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The Girl Who Played With Algebraic Number Theory

There are some dicey moments in the second of Stieg Larsson's 'Millennium Trilogy' crime thrillers. The one that really got me was when his arguably-too-brilliant social-misfit-hacker protagonist Lisbeth Salander finally (after several weeks of intermittent, amateur endeavour) solves the riddle of Fermat's Last Theorem -- a conjecture which famously remained unproven for 358 years after the tantalising claim by the eponymous theorist, scrawled in the white space of his copy of the ancient Greek mathematical text Arithmetica, that he knew a proof but that it was too large to fit in the margin.
Salander began her advance towards the house, moving in a circle through the woods. She had gone about a hundred and fifty metres when suddenly she stopped in mid-stride. 
In the margin of his copy of Arithmetica, Pierre de Fermat had jotted the words I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain. 
The square had been converted to a cube, (x^3+y^3=z^3), and mathematicians had spent centuries looking for the answer to Fermat’s riddle. By the time Andrew Wiles solved the puzzle in the 1990s, he had been at it for ten years using the world’s most advanced computer programme. 
And all of a sudden she understood. The answer was so disarmingly simple. A game with numbers that lined up and then fell into place in a simple formula that was most similar to a rebus. 
Fermat had no computer, of course, and Wiles’s solution was based on mathematics that had not been invented when Fermat formulated his theorem. Fermat would never have been able to produce the proof that Wiles had presented. Fermat’s solution was quite different. 
She was so stunned that she had to sit down on a tree stump. She gazed straight ahead as she checked the equation. 
So that’s what he meant. No wonder mathematicians were tearing out their hair. 
Then she giggled. 
A philosopher would have had a better chance of solving this riddle. 
She wished she could have known Fermat. 
He was a cocky devil. (The Girl Who Played With Fire, Stieg Larsson, 2006)
Suffice to say, Larsson does not elaborate on the details of 'Fermat's original proof' within the novel. Nor have I been able to find a 'full ePrint version' online anywhere. So we are abandoned to our own mathematical imaginings.

More than a little bit ridiculous. I am astounded, really, that it got past his editors and publishers and friends and family without somebody tentatively suggesting that some sort of mathematician-type person be consulted for an expert second opinion on the feasibility of such an accomplishment, and/or to correct his quite embarrassingly mistaken take on Wiles' own solution to the problem (150 pages in which he painstakingly consolidates and builds upon the past work of many, many specialists, with (as far as I can discover) no particular reliance on any sort of computer programme, advanced or otherwise). Fortunately for Larsson, The Girl Who Played With Fire is sufficiently exciting and engaging, and this particular occurrence happens late enough in, for me (and, I'm guessing, many others who might otherwise object) to let it slide and enjoy the rest of the story without too much supercilious contempt.

Overall, I've enjoyed what I've read of the Millennium Trilogy (one to go). The books make some interesting comments on 'women's issues' (though I'm pretty weary of their perspective on sex -- but that's a story for another time) and sustain a generally genuinely exciting and enjoyable and surprisingly surprising plot progression. But, as characterisation goes, Salander is...an interesting case. I've heard her described as a 'Mary Sue', though she lacks, I think, the author-representational traits typically associated with that epithet. But certainly, she comes across as 'too good to be true' -- an idealised, wish-fulfilment version of the underestimated-downtrodden-genius-made-good. There doesn't seem to be a system that she can't hack, or a criminal mastermind that she can't defeat, or (*ahem*) a long-running open problem in mathematics that she can't master in her spare time. She's precisely the quirky hacker heroine that you would make up. And the books seem, well, all the more fictional for it.

Which gets me to thinking. In recent years, the "that's not how I would have had it" aspects of Christianity have increasingly entered among my reasons for faith rather than my frustrations with it or my grapplings with doubt. The Bible just doesn't seem like the religious text that humans would write if we were bent on starting a religion. God, and His dealings with us, don't fit human ideals of what a deity should accomplish on our behalf. And Jesus was neither the Messiah that a first century Jew was likely to invent, nor the "lifestyle guru" that anyone with a personal agenda to "get ahead" in the world would be likely to set up for themselves. These are just some of the reasons why I think that it's all...well...that it's real...

