Theorem: General Intelligence Entails Creativity, Assuming - Computational Creativity Research: Towards Creative Machines - page 57

T gi is composed of truths of arithmetic; that is, of members of the set
TRUE A / I , with an accompanying request for a supporting proof. In the case of PA I ,
there can be a supporting formal proof in a standard, finitary, mechanizable proof
theory (our

The test

˄ ). This will not be possible for members of the set

PA ˄
I

TRUE A / I −

I further define creativity as passing beyond such proofs in basic arithmetic in order to
reach at least one truth. Again, Goodstein's Theorem is currently an ideal example.

3.3 The Proof-Sketch Itself

Theorem : Suppose that an agent a is alien-fair gi. Then a is creative.

Proof-Sketch : Trivial, given our setup. Assume the hypothesis of the theorem. By definition,
a , since its gi includes command of all of basic arithmetic, knows at least one ∈ TRUE A / I

on the strength of a proof as a matter of

mathematical fact exceeds the mechanical type of proof that characterizes our

ˀ discovered and confirmed by a . But since ˀ

, a has left
behind mere mechanical, first-order techniques, and is by definition creative. QED

˄

3.4 Objections

Some objections can be anticipated; I discuss two.

3.4.1 Begging the Question?

Objection: “As you yourself note, given your setup, the theorem is easily established.
So you have simply begged the question. Why would anyone accept your setup in
the first place?”
My reply: Well, every theorempresupposes background machinery, and some of it
will be objectionable to some; the present situation is no exception. I cheerfully admit
that anyone unwilling to accept that alien-fair gi must include significant command

(Footnote 12 continued)
the other hand it's undeniable that it stands ineliminably at the very heart of mathematics, and the
teaching thereof, for us. Specifically, we find it impossible to forego language in which we speak of
whether or not a young student understands some aspect of arithmetic, or whether or not a college
student understands (differential and integral) calculus. Not only is this linguistic practice standard
and unavoidable, but so is (at least in pedagogy) the need to try to confirm that understanding is in
place via giving tests. In the present paper, my use of the concept of understanding, and recourse
to tests designed to confirm its presence, simply follows a suit played in the real world.