Theorem: General Intelligence Entails Creativity, Assuming - Computational Creativity Research: Towards Creative Machines - page 59

logics . Apparently Church, Turing's advisor, who already had on hand his system
O ʱ

, encouraged Turing to pursue
the ordinal-logic direction. 14 Turing acceded, specifically by exploring sequences

, a set of constructive notations a for ordinals

composed of sequences of logics L a where a is in . Mechanisms for generating

larger and larger elements in such sequences are known as reflection principles [ 9 ].
Today we know that such principles, which are pretty much just mechanical, do over-
come the key boundary in your case (and specifically in your Fig. 3.1 ), and hence we
have a counter-example to your so-called theorem.”
Reply: This objection is impaled by either of two unavoidable horns in a dilemma.
Horn 1 : It should be pretty obvious that while all of the creativity in this approach
is at the meta-theoretical, not the object-theoretical, level, there is certainly a lot
of creative, far-from-obviously-mechanical thinking and reasoning going on. This
holds true starting at the genesis of the approach, when Church, armed with

O

O (and

other machinery, e.g., his -calculus), encouraged to Turing to “take the plunge,”

and then as Turing ran into negative, not positive, results. While Turing showed that
p is

λ

0 0

2 formulae were out of reach. But notice that this result is
completely independent of any object-level results: that is, of any proof or disproof
of a particular

1 -complete,

0 0

2 theorem of natural and specific interest to mathematicians.
The situation is no different with respect to more recent work on ordinal logics:
elegance, insight, and creativity abounds at the meta-theoretical level; but results at
the object level isn't part of the game. Therefore, my case is firmly intact.
Horn 2 : The skeptic could offer a rebuttal here in which he concedes my obser-
vation, but points out that he is in fact talking exclusively about the object -level,
and therefore the merely mechanical is in operation, only. But now the second horn
rears up. For the fact of the matter is that [ 11 ]'s proof, like, as a matter of brute
empirical fact, all proofs of particular theorems at this point or above, is stunningly
non -mechanical. In fact, the reasoning is patently infinitary in nature, and hence the
idea, which is needed here by my critic, that the reasoning is mechanical in the sense
of being in line with a Turing-equivalent process, is no more than an article of faith. 15

1 or

3.4.4 Additional Objections

Life, including specifically intellectual life, being what it is, I'm quite sure that addi-
tional objections will occur to those wishing to resist the entailment I've defended
herein. For example, some may urge me to consider alien beings who directly appre-
hend via intuition, without proof, all arithmetic truth. But the point of my essay is
to defend my “theorem” in a context short of the divine! I'm concerned with gi,

14 A marvelous non-technical and historical overview is provided in [ 10 ].
15 I'm not making here any such claim as that the reasoning in Goodstein's case is irreducibly
infinitary (though I do happen to believe that). On the other hand, I do remind readers that a way to
surmount Gödelian incompleteness at the object level is via the -rule. And for all we know from

the alien-fair perspective, there may be entirely new proof theories (including perhaps diagrammatic
ones) that allow aliens or future humans to move beyond incompleteness.

ω