Centered Sets vs Bounded Sets – 40 Years Later

Forty years ago, Dr. Paul G. Hiebert wrote an article called “Conversion, Culture, and Cognitive Categories” in Gospel in Context that introduced a pair of abstruse mathematical concepts from non-cantorian set theory to the world of missiology. These two concepts, the Bounded Set and the Centered Set, sparked a quiet revolution in missions as pastors, theologians, and missiologists confronted Hiebert’s question. “Should Christianity be viewed as a bounded set or a centered set?”

Forty years later, it is worth trying to clarify and expand upon the discussion of bounded and centered sets that Dr. Hiebert so innovatively introduced. The concept of bounded and centered sets are amazing tools. As with all tools, they are good at some things, and bad at others. A hammer is great for pounding in nails and for tearing down walls. It is terrible for painting. We will start out by defining the terms, and then by looking at the strengths and weaknesses of each tool.

Bounded Set – A Cat is Not A Dog

A bounded set is a collection of all objects which possess the defining characteristic(s) which determine the membership of the set.

A bounded set is a very easy concept to grasp. It is humanity’s default conceptual grouping. “Dog” is a bounded set. Every object in the universe can be classified as either a dog or not a dog. Different people (and different ages of people) may classify things differently, but everyone has a conception in their head of what a dog is. There are some “defining characteristics”. We do this every day in very complex ways. For example, a 3 legged dog is still a dog while a four legged fox or wolf is not. “Food” is a bounded set.

A bounded set is defined by its boundaries. The boundaries define what is in the set, and what is not in the set. This is simplest example to see is math. A set of numbers is bounded if it has a maximum and a minimum value, which make up the boundaries. The set of all numbers between 1 and 3 is a bounded set. Any number you can think of is either between 1 and 3, or it is not. That’s a very simple boundary. Think of another set “friends”. Take a moment to think of the people who are your friends. Could you write a list of characteristics that make those people and their relationship with you different from all other people who are not your friends? That is a much more complicated boundary!

There are a few important facts to notice about bounded sets. First, a bounded set is defined by its boundaries and the position of an object with relation to those boundaries. We will often use spatial / locational metaphors as we talk about membership in a bounded set. An object is either “in” or “out”. It is very important to note that there are no objects that are both in or out. There is no gray area in a bounded set, nor is there partial credit. A cat is not “almost” a dog because they are both small, furry mammals, omnivores, and have four legs most of the time. A cat is never a dog.

Second, a bounded set is a static set. That’s a corollary of being based on position. Locations change over time, but at any one moment, they are static. The bounded set “children living at home with their parents” is filled with the children who live at home right now. Children who moved out yestereday or who will move back tomorrow are not in the bounded set today.

Third, the friendship group earlier is a great example of the possible complexity of the defining characteristics of a bounded set. The characteristics are limited only by the ingenuity of the set’s creator. Facebook advertising campaigns create bounded sets called marketing segments. A marketing segment for a Valentine’s Day promotion might be based on gender, relationship status, profession, ethnicity, and religion.

That leads us to point 4, which is closely related to point 3. Regardless of how complex or simple the defining characteristics are, if the boundaries are unclear or ambiguous, the set becomes ill-defined and contentious.  The classic example of the failed bounded set is the category called “race”. How does one define “race”? By appearance? By heritage? How far back does one have to go? A bounded set is defined by its boundaries. If the boundaries aren’t clear, neither is the set.

Finally, bounded sets emphasize similarities and differences in essential characteristics. They emphasize the differences between the “in” group and the “out” groups. Bounded sets also emphasize the similarities among the members of any group. This is very apparent when people talk about politics. “Oh, that person is a Conservative. All conservatives support that issue, I’m sure they do too.” Or “That person is part of the Labor party. I don’t understand how they can believe that. They are so different from us.”

