Like most of
the great mathematical discoveries by the great mathematicians, the famous
incompleteness theorems published by Kurt Gödel in 1931 almost completely
escape the comprehension of the average man on the street. Nonetheless,
scholars familiar with the work of Gödel and his theorems have gone to the
trouble of translating his texts – not only from the original German, but from
the abstract language of logic and high-level arithmetic. What they describe is
a powerful insight with profound limiting implications for otherwise seemingly
unbounded areas of research, such as artificial intelligence and theoretical
cosmology. I suspect they also have implications for theodicy.
Using
sophisticated mathematical and logical machinery, Gödel managed to prove with
the incompleteness theorems that in most any formal and consistent axiomatic
system, there will be a true statement derivable from the system which
nonetheless cannot be proven within the system.[1] The statement in question
can be proven in principle (as it is true), through the addition of more
axioms, but this expansion results in a larger system in which the principle of
incompleteness again holds: New statements will be derivable from the new
system, which cannot be proven within the new system.
To
illustrate the theorem I will take the liberty to borrow an analogy from Rudy
Rucker, that of a truth machine which houses all known truth and can answer all
questions asked of it with only true statements.[2] A truth machine operator
approaches the machine and types in the following sentence:
"The
truth machine will never say that this sentence is true."
Then the
operator asks the machine if the above sentence, as stated, is true or false.
If the truth machine decides the sentence is true, it cannot say so (because
the sentence states that the truth machine will not say it is true). If the
truth machine decides the sentence is false, then again it cannot say so
(because it only answers with true statements) – yet its failure to say so is
precisely what the sentence says of the truth machine. It is true, then, that
the truth machine will never say that the sentence is true. Though true in
itself, the undecidability of the sentence for the truth machine means that its
truth cannot be recognized by that same machine.
All this
implies that as outside observers, we can somehow ascertain a truth that even a
perfectly programmed truth machine cannot. This implies in turn that we, along
with this special insight that only we can see, in some sense transcend any
programmed system – even a system that houses all known truth. How can this be?
Well, for one thing we have not been programmed. Human beings are evidently not
reducible to machines, any more than our thoughts are reducible to abstract
statements derived from formal systems of logic or mathematics. Often the
undecidable statement in a proof of Gödel's theorem is termed self-referential,
and this is telling; for what a machine lacks by its classical definition is
self-awareness. Penrose argues that with this ability to reflect human beings
alone can see both sides of a paradox, whereas a machine can only process
inputs given it from outside itself.[3] In a brilliant stroke of genius
eminently logical and equally paradoxical, Gödel managed to establish the
critical distinction between God-given reason and mechanical computation.
Technically
Gödel's theorems only hold in the context of consistent systems featuring
formal language, system-specific axioms, and rules of inference. Peano
Arithmetic is thought to be the ideal such system. Euclidean geometry is also
said to suffice. But the principle appears to apply more generally. For
example, Stephen Hawking has argued that the eclipse of classical Newtonian
physics by the mutually incompatible theories of quantum mechanics and general
relativity suggests incompleteness of the physical universe. Though
mathematical models can be created which approximate the fundamental structure
of the universe, they cannot be proven in principle because human observers are
entities within the very system under observation:
But
we are not angels, who view the universe from the outside. Instead, we and our
models are both part of the universe we are describing. Thus a physical theory
is self-referencing, like in Gödel's theorem. One might therefore expect it to
be either inconsistent or incomplete. The theories we have so far are both
inconsistent and incomplete.[4]
Even more
so, theological explanations for evil in a physical universe whose theories are
inconsistent or incomplete should be expected to appear similarly inconsistent
or incomplete. Pressing the idea yet further, Thomas Nagel maintains that in
light of the unavoidable subjectivity of human perceptions, "any objective
conception of reality must acknowledge its own incompleteness."[5]
A less
formal but no less baffling undecidable statement facing any theodicy project
might go something like this: "God's act of creating humans free to choose
between good and evil is morally justifiable." If we say that the sentence
is true, we imply that the freedom to choose evil is morally justifiable
(though evil by definition is not morally justifiable). If we say that the
sentence is false, we imply that the freedom to choose good is not morally
justifiable (though good by definition is morally justifiable). The former
means leaving the floodgates open to various forms of evil and its painful
consequences. The latter means closing the door to love, friendship, adventure,
growth, discovery, and personal accomplishments – in short, an absence of any
meaningful experience of good. Either situation could rightly be described as
evil. Theodicy in one sense remains woefully incomplete.
In another
sense, however, the Scriptures supply a complete and coherent solution to the
problem of evil. As Eleonore Stump suggests, certain Christian beliefs speak
uniquely to the problem of evil: The fall of Adam (and by extension all of
humanity); the onset of natural evil ("a curse upon the earth")
through Adam's fall; and the eternal destination of either heaven or hell
awaiting all people, depending on the state of their relationship to God,
principally through faith in Christ (or willful lack thereof).[6] Indeed, a
thoroughly biblical Christian response to evil alone seems capable of answering
the questions still confronting us:
1.
How can God create an eternal paradise, given the priority he places on moral
free will?
2.
Why has God not already created an eternal paradise complete with morally free
beings, given that he has the ability to do so? (Or, why is this-worldly
existence even necessary?)
