More from the oracle.

I've been rather busy for a while, so I haven't been posting here, but I have still been getting emails from the sibyl. I've got 3 more since I last posted, and I'm not sure what to make of them.

Here's the first:

Date: Sun, 9 Nov 2008 13:26:03 +0100
From: sybil.cumae@googlemail.com
To: sfwc@hotmail.com
Subject: I thought you might want to know...

World peace would be a good thing.

I didn't pay too much attention to this one when I first got it. I thought it was just a return to the simple truths of the earlier emails. I didn't notice the obvious problem until I read the next email:

Date: Tue, 11 Nov 2008 20:13:26 +0000
From: sybil.cumae@googlemail.com
To: sfwc@hotmail.com
Subject: I thought you might want to know...

The Badshahi Mosque is a beautiful building.

Again, my first reaction (after finding out what the Badshahi mosque is) was to accept this as a standard fact about the world, easily determinable by a wiki-search. Then I changed my mind. After all, whether a building is ugly or not isn't an objective fact; it is subjective. I guess that there are people somewhere who think this mosque isn't all it's cracked up to be. We don't have an objective standard we can go to to show that they are wrong: It's a matter of taste.

On the other hand, the emails from the sibyl have all been true (I've given up on explaining how) so I'd expect the content of this one to be true as well. Is the accuracy of the emails so far a good reason for me to accept the idea that there is an absolute aesthetic standard? Is it good enough to accept that the Badshahi mosque is (absolutely) beautiful? Is it better evidence than my own direct perception of that beauty in pictures of the mosque?

This train of thought led me to reevaluate my thinking about the previous email. After all, I don't normally ground my moral decisions in an objective moral standard. But for the content of that email to be true there would have to be such a standard. Is this good enough evidence for me to accept that some things are good in an absolute sense, or that world peace is one of those things? Is this a better reason to believe that world peace is good than my own direct perception of that fact?

Things got even worse this morning, when I got the following email:

Date: Thu, 13 Nov 2008 08:11:19 +0000
From: sybil.cumae@googlemail.com
To: sfwc@hotmail.com
Subject: I thought you might want to know...

Humans have souls, which survive the death of the body.

This is a statement for which (apart from this email) I had negligible evidence. Is the accuracy of the previous emails on matters of fact enough to make this email into good evidence of life after death?

I don't know quite how to think about these emails. First, I have no explanation for their earlier accuracy. Second, I have no idea whether or how far it is sensible to extrapolate that accuracy to statements about beauty, morality or spirituality.

An extreme oddity.

Today I got yet another email from the sibyl, and it has completely freaked me out. It read as follows:

Date: Thu, 6 Nov 2008 07:37:41 +0000
From: sybil.cumae@googlemail.com
To: sfwc@hotmail.com
Subject: I thought you might want to know...

When you read this email you will have 3 20p coins in your pocket. Toss one of them 30 times and you will get the sequence THTHHHHHTHTTTTHTTHTTTTHHTTHTTT.

I had 3 20p coins in my pocket and I tossed one 30 times. I got exactly that sequence of heads and tails. The chance of this is less than 1 in a billion. There's no way any scammer, however sophisticated, could beat those odds. Dedicated googling has turned up no information at all about where these emails could be coming from. I can think of no explanation. Maybe I'm going mad.

A worrying development.

I've been avoiding checking my inbox for a couple of days, but today I finally had a look. I've only had one email from the sibyl, which was as follows:

Date: Mon, 3 Nov 2008 00:17:24 +0000
From: sybil.cumae@googlemail.com
To: sfwc@hotmail.com
Subject: I thought you might want to know...

When you read this email you will be wearing glasses, beige trousers, a black T-shirt, a brown patterned jumper and gray boxer shorts. You won't be wearing shoes or socks.

This prediction is spot on, but I'm not sure what it means. How do they know what colour my underwear is? Is this supposed to be some kind of veiled threat? In any case, if they know so much about me then they almost certainly read this blog. If so, I'd like to make the following statement:

I did not buy a lottery ticket, and I do not believe anybody has the ability to forecast lottery results. I cannot and will not reimburse you for any money you have expended in the course of your scam. If you threaten me in any way or ask for money from me, I will contact the police.

The oddity again.

I've just had a look at the national lottery results from last night, and they matched the email from the sibyl. I guess this means that, if this is a scam, I'll soon be targeted. If I'm right in my guess about how the scam works then whoever is running it has lost a reasonable amount of money and is hoping to make it back from me. Of course, all of this is built on a tissue of conjectures which I'm now beginning to hope are false. I'm not really sure what to do; I need to take a little time to think.

Reviewing ideas.

The second birthday book I read was a mini-tome called 'Ideas - A history', by Peter Watson. The aim of this book is to provide a condensed overview of the history of ideas and their development, and it does this in a remarkably thorough and wide-ranging way. I was enthralled with the scope of the development and continually surprised by two particular aspects of this history.

The first thing that struck me was how recent many of the ideas were. In particular, ideas that I hadn't recognised as such because I take them for granted are in some cases just a few thousand years old, or even less. For example, the key idea that we can distinguish between subjective and objective kinds of knowledge (on which I'll say more when I discuss one of the later books) seems to have sprung up in the last millenium, and the distinction between personal and physical explanations of events is almost as recent.

The second big surprise was the number of ideas involved in the development of things which I normally think of as individual ideas. A good example is writing, which is a skill I have known for so long that I did not realise how many ideas (apart from the obvious development of language itself) must be developed before it is possible. I can think of at least 12 steps:

1. The use of tools external to the body.
2. The preparation of tools in advance of when they are needed.
3. Using the state of the tools as a memory aid.
4. Creating tools to be used purely as memory aids.
5. Drawing pictures as a reminder of the things they resemble.
6. Associating the pictures with particular words.
   
7. Representing abstract ideas by the pictures of words with similar sounds.
8. Representing ordered strings of words (as from speech) through strings of pictures of the individual words of which they consist, to achieve a kind of frozen speech.
9. Representing particular sounds consistently by abstract symbols.
10. Developing conventions for which collections of symbols correspond to which words.
    
11. Representing other aspects of speech via punctuation marks.
12. The development of forms of language appropriate to written, rather than spoken, communication.
    

Each of these steps happened surprisingly recently. Further, although each seems trivial given (or even indistinguishable from) the last, the gaps between the developments of these ideas were remarkably long, being at least a generation in even the shortest case. The fact that this process was so painstaking makes it clear that we are currently missing some developments which, with hindsight, will seem equally obvious.

There were a couple of problems with the book, though. The first is related to the fact that the book is one of history, and often had to deal with periods about which we have very little evidence. In these cases, Watson would give the latest scholarly opinion, but without treating it as opinion. He reported it (in many cases) as if it were established fact. This meant that he was often reporting current, rather than past, ideas.

