Proofs and Refutations

Forewarning: This is one of those posts where I ramble about something unrelated to religion and then make a tenuous theological point at the end. Today’s topic: math…. no don’t go, it won’t be that bad.

I was a counselor on a summer camp. A friend of mine knew I was into math and puzzles and set me this one.

There are three houses, each needs to be connected to three utilities. Can you connect them in 2d without any pipes crossing?

I sketched a couple of attempts first and it looked unlikely, so I resorted to a strategy that served me well in Math Olympiad type problems: try to prove it impossible. I remember my line of reasoning.

First I figured that the solution didn’t depend on where the utilities or houses were. Any solution would be essentially the same solution if you dragged the end points around, or stretched or bent the pipes. In math terms, this is topology – you can move stuff about as long as you do so with continuous deformations. Locations or pipes could be moved and stretched and reshaped, but could never jump.

Then I started from a simple problem – the 2 house 2 utility problem, and I played around until I saw there was only one solution to that. Any other routing would be stretchable back to that one solution. (I didn’t, if I remember rightly, prove this properly, I did it mostly by intuition, but I was right — from this point on however, my reasoning does form the outline of a valid proof)

The solution to the 2×2 problem divides space into two regions. It is impossible to get from one region to another without crossing a pipe. So when we go to the 3×2 problem, the next house has to be in one region or the other. And whichever region it is in, the pipes connecting it to the two utilities must be in the same region. So the result is the same in either case. All solutions to the 3×2 problem are topologically identical to this.

Which splits space into three regions. But notice that each region only borders two houses.

Whichever region the next utility is placed in, it will not have access to exactly one house. If it is in the red region, it won’t have access to the red house, the green house is inaccessible from the green region, and the blue house from the blue region.

Therefore, the 3×3 problem is impossible. As long as my intuition about the 2×2 problem was correct, I had a watertight proof that the problem couldn’t be solved.

And I said so. “It’s impossible, I can prove it.”

“No it isn’t”, said my friend, and drew the solution.

All utilities are connected to all houses, no pipes cross any other pipe. I was wrong.

I was wrong in a very important way. I had invented constraints that weren’t there. I had set myself a different problem to the real one.

I had proved my artificial problem was impossible, but who cares if the artificial problem you invented turns out to be impossible? Who cares if you have enough advanced math to construct a proof of it? (I’ve since figured out several other ways to prove the same result, it has become a favorite brain game of mine to find new ways to prove this irrelevancy).

Five years later, after describing this to my PhD supervisor, he pointed me at “Proofs and Refutations“, an excellent book on the philosophy of math by Imre Lakatos. Which discusses this phenomenon in detail. I enjoyed learning this lesson, and it has stayed with me ever since.

… and hence to the theological point …

I’ve been thinking about the “Argument between Science and Religion” recently, and reading around it. Everything I read seems to be on either the science side (which is usually but not exclusively anti-religion) or the religion side (which is almost always in favor of there being no conflict), and everything I read is unsatisfactory. And today I realized why. Both sides have manufactured their own (slightly different) imaginary problems. This usually happens in the prologue or first chapter. The authors then spend hundreds of pages arguing very powerfully and convincingly for why they are right. And pretty much, they persuade me. But even if they could be said to prove they were right, it doesn’t change the fact they’ve essentially invented an imaginary question. One that is, as far as I can see, utterly irrelevant. Nobody seems to want to step up to the really hard challenges on the other side.

As to what they are, in my opinion, I’ll leave for another post. But feel free to pre-empt me with your thoughts…

Inoculation

An observation: I know a few people who were proper atheists then came to faith, I know quite a few people who left their religion and became atheists, I know one person who was an atheist, was converted, then later returned to their atheism, I know people who’ve been through various religious identities and ended up either as religious or as an atheist. But I don’t know anyone who has left a strong religious commitment for atheism who’s then later been converted back to faith.

I’m sure there are many such folks, but I don’t know any.

Changes in religious identity are common. It is common to change within a religion (many of my friends have moved from evangelical to liberal Christian, for example), and to a slightly lesser extent between faiths (Buddhism seems to be a popular destination for ex-Christians here in the west). I think it is important to remember that these are changes of identity rather than necessarily belief. The beliefs of many Christians are indistinguishable from those of many atheists. The difference is where one chooses to find one’s identity.

