bounded and centered sets

Some are developing a new way to think about church; instead of thinking in terms of a solid wall or fence with “these persons are in – those persons are out”. Lots of people are doing this re-thinking. “Missional” and “Emergent” approaches are part of this.

We are being reminded that in some places (eg. farms in the outback of Australia) the focus is not on fences – the focus is on wells.

In the early church, the pre-Constantine church, it was clear who was in; and who was out. Christians met furtively and had a guard at the door. Beyond the “fence” of the Christian community the environment was hostile and persecuting.

The “Constantinian” approach (from the time of Constantine) had a similar clarity. Everybody in the “West” was “in”. Western civilisation was Christian civilisation. People spoke about a “Christian nation”… Beyond the “Christian nation” the world was resistant – that’s where you went on mission…

We live in a new context. Some still want to understand church using the pre-Constantinian and/or Constantinian models, making a list of beliefs and values, and if you tick all the boxes – you are “in”. The post-Constantinian context we find ourselves in, however, generally has more fuzzy edges. People are less “fixed-menu” Christians; and more “pick-and-mix from the spiritual deli”. There is some hostility, indifference, and supportiveness “within” and “outside” the fence-paradigm church.

People like Paul Hiebert, John Wimber, Miroslav Volf, the Vineyard churches, and others, are rethinking the paradigm. Paul Hiebert, in Anthropological Reflections on Missiological Issues, is a good example of using a mathematical model that has been adapted to sociology and applying it for mission.

He writes of a “bounded set”, where things are either “in” the set or “outside” the set. He contrasts this with a “centered set” as being defined not by its boundary but by its centre, and objects are “in” the set if they are moving towards the centre. Jesus is the centre. We hope to be turned towards Jesus, moving towards Jesus – the “distance” from Jesus is not important; the direction we are moving is.

I think there is a lot of value in the concept; the focus on wells and nourishing rather than fencing in, the inclusiveness of encouraging all to move towards Jesus – including those who may not even realise that is the direction they are heading.

But I am suspicious of the Maths

An upwards centered set A is a subset of a Partially ordered set P when any finite subset of A has an upper bound in P (and a downwards centered set has same but with a lower bound).

And a partially ordered set P is one that has a relation ρ if the relation is reflexive, antisymmetric, and transitive:

aρa ∀ a∈P (ρ is reflexive)
if aρb and bρa then a=b (ρ is antisymmetric)
if aρb and bρc then aρc (ρ is transitive)

For example: positive integers are partially ordered by the relation “is divisible by”. A second example: positive integers are partially ordered by “is greater than or equal to”.

So if P is, for example, the positive integers ordered by “is greater than or equal to”, then the even numbers would form an upwards centered set because for any finite subset of even numbers one can find a positive integer (ie in P) that is greater than or equal to the highest even number in the subset.

Now for the concept of a bounded set.

Let S be a set with the partial order relation ρ. A subsest T of S is said to be bounded from above (or below) if there is an element b∈S such that tρb ∀ t∈T.

For example, let S be the integers, and ρ the relation “less than or equal to” (≤). If T is the set of all integers whose square is less than 25, then T is bounded from above by 5 (or in fact any integer greater than 5).

A set S is bounded if it has both upper and lower bounds. Therefore, a set of real numbers is bounded if it is contained in a finite interval.

Clearly, and unfortunately for the theological stuff expounded first, a bounded set is a subset of a centered set. Ie. all bounded sets are centered sets.

So the theological shift from a bounded set to a centered set makes little mathematical sense as a bounded set is a centered set.

I thank Mr Robert Sharp for checking my Mathematics. Any of the correct bits can be credited to him. Any of the errors, I’m sure, are totally my fault.

The contrasting of centered sets and bounded sets has now become accepted in theological circles. Maybe there just aren’t many Christian mathematicians? [See here and here and here and here and here and here…]. Mathematically they just don’t contrast.

In conclusion

I like and support the idea of a new paradigm for church. I am very wary of a long list of beliefs and values where you have to tick every box before you are “in”. I think we live in quite a new context. I like models which are about wells and nourishing rather than fencing in/out. I see evangelism and lifelong metanoia as more about moving towards Jesus – even when people do not acknowledge Jesus consciously/explicitly.

[but please take care in using a mathematical model unless you really are sure it’s correct]

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