The second problem was the provincial focus of the later parts of the book. The only serious ideas mentioned for the last 800 years were from Europe and North America. Though other countries were mentioned, it was only in terms of what effect consideration of their cultures had on the West. It gave the impression that everyone else had been sitting on their hands for the last millenium.

The oddity explained.

I've continued to get odd emails from sybil.cumae. I've worked out where the name comes from (By the power of google!!). It seems that there was a mythical oracle called the Cumaean sibyl, about whom there are various legends. I guess this is a sidelong claim that the content of the emails is true. This isn't my only discovery, though. In fact, I now reckon I know what's going on, as I'll explain in a minute.

Although the emails are starting to get a little longer, they are still terse by comparison with other emails. I've started to enjoy getting them; it's fun to check the random odd facts, which are beginning to be a little more interesting, and introduce me to areas I wasn't aware of. A disproportionate number deal with mathematical facts, though, and I think I've finally found one email which is in error:

Date: Tue, 28 Oct 2008 09:59:25 +0000
From: sybil.cumae@googlemail.com
To: sfwc@hotmail.com
Subject: I thought you might want to know...

Any even integer greater than 2 is a sum of two primes.

This is, of course, the Goldbach conjecture. The reason it is called a conjecture is that, although it is widely believed to be true, nobody knows of any way to prove it. I guess whoever is researching the facts in the emails didn't do their homework properly.

However, the content of the email that has finally cleared up this whole mess was as follows:

The winning numbers in the UK national lottery lotto game on Saturday 1st November 2008 will be 18, 19, 25, 30, 33 and 43.

Of course, there is only one way to reliably predict such things in advance: The emails I've been getting are part of an elaborate 'football picks' scam. The way that this usually works is that letters or emails are sent out to (initially huge numbers of) victims with betting tips about who will win particular football matches. For each match, about half of the victims are told that each team will win. Of course, this means that a lot of incorrect predictions are made. Any victim to whom an incorrect prediction has been sent will be abandoned. Eventually, there will be a very small number of victims left, for each of whom all of the predictions so far have been correct.

All that any of these victims are aware of is the sequence of correct predictions which has been sent to them. So they think that the tipsters are giving sound advice. Playing on this mistake, the tipster contacts these victims offering to give further tips in return for a fee, now that their expertise has been 'demonstrated'. The victims pay the fee hoping to win their money back by gambling on the basis of the tips they are given. The tipster, no longer caring whether the victims win or lose, disappears with the fee. Something of this kind was the basis for Derren Brown's 'the System'.

This scam has now become rather well known, (even referenced in 'the Simpsons'), so that running it in this iterative way has become impractical. Accordingly, the scammers who are sending me these emails have modified it to make it less recognisable in at least three ways:

• Numerous other, easy to obtain, facts are also sent. This encourages the user (me) to check those facts, and to imagine that that whoever is sending them has researched all of them. This subtly hides the fact that the final 'fact' is not researched at all.
• All the eggs are put in one basket: Only one prediction (of lottery results) is made, but the prediction is improbable enough to be highly convincing. This removes the repeated predictions which are a giveaway for the usual scam.
• I'm not completely certain about this last point, but it is rather likely. The chances of winning the lottery are about 1 in 14 million, and there are only 60 million people in the UK, so normally to pull this off the scammers would have to email about one person in four. Nobody I've spoken to is getting similar emails, so I think it is very unlikely that they have done so. Instead, my guess is that they have covered the numbers they didn't send by placing a bet on the results of this weeks lottery draw betting that none of their predictions were right.

I've worked out the numbers for this last point and I reckon it could be done. For example, they may have placed a bet of £100,000 that none of the numbers will be 25, thus reducing the number of people they must email to about 1.7 million, which is a much more manageable number (I'd guess about double this number were emailed to give the system some slack). If none of the numbers are 25, then they win their bet and so make a profit of about £14,000. Otherwise, they have lost £100,000 which they must recoup from somebody they have emailed. But one or two of the victims believe that they can predict lottery numbers (and may even have won) and will now be squeezed in the hopes of recouping the lost £100,000 together with a large profit. The money involved here is not precise: It can be scaled up or down depending on the resources available to the scammers and how much they think they can practically squeeze out of the victims.

Needless to say, I haven't bought a ticket.

To swallow meditation...

Over the summer, I regularly attended a meditation class run by the Cambridge Buddhist Centre. Unfortunately, it clashes with the category theory seminars so I've had to stop. Each class lasted about an hour, and consisted of a short introductory talk (to let complete newcomers like me know what we would do) followed by about half an hour of meditation. The meditation was of two kinds: The 'mindfulness of breathing', in which we tried to be completely aware of our breathing, and the 'development of loving-kindness' in which we pictured a variety of people and tried to feel love for them.

These practices came out of a Buddhist tradition (the good folk at the Buddhist Centre claim that they were developed by the Buddha himself, but it doesn't really matter either way). In order to explain a bit about what they are about, I shall show my complete ignorance of Buddhism by talking about it a little. The first surprising thing about Buddhism is its diversity. The first big split seems to have come a few centuries after the death of the Buddha, when the Mahayana school split off. When Buddhism (mostly Mahayana, as far as I can tell) spread across Asia it mutated and incorporated local traditions and pantheons wherever it went, getting mixed up with Taoism and Bon and producing anything from the paradoxical Zen to the flamboyant Tibetan style. Each of these schools, and the many others, are further subdivided, in a way reminiscent of protestant denominational fertility gone wild.

I've followed this trend by producing my own slant on what the Buddha (or at least those who came up with the words attributed to him) was on about. What follows is by no means orthodox Buddhism of any kind, though the ideas it is based on mostly come from the Theravada school. Think of it as an improvisation in a Buddhist key.

As human beings, we are subject to persistent and pervasive delusion. One of the nastier aspects of this is that it is built into the way that we think, reason and communicate so that any attempt to deal with it is liable to suck us further into the quicksand. Here are several specific areas where we are deluded:

• When we use language, we divide the world up into neat conceptual chunks. We'll see a building as a single entity, distinct from the rest of the world. This distinction is sensible, but it isn't as precise as we habitually assume. There are gaping grey areas. Is the furniture part of the building? What about the built-in furniture? The paint on the walls? The pictures on the walls? The hooks holding them up? The water pipes? The water? The limescale? The wiring? The light bulbs? The walls? The curtains? The windows? The holes in the walls in which the windows are placed? Some things (like the bricks in the walls) are a relatively stable part of the structure and change only rarely. Others (like the newspaper on the table) are replaced much more often and we don't count them as part of the building. In between it isn't so clear.
  