I wonder if the often painful process of losing one’s faith inoculates to some extent against finding another faith attractive in the future.

Being an atheist isn’t any guarantee that one won’t be converted later on (much to the chagrin of some atheists I’ve met who like to suggest — in a irony of epic proportion — that the converted person wasn’t a ‘real’ atheist in the first place). There are plenty of people raised religious, who declare themselves atheists in their late teens then find faith later on. But maybe the process of becoming an atheist, particularly if that becoming is a painful process, makes it more likely that you will struggle to give yourself wholeheartedly to something later.

I’m not saying it is impossible, and it may be a quirk of my circle of friends, but it does seem notable.

Does it chime with your experience? Are you confident you won’t throw over your current religious identity at some point in the future? If not, what could sway you?

Thanks John, Goodbye.

John Hick passed away on 9 February (Telegraph obituary). He was intimately connected with University of Birmingham, where I did my undergrad theology. Although he was an emeritus professor with a light teaching load (if any, I didn’t take a class he taught), his influence was definitely felt down to the lowliest depths of the department.

I found in Hick some of the sanest facing up to the collision between theology and consensus reality. I owe him a lot from where I’ve ended up theologically, though I did not end up exactly where he did.

If you’ve never read Hick, might I suggest “The Myth / Metaphor of God Incarnate“?

My iPhone App

Following on from the previous post, about some of the more-or-less spiritual things I’ve found that help me induce pleasant states of mind (states previous associated with religious phenomena)…

I put together a simple iPhone app. Called Calmer, with the help of my wife.

It is vaguely game like, but isn’t a game. It was designed in the midst of our family crisis, to help calm down. To take stock, to slow, to focus and to distract the mind from spinning wheels of worry.

To use the app, you trace shapes with your finger, slowly. When you complete a shape, the app will describe a little more about what you’re doing, building into a set of interesting things to think about. There is enough challenge to engage your brain, but not so much it becomes a puzzle to worry about.

As you make the shape, the app uses one of my blended white noise mixes to mask language production and calm you down.

I know I designed it exactly for my very personal preferences, but I do find myself ‘playing’ with it quite regularly.

If you have an iPhone, and want to give it a whirl, let me know what you think.

Cynicism and Spirituality

You might find prayer rather silly. I find meditation rather silly. We both might agree that taking a pilgrimage to a rock where someone once had a vision of God was rather silly.

But clearly some people find prayer edifying, others find value in meditation and others in pilgrimage. Why?

I watch some sports, like soccer, and it just seems rather daft. I watch a good baseball game, and I am transported by it. Why?

I’ve been thinking about why some experiences are meaningful and ‘deep’ and others trivial and ‘shallow’. There may be many answers, but I’m pretty sure one of the most important is investment. We find experiences to be edifying if we invest in them: if we give ourselves over to them without cynicism.

I still pray from time to time. Not to anything. Prayer is not an transitive verb for me (like singing, I only do it for and to myself). Sometimes it is good, sometimes not. When it isn’t good, that’s because I catch myself doing it and my cynical brain decides it wants a debate.

Cynicism is the opposite of investment when it comes to these things: at least for me.

Investment is about choice. Lack of cynicism is about choice. I realize I can choose to engage with anything and find it deep. I could choose to invest uncynically in prayer, or the laying on of hands; in a tea ceremony or a shabbat meal. And it would be meaningful.

But!

I have other reasons for not participating in some things. I won’t join in Communion if I go to a church. Not because I couldn’t choose to invest in it and make it meaningful. But because doing so would be deceitful: I would be adopting an identity that I don’t want. I wouldn’t take part in a Peyote ritual, because I don’t do drugs. I have political and credal objections to a lot of the spiritual practice of organized religions. But yet, the mental states associated with them I would like to experience from time to time.

So over the last year or so I’ve been looking for things that I can do. Things that I find morally neutral, but open to investment, and capable of inducing the same experience of transcendence (by which I only mean the sensation of loosing one’s sense of self or self-importance). Its been hard, but there are a few that work well (the list is personal, YMMV).

