Yet one can’t do any mathematics at all, not even
basic arithmetic, without referring implicitly to the infinite.
(Rebecca
Goldstein, Incompleteness: 186)
In 1931 a young
logician, Kurt Godel, transformed the study of logic, especially mathematical
logic, and epistemology, by proving that all formal systems of human thought
are inherently incomplete. His
groundbreaking, paradigm-busting, work was titled: On Formally Undecidable Propositions of
Principia Mathematica and Related Systems. It yielded two famous theorems, called Godel’s
incompleteness theorems.
The Principia Mathematica was a work of
mathematical logic by Bertrand Russell and Alfred North Whitehead that
attempted to set forth a complete and consistent formal, logical foundation and
the essential principles of mathematics.
They claimed that they had done so, but upon closer examination it was
found that to achieve consistency Russell and Whitehead had introduced some ad
hoc principles. A formal system, in mathematics
at least, is one that is divested of all appeals to intuition. It is a closed, axiomatic system of reasoning
with primitive givens, stipulated rules of inference and proved theorems. Because it is closed, it is finite. There can be nothing ad hoc about it.
An axiomatic system is consistent when employing the rules of the system generates no contradictions, and
consistency itself is one of the propositions of the system. Thus, every axiom, or true statement, must be
derived from the basic principles and rules of the system. Too, no statement can be both true and false
at the same time. It is a real
left-brainer construct. Godel discovered and
proved that neither the truth nor the falsity of some propositions of any
formal system of thought can be proved.
That is, a proposition may be true, contextually, but it can’t be proved
to be true. It can only be proved to be
unprovable. Thus, to prove the truth of
these formally undecidable propositions requires another, richer kind of
thought-structure. Thus the formal system
is incomplete.
Mind can never embrace the totality, because as Hamlet
told Horatio: “There are more things in heaven and earth, Horatio, Than are
dreamt of in your philosophy.” (Hamlet 1:5) If any contradiction arises, or
if a true proposition or statement cannot be proven using only the rules of the
system, then the system is inconsistent (a bad thing for formalists) and
incomplete (that’s OK, so long as a higher system of thought can complete it). How to get out of this? Think—go right-brained. The problem comes with self-referential
statements. These operate at the limits
of formal systems, defining their boundaries of truth, sort of like the complexity
or chaos sciences.
For example take
the famous liar’s paradox: This statement
is untrue. Taken by itself, with no
reference to anything else, if it is true, then it is a contradiction, because
it says that it is untrue. If it is
untrue, then it is also a contradiction because that would mean that it is true. Consistency is broken, and the only way out
is to introduce something new into the system of thought, or to refer outside
to the system to a context. Godel proved
that no axiomatic system can be sufficiently rich to formally capture even
arithmetic. All formal systems are incomplete, deriving their consistency from
appealing to a richer formal system.
He proved this for
all formal systems by showing how the truth or falsity of some propositions is
not formally decidable using the rules of the system. That is, the question, can a particular proposition
be proved within the system sometimes yields the answer that it can’t be
proved. How?
A particular
proposition (A) is unprovable in the system if the negation of A is another
proposition: A is provable in the system.
But if A were provable then its negation—which says that A is
provable—would be true. But if the
negation of a proposition is true, then the proposition itself is false. So if A is provable then it is false. But if A is provable, then it is also
true. So, assuming the consistency of
the system, if A is provable then it is both true and false—a
contradiction—which means that A is not provable. But that is exactly what A said in the first
place: that it is not provable. So A is
true. Therefore, A is both unprovable
and true and this is expressible within the system. The final conclusion is that the formal
system is either inconsistent or incomplete, which are the two-sides of one
coin. The proof rests on the method of
assigning a number to statements so that a blending of voices—arithmetical and
metaarithmetical—is accomplished such that arithmetical statements are also
making metaarithmetical statements: i.e. statements about arithmetic made in
numbers, each statement assigned a number.
This is possible because language and number are both symbol systems--i.e.
numbers can be assigned letters and letters numbers, as in numerology.
In using the
same language of arithmetic, i.e. the map, to express both meanings and
metameanings, two different sorts of descriptions will be collapsed into one
another. It is introducing an element of
the organization of events on the level of events itself, as if the thoughts
organizing your physical behavior were themselves part of your physical
behavior. In arithmetic, arithmetical
descriptions setting forth relationships between numbers are, of course,
expressible within the formal system e.g. 2+2=4. But if, as Godel did, you assign a number to
a metastatement about arithmetic (e.g. the rule that the sum of two integers is another integer could be given the number 5) then
metadescriptions about the logical relationships holding between numbers (i.e.
the syntax of the language) can be described within the formal system
itself. Metastatements are purely
syntactic, i.e. rules about rules. They
can not be proved to be true or false using the rules of the system. They are simply unproveable. That is, it is not about the truth or falsity of any statement, but about its proveability to be one or the other. Rebecca
Goldstein sums up the results: “Godel’s first incompleteness theorem tells us
that any consistent formal system adequate for the expression of arithmetic
must leave out much of mathematical reality, and his second theorem tells us
that no such formal system can even prove itself to be self-consistent.” (Incompleteness: 192)
That is, the
syntactic features of formal mathematical systems (e.g. the rules of combining
numbers in arithmetic) can’t capture all the truths about the system, including
the truth of its own consistency.
Consistency, which defines the system, transcends the grasp of the system itself: consistency is not completeness. Completeness is something else. There is something within all formal system
that is not “of itself’ but is the representation of a higher entity to the
system: paradoxically, what makes the system a system by being something else. This "something else" completes the system, or, at least, is the means of its completeness. Consistency is not within any system, but
only as part of a richer system, which is also only fully consistent within yet
another, higher system.
In any universe
of discourse, whether language or number, this kind of confusion of realms that
arises with self-referencing—where the metasyntactic and the syntactic
statements collapse into one another, so that the logical relations that hold
between units of meaning (e.g. numbers and their relations) in the formal
system themselves become relations expressible in the language of the system—is
the mechanism of encoding complexity, for it allows an entity of a higher level
of reality to exist in some form in a lower level of reality. Potentially, infinite meanings can then be
expressed in finite language if they become self-referential in a higher sense. OK, so what?
The implications are terrific. But that is for next post.
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