Key To Rule-Generating Numbers—Gap Between Self And Self-Knowledge

Mystery Continued
Searching For A Philosophy

According to the logic school of mathematics, number is defined in terms of
the concept of sets and cardinal numbers. According to the Frege-Russell
definition, "two sets are said to have the same cardinal number if
there exists a one-to-one correspondence between them." The cardinal
number of a given set is defined as the set of all sets that have the
same cardinal number as the given set. However, it was later shown
that a contradiction arose from this number concept. Even before the
contradiction arose, this definition was received poorly among
intuitionists who did not consider it necessary to reduce the concept
of natural number to simpler concepts. For them number was simply the
result of the notion of an abstract entity plus the notion of an
indefinite sequence of those entities.

An advocate of the formalist school of mathematics, David Hilbert,
attempted to formalize mathematics in a way that would satisfy both
the logicists and the intuitionists. Hilbert proposed to formulate
classical mathematics as axiomatic theories and then prove that these
theories were free from contradictions. This attempt came to an abrupt
halt when Kurt Godel published a theorem that demonstrated that any
formal number-theoretic system, if consistent, contains an undecidable
formula; that is, a formula that can neither be proved nor disproved.
In other words, Godel's theorem tells us, "that it is
self-contradictory to suppose that mathematics can be proved free from
self-contradiction—that, in fact, there must always be true but
unprovable theorems" (Pledge, 1959, p. 190). This striking result--
Godel's theorem, suggests that the source for the certainty of rule
generated information will not be found in the "language of the
universe," which is what Galileo Galilee once called mathematics. If
we can't look to mathematics for certainty then where can we look?

Certainty is hard to find no matter where we look, but finding the
origin of number shouldn't pose so great an obstacle; at least that's
what the philosopher, mathematician, and linguist Ernst Cassirer
thought. According to Cassirer, "…in many languages, the etymology of
the first numerals suggests a link with the personal pronouns: in
Indo-Germanic, for example, the words for `thou' and `two' seem to
disclose a common root…we stand here at a common linguistic source of
psychology, grammar and mathematics; that this dual root leads us back
to the original dualism upon which rests the very possibility of
speech and thought" (Cassirer, 1957, vol. 1, p. 244). The
mathematician Dedekind traced the concept of number back to an even
more fundamental origin. He ended up reducing the system of natural
numbers to a single basic logical function: he considered the system
to be grounded in "the ability of the mind to relate things to things,
to make a thing correspond to a thing, or to image a thing in a thing
(Cassirer, 1957, vol. 3, p. 257). If the origin of number is located
in the mind's ability to relate things to things, then the
self-limiting theorems of mathematics, it seems to me, have something
to say about consciousness itself, something strange, and, ultimately,
something that will remain strange. Douglas Hofstadter seems to agree.
In his book, Godel, Escher, Bach: An Eternal Golden Braid, he echoes
this sentiment when he states:

"All the limitative Theorems of metamathematics and the theory of
computation suggest that once the ability to represent your own
structure has reached a certain critical point, that is the kiss of
death: it guarantees that you can never represent yourself totally.
Godel's Incompleteness Theorem, Church's Undecidability Theorem,
Turing's Halting Theorem, Tarski's Truth Theorem—all have the flavor
of some ancient fairy tale which warns you that "To seek
self-knowledge is to embark on a journey which…will always be
incomplete, cannot be charted on any map, will never halt, cannot be
described" (1979, p. 697).

Perhaps, it is that critical disjuncture between self and
self-knowledge that holds the key to the rule-generating phenomenon
that we are looking for!