them have to do with side
   >    effects, and since when are there any side effects on numbers?  So
   >    they don't apply.
   > 
   > Granted.  "Plus ca change, plus c'est la meme chose", as Sussman and I
   > quoted in "The Art of the Interpreter".  The more things change, the more
   > they remain the same; that is, the notion of object identity may be linked
   > to the notion of side effect.  And yet, if numbers are not subject to
   > side effects, in what sense can we say that 1 = 1, or that 1 /= 2 ?

   I think this observation of numeric identity shows that identity can
   be linked to side-effects, but also that identity doesn't require this
   link.  Indeed, I don't find the argument that there's some sort of
   implicit potential side-effect involved in asking 1 = 1 very
   convincing.

How is it, then, that 1 /= 2?  What is there that makes them different?
How is it that "+" can distinguish between them?  I find it very
unsatisfying to say of "+" merely that it is primitive, it does it
because that's what it does, and I should stop asking so many silly
questions.  The same is true, by the way, of symbols: how is it that
"FOO" is distinguishable from "BAR"?  

   > I think that when we say 1 = 1, we are temporarily entertaining the
   > possibility that the two instances of "1" represent *different* things,
   > precisely so that we may then make the assertion that they are the same
   > after all.

   When I say "Nixon is the author of Six Crises", do this mean I'm
   temporarily entertaining the possibliity that someone else wrote it?
   Well, perhaps I'm at least entertaining the possibility that the
   person I'm speaking to thinks so.  

Exactly so.
				      But what if I say just "Nixon is
   Nixon"?  It may seem that I must have some possibility in mind that I
   am trying to deny.  But I don't think it's clear what this possibility
   is unless I say something more.  For example, I might say: "The person
   you call 'Nixon' is the same person I know as 'Nixon', the author of
   Six Crises, etc., and not two different people as we had once
   supposed."  

Again, exactly so.  See below.

	       But it's hard to see how there could be this kind of
   confusion about "'1' as defined by Scheme".  

Only because "1" is a little more famous than Nixon; it's a matter of
degree rather than principle.
						Indeed, it seems to me
   that once we agree about what "Nixon" refers to, we can say "Nixon is
   Nixon" without necessarily entertaining any possibility that Nixon
   isn't Nixon.

   So I don't think it's straightforward to conclude that when we
   say 1 = 1, etc.

   > If one of the 1's were not 1 after all, then things would be
   > different, including the truth of the equality.  This subjunctive
   > hypothesis amounts to a side effect, for we are considering, hypothetically
   > and however evanescently, a world altered from our own (whether by SETQ
   > or by adding an entry to the head of an a-list).

   But is every difference the result of a side-effect?  Perhaps we're
   just entertaining the possibility that "=" doesn't mean what we
   thought it meant or that tokens like "1" refer to something different
   each time.

The latter is a possibility, however temporarily it is entertained or
however trivially resolved.