June 25, 2012 12:25 PSP Book - 9in x 6in 04-Junichi-Takeno-c04
Language and Formal Logic 33
if it will finish running or will run forever. Any given problem is
solved when the Turing machine finishes running.
Turing’s attempt to create models of human thought paved the
way to the development of computers, on the one hand, and the
study of artificial intelligence, on the other.
4.2 Language and Formal Logic
We will start with a description of the study of formal logic before
discussing mathematical ways to model human thought. According
to formal logic, human thought arises from languages, and is an
inference in languages. Formal logic was the basis of the study of
languages and inference at that time. The development of modern
formal logic is credited to Gottlob Frege (1848–1925), (Frege, 1892)
and Bertrand Russell (1872–1970), (Russell, 1959). I believe that
their objective of research was to unravel the intelligent functions
of humans from the standpoint of formal logic. Their attempts,
however, deviated from and became irrelevant to human thought
and shifted to a pure pursuit of formal logic itself. They attempted
to develop a method to derive mathematical knowledge from basic
mathematical hypotheses using logic. Their efforts were in vain,
however, when in the 1930s, Czech-born Kurt G
¨
odel (1906–1978)
mathematically proved that such an attempt was impossible (Fig.
4.2). These are G
¨
odel’s famous incompleteness theorems (G
¨
odelsche
Unvollst
¨
andigkeitssatz). The theorems say that formal systems are
incomplete because they inherently include arithmetic equations
that are defined to be true by the system but that cannot be
proved by the basic assumptions of the system. These theorems
thus negated the clarity of mathematics. The meaning of these
mathematical theorems also applies to the relationships between
human thought and the computer as a representative of formal
systems. Namely, there are some arithmetic equations that human
thought believes to be true but that cannot be proved by computers.
Based on this recognition, Nagel and Newman (1958) held that
computers could never reach the level of human thought or the
human mind. In 1994, Roger Penrose asserted that human thought
could clarify mathematical truths better than computers (Penrose,