## Banach-Tarski and the Trinity

Yesterday’s XKCD cartoon was a treat for maths fans, referencing as it did the Banach-Tarski “paradox”: the jaw-droppingly counter-intuitive theorem proving that a solid sphere can be cut into five pieces that, when rotated and rearranged, can be fitted together to make two solid spheres of the same size as the original. (See the mouseover text for the cartoon, which also includes a cunning biblical reference.)

I’d heard of the Banach-Tarski theorem, but had assumed I’d never be able to understand how the pieces of the sphere are constructed. However, the Wikipedia entry had a link to this entry from the Irregular Webcomic which presents an outline argument for the theorem that is, quite simply, one of the most brilliant pieces of mathematic explanation I’ve ever read. As the writer says at the end of the argument:

Now, either you’re blinking in amazement that you understood everything above and going “wow”, or you got a bit lost somewhere (in which case I apologise), or you’re thinking there must be something wrong with this because it can’t possibly be true.

And of course, in the physical world it can’t possibly be true. As the writer points out:

This sounds crazy, and for good reason. If you take a rubber ball and cut it up, there’s no way you can reassemble the pieces into two balls, each the same size as the original. Assuming you’re not doing anything tricky like stretching the pieces.

It’s important to point out up front that the Banach-Tarski theorem only applies to mathematically ideal objects, not to physical objects. [Not even pumpkins, we might add.]

Broadly speaking, there are two ways you can respond to that statement. One is to say, “OK, gotcha, now can you get on with your argument?” The other is to say, “Right, that’s all I need to know. You can’t actually do it. Now if you’ll excuse me, I’m going to focus my attention on things that can actually happen.” In other words, it brings out the distinction between those who tend to think in abstract terms and those who tend to think in concrete terms.

A taste for abstract thinking and youthful exposure to mathematics can come in handy in defusing certain theological objections to (or within) the Christian faith. For example: “You can’t contain the infinite in the finite!” is an objection made by some Christians to the doctrine of the real presence of Christ’s body and blood in the elements of the Lord’s Supper – though I first encountered it as an argument used by a Muslim against the doctrine of the Incarnation. (In other words, Calvinists: careful where you’re pointing that thing. 😉 )

From the moment I first heard that argument, I found it slightly baffling, because I just thought of the real numbers between 0 and 1. The line segment [0,1] is finite in length, but it contains an infinite number of points – and, more significantly, those points can be put into a one-to-one correspondence with the entire number line (-∞,∞). So there are “as many” points between 0 and 1 as between -∞ and ∞. So the finite can contain the infinite, dude.

Now no doubt I am missing no end of subtle philosophical points here and “finitum non capax infiniti” isn’t as easily dismissed as all that. And I’m certainly not claiming that there is any parallel between the Incarnation or the real presence and the mathematical concepts of infinity or real numbers. The point is that my exposure to mathematics gave me the mental toolkit to ensure that the “non capax” argument, on the level at which I encountered it, was never a threat to my faith.

A similar point applies to the doctrine of the Trinity. I never found this a problem, because I was used to dealing with concepts that you couldn’t quite picture but which still hung together on an abstract level. Plus there are some specific concepts from maths which can be useful as analogies for “three persons in one God”: for example, the [four-dimensional] hypercube as an object whose “faces” are eight cubes. Eight cubes in one hypercube: you can’t picture it, but it hangs together as a concept. Three persons in one God: you can’t picture it, but it hangs together as a concept. Job done.

The Banach-Tarski theorem strikes me as another example: “Three into one can’t go!” “Oh yeah? Well here’s two into one…” (The obvious retort then being: “That’s a load of balls!” But let’s move on…)

This is all very well, if you are inclined towards abstract thinking (or if, like me, you are like the stereotypical Frenchman who says: “That’s all very well in practice, but does it work in theory?“). However, it has significant dangers. In particular, it can lead you to think that the essence of these Christian doctrines is to be found on level of the abstract, and that the way to deal with people’s difficulties is to give them better mental toolkits for dealing with abstract concepts.

However, God did not make foolish the wisdom of this world by pre-empting Georg Cantor, Stefan Banach and Alfred Tarski. The message of the gospel – its foolishness that is wiser than human wisdom – is that the Word became flesh and dwelt among us; that is, became concrete rather than abstract. And that gospel comes to us today not as an abstract set of propositions, but in the concrete forms of word and sacrament.