  In a sense, there's nothing wrong with this world-chunking: To an excellent approximation our environment can be divided up in this way. The division itself is no more of an illusion than is Newtonian physics; both are great approximations (though quantum theory suggests that the ladder of approximations never bottoms out in precision: The world is radically and ineradicably approximate). The illusion comes in when we forget that that is all that is going on, and see the divisions as precise rather than approximate. We imagine, for example, that there is some kind of clearly delineated 'substance' of the building, to which the various parts stand in some clear relation. We even conceptualise entities with no sensible physical correlate, such as rainbows, in this simplistic way.
  
  
• The first problem is particularly serious for the way that we see ourselves. First, we are especially prone to seeing ourselves as simple coherent wholes, and to seeing our chunking of ourselves off from the rest of the world as exact. This is made all the worse by the fact that this approximation is much less sensible than many of the others that we make. I've discussed one aspect of this in a previous post. Essentially, the idea that 'me, right now' and 'me in a few years' correspond to different instances of the same self is no more than a useful convention.
  
  I must pause at this point to explain why this ambiguity of the self unifies the idea of reincarnation with the golden rule. The trouble with the idea of reincarnation, as with many hypotheses about life after death, is that we typically identify beings by the continuity in time of the bodies to which they are associated. The death and decomposition of the body prevent such identifications being made, so identifications of newly born beings with past dead beings seems arbitrary and unjustified. On the other hand, we have a habit of identifying various beings with the beings associated to the same bodies at slightly later times, or after a good night's sleep. There is a strand of Buddhist thought which emphasizes the arbitrariness of this by talking about life as a process of continual reincarnation, with every moment seen as a small death and rebirth, sleep seen as a more emphatic instance of the same, and death itself as a qualitatively similar process.
  
  This has moral implications. I've discussed morality already here, and this comment is an expansion on the third paragraph from the bottom. We have clear moral beliefs related to our own benefit; for example that pain is bad. In fact, these beliefs are mostly related not to ourselves as beings now, but to ourselves in the future: The beings that will in the future be associated to our bodies. But the identification of these beings as special is pure convention, and singling them out for special moral attention is so arbitrary that the conscience rebels against it. I've heard Buddhists argue that I should treat other beings ethically because (modulo reincarnation) they may be, for example, my mother. The perspective I have outlined is even more radical. I should love my neighbour as myself because (modulo a useful convention) they are myself.
  
  
• Closely related to the above two problems is the way that we think about the words we use. We imagine that each word captures a clearly definable concept with some correlate in the world. A major activity in philosophy is exploring to what extent this is true. Alarmingly, many of the words we use (such as 'self', 'be', 'know', and so forth) don't have sensible definitions corresponding to the way that we naturally use them. This hasn't stopped some from mistakenly trying to find out what, for example, knowledge 'really is'. I'm told that pointing out and avoiding such mistakes was a big part of what Wittgenstein was doing later in his life, but this may be just a rumour; I've never checked.
  
  
• Very often we find that the world is not as we wish it to be. Such problems are exacerbated by our habit of fooling ourselves into thinking that the world could be other than, in fact, it is. For example, if I break my arm, then I am likely to find myself imagining a world in which my arm is suddenly no longer broken as if it were possible, when it is not. Most of our desires (not all: Hunger and thirst are almost always free of this) are of this illusory form; when we desire a thing we imagine a world in which we possess that thing without undergoing the sacrifice necessary for that possession (if it is even possible).
  
  
• Sometimes we desire the world to be different in a way that we can legitimately bring about. For example, you may want to own a TV, and be able to afford one. Even in these circumstances, however, our desires often fool us by presenting as possible a world in which we obtain the desired in an ideal way. You may ignore the fact that the television will break down soon enough, or that the image will jump when old motorbikes pass your house, or that the television will bring disappointment through expectation of programmes which you miss or which are over-hyped. When we desire a thing we imagine a world in which we possess that thing without the inconveniences (both trivial and serious) that follow from the nature of that thing.
  
  
• We have the same blinkered attitude to the things that we already have and take joy in. We can not recognise without effort that they are transitory: Gone before you can say 'where moth and rust doth corrupt'. It is hard to enjoy a thing without imagining a world in which that thing is enduring and innocent of the inconveniences (both trivial and serious) that follow from the nature of that thing. There's a neat story illustrating a good attitude here: A man had a beautiful glass which he would often drink from. One of his friends asked him 'aren't you worried that if you use that glass all the time you will break it?'. Holding the glass up so that the light sparked intricately through it, the man replied 'I know that it is already broken. That is why I make such use of it.'
  
  
• This mistake is also particularly serious for the way that we see ourselves and those we love. We do not recognise that we are transitory. Tomorrow we will not be the same as today, and the pleasure we have now will have passed. In a year or two we may be embarrassed by who we were, just as now we would be embarrassed at who we will become. As we grow older, our bodies and minds will gradually break down, until sooner or later they fail catastrophically in death. After that, we will be inaccessible to all who seek us. Those who love us will be left with small reminders of us which mock them with false hope. It is painful even to consider these things, and we typically simply ignore them. This cocoons us from seeing how precious our passing bounties are and multiplies our grief when they are inevitably lost.

These are just a few of the veins of illusion running through our minds. Even in talking about them I've had to use words and concepts, which introduce their own mirages. This all seems pretty hopeless and gloomy. However, the Buddha was extraordinarily optimistic about our ability to overcome all of this.

He claimed that it is possible to become free of all of this illusion, and to see the world as it is. So, for example, we would not see ourselves as simple unified souls, or distinguish ourselves fundamentally from other people. We would go beyond words by not attributing to them any power to divide up the world. Since the delusions we suffer are so fundamentally part of us, and form much of the way we conceptualise ourselves, this freeing can be thought of as a kind of loss of self, like the blowing out of a candle: An unlightenment, if you will. On the other hand, since much of what would be lost is associated to the 'autopilot' systems in our brains which guide us unthinkingly through daily life, there is also a strand of thought which sees this as a kind of awakening.

He also made some particular claims about the way to achieve this unlightenment, all of which were thoroughly practical. Over the years, Buddhists have come to think of this Way in sufficiently exalted terms to merit a capital W. Much of the Way consists in a particular style of living, modelled on how a person would live if they were unlightened. Much of this corresponds to sensible ethical teaching. Other bits of the Way concern meditation, and in particular meditation of the two kinds mentioned above.

