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bounded and centered sets

Good theology weak maths

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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14 thoughts on “Good theology weak maths”

  1. Statistician rather than pure mathematician, but I think you’re pretty well on the right track. A bounded set is a particular form of centred set. However, the problem seems to be that, as you point out, the mathematical concepts don’t match the theological ones. What Hiebert has essentially done is take some mathematical terms (bounded set, centred set) and used those terms to describe aspects of ecclesiology / missiology. In doing so he is changed the meaning of those terms from the mathematical meaning, so that when he then attempts to impose the mathematical meaning of those phrases onto the theological meaning he has assigned them, there is a clear mismatch in meaning.

    Making a word or phrase mean whatever you want it to mean – Alice would be proud.

    1. Thanks, Paul. Like you, I also saw the application of this reflection on discussions around the “Anglican Covenant”. I also think that the denominational “fences” are applying less and less. Blessings.

  2. I am now sure that maths is a language I have never mastered nor am I likely to do so but I heartily applaud the concept of centred churches, wells and nourishment as one to which we should all be aspiring. The struggle centres on those who feel safe within the boundaries, on ways in which to encourage them to break out.

    1. Thanks, Jennifer. You do not need to master the mathematical language – because the maths being used by those advocating for the shift that you and I agree on is incorrect. Peter, in the comment above, says it well – the mathematical concept is being ab/used to express a sound theological concept. I think we need to work with the paradigm shift and be honest and abandon pretensions of sophisticated mathematical connections. As for those who feel safe within fences – I think that in most places no encouragement is needed. The tectonic plates are shifting. Blessings.

      1. I pity it’s taken a tectonic shift.

        I was encouraged by reading in The Press this morning about the Church leaders in many other denominations in this City offering support and encouragement to Bishop Matthews. (http://www.stuff.co.nz/the-press/news/christchurch-earthquake-2011/6682160/Church-leaders-back-bishop)

        Perhaps its when we all start looking outside the perspective of our own traditions and denomination that we can start experiencing the Church as one body working together, learning from each other and appreciating the strengths that each tradition offers. For example, I’m not Anglican, but I’ve been finding lately in the Anglican liturgies and traditions something I’ve been missing in my more contemporary background, and by exploring them I’ve found my faith grow. (Which, BTW, is how I stumbled onto this site in the first place.)

        Thank you for your posts and comments, Bosco. Many blessings to you.

        1. Thanks, Claudia. And this post, you’ll notice, draws on and affirms emergent and missional perspectives about which I am particularly enthusiastic – Christian communities that have leaped over a lot of the fencing-in denominational differences and distinctions which have less and less meaning in our new context. Blessings.

  3. I think that Paul Bagshaw’a Globular Communion’ is a real possibility for the future of Anglicanism. In a way, it already exists – with various parts of the Church doing their own thing without too much reference to a R.C. style ‘Head Office. However, the Roman papal style doesn’t seem to be working too well either. North America already has women priests (albeit unrecognised as yet by Rome) – something unimaginable not so long ago.It will soon have to ordain married men, or become ministerially under-staffed.

    Gafcon will have it’s own ‘centrum’ – possibly in Kenya or Nigeria. Gay-ordaining and blessing Provinces may revolve around North America – or the U.K., depending on the effects of the next GAFCON there. A small part of Australia (Sydney) may become a globule of GAFCON, with the rest of us allied to either U.K. or North America. In some ways, the tendencies are already emerging.

    I agree with Paul Bagshaw, that the future is unpredictable, but that the Covenant has no real answer to the problem. What really matters is for each part of the Communion be true to Christ and itself, in situ. Maybe God can work with that.

  4. Richard Catterall

    Bosco, It’s nice to see your years of study of mathematics were not wasted – one of the beauties of mathematics is the precision of expression it offers. Contrariwise, since we are not in Alice’s Wonderland, it is also nice to see theologians unafraid to mix good ideas with bad maths because it gives us old pedants something to write about! Someone once summed up the Law and the Prophets in two rules: Love God, love people. There’s probably enough in there to distract us from mathematics for quite a while. 🙂 Happy Easter to you and your family. Love and God Bless and keep up the good website.

    1. Thanks, Richard. It’s reassuring that you think my maths is holding up. Sadly, in this country, set theory has gone the way of theology and is not part of the national curriculum. The loss of understanding venn diagrams is a significant loss to careful thinking, for example, IMO. Blessings.

  5. I will out myself as a former pure mathematician. Bertrand Russell once said:

    “Pure mathematics consists entirely of such asseverations as that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing… It’s essential not to discuss whether the proposition is really true, and not to mention what the anything is of which it is supposed to be true… If our hypothesis is about anything and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.”

    I would be more convinced by an appeal to the Possible World Semantics of Saul Kripke. But that is just a way of saying “I can see why those people call themselves Christian (group A), that makes sense to me – but this others, that doesn’t (group Z)” That’s the simple bit. But Group A think Group B are Christian … and Group Y think group Z are Christian. And suggesting a connection with Kripke merely adds spurious confusion to something which is really quite simple. “Proof by mystery” is not part of pure mathematics.

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