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From: jeff@aiai.ed.ac.uk (Jeff Dalton)
Subject: Re: Putnam on Penrose
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Date: Tue, 6 Dec 1994 17:40:33 GMT
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In article <3bod1e$jrq@newsbf01.news.aol.com> jrstern@aol.com (JRStern) writes:
>WARNING: LONG
>
>In <D01nJB.J6E@cogsci.ed.ac.uk> jeff@aiai.ed.ac.uk (Jeff Dalton) writes:
><
>In article <3bdkfs$4dg@newsbf01.news.aol.com> jrstern@aol.com (JRStern)
>writes:
>>In <CzsDKp.3tu@cogsci.ed.ac.uk> jeff@aiai.ed.ac.uk (Jeff Dalton) writes:
>><
>>>        But - even
>>>apart from the totally unjustified way this latter possibility is
>>>dismissed - there is an obvious lacunae: the possibility of a program
>>>we could write down but not succeed in understanding is overlooked!
>>>
>>>This is the mathematical fallacy on which the whole book rests.
>>
>>Now this puzzles me a bit.  There are some cases, and Penrose has
>>overlooked one.  Suppose his conclusion follows in all the other
>>cases.  Don't we then get this: if we understand the program, 
>>Penrose is right?  Isn't that a significant conclusion in
>>itself?
>>>
>>
>>It's very significant.  It's just that Turing and Godel beat him
>>to it by fifty years.
>
>To what conclusion, exactly?  I just want to be sure I understand
>what you're saying.
>>
>
>I joined comp.ai.phil to try to figure out what I'm saying, too <g>.
>Let me see if I can present this same old argument one more time,
>with increased clarity as well as volume.
>
>First, the question is actually meaningless.  What the heck does
>it mean, "a program we can (not) understand" ?  Penrose claims
>that he can "understand" things.  He claims that computers cannot
>"understand" things.  So, can Penrose "understand" even a trivial
>program?  Apparently yes.  Are there some programs Penrose cannot 
>"understand"?  Apparently yes.  Which ones are "intelligent" ?
>
>Let us do as Penrose does, and equate "understanding" with 
>"decidability".  (I do not for a moment really believe this,
>but it seems to be what Penrose bases his arguments on).

Well, as I said before:

  Penrose talks about whether algorithms are unsound, unknowable,
  not knowably sound, and so forth.  Putnam's language doesn't quite
  line up with Penrose's, because Putnam talks about whether we
  can understand (the program?).  So it's not clear that Penrose's
  way of dividing things up leaves the gap that appears in Putnam's.

Since I haven't read all of Penrose's new book, I not sure how
he deals with "understanding", if at all.  Where does he equate
understanding with decidability (so I can try to look it up)?

I'm also not sure what it means to say a program is decidable
or not.  (It's been over 10 years since I studied any of this
stuff.)

>Putnam makes two mistakes.  One, he accepts Penrose's equation of
>"understanding" and decidability.  Otherwise, Putnam's statement
>of "a program we cannot understand" is meaningless.  Once you 
>try to play the word game by Penrose's rules, you end up juggling
>tautologies, or arguing against them.  Two, he is mistaken when
>he claims that Penrose has overlooked the case of "programs we
>cannot understand".  He did consider this, and rejected it:
>
>(in discussing Platonic reality and Godel undecidability, p. 418 
>of Emperor:)
>   "Thus we are driven to the conclusion that the algorithm that 
>mathematicians actually use to decide mathematical truth is so 
>complicated or obscure that its very validity can never be known 
>to us.
>   But this flies in the face of what mathematics is all about!  
>The whole point of our mathematical heritage and training is that 
>we do not bow down to the authority of some obscure rules that 
>we can never hope to understand. ... It is something built up 
>from such simple and obvious ingredients  and when we comprehend 
>them, their truth is clear and agreed by all."
>
>This is one of a dozen similar quotations in Emperor.

Can you give me some others.  The one you give above is the one
I had in mind when saying he discussed the missing case in his
earlier book "though very briefly and not (to my mind) very 
convincingly".

-- jeff


