The Quest for a Quantitative Social Science:
A Problem for Christians?
James Bradley
Dept of Mathematics & Statistics
Calvin College
Much of this paper was originally
published in Mathematics in a Postmodern Age: A Christian Perspective,
Russell W. Howell and W. James Bradley, eds., ©2001
Wm. B. Eerdmans Publishing Co., Grand Rapids, MI, 800-253-7521, www.eerdmans.com
<http://www.eerdmans.com>
Historical background
Perhaps the most
influential mathematician of the twentieth century was John Von Neumann. With Oskar Morgenstern, Von Neumann
co-authored one of the most formative books in twentieth century social
science, The Theory of Games and
Economic Behavior. In the first chapter, Von Neumann and Morgenstern ask
why mathematics has not been as successful in economics as it has been in
physics, chemistry, and biology.
They argue against several commonly offered reasons, such as the
presence of psychological factors in economics, the absence of measurements of
some important factors, and the discrete nature of economic quantities. They conclude that
The reason why mathematics has not been more successful must, consequently, be found elsewhere. The lack of real success is largely due to a combination of unfavorable circumstances, some of which can be removed gradually.
The unfavorable circumstances they list are that
"economic problems were not formulated clearly," the
"mathematical tools were seldom used appropriately," and "the
empirical background of economic science is definitely inadequate." Thus Von Neumann and Morgenstern lay
out their vision of a mature discipline of economics, characterized by careful
empirical observation, precisely formulated questions, and the fruitful use of
appropriate mathematics.
The desire to reproduce the success of
mathematical physics in the realm of the human and social sciences goes back as
far as 1630. That was when the
political philosopher Thomas Hobbes first encountered EuclidÕs Elements.
He was impressed that one could begin with the simplest propositions and
end, after a lengthy train of deductive reasoning, with results that were far
from obvious. Galileo used the
same methodology to develop his new science of motion. This provided Hobbes both with a model
of how to proceed and with ideas he could use to attempt a fully mechanical
explanation of sensation. His
views on the nature of man in society were published in the 1650s.
Hobbes claimed that it is in the nature of
human beings to act in ways that seek to maximize pleasure and minimize
pain. This quasi-quantitative
axiom underlies his science of man.
Seeking their own self-interest in society, individuals interact with
one another much like colliding particles. A competitive mechanistic outlook on society thus came to
replace the earlier organic view of society as an ordered community and became
axiomatic for later social theorists and political economists.
Following Hobbes, in England about 1660,
William Petty and his friend John Graunt began to analyze a wide range of
economic and demographic data quantitatively as background for forming rational
public policy. Mathematical
economics continued to develop in the work of various seventeenth- and eighteenth-century
thinkers. Particularly in France
and England there was a move to develop a mathematical theory of society and
political economy. Perhaps most
notable is Adam Smith, author of Wealth of Nations (1776), the Bible of all Western
industrial economies. From this
perspective, there are natural laws at work in societyÑpursuit of individual
self-interest and a self-regulating marketÑthat make economics as much a
deterministic science as Newtonian mechanics. Using these basic principles, social theorists explained a
wide variety of economic phenomena, thus providing a measure of justification
of these principles.
Perhaps the place these ideas were most thoroughly developed, however, was in the Soviet Union. The Soviets believed that natural laws underlay all social phenomena and that all decision making should be done "scientifically," that is based on such laws. For example, they believed they knew the "laws of war," and they used these laws to justify building an enormous nuclear arsenal during the Cold War.
What remained constant throughout these examples was the Enlightenment belief that our world, with its natural, social, and personal realms, is the sort of reality that is amenable to rational scientific and mathematical treatment. As one historian has observed,
... the essence of the Enlightenment was the belief that the
world could now be seen in mechanical terms and defined in mathematical language...all
reality...acts by natural laws, if we had eyes to see them. The world of human affairs, in sum, is
the same as the natural world because the same laws that govern each govern
all. The task for Enlightenment
thinkers, therefore, was to ascertain those general laws that governed reality
and then apply them to the various cases that came up, whether political,
economic, social, or religiou1]
Our
problem
The Enlightenment perspective on mathematized social science is problematic
for Christians. It posits that
the laws underlying human affairs can be discovered without reference to God.
These laws can then be used to order society.
In fact, the Enlightenment perspective implicitly sees human beings
as better off without religious ideas at all, at least in the public arena. Today, in spite of the fact that we have
moved to a "post-modern" era, this perspective still predominates
in most social science. Christianity,
on the other hand, posits that this world is not the way it is supposed to
be. Further, the reality of this
fallenness and some characteristics of its alternative are knowable only by
divine revelation Ð not by observation and induction.