The first good reason to do the 'mindfulness of breathing' meditation is that it is a simple context in which to attempt mindfulness; that is, a direct awareness of how a thing is without use of the usual mental filters and shortcuts. The breath is so comparatively simple and regular that it is a good place to start in attempting to see things as they really are. The second good reason comes from one of the direct effects of making such an attempt: Namely that you fail, and fail in a particular way. The mind becomes distracted, and thoughts of unrelated matters slip in unnoticed. Observing this process allows you to see how busy the mind is, and (to a small extent) in what its activity consists. Not only does this allow you to see how automatically illusory thought patterns are produced, it also reveals how inadequate our ideas of the self as simple and continuing are. Hume put it very well:

For my part, when I enter most intimately into what I call myself, I always stumble on some particular perception or other, of heat or cold, light or shade, love or hatred, pain or pleasure. I never can catch myself at any time without a perception, and never can observe any thing but the perception. When my perceptions are removed for any time, as by sound sleep, so long am I insensible of myself, and may truly be said not to exist. And were all my perceptions removed by death, and could I neither think, nor feel, nor see, nor love, nor hate, after the dissolution of my body, I should be entirely annihilated, nor do I conceive what is further requisite to make me a perfect nonentity. If any one, upon serious and unprejudiced reflection, thinks he has a different notion of himself, I must confess I can reason no longer with him. All I can allow him is, that he may be in the right as well as I, and that we are essentially different in this particular. He may, perhaps, perceive something simple and continued, which he calls himself; though I am certain there is no such principle in me.

The second kind of meditation is aimed at dealing particularly with the issues mentioned after the second bullet point above. The first part of the meditation (once you have settled down and relaxed) consists in regarding yourself, and exploring how it is that you love yourself. Then various other people are brought to mind (a friend, a passer-by and an enemy) in turn, and for each person you take the time to recognise that they are a person in the same way as you are, and explore how this allows you to love each of them in the same way. Finally, you regard all of these people together, and try to equalise up the love that you feel for them (and, of course, yourself) which of course involves seeing the distinctions between these various beings for the conventions that they are. This second meditation is therefore aimed at a particular illusion only, but one which there are strong ethical reasons for overcoming.

That is the theory, at least. Do these practices work? That is, do they lead people towards unlightenment (and, along the way, love of neighbour)? I don't know. The theory sounds plausible but so do many false theories. Tests so far have established that there is something going on (that is, meditation involves real structural mental change), but as far as I know they have not shown that these changes are as claimed. Since the claims seem perfectly testable, however, I'd be very much in favour of tests exploring whether meditation does what it says on the tin, and whether any other strategies are equally or even more effective.

An oddity.

I've been getting some very odd emails, which I can't really explain. Here's a typical example:

Date: Mon, 13 Oct 2008 11:52:04 +0100
From: sybil.cumae@googlemail.com
To: sfwc@hotmail.com
Subject: I thought you might want to know...

Alaska is the largest state in the USA.

Others (from the same address and with the same subject line) have had similarly terse content:

• The 500th digit of pi after the decimal point is 2.
• The reciprocal of the fine structure constant is about 137.036
• Gray whales migrate at an average speed of 3 mph.
• King George III of the United Kingdom died on a Saturday.

I've checked these statements (wikipedia is my friend) and they all seem to be accurate. I can't see anything else they have in common. And I have no idea why anyone would send them to me. I don't know anyone called anything like Sybil Cumae. They don't have any kind of attachments that could be harbouring viruses. I've tried telling hotmail that messages from sybil.cumae@googlemail.com are junk, but it still puts them in my inbox. I'm beginning to suspect that this is some kind of viral marketing thing for a microsoft product, but I haven't heard of anyone else getting similar emails and I can't find any reference to it online.

I'm stumped.

Strange loops 'R' Us

I got a heap of books for my birthday, which I want to get my thinking straight on. So I'll be writing about them here. The first to arrive was a book by Douglas Hofstadter (of 'Gödel, Escher, Bach' fame) called 'I am a strange loop'. In this book, Hofstadter revisits his previous musings on selves, souls and strange loops.

The jumping-off point is a paradox. Careful examination of the brains of human beings has shown no violations of the laws of physics, and all the physical evidence we have points to the idea that the matter of which we are formed follows the same rules as all the other matter we come across. So it looks like we could, in principle, explain all the behaviour of any given person in standard physical terms. So, for example, the answer to the question 'why is this person angry' would be 'well, you see, these various atoms are banging about in such-and-such a way'.

But an odd thing happens; our explanations very often go the other way. We ask 'why is it that this collection of atoms is moving about in such-and-such a way?', and reply 'well, because this person is angry/sad/happy/thankful'. The explanation here looks like it is going the wrong way; the lower levels (in a reductionist sense) are being explained in terms of the higher.

Hofstadter summarises this paradoxical aspect of our selves by saying that we are 'strange loops'. The loopiness is clear; our thoughts can be explained in terms of physical events, and physical events can be explained in terms of thoughts. It is this second kind of explanation which makes the loop strange; it goes the wrong way - events at a lower level are explained in terms of those at a higher level.

As an analogy, he points to the proof of Gödel's theorem. In this proof, a statement G of number theory is concocted which cannot be proved or disproved. The (lower level) fact that G cannot be proved is explained in terms of higher-level properties relating to a subtle scheme for coding statements of number theory as numbers. Here, too, the explanation seems to go the wrong way.

Hofstadter goes no further than this in defining just what a strange loop is, and he certainly does not give a precise definition. Instead, the idea serves as a powerful analogy bringing out an aspect of the strangeness we often overlook in our all-too-familiar selves. The puzzle has been sitting under our noses for a long time, and there have been many proposed resolutions.

For example, it has been proposed that it is impossible in principle to reduce the soul to the usual behaviour of matter. Instead, our souls are made up of a different, more ethereal stuff which interacts with the matter in our brains in a deeply mysterious way. This removes the strangeness and sees our consciousness as involved in a much more mundane causal loop.

The trouble with this proposal is that if it is right we ought to see some violation of the laws of physics happening inside people's brains. The fact that we see no such thing has led to the suggestion that the link may not be causal; that the matter may in fact be carrying on in the usual way of matter and yet still have a mysterious 'self', made of spirit-stuff, associated to it. The trouble with this idea that the soul doesn't affect the body is brought out in a beautiful though experiment by Raymond Smullyan.

Another proposal is that there is no such thing as a soul; that such things as minds, ideas, consciousness etc. are pure illusions, and there is really nothing but physical stuff. Now, given how obvious it is that we are, for example, conscious beings, it is hard to argue for the idea that this is just an illusion. The main argument presented here is that, after careful investigation, we have not found the soul anywhere. But things need not be localised to particular collections of matter in order to exist. Think of the colour red, for example.

Another argument sometimes given here is as follows: Carefully examine the many things you associate with yourself. You find that your body is not yourself. Nor is any particular thought or feeling or sensation. In fact, close introspection reveals that in our multitudinous mental life there is not one simple thing we can point to and cry 'There it is: The self!'. The absurdity of this argument can be seen by applying it to chairs. Examine a chair closely, and study its parts. The legs are not the chair. The flat surface for sitting on is not a chair, and nor is the back. In fact, close examination shows that no simple part of the chair is the chair itself. Are we forced to conclude that there are no such things as chairs?