The Bible: not the neat doctrinal basis we would write

Way I see it, right -- you get the finer points of your theology ironed out in your head before you write it all down. And obviously, you make sure everything comes together to support your premise. Give it a few good read-throughs before you unleash it on the public; get rid of anything which jars with or confuses the central 'message'.

Well, that's not what you get from the Bible. How often have I been right in the middle of a passage which would otherwise happily serve as a supporting text in any 'how to become a Christian' tract, only to suddenly find myself in the realms of baptism for the dead (1 Corinthians 15:29), or women being saved through childbearing (1 Timothy 2:15), or the 'unforgivable sin' (Matthew 12:31-32), and I'm like totally "aargh!" I don't know what to 'take' from such passages; some of them genuinely scare me from time-to-time, and it is tempting to just wish they weren't there at all. [1] But whatever the 'right' response, their inclusion at least reassures me that the Bible is not just a collection of supporting evidence for the nice, neat, prefabricated manifesto of a person or a committee.

Rather, it's truth emerging from history, from human experience -- sometimes remarkable, sometimes remarkably typical; it's people writing letters and songs and expressing what's inside them without (often) realising the ways that God is going to weave their words into His own message to others in many places and in different times. We're so used to "smoothed" representations of reality -- idiosyncrasies all pre-rationalised away -- that we forget that it is seldom quite so simple and digestible in "raw" form, which is what I would argue we find in the Bible: "raw" reality.

God: 'inconveniently' beyond our worldly, self-interested schemes

If you were going to 'invent' a deity for yourself, you'd want his interests to align with yours. You'd want him to do everything you wanted him to do, or at the very least to sanction you in doing everything that you wanted to do.

That's not what you get from "the God of the Bible". From His dealings with Israel, whom He calls His own people (e.g. Isaiah 43:1-2), it is clear that having God 'on your side' does not equate to having Him 'on your own terms'. Take, for example, the laws that He imparts to them (see especially Exodus, Deuteronomy and Leviticus) in order to set them apart from the surrounding nations and to prepare them for a right relationship with Himself. So many of these appear to work to Israel's worldly disadvantage: the prohibition of inter-marriage with neighbour nations [2] hardly seems geared towards the rapid population growth and diplomatic relationship-building desirable in a largely tribal, conflict-heavy culture; the requirement to 'keep' the Sabbath -- no work, no trade -- is most awkward from an economic perspective. In short, they look quite different from the types of laws I'd expect an emerging people group to have naturally invented for themselves. Likewise, the many protections for the disadvantaged and the destitute -- where even in our own society it all too often seems that, guided by the principles of power preservation, ruling parties shirk the cost and trouble of serving those in need who can offer them little in the way of political leverage.

The Old Testament is full of statements to the effect that God is "for" His people, or "with" them (e.g. Psalm 46:7, Psalm 118:6, Isaiah 41:10) -- but it is unambiguously clear that He will not be commanded, nor yield to our self-serving agendas, as Jonah discovered to his dismay and fury when He shows mercy to Ninevah, the enemy of Israel:
When God saw what they did, how they turned from their evil way, God relented of the disaster that he had said he would do to them, and he did not do it. But it displeased Jonah exceedingly, and he was angry. And he prayed to the LORD and said, “O LORD, is not this what I said when I was yet in my country? That is why I made haste to flee to Tarshish; for I knew that you are a gracious God and merciful, slow to anger and abounding in steadfast love, and relenting from disaster. Therefore now, O LORD, please take my life from me, for it is better for me to die than to live.” And the LORD said, “Do you do well to be angry?” (Jonah 3:10-4:4)
Much of the book of Jonah up to that point is about God's radical mercy and patience and love towards Jonah in all his stubborn-hearted disobedience. How quickly he forgets that once confronted with the inconvenient reality that he can't control God, that he doesn't get to choose the objects of the mercy which has rescued him...

So, whilst others may be content to dismiss God as the invention of human minds, for me this doesn't entirely square with the Biblical representation of Him as very much having "a mind of His own" -- in ways which do not always work to any human being's obvious advantage.

