1≤3 : A Meditation

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One is less than or equal to Three — taking theological truth from mathematical mundanity.
“Count the stars — if indeed you can count them.” (Genesis 15:5)

This post is a little exercise in hearing God’s voice — using a simple mathematical equation as a springboard for spiritual contemplation.

I realise that there may not be many others who find this as inspiring as I have — and when I recently tried to share this profoundly meaningful triad of mathematical symbols with a few friends we quickly ran aground in a cross-cultural debate about the proper way to write the less-than-or-equal-to symbol. So I share my thoughts here in an attempt to help whoever’s interested to squeeze some theological revelation out of this self-evident piece of mathematics. (Perhaps this sort of thing was the point of my having studied Mathematics and Theology at Cambridge.)

Before starting though, it is necessary to address the comment occasionally voiced that one is not ‘less-than-or-equal-to‘ three it is simply ‘less-than’ three. For this comment is merely mistaken. And while I admit that it is peculiar to actually write 1≤3, the fact is that it is a perfectly legitimate mathematical sentence, for if we were to consider the set {x≤3}, then 1 would certainly be a possible value of x. Alternatively, we could put it like this: one may not be equal to three, but since it is certainly less than three, it is consequently logically true to say that it is less-than-or-equal-to three.

Okay — trivialities completed, we press on in hope of theological profundities.

One is less than Three — obvious truths point to ultimate Truth.
What can be known about God is plain to them, for God has shown it to them. (Romans 1:19)
“…and you will know the truth, and the truth will set you free.” (John 8:32)

The first nugget of theological gold that we can mine from this equation (and indeed, from any equation) is that for equations to function at all depends on the existence of Truth. Obvious truths like this point us towards the existence of ultimate Truth. And in a world beset by relativism and all manner of postmodern pandering to insufferable nonsense, it is encouraging to step back from the fray and take solace in the fact that mathematics at least can provide us with sound, reliable, unquestionable Truth.

And a mathematical mind could even easily construct a little proof to refute the pernicious doctrine of absolute relativism: ‘Suppose there are no absolute truths; then it would follow that There are no absolute truths would be an absolute truth. Contradiction! QED.’

But we can go further than arguing the mere existence of mathematical truth — we can follow this trail of revelation by noting that mathematical truths consistently and inexplicably describe with impeccable precision the behaviour of the universe in which we live. As the Nobel Prize-winning scientist Richard Feynman once commented, “The fact that there are rules at all to be checked is a kind of miracle; that it is possible to find a rule, like the inverse square law of gravitation, is some sort of miracle. It is not understood at all, but it leads to the possibility of prediction”.

I have elsewhere referred to this inexplicability as ‘The Orderliness Argument’ for the existence of a consistent God who enables and sustains the natural laws of the universe. And in a similar vein, one could use the Cosmological and Teleological Arguments to argue for (but not ‘prove’ — note my epistemological caution!) the existence of a spiritual and eternal Creator and Designer; the Moral Argument then shows that even evil points towards the existence of absolute moral Goodness; finally, the Historical Argument confirms our faith in the God of the Bible and leads us to identify the biblical God with these other philosophical accounts of Deity.

One is equal to Three — the mystery of the Trinity, and the dignity of ‘The Person’.
…he was even calling God his own Father, making himself equal with God. (John 5:18)

Thus far we have simply contented ourselves with the trivial observation that if one is less than three, then it is necessarily also true that one is less-than-or-equal-to three. But now that we have mentioned the God of the Bible, whom we believe to be One Holy Undivided Trinity, the One and the Three now grow in significance, causing our philosophical courage to rise and our analytical boldness to grow.

For the biblical God reveals himself through the Scriptural narrative as uniquely One — the God who alone is worthy of worship, and also as Three — Father, Son, and Holy Spirit. Now to be one God and yet three Persons implies no necessary contradiction: Trinitarian dogma does not state that God is one God and three Gods, or one Person and three Persons, but rather merely one God and three Persons.

But the fact that each of these Persons is fully God means that it seems somewhat meaningful to suggest that ‘1=3’ in the context of the Trinity. Properly, what we mean is perhaps something more like V(1P)=V(3P), where V(x) is the function assessing the Value of x, and xP is the number of divine Persons. And since the infinite value of the divine Persons follows from their divine nature, it is independent of their number.