The option I think is correct works like this: We can (in principle) describe a person in terms of the physical stuff of which they are formed (though such descriptions are likely to be tedious). We could alternatively describe them in terms of their personality, and of what they are thinking and feeling. These are just two different descriptions of the same thing, with the second description much more useful to us because of its coarser grain. Anger does not cause the mental processes we associate with it, nor is it caused by them. They are two descriptions of the same process. The explanation of each in terms of the other is not causal; it is a matter of translation.

A helpful analogy is that of temperature and pressure. We can (in principle) describe a gas by specifying just where all the particles are and how fast they are moving (though such descriptions are likely to be tedious). We could alternatively describe it in terms of its temperature and pressure. Once more, these are just two different descriptions of the same thing, with the second description much more useful to us because of its coarser grain. The temperature does not cause the particles to move at the speeds they do; it is a redescription.

Scientifically, we have a pretty good idea of what temperature and pressure mean in terms of the behaviour of atoms. At one time, though, there was no understanding of what temperature and pressure meant in terms of more basic physics. We are in a similar position today with regard to the soul. This does not mean that there are no such things as souls, any more than their ignorance meant that, for those long-gone scientists, there was no such thing as pressure.

This gives a useful perspective on the second resolution. It may be that, one day, we will understand that some of the vocabulary we use to talk about ourselves (such as the word 'ourselves') is misleading, and reject it, just as the theory of phlogiston was rejected. At that time we will be able to say with confidence that the notions we reject are illusions. We are currently very far from that stage: We may never reach it, and almost certainly won't in my lifetime. For the forseeable future, we won't not know what is illusory and what isn't.

Hofstadter's next point is that our idea of the soul is a little blurred around the edges (just as all our ideas are). For example, each person has models in their own head of the other people they know well. For people in strong relationships these models can become intertwined in feedback loops with the self-models of the other people (Hofstadter repeatedly confuses these self-models with the selves themselves). Accordingly, it isn't possible to completely localise a person within their own head.

The punchline is that, when a person's body dies, the person lives on to a small extent in the heads of those who loved that person. The trouble is that the extent to which the person lives on is negligible by comparison to what is lost. It is analogous to the extent to which the person lives on via the atoms from their body which have been incorporated into other living organisms. When a person dies, the loss to us is irreparable and complete. This is why we grieve so strongly, and why we reach out for comfort in the smallest scrap of hope that the person still, in some sense, lives on.

Swift meditation.

In ten or twenty years, when the person I have become reads this blog, I have no doubt that he will be shocked and outraged that he could be associated with a person like me. I feel the same way when I read the brief diary I kept on a short trip I and some friends took to Bolivia just after our GCSEs. We spent most of the time touring the diverse and spectacular landscape, from salt plains to rainforest, by boat, bus, plane and on foot.

The trip had two large impacts on my life; first, it helped me recognise in a small way how the same humanity I see in myself and those I know well is also found, radically transformed, in other cultural settings, and how the simple joys of life do not rely on the material securities in which we often ground them. Second, and more mundanely, to get fit in readiness for the high altitudes, I began to run. Running has now expanded to become an integral part of my life, and it is this much easier topic that I'll talk about here.

When I first started, I could barely keep going for 2 minutes together, even at a jog. I've been steadily improving since then and I'm pleased that I haven't yet reached my limit. I now normally run 3 times a week, for about an hour, on a variety of routes. I only occasionally run competitively, and though I'm encouraged by my clear improvements I'm still nowhere near the front of the pack.

I run, in increasing order of importance;

• To keep fit. As ways of staying fit go, running is simple, flexible and cheap.
• For the beauty. This consists not just in the magic of the landscape but the freshly-patterned sky set free from the buildings which cage it in in the town.
• To get away. When I'm running, I can't worry about work or life, and I don't have to.
  
• For the buzz. As ways of getting high go, running is simple, flexible and cheap.
• For the willpower. By experimenting in this toy setting, I've learned how to 'force my heart and nerve and sinew to serve their turn long after they are gone'.
  

A final aspect of running raises an odd moral problem. When I run, I'm often in pain. This is my body's way of saying it wants to slow down. So I lie to myself, saying that the pain is unrelated to the running, and would carry on in any case. This barefaced lie is enough to fool the system in my brain which otherwise would override my desire to keep going and bring me to a wheezing halt. I'm not too worried by this; I think the bit of myself that I'm fooling is non-conscious and so I have no obligations to deal honestly with it.

Where, though, is the line to be drawn? How deeply can I delude myself without hypocrisy? Do I have an obligation to correct my natural tendencies to fool myself (for example, about how likely a marriage is to last for life)? I don't know. As with other moral issues, I err on the side of caution, and of doing no harm. What I do to myself when I run is about as far as I'd be prepared to go.

A multitude of books distracts the mind.

Books: The crack cocaine of the literate classes.

This (stolen) quip captures with uncanny precision the way I relate to agglomerations of written words. I read compulsively and obsessively, to relax and to work, to exercise my mind and soul and to calm them. I'm currently partway through reading 5 books:

Sheaves in geometry and logic - Saunders MacLane and Ieke Moerdijk
The audacity of hope - Barack Obama
Surprised by hope - Tom Wright
Ethics - Baruch Spinoza
Conceptual foundations of quantum mechanics - Bernard D'Espagnat

This typical collection of books, together with the various papers and websites I'm reading my way through, takes a significant chunk of each day, but it is time I delight to give. The words I read connect me to people far removed in time and in space. When writing, they were able to carefully compose expositions of insights of a depth that we are too impatient to plumb in everyday conversation. By their radically different assumptions, they implicitly challenge the concepts that form the comfortable furniture of my worldview, undermining ideas that are so familiar they seem inevitable.

As I type this, I am sitting within 200 metres of one of the largest collections of books in the United Kingdom; the Cambridge University library. I find the thought of the shelves and shelves of books (over 5 million in all), most of which are filled with ideas and thoughts which will never have the chance to change me, impossible to capture. The simple awareness of it sucks away my mind like a sky filled with stars.

Of the many books I have read in the last few months, one stands out: 'The Road to Reality' by Roger Penrose. Penrose explains experimentally-based physics as it stands today, first laying the necessary mathematical foundations then swiftly assembling a structure which only just has sufficient rigidity to reach into the cloudy obscurity of general relativity and quantum field theory. Along the way he repeatedly points out subtle but illuminating perspectives on complex ideas. I was repeatedly left kicking myself over bits of mathematics that I thought I knew: 'Why did I never look at it like that? It's so obvious'. If you can get your hands on this book, and are prepared to fight to get through it, then you must read it.

I currently have a small stash of books sent to me by my relatives for my birthday, sitting in a knee-high pile on the floor of my room. I'll write a bit more about each of them here as I read them.

Long time, no say.

Since I've hardly posted anything here since before the summer holiday, I'll give a summary of some of the stuff I got up to.