Jesus: not the Messiah his early followers had thought they were waiting for, nor the Lord any of us thought that we wanted

If you were going to promote a made-up Messiah, you'd first of all want to make sure that he fitted the bill. You'd also, I expect, begin to waver in your allegiance to something you knew to be untrue once you came up against sufficient opposition. The threshold for renunciation may vary from person to person but for most I imagine it would come considerably before the point of death.

From what is known about Jewish expectations at the time of Jesus' ministry, it seems he really did not fit the criteria typically associated with the Messiah.  As N.T. Wright observes:
"There were, to be sure, several variations on Jewish messianic belief in this period. None of them envisaged a Messiah who would die at the hands of the pagans. On the contrary, where Jewish expectations of a Messiah did exist, they regularly possessed a dual focus. In a line of tradition stretching from David to Bar-Kochba, including the Maccabees and Herod, we find that the king would have to defeat the pagans, and that he would have to rebuild (or at least to cleanse) the Temple. The two actions would, of course, go together: as long as the pagans remained undefeated, YHWH had not returned to Zion, presumably because his house was not ready. If a messiah was killed by the pagans, especially if he had not rebuilt the Temple or liberated Israel, that was the surest sign that he was another in the long line of false messiahs." (N.T. Wright, Christian Origins and the Resurrection of Jesus: The Resurrection of Jesus as a Historical Problem, Sewanee Theological Review 41.2, 1998.)
And yet, something happened to make large numbers of Jews, and of their gentile contemporaries, and of increasing numbers worldwide thereafter, willing to follow him even to their own deaths. "For me, to live is Christ and to die is gain" (Philippians 1:21), wrote Paul, who -- having initially approved the matyrdom of others of Jesus' followers -- fell in with the most fervent, life-forsaking of them following a (well, the) Damascus Road experience (Acts 9).

N.T. Wright quite happily argues, and I'm rather inclined to agree, that the mysterious "thing which happened" to cause all this was none other than the explanation which the Bible offers: that Jesus really did rise from the dead (Matthew 28). This possibility, once considered, opens up a whole new perspective on the world and human existence, from which suddenly even death appears a somehow 'smaller deal' than hitherto.

But even so, self-sacrifice does not come naturally to human beings -- not that I have witnessed in myself, at least. And Jesus' pattern, and his instructions, are hardly a model for 'how to get ahead in life': "If anyone would come after me, let him deny himself and take up his cross and follow me", he invites (Matthew 16:24). "If anyone slaps you on the right cheek, turn to him the other also. And if anyone would sue you and take your tunic, let him have your cloak as well." (Matthew 5:39-40). Anyone can see that this is not the way to 'win'. Not by the world's assessment, anyway. Nor is that what Jesus claims to offer: "If you were of the world, the world would love you as its own; but because you are not of the world, but I chose you out of the world, therefore the world hates you." (John 15:19)

Whatever the many 'Jesus theories', it seems clear to me there must be more to all this than the invention or false elevation of a convenient figurehead for a movement. There were way too many more convincing forms for such an invention to have taken, and there are way too many more appealing human-motivated agendas to pursue. For a more coherent 'theory', I propose a closer look at the one explicitly elaborated in the New Testament....

To summarise, the Bible doesn't deal in ideals -- and this is one of the reasons why, as I 'deal' with the Bible myself, I am increasingly confident that it does deal in realities... Realities which are messy, and inconvenient, and require grappling with, but which, on closer investigation, turn out to be far more bewilderingly wonderful than any neat idea which we might have idealised for ourselves.



[1] If I believe that the Bible is true, though, I have to recognise that, whatever the problematic parts mean, they do not invalidate the rather more straightforwardly exciting promises and encouragements of the larger, clearer picture which emerges from the whole. I suspect that it is religious anxiety which causes me to focus (sometimes obsessively) on the few small, scary, confusing bits to the exclusion of the many big, reassuring, clear bits, like Romans 10:13, or John 3:16, or John 6:37 to name some oft-cited examples.

[2] That some have been determined to shoe-horn these incredibly context-specific instructions (geared around the dangers of idolatry) into various racist agendas, baffles and horrifies me, and I think you only have to actually read them (and maybe, for example, Colossians 3 too for good measure) to see that such 'use' of the passages is patently unjustified.

[Thumbnail image CC by iwannt on flickr].

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