How are we as Christians to function in such a climate? Post-modern thinking strongly repudiates
the hubris of Enlightenment rationalism and this repudiation makes it very
attractive. But post-modernism
in its anti-rationalism and its rejection of any sort of transcendence poses
equally serious problems for us. What can we do? I
am convinced that a Christian perspective can provide us, if not a map, at
least some clear guidelines. I'll
proceed like this. First, I'm
going to sketch some examples of the positive accomplishments of mathematization
over the last several hundred years. Next I'm going to examine some difficulties
with it. Lastly, I'll sketch
a Christian framework.
Some
accomplishments of mathematization
By "mathematization," I don't only mean the use of sophisticated mathematics to solve complex problems in the natural sciences. I also mean the adoption of a set of ideals and values such as the precise use of language, standardization and formalization of methods and procedures, abstraction away from particulars, careful reasoning, and impersonality in the sense of intersubjectivity Ð universal truths independent of any individual or group's experience. It's easy to find many examples of mathematization in the social sciences. So I think it may be more instructive for us to look at examples of mathematization in the everyday life of the Western world.
In Washington, DC, the streets are laid out in a rectangular grid. The east-west streets are numbered, the north-south streets lettered. The main entrance to the State Dept., for instance, is at the intersection of 22nd and C. This is quite different from older cities like Philadelphia or Boston. But Washington was designed by Pierre L'Enfant, trained in France and steeped in Enlightenment principles. The design of Washington is an important symbol Ð it embodies the vision of people like Jefferson, that this new democratic nation would be built on the basis of reason. And it certainly is a lot easier to find a location in Washington than in Boston.
Today, even in places that are not laid out using a rectangular grid, property boundaries are precisely defined and maps of them are included with deeds that reside in a government office. In fact, the entire globe has been laid out using coordinate geometry (originating with Descartes, a key pre-Enlightenment thinker) and, at least within North America, every property boundary is located within that grid. While this does not eliminate disputes over property boundaries, it does provide a means that can be used to resolve them peacefully.
Standardized units have been developed for length, weight, volume, and many other measures. Today we take this largely for granted. However, two hundred years ago uniform standards did not exist. For instance, there were separate units for measuring linen and silk. In France, each local geographical area had its own bushel. Standards varied (and disputes arose) as to how to handle the heap at the top of the bushel. The mathematics here is not difficult, but decimal units for currency, accounting, and standardized measures incorporate mathematical values of impersonality, abstraction, and precision.
Economic expansion drove most of these developments in the history of measurement. In order to do business with strangers, whose character was unknown, a basis for trust was needed. Standardized measures, quantification, and means to enforce honest measures provided such a basis. A medium for communication across languages and cultures was also needed; standardized measures and quantification provided this as well. Thus the distance that quantification bridged was both geographic and interpersonal. In short, in a large society where personal trust is hard to develop or where uniformity is needed for comparison purposes or cooperative ventures, mathematization provides a common impersonal basis for public decision-making[2]
Another area in which the ideals of mathematization have been felt is that of politics and government. Note that the ideals of mathematicsÑopenness, independence of knowledge from social class or positions of authority, impartial impersonalityÑcomport very well with those of democracy and the rule of law. Numbers have substantial rhetorical power. They appear independent of the passions so prominent in political debate, and they represent an ideal of diligence and careful analysis. Thus they are very attractive to decision-makers in a democracy; they provide a means for leaders to protect themselves from charges of bias, self-interest, or incompetence. Moreover, the impact of numbers has extended beyond rhetoric; they have advanced the very application of democratic principles. For instance, advances in accounting have helped root out corruption. Careful reporting of employment and housing statistics have called attention to systematic differences in treatment of women and minorities and have led to major changes. Thus, mathematization has contributed to a shift in the social basis for authorityÑaway from powerful elites and toward abstract concepts of equity. Mathematization and democracy have mutually advanced each other.
Governments and organizations now depend on enormous amounts of data for making informed decisions. Collection and analysis of such data has become a major concern of our modern Western world. The Inter-University Consortium for Political and Social Research is one group that provides an on-line archive for social science data. This is how it describes its holdings.