First, I went with some fellow category theorists to Calais, for a conference. This was a really great experience. I was pleased to find that I could (with a little effort) follow most of the talks, and reconstruct those that I hadn't followed at the time. The range of topics was extraordinary, and all were extremely interesting. The main problem I had was with seeing the motivation of some of the talks; the speakers were addressing problems which had arisen slightly before I got involved and which I hadn't come across.

The other great thing about this conference was the opportunity to get to know some of the other students in the category theory community. Some were friends with whom I'd lost touch, others were completely new. It was good to spend time with people who care about the same mathematical issues. I can't wait until this conference rolls around again. Excitingly, there will also be a major category theory conference in Cambridge next year, in honour of Peter Johnstone and Martin Hyland.

Shortly after this, I visited Wales with some friends. We stayed in a youth hostel, and went on several enjoyable walks. The scenery was beautiful, and the weather mostly stayed fine, though we did get lost in the mist at one point, where the path disappeared. Thankfully, the land was sufficiently bumpy that we were able to find our way by navigating by the contours.

From there I went on to join the trinity lake hunt. This is a glorified game of tag over an area of a few square miles in the heart of the lake district. A few runners, designated 'hares', are given hunting horns and a half-hour head start. They have to blow the horns every couple of minutes and try to avoid getting caught by the 'hounds', that is, everyone else. Since I was only able to be there for a couple of days, I was made a hare: An exhilerating experience.

Finally, after a little time at home, my family and I spent 2 weeks in Scotland (my brother was only able to make the second week). We spent our time on gentle walks through the beautiful and diverse landscape, with some pauses to admire the birdlife and other wildlife. We saw a glut of eagles on Mull and were particularly thrilled by a close-up sighting of red-throated divers fishing in the sea.

Since then I've been in Cambridge, working on my PhD and amusing myself in between times with various other activities, which I'll discuss in the next couple of posts.

Do you know?

Strong agnosticism can be summed up on a bumper sticker: 'I don't know and you don't either.' With respect to Christianity, a strong agnostic would typically believe that nobody can know whether it is true or false. Initially, this seems possible; most major religions have spread by politics or violence, rather than by rational argument. But doesn't a little thought show that strong agnosticism is inconsistent?

Imagine a strong agnostic, Steve, contemplating a Christian, Dave. Steve says 'Dave doesn't know whether Christianity is true or false'. But then, looking in a Bible, Steve sees that it claims that people can know that it is true. Steve could then conclude that the Bible is false, and so that (at least evangelical) Christianity is false. He could go so far as to say 'I know Christianity is false,' thus refuting himself, for he believes that nobody knows whether Christianity is true or not.

It is just such an inconsistency that Alvin Plantinga points at in the book 'Warranted Christian Belief'. However, he goes further. After all, faced with the above argument, Steve is more likely to conclude that Christianity is false than that it is possible for people to know that Christianity is true. To escape this possibility, Plantinga sets up a model of how Christian belief could have warrant (for a discussion of warrant, see my previous post). He claims that none of the current arguments for strong agnosticism exclude his proposed model, so they don't show that nobody knows Christianity is true.

Of course, it wouldn't be inconsistent for Steve to say 'You propose that Christian belief is warranted in the following way. Although I don't know whether Christian beliefs are true or not, I do know that your account is flawed. I have an argument showing that Christian belief does not get warrant in the way you suggest.' The cunning argument above would not apply in this case, unless it could be shown that if Plantinga's account is incorrect, so is Christianity. Plantinga makes a claim of this kind ('if Christian belief is true, then the model in question or one very like it is also true'), but he provides little justification for this claim.

If Plantinga's account were to turn out to be fundamentally incoherent, Christians would be more likely to jettison Plantinga (together with his claim that if his argument falls so does the whole of Christianity) than to abandon Christianity itself. Indeed, this is true of any account of warrant. Dave would not be too worried if Steve were to show that any particular account of how Christian beliefs get warrant fails. He could just abandon that account, which is after all not fundamental to Christianity.

Surely, however, a similar thing would happen if Steve were to come up with an absolutely watertight argument showing that Christian belief, irrespective of whether it is true, cannot have warrant. Dave, forced to accept the argument, would not abandon Christianity itself. He would instead reinterpret the claim that Christians can know that their beliefs are true. Perhaps he would say 'This knowledge is not the petty kind that philosophers are able to tie up with words and play around with. This is a higher, a deeper kind of knowledge; in a sense it is more real, rather than less.'

Most strong agnostics are aware of this ploy; it has been used throughout history when part of Christian belief has been demonstrated to be false. The damaged part is abandoned, and the related claims are reinterpreted. This awareness means that they can't claim a proof that nobody knows Christianity is true as a proof that Christianity is false. For this reason, strong agnostics are not guilty of the inconsistency cited above. The argument is neat, but fails to take into account the realities of human nature.

Of course, this by no means shows that strong agnostics are justified in their claims. Indeed, Plantinga's model was constructed to show that they are not. Using the model, and other arguments, Plantinga considers several arguments for the strong agnostic conclusion, and finds them all wanting. Here Plantinga shows his skill as a philosopher: If the arguments are as he reports then they fail.

How does the model work? Well, Plantinga has defined warrant in terms of proper function. For example, it is through the proper function of my visual system that my beliefs about what I see have warrant. He claims that, similarly, there is a system (the sensus divinitatis) the proper function of which is to give us true beliefs about God. Accordingly, when we get beliefs about God in this way (such as that He exists, is omnipotent, omniscient and perfectly good), those beliefs have warrant.

A typical case of the operation of this sensus would be the arising of the belief, in a person looking up at the stars, that they must have had a majestic creator. It is also intended to include more specific religious experience, as well as the 'sense of the presence of God'. The claim is not that these experiences are evidence for Christian belief, but that the fact that they arise from a properly functioning cognitive system aimed at producing true beliefs gives them warrant.

It is therefore necessary to examine the idea of proper function. I believe that the principal system whereby I am able to know about my past, namely memory, is functioning properly and is aimed at giving me true beliefs. This is the case even though I have hardly any idea how it works. I would believe that my visual system functions properly and is aimed at giving me true beliefs even if I didn't have the slightly greater, but still extremely weak, grasp of how it works that I do have.

Why is that? Well, when I see something red, and I talk to other people, they also say it is red. Where I see a tree, others also claim that they see a tree. In much more complex systems, involving complicated interactions with other people or other senses, my visual system is vindicated time and again. This vindication is not the same thing as proper function, but it is certainly evidence of it. Accordingly, I have a second-level model of the world in which I model myself as seeing things via a visual system which gives me true beliefs about the world (for a discussion of this kind of model, see this post).

This higher-level model does not justify my visual beliefs; they do that for themselves. It explains them, and explains them in such a way that I believe that they have warrant. The model does not give the beliefs warrant (it is the proper function of the visual cortex which does that). Instead, the model justifies my belief that my visual beliefs have warrant: That they constitute knowledge.

