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James B. Miller, Ph.D.
AAAS Dialogue on Science, Ethics, and Religion
Washington DC 20005
Introduction
This paper is not a formal theological analysis of the
Christian doctrine of sin much less an exploration of theological anthropology
generally. Its more modest aim is to demonstrate how the use of mathematics
as metaphor can be of heuristic value in an effort to understand a theological
concept which often seems paradoxical when described in ordinary or traditional
language. Given this disclaimer, it is perhaps also useful to make several
significant distinctions.
Mathematics functions in science as the language of models.
It is important for present purposes to distinguish between models and
metaphors. Ian Barbour identifies three types of models used in science:
experimental, logical and mathematical.[2]
The last, he suggests, falls between the first two. A mathematical model
"is a symbolic representation of particular aspects of a physical
system, and its chief use is to predict the behavior of the latter."[3]
As a subclass geometric models are symbolic "pictures" of the
relationships of the system. The relationships which are symbolized may
be physical or conceptual.
A metaphor may be defined as "a figure of speech in which a word
or phrase literally denoting one kind of object or idea is used in place
of another by way of suggesting a likeness or analogy between them."[4]
It is the nature of the metaphor to be both like and not-like the other
to which it relates. Part of the power of insight which a metaphor provides
derives from this tension of likeness and unlikeness. I am confident that
this tension will be evident to theologians and mathematicians/scientists
alike as they consider the following.
In light of these distinctions this paper is not proposing
a mathematical model of sin by which we might be able to predict the moral
quality of human behavior. Instead, this paper is proposing a mathematical
metaphor which may illumine the conceptual understanding of sin.
Issues
The biblical account of the nature of sin and the relationship
of the sinner to God exhibits a number of apparent paradoxes. It is said,
on the one hand, that the sinner is a slave to sin and, on the other hand,
that absolute obedience to God is freedom (Rom. 6:16-18, 22; Gal. 5:13).
It is said that God's way is narrow, that the way of sin is wide; and
yet Gods' way is said to lead to abundant life (Matt. 7:13,14; John 10:7-10).
To the extent that one can associate freedom with creativity, these teachings
about sin also seem to imply that being committed to God's"narrow"
way makes possible the greatest creativity.
It is my judgment that these apparent paradoxes are in
part an artifact of the natural languages by which these moral insights
have traditionally been expressed. It is my proposal that the use of a
particular mathematical form as a theological metaphor can provide a more
coherent expression of these insights.
At the same time it is also clear that mere error is
not sin. That we are finite creatures, limited in our knowledge, and therefore
prone to make mistakes, does not in itself account for our state of sin.
Yet it is also said that we have no excuse for our sin because the way
of God is clear (Rom. 1:19,20). The suggestion is that there is a natural
morality does not require any "special" revelation in order
to discern the good that God would have us do. Again, I will propose that
the mathematical metaphor discussed below allows for both a distinction
between error and sin as well as illustrates how moral discernment can
function without recourse to specially revealed moral injunctions.
Lastly, the biblical account of the presence of sin in cosmic history,
especially as it has been interpreted through the Augustinian tradition,
involves both a particular understanding of natural history and a "genetic
theory" of the procreative transmission of moral flaw. These understandings
are no longer viable in the light of contemporary natural history and
biological science, particularly the history of human origins and development.
The tradition assumes a pair of original ancestors for all humanity who
in some primordial time by virtue of their disobedience to God acquired
a Lamarckian-like genetic flaw which has since been passed on to all future
generations thereby morally disabling the human character.[5]
Further, when human's "fell" from a state of primordial moral
purity it is said that they dragged the rest of nature down with them,
the chief mark being death and bodily decline and decay (Rom. 8:18-22).
Yet, even a cursory understanding of contemporary evolutionary history
and anthropology indicates that humanity emerged as a group not as a pair
of individuals to whom all subsequent humans are related genetically.
Further, aging, physical decline, death and decay did not come on the
scene only with the appearance of humankind.[6]
This suggests that whatever their value as contributions to theological
anthropology, the biblical and Augustinian traditions cannot be taken
today as adequate accounts of the origin of sin. Again, I propose that
while the mathematical metaphor to be considered does not completely resolve
all questions concerning "original sin", it does provide a conceptual
context amenable to contemporary natural history and within which such
questions can be explored more coherently.
The Gaussian Distribution[7]
The mathematical form which is the basis of the metaphor
I am proposing is the "Gaussian distribution" or what is often
called the "normal distribution" curve (see Figure 1). It is called "normal"
in the sense that it represents the norm or idealized form of distribution
for a set of random or complex variables (e.g., the set of heights of
all living men). The curve is bell-shaped in that it is symmetrical about
its mid point.