Beginning with a few major surveys of the American electorate, the holdings of the archive have now broadened to include comparable information from diverse settings and for extended time periods. Data ranging from nineteenth century French census materials to recent sessions of the United Nations, from American elections in the 1790s to the socioeconomic structure of Polish poviats, from the characteristics of Knights of Labor assemblies to the expectations of American consumers are included in the archive. Surveys, aggregate data, and computer-based teaching packages in various substantive areas are continually deposited in the archive by leading scholars around the world. The content of the archive extends across economic, sociological, historical, organizational, social, psychological, and political concerns. Topical expansion is taking place to include urban studies, education, electoral behavior, socialization, foreign policy, community studies, judicial behavior, legislators (national, state, and local), race relations, and organizational behavior [3]
Note that all of these areas are described
by data and the
collection of data minimally requires the precise definition of
categories. It may also require
quantification and a fair amount of abstraction.
In short, mathematization has become deeply entrenched in Western
culture over the past 500 years or so.
It is a significant part of various dimensions of our daily lives as
well as of technologically sophisticated devices. Mathematical values of abstraction and impersonalness have
become major dimensions of contemporary culture. In addition to the widespread application of mathematical
content in science, technology, and everyday life, the methods and values of
mathematization have contributed to the way in which we think about issues and
seek for solutions to problems.
Not surprisingly, some deep and difficult issues accompany such an
extensive cultural impact. So
letÕs turn from successes to difficulties.
Some Issues Accompanying Mathematization
Our first issue is that using mathematics to address a real world problem shapes our perception of that problem. In other words, the use of mathematics in studying anything is neither epistemologically nor metaphysically neutral. Thus mathematization does not possess the neutrality and total objectivity for which Enlightenment thinkers had hoped. Consider the distinction that some social scientists (especially economists) frequently make, between "positive" and "normative" approaches to their discipline. The positive approach in economics means data collection and mathematical modeling (frequently based on regression) and analysis based on these techniques. That is, the positive approach seeks to merely describe "what is," using the tools of mathematization. The normative approach, on the other hand, seeks to prescribe "what ought to be." For instance, an economist in the positive tradition might approach the study of inflation by gathering large quantities of data and seeking to understand what factors influence inflation rates. Nevertheless, she would leave for the political process the normative question of what level of inflation in society is tolerable. Most social science research uses the positive approach. The positive approach lends itself well to the study of replicable events and concepts that can be well defined. But aspects of a situation that are hard to quantify are easily neglected. The fact that most results about human beings involve some interpretation by the researcher may be obscured. The element of intentionality Ð individual free choice Ð is usually ignored. Unique, non-replicable events are automatically excluded, as is any insight that expresses the observerÕs unique perspective. Thus mathematization is neither epistemologically or metaphysically neutral.
Note also that use of the positive approach is often accompanied by the value judgment that only knowledge accessible by this method is legitimate. When such a judgment is made, the post-modern criticism that claims of objective knowledge are often accompanied by exercises of power is clearly applicable.
Our second issue is that mathematization often implicitly introduces norms into situations. In democratic societies, norms are among the most powerful forms of social control that societies exert over individuals. [4] Quantification of human qualities inevitably introduces norms. The intelligence quotient or "IQ" is a good example; however, the same principle applies to all of mental measurements. The IQ test provides a means to map the intelligence of individuals (operationally defined by the test questions) into the positive real numbers. This mapping automatically gives us a norm for intelligence: if a is greater than b, then a is better than b. Measuring IQ by a single number implicitly assumes that intelligence is best represented by a linearly ordered set of numbers rather than by a more complex structure such as a partial order. In contrast, Howard Gardner, a leading psychologist, asserts that intelligence is multi-dimensional. However, even his approach still yields linear norms for each dimension.
Such norms are often helpfulÐfor instance, a low grade may motivate a student to study more effectively. A low quality of life indicator may stimulate a city to improve the environment for its citizens. However, those making such assessments are assuming a great deal of responsibility. It is clear that measurements prioritize aspects of a situation and alter people. It is not clear, however, what meta-normative principles ought to be applied to assess these mathematically generated norms.
Thirdly, mathematization has limits that are often forgotten. Suppose, as we have argued above, that mathematization is indeed the principal method that large democratic societies use to establish a basis of trust enabling transactions to occur among strangers. Mathematics limits affect the extent to which it can serve in such a critical role.
Specifically, mathematization usually (but
not always) requires the collection of data. However, in recent years, scholars have increasingly recognized
that sense data are an inconsistent and not fully reliable guide to knowledge.
One's prior understandings affect even perception itself. This situation is commonly described by saying that perception
is "theory-laden." For
example, if we look at the object on which we have been sitting, we perceive
a chair. Note we do not say that
we perceived certain colors, shapes, or materials. We report our perception as being a chair. However, imagine a person from a culture
that did not use chairs and had never seen one. In spite of the fact that the object being
observed has the same colors, shapes, and materials for him as for us, his
perception would
be very different. The reason
is that the concept of chair is culturally formed. So even our perceptions of social entities are culturally shaped. (Some commentators have captured this
idea with the phrase, "There are no immaculate perceptions.") Thus, empirical observations of social
entities can never be totally free of cultural influences, and thus neither
can our models.
