[...]

>  … The true explanation is very simple. The power of logic and
>  mathematics to surprise us depends, like their usefulness, on the
>  limitations of our reason. A being whose intellect was infinitely more
>  powerful would take no interest in logic and mathematics. For he would
>  be able to see at a glance everything that his definitions implied, and,
>  accordingly, could never learn anything from logical inference which he
>  was not fully conscious of already.

>Nik Silver.

In fact, this is not correct, if means of expression expand as well as
means of reasoning.  Analogues of Godel’s theorem would continue to
hold, and there would still be things which were "easy" and "hard" for
the hypothetical infinite intellect to prove; it would simply be the
case that everything that we can express would fall in the (trivially)
"easy" category.

For the analogues of Godel’s theorem not to hold, the infinite
intellect X would have to be unable to refer to his own techniques of
proof (among other limitations).

–
The opinions expressed          |     –Sincerely,
above are not the "official"  |     M. Randall Holmes
opinions of any person          |     Math. Dept., Boise State Univ.
or institution.                 |     hol…@math.idbsu.edu

>Some of the logicians who have discussed this issue in a traditional
>context are CS Peirce, JS Mill, G Frege, M Dummett and W Salmon.

You can also add A. J. Ayer’s name to the list. Ayer refused to
believe the paradox, offering a rather down-to-earth solution. This is from
his “Language, Truth and Logic” pp. 81–82 (reprinted by Pelican Books).
After suggesting that proving 7+5=12 does not tell us anything we didn’t
(at least implicitly) know already, because 12 is defined precisely to be
7+5, he says….

  What is mysterious at first sight is that these tautologies should on
  occasion be so surprising, that there should be in mathematics and logic
  the possibility of invention and discovery. As Poincare says: `If all the
  assertions which mathematics puts forward can be derived from one another
  by formal logic, mathematics cannot amount to anything more than an
  immense tautology. Logical inference can teach us nothing essentially new,
  and if everything is to proceed from the priciple of identity, everything
  must be reducible to it. But can we really allow that these theorems which
  fill so many books serve no other purpose than to say in a roundabout
  fashion “A=A” ?’ …
  … The true explanation is very simple. The power of logic and
  mathematics to surprise us depends, like their usefulness, on the
  limitations of our reason. A being whose intellect was infinitely more
  powerful would take no interest in logic and mathematics. For he would
  be able to see at a glance everything that his definitions implied, and,
  accordingly, could never learn anything from logical inference which he
  was not fully conscious of already.

Nik Silver.

   Valid deductive inference is subject to a seeming paradox.  (1) The
   content of the conclusion is already stated or ‘contained’ in the
   premises … , and so such arguments are ‘non-ampliative’. But (2) such
   arguments can give us something new and useful and
   so would seem to be in some sense ampliative.

From an AI point of view there is no paradox here – the theorem that
is "contained" in a set of premises is only there *implicitly*, and is
made *explicitly* available only if and when it is deduced.  For many
real-world purposes and real-world information users (e.g., people or
programs), information cannot be used unless it available in some
particular explicit form.  Even if the information users have some
limited deductive ability, and can on their own make *some* implicit
things explicit, they rarely have the ability to compute the full
deductive closure of a theory (at least, to do so fast enough to be
useful).  Thus, the theorem may not be usable while "contained"
implicitly in the premises, but it may be very usable once it is
explicitly present in the derivation, or at least once something from
which it can be derived by a specific limited deductive process is
present.

Traditionally logic has dealt with issues where a theorem implicitly
present was just as powerful as one explicitly present (e.g. in
defining terms like "consistent", "complete", "model", …), Perhaps
this assumption has become so embedded in your worldview that you no
longer remember you are making it.  (It has become implicit
—
                                        Lou Steinberg

uucp:   {pretty much any major site}!rutgers!aramis.rutgers.edu!lou
internet:   l…@cs.rutgers.edu