These
questions really turn on one another. God can create an eternal paradise
featuring sheer moral goodness only if its inhabitants are free to choose the
good. But such a paradise requires that its inhabitants never choose evil,
which implies a restriction on freedom. Just what is it, then, that makes it
possible to retain human volition and at the same time ensure uncorrupted
goodness? Jesus preached the answer consistently: the coming of the kingdom of
God. The theology of the kingdom, especially its eschatological and eternal
aspects, depicts a gradual but final and irreversible, i.e., complete, triumph
of good over evil. As Jesus preached it and as most New Testament scholars
acknowledge, the kingdom can be best viewed as having already arrived in one
sense and yet awaiting its complete fulfillment in another. This is the
"Already-Not Yet" paradigm, which suggests incompleteness in
theology.[7]
From this
perspective, the creation of the world as described in Genesis was not the end
of God's work of creation, but only the start of a much more expansive
creative-redemptive program with ultimate, everlasting joy in view. This
creative-redemptive program, as I have called it, consists of three distinct
phases. During the first phase in the paradise of Eden, human free will was
unrestricted with respect to choosing among certain moral and relational
options. Among the numerous fruit-bearing trees in the garden were both the
tree of life and the tree of the knowledge of good and evil. Despite God's
warning that death would result, Adam (following Eve) ate of the fruit of the
knowledge of good and evil; and of course to know good and evil is to know
evil, and to know evil is to experience it. Given the basic truth of the
doctrine of original sin or universal depravity, that all men have shared
significantly in the transgression of Adam, all humans have experienced evil
directly both as perpetrators and victims.
Inhabitants
of a world fallen and cursed by sin, we are now in the second phase of God’s
creative program. Having partaken of the knowledge of good and evil, we still
operate with free will but with the added "advantage," so to speak,
of being better (but still not completely) informed. Experience has taught us,
i.e., Christian believers, that sin breeds more pain than pleasure in this
life, and death at the end of it. Equally we have tasted of the forgiveness of
sins, the liberating life of God in Christ, and the comforting ministry of the
Holy Spirit. For believers, then, the innate human appetite for evil has been
weakened and becomes ever weaker with our growth in the faith. Replacing that
old craving for transitory pleasure is a desire for the eternal knowledge of
God himself, the very source of all good things. On such a view, this-worldly
existence is necessary as the arena in which eternally binding choices are
made, and where evil – especially the irrational, excruciating sort we prefer
to call pointless and gratuitous – serves as a powerful inducement to seek God
rather than sin. "So things that contribute," says Stump, "to a
person's humbling, to his awareness of his own evil, and to his unhappiness
with his present state contribute to his willing God's help." She then
concludes that "moral and natural evil make such a contribution."[8]
Jesus said simply, "Blessed are the poor in spirit, for theirs is the
kingdom of heaven."[9] In a fallen world, if no other, we are able to
hear, freely and clearly, the divine call to repentance from sin and ongoing
faith in Christ. Evil in that case might not be a senseless aberration from
God's creative-redemptive plan, but an essential part of it.
Nonetheless,
the third phase of God's creative-redemptive design alone will bring about the
completeness we seek. Only in the future, final consummation of God’s plan will
we realize how one can remain ever free to love God and others but never free
to become evil. Although the logical compatibility of evil and divine
benevolence, of free will and eternal blessedness, cannot be strictly proven
within the system of this world, Scripture posits its provability in the larger
transcendent system of the kingdom. In the eternal kingdom of heaven theodicy
will be completed. But of course no theodicy will be necessary. God will wipe
every tear from our eyes and every trace of evil will have vanished away
forever, not in violation of our free will, but in the divine response to it.
This may explain why there is no tree of knowledge of good and evil in the
heavenly paradise of Revelation 22 – only a tree of life. Having already tasted
the bitter fruit of the knowledge of good and evil, and as a result having
freely renounced sin and embraced eternal life in Christ by faith, we will
enter the New Jerusalem prepared to joyfully partake of the tree of life
forever. Only then and there, in the eternal kingdom of heaven, will we
experience the culmination of both genuine freedom and everlasting joy.
[1] The idea
goes something like this: For any system based on formal language L, there will
be a self-referentially true statement G coded in L such that neither G nor
not-G is provable in the system. G, then, is true but formally undecidable.
Either the system is incomplete with respect to the truth of G, or the system
is inconsistent (consistency here means that in principle no statement can be
derived that is both proven and disproven via the axioms of the system). But
since G is true (as can be proven in principle by expanding upon the system to
include true axioms bearing on the truth of G), the system must be incomplete
with respect to the truth of G.
[2] The illustration as I describe it is a
condensed and modified version of Rucker's step-by-step explanation depicting a
"Universal Truth Machine" and Gödel himself as its operator in Rudy
Rucker, Infinity and the Mind (New York: Bantam Books, 1982), p. 174.
[3]
"Reflection principles provide the very antithesis of formalist reasoning.
If one is careful, they enable one to leap outside the rigid confinements of
any formal system to obtain new mathematical insights that did not seem to be
available before." – Roger Penrose, The Emperor's New Mind (New York:
Oxford, 1989), p. 144.
[4] Stephen
Hawking, "Gödel and the End of Physics," lecture given at the Dirac
Centennial Celebration, Cambridge, UK, July 2002, http://www.hawking.org.uk/godel-and-the-end-of-physics.
[5] Thomas
Nagel, The View from Nowhere (New York: Oxford University Press, 1986), p. 26.
[6] Eleonore
Stump, "The Problem of Evil," Faith & Philosophy, Vol. 2, No. 4
(Oct. 1985), p. 398.
[7] For a
comprehensive survey of historical and contemporary theology of the kingdom of
God, see Mark Saucy, The Kingdom of God in the Teaching of Jesus (Dallas: Word,
1997).
[8] Stump,
"The Problem of Evil," p. 409.
[9] Matthew
5:3, New King James Version. The New Century Version describes these poor as
"they…who recognize their spiritual poverty." Presumably they belong
in the same spiritual category with those who mourn, the meek, those who hunger
and thirst for righteousness, the merciful, the pure in heart, the peacemakers,
and those who are persecuted for righteousness' sake, Matt. 5:4-10.