The "mean" of the distribution is the arithmetic
average of the set of variables. It forms the center-line of the curve
and intersects the curve's maximum point.
The "standard deviation" is a measure of the variance
of particular values of the variables from the mean. Approximately 68%
of all the set falls in the region which is plus or minus one standard
deviation from the mean.
The Gaussian Metaphor
In order to transform this well established mathematical
form into a theological metaphor let us begin by assuming that the curve
is drawn in what might be called "decision space." By this I
mean that it encloses the set of real possible decisions which lie before
an individual at any particular decision moment in life. The area above
the curve would be the region of imaginary decisions; that is, decisions
which are conceivable but not actually realizable (e.g., a decision which
would require the actualization of both x and not x). (see Figure 2)
Now let us add a moral dimension by identifying the x-axis
as a measure of variance in the moral quality or value of the decisions
(see Figure 3). The y-axis is taken to represent the total number of particular
decisions which could be made exhibiting a particular moral value. This
assumes that two or more decisions may be morally equivalent.
Let us now define the mean as the qualitative moral intention
of God for any particular decision moment or, in other words, the "will
of God" (see Figure 4). The value of other possible qualitative sets
of decision then vary plus or minus from the divine will. Aristotle argued
that moral action lay between extremes of excess and deficit, at the "Golden
Mean." In the Christian context it might be said that distinctive
human moral agency lies between sheer animality and the hubris which assumes
a divine-like status.
One further characteristic of the distribution needs
to be noted. The standard deviation is a measure of variance related to
the distribution of the whole set of variables. Let us define the area
plus or minus the standard deviation about the mean to be the "domain
of ignorance" (see Figure 5). By this designation I mean to suggest
that it is the domain in which the limits of finite human understanding
do not admit to a clear or unambiguous assessment of the moral quality
of a particular decision. It is the domain of moral error.
A Reflection on the Issues
In the light of this metaphorical use of the Gaussian
distribution several observations may be made in relation to the issues
identified earlier. First, you will note that there are several elements
of the metaphor which are fixed or determined prior to any decision being
made: the shape of the curve or configuration of the decision space, a
certain finite set of possible decisions, a set of decisions which are
qualitatively the "will of God." At the same time there is no
prior determination of which particular decision will be made. Even if
one supposed that that decision would be congruent with the divine will,
that still would not determine which particular decision would be made
since there is a set of decisions which are all morally congruent with
the divine will. Thus, one can see how the divine establishment of a moral
order does not preclude human freedom or creativity.
But secondly, even more, it is evident from this metaphor
that obedience to God, seeking to enact the divine will admits to the
greatest freedom. Of all of the qualitative sets of decisions, the one
which corresponds to the divine will contains the greatest number of members.
Since the set of decisions which is the will of God is but one set among
many it can be said that the divine way is narrow while at the same time
it permits the greatest freedom by virtue of containing the greatest number
of possible decisions.
Third, the metaphor makes clear that there are sets of
decisions so variant from the divine will that any person of good intent
would have to acknowledge their perniciousness (see Figure 6). I would
suggest that to maintain that such decisions are "moral" requires
substantial willful rationalization. At the same time these sets represent
a smaller proportion of decision space than those sets which fall within
the domain of ignorance. It is in this latter domain that finite human
moral insight is more ambiguous and that people of good intent will differ
concerning what is qualitatively the best decision. But if there is this
ambiguity due to moral ignorance, how can one be held morally responsible
for one's actions? In a mundane sense just as one can be held legally
accountable for the consequences of an accident, so one can be held morally
responsible for the consequences of moral error enacted out of ignorance.
But it is important to note that such error is not in itself sin. In the
light of careful reflection on a decision which has been made and its
consequences, one should gain a sense not only that it is at variance
from the divine will but also the "direction" of the error so
that there is a clue as to what moral quality needs to be enhanced in
future decisions. It is only if one rejects the insight gained from such
reflection that one is engaged in sin. Thus, I am taking sin to be the
willful rejection of the will of God, not simply moral error.
If one commits oneself to the judgment that one's moral insight is unambiguous,
that one perceives with absolute clarity the divine will, then it is likely
not only that one is engaged in sin but further that one has established
a skewed perception of decision space, a perception which is morally distorted
(see Figure 7). The metaphor assumes that there is at least a tacit apprehension
of the Gaussian form of decision space. But if that is the case then the
moral distortion which results exhibits two forms of captivity or the
diminution of freedom. On the one hand, there is the loss from perception
of some real possible decisions including particular decisions which would
correspond to the divine will. But, in addition, such a skewed perception
would include imaginary possibilities which can not in principle be actualized.[8]
Lastly, although this metaphor assumes that there are
creatures (Homo sapiens) that have emerged in the process of the
history of the cosmos to possess a moral sense that functions in relation
to such decision space, the metaphor is not necessarily restricted to
the human context. In one respect it could be applied wherever there is
some degree of moral freedom; thus, it could be applicable to other terrestrial
life forms to the degree that they possess some form of moral sensibility.