Yet although our argument thus far has minimised the significance of the personal, one cannot meditate on the very fact of the existence of these divine Trinitarian persons, without also coming to a weighty sense of the dignity of ‘the Person’ in general, even and perhaps especially as that applies to non-divine Persons. Persons like you and me, ordinary everyday human persons, with idiosynchratic mannerisms and annoying habits. You do now have to have read Emmanuel Mounier to appreciate the potential significance of this. And the fact that the second Trinitarian person actually became an incarnate, embodied human person only strengthens and confirms our intuited sens of the awesome value of each and every individual person.

One is less than the Triune God — wonder, consolation, and other theological implications.
…what is man that you are mindful of him? (Psalm 8:4)
…for whenever our heart condemns us, God is greater than our heart… (1 John 3:20)

But to say such things without qualification is dangerous in a world so full of commercialized self-help coaches and glib feel-good gurus. So no sooner have we asserted the worth of the human person, than we must immediately proclaim again the matchless glory of the invisible God and lift our eyes away from our poor and broken selves, up towards the heavens.

As we begin to do so, we realize that even our entire planet is but a mote of dust suspended in a sun beam. But rather than overwhelming us with existential dread in the face of our apparent insignificance, the immensity of the universe should rather fill us with an unspeakable wonder that releases a wordless joy. It is perhaps the joy of being reminded that we are not God, while simultaneously knowing prior to all analysis that the heavens demonstrate that nevertheless Someone is! And that Someone is glorious.

And in this revelation there is supreme consolation, for it means that for all the dignity and responsibility that comes with being a person made in God’s image, still we can be justified in (and by!) confessing that our sins are manifold and our problems are beyond our abilities to solve.

One Heart — no Love without the Trinity.
God is love. (1 John 4:8)
We love because he first loved us. (1 John 4:19)

This is a good point for me to freely admit that we are now springing rather creatively from our simple equation into the realms of revelation. For I now want to draw your attention to the fact that the ‘<3' looks remarkably like a heart (indeed, it is a standard emoticon).

So we note next that God does not call us just to reverently tremble before him. Rather, He invites us to love Him with all our hearts, minds, souls, strength. And, as all the pop songs on all the radio stations in all the world testify, it is Love that the human heart longs for. This is what we were made for!

And here we make the controversial claim that unless the Trinity is acknowledged to be actual reality, then in fact Love is deprived of its ontological foundation and necessary rationale, and becomes nothing “but a second-hand emotion”. For only if we can affirm the existence of a plurality of eternal Persons can we conceive of and invoke the reality of eternal necessarily-interpersonal Love. Without God, love might be a fleeting feeling or a chemical contingency, but love cannot be the capital-L Love that the inspires the poets, commissions the prophets, and promises to heal the world’s wounds and solve the planet’s problems.

The Heart on its side — Love laid down is the greatest Love.
This is how we know what love is: Jesus Christ laid down his life for us. (1 John 3:16)
Greater love has no one than this, that to lay down his life for his friends. (John 15:13)

That established, we look again at our mathematical muse, and note that in our equation’s emoticon the heart lies fallen on its side. What, we ask, might this signify?

And in answering we remember that it is a truth universally acknowledged that there can be no greater demonstration of love than to lay one’s life down for one’s beloved. Thus a lover irreversibly gives up his everything for the sake of her whom he loves; thus the story of Romeo & Juliet captivates our imaginations with its two lovers each confirming their supreme love for the other by willingly embracing Death.

Without being distracted by the tragedies of Shakespearean romance, we press on to identify the true and final fulfilment of this ‘love laid down’ — and, in Jesus Christ crucified, we find it. As the apostle John sums it up in his First Epistle: “This is how we know what love is: Jesus Christ laid down his life for us”.

‘One’ opens his mouth — Love will not be silent.
For Christ’s love compels us, because we are convinced that one died for all, and therefore all died….We are therefore Christ’s ambassadors, as though God were making his appeal through us. We implore you on Christ’s behalf: Be reconciled to God! (2 Corinthians 5:14,20)

And so the contemplation
of this simple equation
has led us to the revelation
that Jesus died for our salvation.