Furthermore, many important social concepts are value-laden. As such, they do not lend themselves well to approaches that require impersonality, precision, and abstraction. Consider important concepts such as mental health, friendship, hostility, aggression, and even religion. It is hard to imagine how one could study mental health, say, without categories that distinguish healthy and unhealthy characteristics of personalities and/or behaviors. Such criteria necessarily entail value judgments. Consequently, in gathering data or "facts" about mental health, such facts (as well as the models based on them) are inseparable from values.
These are technical limitations. However, there is another type of limitation. For example, consider contemporary mathematical economics. Most economic models are built on the concept of "preferences," represented either as ordinal lists (a simple ranking) or as cardinal utility functions (numerical values indicating the strength of preferences). These notions have been made mathematically rigorous and powerful theorems have been proven about them. However, in the process of such formalization, the critical edge is usually lost. That is, preferences are taken as givensÐthey are not critiqued. When preferences are taken as givens, greed becomes undefinable. More generally, mathematization can easily be used to avoid consideration of issues of personal responsibility, ethics, virtue, and caring.
The last problem I want to consider is one noted by a French social thinker, Jacques Ellul. Ellul defines "technique" as a way of thinking that seeks to reduce activities to sequences of routine activities and then optimize them. While not explicitly mathematical, it instantiates the mathematical ideals of impersonality, precision, formalization, etc. Ellul not only sees technique as widespread in Western culture, but also views it as harmful. This conclusion of Ellul's may seem surprising, as technique appears to be morally neutral. That is, it is used simply to identify and optimize routine processes. However, technique is non-neutral because of the ways it shapes how people formulate and solve problems. Ellul argues that its appearance of neutrality is precisely what makes technique such a serious problem. That is, because technique explicitly excludes consideration of moral and religious values, its implicit values, productivity and efficiency, become the only values of its users, and as the only values, they become elevated to absolutes. Ellul is not arguing that efficiency and productivity are of no value, nor that inefficiency is preferable to efficiency or unproductivity to productivity. Rather he is arguing that values that ought to be regarded as means to an end have been elevated to ends in themselves. In the process, values commonly held to be ends in themselves, such as human dignity and worth, are overlooked and debased. In his book The Technological Society, Ellul provides an enormous number of examples of how such human values have been ignored in the quest for efficiency and productivity.
As we have seen, mathematization of
culture has provided some extraordinary social benefits. Among these are that it has provided a
basis for trust in large heterogeneous societies, has enhanced cross-cultural
communication, has provided some objectivity in political debate, has provided
a means for scholars to order and discipline their investigations, and has
given (via the vehicle of mathematical proof) a degree of certainty and logical
interconnectedness to some pieces of knowledge. On the other hand, it has often been accompanied by a neglect
of ethical and normative issues and has been used in ways that have ignored
limits intrinsic to the mathematical method and excluded the substantive issues
of values, purposes, and interpretation.
These actions have had significant consequences. Valuable sources of knowledge have been
neglected. Dimensions of human
thought and culture that are not primarily rational or empirical (such as
the arts) have been treated with disrespect.
Social sciences have been modeled on the natural sciences and in the
process have tried to avoid the critical matters of interpretation and valuing
of human qualities and behaviors. [5] Intrinsic human values have often been
neglected for the sake of productivity and efficiency.
Can Christian concepts help us assess and respond
to this complex situation? If
we consider only the abstract, formal aspects of mathematics, the answer is
"no" or at most "not very much." For example, the statement of the Pythagorean theorem is the
same for Christians and atheists, and its proofs are equally convincing to
the one as to the other. While
its content might lead a Christian to praise God for how He has ordered His
creation, the theorem itself seems only remotely connected to deeper religious
matters. However, there is another
sense in which the answer is "yes."
Consider this metaphor. For
a carpenter, there is no Christian way to cut a board (beyond the obvious
approach of using the best tools available and doing one's best work). Christians and non-Christians alike use
the same tools. However, it makes
a huge difference whether that board will be used to build a school or a brothel.
In the same sense, most of the technical
content of mathematics is immune from Christian influence. Nevertheless, for a Christian, extending
God's reign is an ultimate concern. For many Christians, this entails a concern with social structures
in this world and their effect upon people. Therefore, the larger question of what
kind of a culture people are building and how they are using mathematics to
do that is of extraordinary importance.