Bounded sets are great tools when the difference between in and out is clear and important. They are most powerful when the differences are essential or ontological differences. Differences that will never change over time. Bounded sets are a great tool when the list of criteria for membership is clear and can be stated without ambiguity. But the bounded set does not function well when similarities between groups are important, when there is a lot of gray area, or when there are many unclear boundary cases. Bounded sets are also not dynamic. They are not the best tools to visualize change.

Centered Set – Trending Stories

A centered set is the collection of all objects moving towards a well-defined center.

The centered set is also relatively easy to grasp. Spatial language makes an intuitive understanding simple. A bounded set is based on location. It is “in” or “out”. A centered set is based on direction. Each person is either moving “towards” or “away”. A bounded set is defined by its boundaries. A centered set is defined by its center. The centered set can be thought of as the inverse of the bounded set.

Forty years ago, Dr. Hiebert struggled to find examples of centered sets, but many are available today. YouTube’s “trending videos” are a perfect example of a set centered on popularity. YouTube has an algorithm that decides which videos are growing more popular. A station with only a few thousand views can be trending up. More and more people are watching the video every day. On the other hand, a massively popular video with hundreds of millions of views may be trending down because less and less people are watching the video every day. That’s the key difference between the bounded set and the centered set. The bounded set is based on location (e.g., videos with more than a 100 million views), but a centered set is based on direction (e.g., videos with 20% more views today than yesterday).

There are several key points to notice about the centered set. First, a centered set is defined by its center and the direction of objects with relation to the center.  The center must be clearly defined, and its position is very important. But the location of the objects are irrelevant. An object may be 10 meters away from the center, or 10,000 kilometers. The only relevant factor is whether the object is moving towards the center. That implies that the direction of the object, in a conceptual sense, must be measurable. If we can’t define the center and measure direction, then the centered set approach does not make sense. This is an easy task when we talk about videos trending towards popularity. It is much harder to do with “Christian orthodoxy” or “artistic perfection”.

Third, the farther a person is from the center, the less precise the definition of the center and the measurement of their direction can be. If a person is in Spain, and the center of the set is the Apartheid Museum in South Africa, the exact address and the exact direction don’t really matter. If the person is heading generally south, they are heading towards the center. However, if a person is already in Johannesburg, then they are going to need the address and the determination of their direction must be much more precise to be useful.

Finally, centered sets emphasize similarities and differences in goals or destinations. Where a bounded set emphasizes similarities and differences based on what something is like right now, a centered set emphasizes similarities and differences in what something will be like in the future. If “stable democracy” is a center, then it may be far more important to understand if a government is moving towards or away from stable democracy than trying to define precisely what a stable democracy is and list which governments are and are not members.

Centered sets are great tools when future changes are more important than current conditions . Centered sets are most powerful when characteristics are mutable or non-ontological. In other words, in situations where change can and will happen. Centered sets function well when the center is clear and unambiguous, or when possible boundaries are unclear and ambiguous. The exact location of the border of Ukraine and Russia may be in doubt in some locations, but the capitals of Russia and Ukraine are not. The centered set does not function well with static categories (there is no change to see), or when group cohesion is very important. Black and white categories do not do well with centered sets.

The Next Step

Forty years ago, Hiebert offered the following words for our consideration.

“What does it mean to be a Christian? Before we can answer this question we must look more closely at our own thought patterns—at what we mean by the word “Christian.” This word, like many other words, refers to a set of people or things that we think are alike in some manner or other. It refers to a category that exists in our minds. To be sure, God, looking at the hearts of people, knows who are his. It is he who one day will divide between the saved and the lost. But here on earth, we as humans pass judgments, we decide for ourselves who is a Christian, and, therefore, what it means to be a Christian. What criteria do we commonly use? Before we answer this question, we must ask an even more fundamental question: what kind of category are we going to use? … Ultimately the question of whether we should see the term “Christian” as a bounded or as a centered set must be decided on theological, not pragmatic principles. But this demands that we think through all of the basic theological terms and decide which of these should be viewed as bounded sets, and which as centered sets.”

That is a significant challenge. In the next post, we will look at the conclusiosn that Hiebert drew and reexamine the implications of viewing Christianity as a bounded or centered set.