But permit me one final observation before we conclude: that the less-than symbol’s triangle somewhat resembles a mouth, or perhaps the stem of a speech-bubble. Which leads us to conclude with the comment that Love cannot be silent, but necessarily must result in vocalized praise and declarative adoration, in joyful summons and unrelenting invitation.

And on that I will end, with a summons for you to consider the claims of Christ, and what it might look like for the shape of your life to be conformed to the cruciform love of Jesus. And if you’re ever around in Cambridge on a Sunday afternoon, then there’s always an invitation for you to join us for lunch amidst a community of people imperfectly attempting to work out the implications of a Love that surpasses understanding.

Sketching Out A Fractal Theology

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This was a revelation that I had a few years ago, which Phil asked me to put into writing and I never did. More recently, at the YWAM Leaders Gathering in Lyon, Carl Tinnion was talking about kaleidoscopes and fractals, and urging us to think through how embracing the possibilities of chaos might allow us to lead more open-handedly. So, with that in mind, here at last is my attempt to begin sketching out a fractal theology.

Preamble: Our Insatiable Desire For Beauty
“He has made everything beautiful in its time. He has also set eternity in the human heart; yet no one can fathom what God has done from beginning to end.” Ecclesiastes 3:11

The human heart has an insatiable appetite for beauty. We look and we see and we like what we see — and so we grasp and we take and we taste and see whether or not that which looked good really is good. But our hunger is for an ever deeper taste of goodness, our desire is for an ever richer experience of beauty. Our heart refuses to be satisfied with any finite manifestation of the good, the beautiful or the true. We are curious, like questioning children infuriating their poor parents with endless requests for deeper understanding.

Sometimes we are told that this desire, this curiousity, this lust for the infinite is a bad thing. But to deny this is to deny our humanity. And to deny this is to blaspheme our Creator’s divinity. It is God who has set eternity in the human heart.

Fractals: Delightfully Complex In Their Simplicity
“Being a language, mathematics may be used not only to inform but also, among other things, to seduce.”

We come then to the fascinating mathematical entities known as ‘fractals’, which are basically patterns that are similar at every level of magnitude. Such a definition might not satisfy real mathematicians, who discuss such things using words like ‘topological’ — but it communicates the idea well enough for us to look at some examples.

Our first specimen is a thing called ‘the Koch Snowflake’. The construction of the Koch Snowflake is very simple. First you take an equilateral triangle. Next, you divide each side up into thirds and turn the middle third of each side into the base of a new equilateral triangle. Then you repeat. Infinitely. At the age of fifteen, my maths teacher thought it would be amusing to set me the task of working out its perimeter and area. Which I did by working out a recursive formula and then applying it. Fascinatingly, the length of the perimeter tends to infinity, while the area never exceeds eight fifths that of the original triangle. The key thing to notice is the exact self-similarity of this object at every level of magnification.

In contrast, the Mandelbrot set is a more complex thing, possessing only quasi self-similarity–which is to say that at greater zoom it approximates but never quite replicates itself.

Biblical Fractals: The Cherubim
“…their appearance and construction being as it were a wheel within a wheel…”
In the last few decades, fractals have helped further scientific understanding in a huge range of different fields. But I found it utterly astonishing when I saw that the biblical cherubim appear to possess exact self-similarity. Quite a claim, I know. But let’s look at the relevant passages.

In Revelation 4, John talks about how one of the living creatures had a face like a lion, one the face of an eagle, one the face of an ox, and one the face of a man. Then in Ezekiel 1, Ezekiel says something similar–but rather than saying that the four had faces like a ox/human/lion/eagle respectively, he says that they each had the faces of all the four different creatures. And then–to prove the point–in Ezekiel 10 he says that each has the face of the eagle, the lion, the human and–not, contra the NLT, the ox! but rather “the face of a cherub”. Which, we saw in Ezekiel 1, is the four-fold fractal face.

I would love for someone with artistic talent to attempt a visual rendition of these fractal cherubim. It would be something like Michelangelo crossed with Escher. And certainly nothing like the typical depiction of baby-faced cherubs.