Things do not always turn out so well. When I look at the systems that give me my beliefs, I sometimes find that they are not reliable, so that they either don't function properly or aren't aimed at producing true beliefs. Suppose, for example, that I have a particularly malignant kind of cancer. I may find that I nevertheless believe that I will survive. Careful study may show that this belief is common amongst those with terminal illness, and that it is as commonly false. I would have to conclude that the belief doesn't have warrant for me. The system that produced it is aimed at survival, rather than truth.

Of course, no system is perfect. Careful study of a system should allow us to discover under what circumstances a belief produced by it will have warrant for us. The idea that we can have beliefs, but believe that those very beliefs don't have warrant for us may seem odd, but it is a natural concept and one that we express using the words 'as if'. It looks as if the pattern is in motion, but I know that it's just an illusion. It feels as if there must be something I can do to contact my spouse, who is dead, but there is nothing.

Whatever system it is that philosophers use to come to their philosophical beliefs, it can't normally be functioning properly. This can be seen by considering the notorious and extensive disagreements amongst philosophers over the most basic things. Accordingly, you should take all philosophical ideas I present on this blog with a large pinch of salt (particularly since I'm not very good at philosophy).

What about the sensus divinitatis? Does study of the beliefs this system produces suggest that it is functioning properly? Frankly, no. The main reason is that it does not produce consistent results. Jews and Muslims believe on the basis of this sense that there is just one god. Christians believe something similar, though they are led to believe that that god consists of three persons. Hindus believe that there are very many gods, in just the same way. For still others, this sensus seems to give no more than the belief that the universe is a deeply mysterious, beautiful and awesome place.

Of course, Plantinga is aware of this problem, and is able to circumvent it. According to him, in anybody who is not a Christian the sensus divinitatus is malfunctioning as a result of the fall. Only in Christians is it functioning properly. This miraculous proper function is achieved through the proper function of yet another process (faith), which acts in Christians although it is not part of their native cognitive equipment. Since this second process is only given to Christians on the basis of grace, we should not expect to find it operating in non-Christians.

How does this secondary system work? First, there is the existence of the Bible, of which God is the principal author. When a Christian reads the Bible, a system called the internal instigation of the Holy Spirit (IIHS) kicks in, which when functioning properly gives the Christian the belief that what they are reading is true. This belief, coming as it does from a properly functioning system, has warrant. In such a circumstance, we say that the Christian knows by faith.

So now, it is necessary to ask, is faith (described in these terms) a properly functioning cognitive system aimed at the truth? Again, we must answer no. The first reason is once more the inconsistency of the beliefs produced in this way. The history of the Church is full of schisms and disagreements, some over substantial points of Christian doctrine, in which both sides claimed that they believed what they did by faith through the internal instigation of the Holy Spirit revealing to them the truth of scripture.

The second reason is one of inaccuracy. Some passages of the Bible suggest, on a naive reading, that God will answer any prayer whatsoever. Scientific studies have shown that this is false; in all circumstances that have been tested, God answers no more prayers than you would expect to be answered by chance if there were no God. You may not be so naive as to believe that the passages about prayer should be interpreted in a literalistic way, but others have been. They would have said that they believed God answers prayer, if not always, at least in such a way as to be distinguishable from coincidence, and that they believed this by faith through the internal instigation of the Holy Spirit revealing to them the truth of scripture.

Very well, the Christian may say, not everybody who says they believe something by faith is right. Nevertheless, in my case, faith is functioning properly, and so my beliefs have warrant. Even if this were true, the Christian surely does not have good reason to believe it. After careful consideration of the people with whom she disagrees, she concludes that people may have similar experiences to her and may be misled into false beliefs on the basis of those experiences. So she can't conclude with certainty that the same experiences in her are the operation of a properly functioning system. Nor can she claim that she knows by faith that her particular beliefs have warrant. On this model, only truths revealed in the Bible are known by faith, and claims about her particular beliefs aren't there. The best she can say is 'It feels to me as if these things are true'.

Even worse, faith can lead people to believe things in the face of strong contrary evidence. For, faced with strong evidence in favour of evolution, the Christian may reason thus: 'If my cognitive faculties were working perfectly, I would believe what the Bible says, and I would see its truth so clearly that this evidence would pale into insignificance. What's more, my belief would be warranted. So I can be perfectly warranted in believing this. Accordingly, I won't believe the theory of evolution by natural selection on the basis of this evidence'. This may seem like an extreme exaggeration of Plantinga's view, like a disfigured straw man. But in fact, it is precisely this sort of argument which Plantinga gives in response to the problem of evil in the last chapter of his book.

Plantinga's definition of faith, then, draws dangerously close to the caricature suggested by Richard Dawkins:

Faith ... means blind trust, in the absence of evidence, even in the teeth of evidence.

Then again, as Plantinga himself says,

What is supposed to be bad about believing in the absence of evidence?

Show me your warrant.

Imagine yourself on a walking holiday in the Lake District with an optimist. Despite forecasts of heavy rain, the optimist is convinced that the weather tomorrow will be fine, and plans a serious walk. As it turns out, the weather is perfect. The optimist triumphantly says 'You see; I knew it would be fine.' However, there is a sense in which they did not know. They had no good reason even to think that there would be no rain, and were right by pure chance.

Is Christian belief similar? If Jesus were to return tomorrow, and a Christian were to say 'You see, I knew it,' would they be misusing the word 'know' in precisely the same way? This is the question Alvin Plantinga addresses in his warrant trilogy. The main concept Plantinga uses is 'warrant': That quantity, enough of which makes true belief into knowledge. He begins by trying to answer the question 'just what is warrant?' This is a typical philosophical question, and comes with some typical philosophical issues.

We human beings, who spend most of our time interacting at sensible speeds with objects about the same size as ourselves, have developed language to allow us (among other things) to communicate truth about the world in which we live. In doing so, we have given words meanings. A philosopher, seeing that people use a word like 'exist', may wonder precisely what that word means. They will not be satisfied with the synonym-based definitions in dictionaries; they want to get to the heart of the matter.

Unfortunately, although we all get along perfectly well using the word in normal life, we become very confused trying to apply the word in extreme special cases (do possible worlds exist?). There may not be any precise formulation which captures the meaning of the word and corresponds to our intuitions in all special cases. One skill philosophers have honed is finding how close to this ideal they can get. In the course of their investigations, they have discovered that finding the meaning of words is hard, and often impossible.

Sometimes, if you are very lucky, it is possible to find a precise definition of an idea which is both helpful for talking about the world and close to the meaning of the original word. If this project is successful enough, the meaning of the word will eventually change to fit the definition. Something of this kind has happened with words like 'energy' and 'symmetry'. These definitions are often 'definitions in use': Rules for turning sentences involving the word into other sentences with the same meaning not involving the word. Unfortunately, when this is done successfully, the word is normally claimed in the name of science or mathematics.