Listening to noise – I started listening to modern classical, particularly highly minimal pieces. I mean really listening. As an act of investment. But I wanted the sound to be always less structured, so I turned to noise. I mix my own concoctions with a blend of broad-spectrum noise, low frequency hum and static. Sometimes now I find myself fantasizing about the sound of noise and the effect it induces.

Studying Kōans – I solve tricky problems for a living. There is something quite moving about committing to something with an inherent contradiction. Here the dividing line between investment and cynicism is stark. When I first heard about Kōans such as “what is the sound of one hand clapping?” I thought them silly. But investing in them really does induce a different state of mind and a different perspective. Sometimes this just doesn’t work, because my cynical brain won’t be quiet.

Gratitude – Not to anyone (though I am grateful to plenty of people), but spending time investing in being grateful. In deliberately focusing on those things and people that enrich, challenge and protect me. In dedicating time to teasing it out from the shadows of my selfishness.

Mazes, Labyrinths and Shapes – Tracing shapes and symbols I find a really efficient anchoring system. Shapes and paths can suck up meaning, if you invest it. I made a conscious choice to give certain shapes certain meanings. A silly conceit, no doubt. But they repay that investment, because those shapes are everywhere.

These ad-hoc practices, combined with our recent family trauma, prompted me to try something new and a little more structured. Or at least it prompted me to have the courage to suggest it to my wife, who encouraged me to do something about it. It is done, and I’ll share what that is in a couple of days.

In the meantime, for those of you who are post-religious here, or maybe post-theistic but still in a faith community: do you have similar activities you invest in so that you induce the same kinds of mental states that religion used to provide for you?

Creationism and the Courtiers’ Reply

PZ Myers put forward this famous and rather devastating counter argument against theology:

I have considered the impudent accusations of Mr Dawkins with exasperation at his lack of serious scholarship. He has apparently not read the detailed discourses of Count Roderigo of Seville on the exquisite and exotic leathers of the Emperor’s boots, nor does he give a moment’s consideration to Bellini’s masterwork, On the Luminescence of the Emperor’s Feathered Hat. We have entire schools dedicated to writing learned treatises on the beauty of the Emperor’s raiment, and every major newspaper runs a section dedicated to imperial fashion; Dawkins cavalierly dismisses them all. He even laughs at the highly popular and most persuasive arguments of his fellow countryman, Lord D. T. Mawkscribbler, who famously pointed out that the Emperor would not wear common cotton, nor uncomfortable polyester, but must, I say must, wear undergarments of the finest silk. Dawkins arrogantly ignores all these deep philosophical ponderings to crudely accuse the Emperor of nudity.

It was designed to counter those ‘sophisticated theologians’ who criticized New Atheism’s arguments as simplistic and who encouraged Dawkins et al to read and engage with modern theological treatises.

When I first read it I was impressed, but I’ve been hearing it a lot lately, and not from Atheists.

It is, at heart, an argument of anti-intellectualism. It says that for field X there is a direct way of accessing the truth and therefore one can come to the secure conclusion that any academic or scholarly work in that field is wrong without having to consider it on its own merits. Put another way, if you have good reason to believe you know the truth, than you don’t need to look at the evidence.

Which other groups make the same kind of claims? Lots, as it happens. Geocentrists, holocaust denialists, mythicists, and, darling of my debating life: Creationists. And the latter (who I have most exposure to) have begun to quote it to me. When I suggest that they find out about genetics, or learn about radiometric dating techniques. And this week it was quoted by name:

Look up the Courtiers Reply by your beloved PZMeyers [sic] — I don’t need to go to school for ten years in biology to know its all a lot of rubbish — cats don’t turn into dogs — the emperor has no clothes.

I have studied, and continue to study theology. I believe there is no God (for the most common definitions of God). I think those very few theologians who claim their work shows otherwise are wrong, and I can generally say where their reasoning is specious. But such rejections cannot be made on the basis of Emperor’s New Clothes type thinking.

I wince hard when New (or Gnu) Atheists make theological statements the equivalent of “cat’s don’t turn into dogs” and then, when they are corrected hide behind this kind of response: ‘first show me why your field is valid, then I’ll listen to anything you have to say’. Especially when this is most often correlated with communities of non-believers who are more likely to claim they understand religions better than their believers.

And this is why: it is the exact same arrogant anti-intellectualism that endangers good science.