Page 426 - Deep Learning

Notes to Pages 55–57 409

to conclude when a proposition that was once regarded as true turns out to be
false after all (non-monotonic logic: see, e.g., Donini et al., 1990 and McDermott
& Doyle, 1980). In the second half of the 20th century, logic was vitalized by the
creation of computer programs that operate on logical principles (e.g., Amir &
Maynard-Zhang, 2004). Logic remains a dynamic field of inquiry after more than
two millennia of unbroken, cumulative progress (Gabbay & Woods, 2001).
7. Pythagoras was a sixth-century b.c. Greek philosopher and mathematician.
The Pythagorean theorem says that in any right-angled triangle, the square of
the hypotenuse, i.e., the side that stands opposite to the right angle, is equal
to the squares of the other two sides. No text by Pythagoras himself is extant
and very little is known with certainty about him or his life (Riedweg, 2005).
Various proofs of the theorem are available at http://en.wikipedia.org/wiki/
Pythagorean_theorem.
8. The proof of Gödel’s first incompleteness theorem was originally published in an
article titled “Über formal unentscheidbare Sätze der Principia Mathematica und
verwandter Systeme I” which appeared in the journal Monatshefte für Mathematik
und Physik, 1931, vol. 38, pp. 173–198. An English translation is available in Gödel
(1962/1992) and in the collection of papers edited by Davis (1965/2004, pp. 5–38).
Unfortunately, the proof is too complicated and too technical for others than
mathematicians to follow. Nagel and Newman (2001) explain the theorem and
its proof to nonmathematicians. As one might expect, the application of Gödel’s
theorem to matters of human cognition is not straightforward; see Lucas (1961)
and Slezak (1982). My position is that each human being, when operating in a
deductive, symbol-manipulating mode, as in deliberate, conscious reasoning and
problem solving, is subject to the Gödel limitation; in that mode, we can, in prin-
ciple, only reach a subset of all truths. There are two different ways to break out
of this box: First, because different individuals view the world in slightly differ-
ent ways, they can be conceptualized, in their deductive mode, as Gödel-limited
systems with slightly different axioms; collectively, they cover a wider range of
truths than any single individual. Second, and more fundamental for the the-
ory of creativity, the cognitive processes of a person are not limited to deduc-
tive, deliberate thinking. The impact of Gödel’s theorem on cognitive psychology
is thus to prove the importance of nondeductive processes such as percep-
tion, restructuring, subsymbolic learning, etc. for creativity. The micro-theory
of creative insight presented in Chapter 4 is consistent with this view.
9. Fodor’s argument that one cannot learn a more expressive representational sys-
tem is developed in more detail than my brief sketch suggests; see Fodor (1976,
pp. 79–97).
10. Not all computational mechanisms are capable of performing every computa-
tion. If a mental process P maps inputs I = {i 1 , i 2 , i 3 , …, i n } into some output O,
P(I) = O, then any theory for that process must be capable of computing the
function that P implements, i.e., of producing O when given I (and nothing else).
The sufficiency test is a test of the adequacy of a psychological hypothesis, not
of its truth. A sufficiency test does not prove that a theory is true, only that it
might, in principle, be true and hence is worth investigating. Theories that do
not pass the relevant sufficiency test cannot be true and hence can be set aside