So in the Lord’s Supper, we are not invited to contemplate abstract arguments by which an infinite God can be contained within finite bread and wine; rather, through his minister, Christ himself says “this is my body, this is my blood” and bids us come and eat.

As for the Trinity, this is not something we encounter principally as an abstract doctrine, but in the everyday life of the church. Above all we encounter it in baptism, and in the Christian life as a baptismal life. In baptism I stand where God the Son stood at his baptism, God the Father tells me that I am his beloved son with whom he is well pleased, God the Holy Spirit descends on me. That is the concrete reality to which we return every time we “return to our baptism” through repentance and faith; that is why our services begin by invoking the Triune God “in the name of the Father and of the Son and of the Holy Spirit”. It is why we pray to the Father, in the Son, by the Holy Spirit.

As Josh put it in a superb post in 2008:

Without baptism, the Trinity becomes an ideology rather than an identity. Rejecting or accepting the Trinity becomes a matter of to what degree one understands and appropriates a theory. Having a God is understood as subscribing to a certain theory about a deity, thinking about divinity in the right way.

But as a baptismal doctrine – which is how Matthew presents it to us – confessing the Trinity and being baptized into Christ, being part of this New Covenant people, become the same thing, as much as having YHWH as your God and being a circumcised member of Israel were the same thing. The confession and the sacrament are inseparable; having God as your God and being sacramentally defined as a member of his people are identical.

If abstract analogies and arguments help you get your head round the Trinity or the Incarnation (as they help me), all well and good. But first and foremost, confessing the faith is not a matter of intellectual assent, but of being united to Christ as part of his covenant people, the church. Concrete, not abstract; the Word made flesh.

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### 6 Responses to Banach-Tarski and the Trinity

1. Alex says:

For what it’s worth, as a former maths major (and indeed, a maths marker/grader, lab instructor, and teaching assistant) myself, I appreciate this post. The axiom of choice thing kind of breaks down superimposed over the real numbers, but I suppose if we wish to assume it, it all works out… (and I suppose that’s the fundamental paradox at the heart of Banach-Tarski, and why it is somewhat disputed—can you use a well-ordered principle like the axiom of choice over R?)

In that sense, mathematics really does have an implicitly faith-based premise; one must accept/believe certain premises to be true in order to do mathematics. For elementary maths, the basic premises (natural numbers, ordering, sets) are pretty obvious. But when you get into higher maths, then the faith becomes more involved.

I seem to remember having a conversation somewhat about mathematics and theology with JRH whilst in seminary, and we came to the point of acknowledging that maths could make you a deist, but only revelation could make you a Christian.

2. John H says:

I seem to remember having a conversation somewhat about mathematics and theology with JRH whilst in seminary, and we came to the point of acknowledging that maths could make you a deist, but only revelation could make you a Christian.

Exactly. I refer the right honourable gentleman to the Proof from Euler’s Identity… 😉

3. Philip Walker says:

Alex: “The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn’s lemma?” (Jerry Bona, according to Wikipedia)

‘“finitum non capax infiniti” isn’t as easily dismissed as all that.’

Not when you’ve got 1 Kgs. 8:27 to bear in mind, that’s for sure! (I think we have to pin down ‘contain’ quite precisely; in order for finitum non capax to make sense, I expect that we have to mean it in a way which makes it inapplicable to the Lord’s Supper and to the Incarnation. Specifically, ‘contain’ cannot mean ‘circumscribe’.)

4. An old acquaintance, a family friend and a rather notable OT scholar of some international renown once remarked to me: “You can’t really do theology unless you are trained mathematically.”

I did get an A at A level (ordinary, not extended maths) a long time ago. Does that count…?

5. Rick Ritchie says:

“For example: “You can’t contain the infinite in the finite!” is an objection made by some Christians to the doctrine of the real presence of Christ’s body and blood in the elements of the Lord’s Supper – though I first encountered it as an argument used by a Muslim against the doctrine of the Incarnation. (In other words, Calvinists: careful where you’re pointing that thing. 😉 )”

Right. Any argument made against the Real Presence in the Lord’s Supper probably has a parallel argument that could be used against the Incarnation itself. I’ve thought through many of these. I think the Lord’s Supper can remind us how scandalous the idea of the Incarnation really is.

On another note related to the post, I would hear one of the Calvinists complain that evangelicals were watching too much Star Trek. I thought rather that the Calvinists weren’t watching enough. They would confidently assert how things had to be, and it was clear to me that they didn’t have to be that way at all.