Further, beyond the context of moral agency as such, to the degree that
any entity in the cosmos possesses some degree of self-determination,
of freedom, then the curve encloses not "decision space" but
"actualization space." The height and breadth of the curve can
be adjusted to reflect the relative freedom of whatever entity is under
consideration. Although it could be asserted that entities among non-human
nature simply conform to the divine will in a moral sense even if in an
indeterminate manner, the metaphor suggests that there is the possibility
of something like moral variance even for such entities. This would further
suggest that, while the human moral context is an emergent character within
the genus Homo, this character is not not ontologically unique
in the sense that it has no precedence or roots in other species of existence.
Conclusion
What might be concluded, then, from this exploration
of a mathematical form used as a theological metaphor? I believe that
this discussion has demonstrated that such use can have heuristic value
in understanding theological insights which can be expressed only paradoxically
in natural language. However, I also believe that this metaphor reinforces
the idea of humility as a virtue, particularly in the moral realm. If
most of our moral decision making occurs within a domain of ignorance,
if we are all likely to be engaged in moral error, then the assertion
of moral certainty or worse, moral authority, undermines the capacity
for the sort of moral reflection which would tend to overcome error. In
such a way, error can be transformed into sin because it can not be acknowledged
as error. On the other hand, humility does not enervate moral decision
making. Instead, by opening moral reflection to the widest consideration
of the values and consequences of one's decisions, especially as conveyed
by those affected by such decisions, such humility leads toward the possibility
of achieving greater moral value or congruity to the will of God.
Let me end with the suggestion of one possible final
insight drawn from the metaphor which deserves much deeper reflection.
A perennial theological problem has to do with how God can be sovereign
over history and yet allow creatures to have genuine freedom. This metaphor
may be considered in a providential light if it is viewed as a probability
distribution of the actual decisions which will be made in the formation
of cosmic history. The metaphor suggests first that those decisions which
correspond to the divine will have the greatest probability. With respect
to any particular course of action, creatures acting freely are most likely
to act in accord with the divine will (though of all the acts, that set
which is fully in accord with God's will though it has the highest probability
is numerically small relative to all possible acts). At the same time,
all variants are variants in relation to the will of God and so possess
some element of the divine intentionality even if in a diminished fashion.
This would suggest that there may be a kind of averaging among the variants
across the space such that the effective action is that which is congruent
with the divine intent. This would still allow freedom across the system
but also assure an end which conforms to God's will. However, this is
a metaphorical exploration for another time.
Notes
- This paper was originally presented at the 7th European
Conference on Science and Theology, March 31-April 4, 1998, Durham,
England. [Return to Text]
- Ian Barbour. Myths, Models and Paradigms (San
Francisco: Harper & Row Publishers, 1971), p.29f. [Return
to Text]
- Ibid, p. 30.
[Return to Text]
- Webster's New Collegiate Dictionary 6th
ed. (Springfield, MA: G. & C. Merriam Co., Publishers, 1959),
p. 528.
[Return to Text]
- The biblical roots of this view are almost exclusively
found in the New Testament (e.g., Rom. 5:12-19). It has been observed
that "original sin" in this sense is not found in the Jewish
traditions. Likewise, Islam does not have such a tradition.
[Return to Text]
- In fact it has been suggested that it was the massive
terrestrial extinction 65 million years ago which brought the age of
the dinosaurs to an end which made possible the further evolution of
our very early mammalian ancestors.
[Return to Text]
- This mathematical form was discovered by Abraham
de Moivre (1667-1754) in 1733 and studied by Pierre-Simon Laplace (1749-1827).
The name of Carl Friedrich Gauss (1777-1855) has come to be associated
with the curve due to his refinement of the mathematics of the curve
for use in error analysis.
[Return to Text]
- Such an example might seen in the behavior of those
who seek to engender respect for all living things in an environmental
context but are willing to discount the life of a fetus in the context
of decisions concerning the termination of a pregnancy; or conversely,
the behavior of those who would allow the death of innocents in relation
to warfare or industrial development but would then assume the absolute
inviolable sanctity of fetal life. I would suggest that each of these
positions exhibits the sort of moral incoherence that points toward
a skewed moral perception.
[Return to Text]
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