The Management Paradigm for Mission: Providing the Right Answers to the Wrong Questions

Last time, we investigated the introduction of the management paradigm into the world of mission. Here’s a link if you haven’t read it yet.

Dr. McGavran was not a revolutionary thinker. He took an incremental step that was perfectly in line with American culture. It was pragmatic, empirical, and data driven. He continued the relatively recent Western tradition of thinking of the Great Commission as a matter of method, technique, and strategy. As soon as a connection / correlation is made between technique or method and spiritual or church growth, the modern Western mind can’t help but try to optimize the solution. In other words, we jump from correlation to causation. It connects with so many of the deep-seated elements of our worldview: the desire for control and self-determination, the idea of progress and optimization, and the worship of science and data.

We are inexorably drawn by our consciences to the science of management, in particular, project management. To do any less, from our cultural viewpoint, would be the height of irresponsibility, if not outright laziness. The science of project management has changed the world. The scale and efficiency of the modern corporation is unimaginable without a science dedicated to completing projects on time, on budget, and up to standard. The famous triple-constraint requires us to balance quality, budget, and timeframe to find an acceptable and realistic compromise. We must maximize impact, provide a good ROI, increase efficiency, measure indicators, develop strategies and 5-year plans. Management, both as a discipline and a sub-culture, has developed a language all its own. This language is found in the executive committees of corporations, mission agencies, megachurches, and Christian development agencies all over the world.

But is management the right paradigm for mission? If there is one principle that missiological training is supposed to ingrain in us, it is deep suspicion of the places where the gospel and our culture fit too snugly and comfortably together. That is certainly the case here! In fact, this idea is so embedded in our worldview that it is almost impossible to imagine alternatives. And if we had an alternative, how would we decide which one was better? And how would we make that decision? Using return on investment, success, and efficiency, of course!

And that is the problem with the management paradigm. It defines the problem on its own terms, and then provides us with the right answers to, what I believer, are the wrong questions. The questions it asks are management questions. Questions like:

‘What methods of evangelism are most effective at converting Muslims in Southeast Asia?”

“Missionaries from what background and ethnicity will provide the most impact for the kingdom at the least risk?“

“What are the emerging markets for Christianity in Pakistan and how can they be exploited?”

These are great management questions, and they lead to great management answers. But are they the right questions?

The Case Against the Management Paradigm

It is surely noteworthy that the 1974 Lausanne Conference not only marked the global debut of Dr. McGavran, but also the global debut of C. Rene Padilla, the coiner of the term “integral mission”. Dr. Padilla would spend the next 40 years questioning and critiquing the church growth paradigm. Driven by the Latin American experience of evangelism, this group of Latin American theologians couldn’t accept the presuppositions which drive Church Growth Theory. Comparing paradigms is incredibly difficult. But we are given some guidance in the Scripture on how to deal with new teachings. We are taught to compare the new with what has gone before. That is what I will attempt to do in this section.

The Premise that Can’t Stand

One can only manage what one can control. Where there is no causal relationship between method and output, there is nothing to manage. That is the linchpin of the case. Do my methods cause spiritual growth? That is the hardest question in the world for a Western person to answer in the negative. A million objections immediately come to mind:

  • What about bad methods that always fail?
  • Are you saying the method doesn’t matter? That’ ludicrous!
  • Isn’t there proof that seeker-sensitive churches grow faster?
  • I have read the research!

But the questions Paul Hiebert posed to the leadership of his denomination, the Mennonite Brethren, in his concluding remarks evaluating church growth are more relevant and incisive than our knee-jerk reaction..

