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!

These theorems: what are they? How do they come to be available to the likes of us? We are once again thrown up against the mysterious nature of mathematical knowledge, against the mysterious nature of ourselves as knowers of mathematics. How do we come to have the knowledge that we do? How can we? Plato himself had argued that the very fact that our reasoning mind can come into contact with the eternal realm of abstraction suggests that there is something of the eternal in us: that the part of ourselves that can know mathematics is the part that will survive our bodily death. Spinoza was to argue along similar lines.” (Quoted and paraphrased from Rebecca Goldstein, Incompleteness, The Proof and Paradox of Kurt Gödel, 2005, p. 192, 198-199.)