The focus of this talk has been on how mathematics has been used in
building Western culture since the Enlightenment. Christian concepts have a great deal to contribute to a discussion
of this issue. Here are my conclusions:
First of all, in spite of having thought about
this for several years, I can find no dimension of human life, of which I
can say, "This should not be studied with the methods of mathematical
social science." To me,
mathematics means primarily two things: precisely defining one's terms and
using careful logic. I see no
place where these tools are inappropriate as long as their capabilities and
limitations are recognized and respected.
That is, the problems arising from the Enlightenment perspective cited
earlier do not seem to arise from mathematical social science itself but from
its misuse. Consider the kind of harms listed above
- neglect of ethical and normative issues, of interpretation and values, of
limits, of other modes of thought, and attempts to establish the autonomy
of human thought. These are fundamentally
spiritual
matters, involving sins of pride and arrogance. I'm going to have to work with a very broad brush here, but
let me sketch some characteristics of a quantitative social science that is
not being misused:
Aristotle
believed that human beings possess an intuition that enables them to know
first principles about reality with certainty.
Such an intuition when coupled with deductive reason thus led to sure
knowledge. Since Galileo, Western
thinkers have tended to replace Aristotle's intuition with principles inferred
from data. There does seem to
be evidence from recent brain research that some mathematical and linguistic
qualities are indeed built into human brains, so Aristotle's notion here is
not groundless. Of course, the
fact that these qualities are "hard-wired" into our brains does
not in itself establish the fact that they correspond to reality in a meaningful
way. Furthermore, their scope
is limited; data can take us quite a bit further. But, as I argued earlier, our data can never meet the Enlightenment
ideal for objectivity. Thus,
especially in social science, there does not exist a presuppositionless starting
point for our deductions. [6]
For
most of the past twenty four hundred years - back to Euclid - the axiomatic
method has been viewed as a source of absolute truth. That particular belief is gone today, but an aura of rigor
and substance still surround axiomatics.
I believe in the method and its value. But I think we need to deromanticize it. It is simply a means of bookkeeping Ð
of organizing our knowledge so that we clearly identify our assumptions and
lay out the dependencies among their consequences.
We need to respect
other sources of knowledge such as intuition and divine revelation.
Since all of our data are "tainted" by our own experiences
and categories, the incontestable starting point for social planning that
Enlightenment visionaries sought does not exist.
Even our most careful analyses can lead us astray.
An attitude of humility plus a respect for other sources of knowledge
provide checks and balances. That
is, the web of our understanding of human phenomena is complex.
We
need to be more explicit and frequent in acknowledging the limitations of
our methods to ourselves and to those we teach. For example, there are other types of knowledge than the clearly
expressed principles scientists use as starting points for deduction.
For instance, one can certainly write propositions about God's trustworthiness,
but trust in God also has a dimension of "personal knowledge," a
quality that shapes more than our capacity to make logical deductions. It shapes our emotions, intuitions, relationships, and even
our perceptions themselves.
Christians can see both the natural and the mathematical world as having intrinsic value as part of God's creation. But the acquisition of knowledge and the use of reason about them are not ends in themselves. For much of the twentieth century, mathematicians sought to establish mathematics as autonomous knowledge, not dependent on God or even on science. For Christians, however, mathematics and science exist for the purposes of worshipping God and exercising stewardship of God's world. Practitioners are accountable both to God and their human communities.
Thus Christians have a
solid foundation from which they can reject both Enlightenment hubris about
human reason and post-modern anti-rationalism. Instead, by operating out of a posture of humility and service,
they are free to use mathematical social science to explore any dimension
of human life. What makes this
permissible is that they seek to build God's Kingdom, not their own.
1Wells,
Ronald A. History through the Eyes of Faith: Western Civilization
and the Kingdom of God, San Francisco,
Harper and Row, (1989).
[2]See Theodore Porter, Trust in Numbers: The Pursuit of Objectivity
in Science and Public Life, Princeton
University Press, 1995, for a detailed discussion of this matter.
[3] ICPSR. Guide to resources and services, 1994-1994. P.O.
Box 1248, Ann Arbor, MI, p. vii.
[4]See Theodore Porter [1995].
[5] For an excellent discussion of this issue, see
Evans, Stephen, Wisdom and Humanness in Psychology: Prospects for a Christian
Approach, Grand Rapids, MI, Baker Book House (1989)
[6] See, for example, Nicholas Wolterstorff, Reason Within the Bounds of Religion, 2nd edition, Eerdmans, 1984.