Theology: Typology Is Just Fractal Christology
“In the type there must be evidence of the one eternal intention; in the trope there can be evidence only of the intention of one writer. The type exists in history or temporal experience and its meaning is factual, that is, objective…” (Jonathan Edwards)

Perhaps this is true, you finally concede, but what is the relevance of this? Surely all this talk of fractals is nothing but mathematical snobbery and theoretical uselessness!

But I would beg to differ.

For when one begins to train one’s eye to see the fractals around you, one realises that even theologically this is not such a new thing–in fact, finding christological fractals is a standard trick in the arsenal of any competent preacher. It’s just that it usually goes by a different name: the name ‘typology’.

Moses, almost killed by a tyrannical king at his birth, leads his people out of slavery. David, rejected by his brothers but filled with zeal for the house of God, refused to be deterred by political opposition from establishing worship on earth as it is in heaven. Jeremiah, a weeping batchelor, stood boldly against the false prophets of his day to declare the good news of the new covenant. Ezekiel, aged thirty and by a river, suddenly experienced the opening of heaven and the infilling of the Spirit of God, and was commissioned as the Son of Man to enact the word of God to the watching world.

And who do all these remind you of? That’s right — Jesus!

Anyway, this is what we call typology. And this is just a tiny slice of a rich and fascinating subject of infinite depth. Because if Jesus is the Word through whom all creation was made, then actually we should expect all of creation to echo and exemplify various characteristics of the nature and character of Jesus. If you’re interested in more reflection upon these lines, then I highly recommend James Jordan’s book Through New Eyes.

Discipleship: Great Commission as Recursive Equation
“Fractals are normally the result of a iterative or recursive construction or algorithm.” (Paul Bourke)

The Gospel of Matthew famously ends with Jesus commissioning his disciples to go and make disciples. What is less frequently commented on is that this is undeniably a recursive task, as is emphasised by the task of “teaching them to obey everything that I’ve commanded you”–including, most importantly!, this very Commission. This is seen most clearly in Paul’s exhortation to his readers to “imitate me as I imitate Christ”.

And if the Great Commission is a recursive equation, it must follow as a logical consequence that the church should be a dynamically fractal organism of believers grafted into community with God through relationship with each other, and gradually being transformed into the likeness of Christ.

Prophetic Hermeneutics: The Inevitability of Fractal Fulfilment
“Those who study Bible prophecy are aware that many prophetic passages have multiple references.”

I am convinced that thinking in terms of fractals also helps with getting one’s head around the interpretation of biblical prophecy. There are various examples of the double reference of certain prophecies–eg. Isaiah 7:14’s promise that God would give Ahaz a sign by way of a virgin conceiving and bearing a son.

Once one has managed to grasp the strange idea that God’s prophetic word might function in some sort of fractal way, it’s an incredibly attractive suggestion. It’s almost enough to persuade me to be a Six-Day Young-Earth Creationist and a Premillenialist together at once, purely for the neatly symmetrical fractal chronology that would result!

But lest we get too quickly carried away, let us first look a little more closely at prophecy, and see why one might think that a fractal effect might occur.

The thing to notice is that with prophecy we have to hold two things in tension: on the one hand, there is the permanent fact of the prophetic word (as seen clearly by Ethan the Ezrahite in Psalm 89:26-33); on the other hand, there is the dynamic responsiveness of the prophetic word to faith and repentance (as God unequivocally tells Jeremiah). Now, permanent promises that are interacting throughout history with varied responses of faith and repentance (the results of which are themselves affecting subsequent levels of faith and repentance) are again going to function as a sort of recursive equation.

And so to my mind it would seem that the only conclusion is that the various promises of blessing and prophecies of judgement that God has made throughout history must therefore ultimately come together in the sort of apocalyptic end-time scenario of which someone like Mike Bickle speaks. In which case, he seems right to suggest we start praying for abundant revelation and insight into the days in which we live!

Conclusion: The Fertility of Fractal Faith
Now before I go off on too eschatological (and doubtless controversial) a tangent, let me bring my sketch to a close with the modest claim that, whether or not you agree with the details of my suggestions as to where a fractal theology might lead, I think I have successfully shown that looking through the kaleidoscopic lens of self-similarity does at least offer the possibility of contributing to a variety of areas of Christian conviction.

Now I’ll leave you to mull on that while I go enjoy looking through my kaleidoscope…