Page 72 - Deep Learning

The Production of Novelty 55

experience and we cannot think without concepts any more than we can see
without eyeballs or talk without words, so how can we think about an event
or a situation in ways that go beyond experience? if the mind is a system for
processing experience and projecting it onto future situations, then the very
possibility of novel ideas requires explanation.
Formal analyses deepen this puzzle. Logicians have developed a formal cal-
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culus called predicate logic. Some areas of human knowledge can be expressed
in this system, and it possesses a certain generative power. Given a set of ini-
tial assertions – axioms – formal logic enables mechanical deduction of the
consequences of those assertions. Consequences can be remote and surpris-
ing: Pythagoras’s famous geometry theorem about the squares of the sides of a
right-angled triangle does not automatically come to mind when one contem-
plates euclid’s axioms for plane geometry, even though it follows logically from
those axioms. 7
Nevertheless, the generativity of deduction is limited. a chain of logical
deductions from given axioms cannot lead to new or better axioms. Deduction
spins out the perhaps infinitely many implications of a given set of axioms but
cannot override those axioms. Worse, a famous proof by Kurt Gödel shows
that there is no set of axioms that enables the derivation of all true statements,
even within a narrowly defined domain of knowledge. Gödel proved this for
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arithmetic. No matter how a mathematician chooses his axioms about num-
bers, there will always be some true statements about numbers that cannot be
deduced from those axioms. if the axioms are altered, some of the previously
unreachable conclusions become derivable, at the cost of losing the ability
to derive others. There is no set of axioms from which every true statement
about numbers can be derived. Logicians believe that Gödel’s principle holds
for more complex domains of knowledge as well, only more so. any axiom-
atic, deductive system necessarily operates within a bounded perspective that
makes certain potential insights unreachable.
in The Language of Thought, Jerry a. Fodor argued the closely related the-
sis that a symbol system – a language of thought, in his terminology – can-
not contain within it the possibility of constructing a new system that is more
powerful than itself. The gist of his argument is that if the meaning of a new
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concept can be represented in a person’s mental symbol system, that is, if it
can be defined in terms of concepts already known, then learning that concept
does not significantly alter the expressive power of the person’s symbol system.
every thought that the person can think with the help of the new concept he
could have thought with the help of his prior concepts, albeit perhaps in a
more clumsy and long-winded way. every thought that one can think with the