Is Church Growth in danger of over-contextualizing the gospel in a modern cultural setting? The gospel must be contextualized—in other words, it must be understood clearly in each cultural setting. But, we as Anabaptists believe, it must also be prophetic—seeking to transform that culture in line with the standards set by the Kingdom of God. The church is always in danger of letting the context set the agenda rather than of calling that context to change. One of the hallmarks of modernity is a mechanistic, technological approach to reality (Berger 1974, Ellul 1964). In the natural sciences this has led to factories and an engineering mentality that seeks to control nature. In the social sciences this same technological approach has led to bureaucracies and an engineering approach to human beings. This is seen, for example, in the M.B.O. (Management By Objective) style of management found in modern businesses. Goal setting, progress reports and amoral methods are characteristics of this culture. To what extent can the church buy into this culture and still remain the church? At what point, in seeking to contextualize our church planting, have we lost the heart of our message and become a Christian club?

Paul G. Hiebert, “An Evaluation of Church Growth,” International Journal of Frontier Missiology 33, no. 2 (Summer 2016): 81.

I think the Bible has some fairly clear things to say on the subject, once we step out of the Western cultural frame of reference. What is the main point of the book of Job? “My ways are higher than your ways, and my thoughts are higher than your thoughts.” Suffering is not always a reward or a punishment for our actions, and those who think it is will be judged.

“I planted the seed, Apollos watered it, but God has been making it grow.” Who does the saving and the growing in the New Testament? God. End of sentence. The Apostle Paul was one the greatest evangelists of all time. He also wrote Romans 9, the gist of which is God is in charge of salvation. I’m not trying to make some ultra-Calvinist, double predestination argument about the eternal destiny of humanity. I believe that is also going beyond what is written. Instead, I’m trying to point out that there is no place in the Scriptures where I have seen a clear causal relationship between methods and results.

ASIDE: According to the management or church growth paradigm, Paul can’t be considered the most successful evangelist in history. That honor is reserved for the great missionary Jonah. With unbelievable efficiency and effectiveness, his inspired message “Forty more days and Nineveh will be overthrown” caused the whole city of Nineveh (including the animals) to repent in sackcloth and ashes in a matter of days. The author knows of no attempts to replicate this hugely successful evangelism strategy, which seems to represent a large gap in the missiological literature.

In other words, mankind is not the cause, we are the means. And it almost seems like God makes sure that the means are all WEIRD, so that we don’t get the idea of trying to replicate them (e.g., when the Israelites tried to replicate the methods of the past by trotting out the Ark of the Covenant to win a battle in 1 Samuel 4). So we must make an important theological move away from “human actions cause salvation” to “humans as being the means by which God causes salvation”. Once we make that move, management for spiritual growth becomes an irrelevant paradigm because we can’t manage God. And that terrifies us because we are not in control.

The same principle applies to the sanctification of believers. We live in this place of fantastic tension between divine sovereignty and human responsibility, and nowhere is this tension higher than in evangelism and sanctification. We know that we must pursue sanctification, but Paul makes it very, very clear in Galatians that to pursue sanctification by the strength of our works is to fall into deadly folly. Our actions again appear to be means which God uses to achieve His ends, our sanctification. Doesn’t that speak directly to your experience? Have you ever become more holy, more like Jesus because you worked harder at it? I know I haven’t, and I’m not super happy about it. Can’t I at least be in control of my own spiritual development?

A Better Paradigm

The final caveat is that, while we don’t control sanctification or salvation, it does seem that we play a role in helping (and hindering) it. That’s God calling us to be ambassadors in 2 Cor 5:20, it’s Paul asking how people will believe if no one preaches (Rom 10:14). We have agency, but we don’t have control. That’s a key difference! The reductionists among us will take that agency and bring us right back to management because it amounts to the same thing, right? We are just taking care of our part as efficiently as possible! But that approach cuts against the grain of Scripture. A better paradigm is proposed by P.eter. “Each of you should use whatever gift you have received to serve others, as faithful stewards of God’s grace in its various forms.” (1 Peter 4:10) When we don’t have control, faithful stewardship is a far better paradigm than effective management. The right metaphor is the master carpenter and the apprentice. The apprentice has to be ready to fetch wood and help with assembly and other tasks, but those tasks don’t determine the outcome. They are valuable contributions which help the master do something He could have done Himself. It is the master, not the apprentice, that sets the pace and determines the outcomes.