More often, as with words like 'self', there is a definition-in-use which is helpful for talking about the world and fits very roughly with the original word, corresponding closely in some situations and deviating wildly in others. There may be more than one such definition, in which case philosophical arguments spring up. Occasionally, the project becomes so difficult that philosophers despair of the task. Something like this has happened with words like 'exist'.

What about warrant? Well, Plantinga is not using the word 'warrant' in a standard way; he instead defines it as the quantity enough of which makes true belief into knowledge. So the word whose meaning is being considered is 'know'. Where on the above continuum does this word lie? It has caused so much trouble that the study of it has been given a name of its own: 'epistemology'. Some have despaired of ever understanding it. Most, whilst agreeing that it is a stubbornly difficult case, have kept trying since it is such an important word.

Plantinga follows current fashion in saying that knowledge consists of true belief together with some other quantity: Warrant. In the first book of the trilogy, 'Warrant: The Current Debate', he tears apart several misguided attempts to give a precise definition of warrant. In each case he does so by considering some extreme special cases, in which the purported knower is suffering from severe cognitive dysfunction.

In 'Warrant and proper function', the next book in the trilogy, he suggests his own approach. Having just demonstrated how likely it is that any precise definition can be decimated, he presents a broad-brushstroke picture about which he is careful to make no claim of precision. Except in the heat of argument, he avoids phrasing his idea in terms of 'severally necessary and jointly sufficient conditions'.

The idea is that a belief is warranted if:

1. It is produced by cognitive faculties that are working properly in an appropriate cognitive environment.
   
2. The segment of the design plan governing the production of the belief is aimed at the production of true beliefs.
3. There is a high statistical probability that a belief produced under those conditions will be true.

In using the concept of proper function, this approach recognises that very often we rely on nonconscious processing within our brains to arrive at our beliefs. When I see a face, I do not carefully consider the raw sense-data being sent up the optic nerve from my eye and conclude that it matches a complex structural pattern I have designated as characteristic of faces. Instead, some pre-processing in my brain lets me know directly that it is a face I can see. I trust this pre-processing implicitly, because I know it functions properly. It is by means of this proper functioning that I get my knowledge that it is a face that I see.

It isn't that I say to myself 'My pre-processors are telling me I see a face, and I trust them, so I really must be seeing a face'. I just know, without the need for such redundant reasoning, and I get the knowledge from that pre-processing. The belief has warrant because whatever part of my brain it is that subconsciously recognises faces is working properly.

Plantinga discusses some of the other ways we get knowledge, for example by memory or empathy. He concludes in each case that we get the knowledge via some such process, and that the knowledge has warrant because the process is functioning properly. He is a little short on detail, which is fair enough, as the way in which our brains actually do these remarkable things is not yet well enough understood to give a detailed account.

He also doesn't give a clear explanation of what a design plan (specifications for proper function) is, or what it means for such a thing to be aimed at the truth. Of course, we have an intuitive grasp of this idea. We can imagine a designer designing something. But we can also recognise that something is working properly if we think it was not designed; if we think it was produced by natural selection, for example. It is this sense that Plantinga intends at:

Here I use 'design' the way Daniel Dennet (not ordinarily thought unsound on theism) does in speaking of a given organism as possessing a certain design, and of evolution as producing optimal design.

Plantinga doesn't give details of how, if you come across a system, you can tell if it is functioning properly or what it is aimed at. He delights in not doing so; after all, he thinks that there was in fact an intelligent designer so no further explanation is needed. In this, he is mistaken. For if the definition of 'proper function' is 'functioning in the way that the designer intended', then we would have no way of determining, by observation of the system itself, whether it is functioning properly or not. However, microbiologists (for example) claim to be able to do exactly that; they are working with the intuitive notion of 'proper function', which Plantinga also uses but does not explain.

He thinks that the idea of an intelligent designer would explain what it is for a thing to be designed. But in that case, to determine the design plan of something, it would be necessary to know the mind of the designer (in this case, of God). But to say that the explanation of a difficult philosophical idea is that it may be found in the mind of God (as Plantinga does with many ideas, such as number, proposition, proper function and therefore also warrant and knowledge) is to say nothing. There is no more content there than in the more colloquial 'God knows'.

Ranting aside, however, the link Plantinga establishes between our ideas of knowledge and of proper function is valuable. The intuitive notion of proper function should suffice for the rest of this post. This account of warrant isn't quite right; it faces some minor technical difficulties. It does provide a strong enough support for a discussion of whether Christian belief is warranted, a topic which he tackles in the final book of the trilogy, 'Warranted Christian Belief', which I'll discuss in my next post.

A surprising connection.

Galois connections are often hidden behind well-behaved areas of mathematics, and they are often produced in a standard way from simple binary relations. Here's one that seems to produce mathematics out of thin air.

Ultrafilters on a set \small \rule[-1.5]{0.1}{0.1} X are a bit like generalised points of that set. Of course, principal ultrafilters correspond to points of the set in an obvious way. If \small \rule[-1.5]{0.1}{0.1} X lives in some model of set theory, and we take an ultrapower of that model by some ultrafilter, then in the ultrapower \small \rule[-1.5]{0.1}{0.1} X gains a generic point, with respect to which the ultrafilter is principal.

Sometimes we might want to associate actual points of the set to these ultrafilters: We are interested in relations \small \rule[-1.5]{0.1}{0.1} R between \small \rule[-1.5]{0.1}{0.1} X and the set \small \rule[-1.5]{0.1}{0.1} \beta(X) of ultrafilters on \small \rule[-1.5]{0.1}{0.1} X. Such a relation is a set of ordered pairs \small \rule[-1.5]{0.1}{0.1} (x, {\cal U}) with \small \rule[-1.5]{0.1}{0.1} {\cal U} an ultrafilter on \small \rule[-1.5]{0.1}{0.1} X, and \small \rule[-1.5]{0.1}{0.1} x a point of \small \rule[-1.5]{0.1}{0.1} X. \small \rule[-1.5]{0.1}{0.1} {\cal U} may be thought of as specifying a set of subsets of \small \rule[-1.5]{0.1}{0.1} X to which some imaginary point belongs. Unless \small \rule[-1.5]{0.1}{0.1} {\cal U} is the principal ultrafilter at \small \rule[-1.5]{0.1}{0.1} x, there will be some sets containing \small \rule[-1.5]{0.1}{0.1} x but not in \small \rule[-1.5]{0.1}{0.1} {\cal U}: Since such sets are witnesses of the fact that \small \rule[-1.5]{0.1}{0.1} {\cal U} isn't \small \rule[-1.5]{0.1}{0.1} x, and so I'll call them inconsistent with the pairing \small \rule[-1.5]{0.1}{0.1} (x, {\cal U}). All other sets are consistent with the pairing.

This consistency relation induces a Galois connection from the set of relations \small \rule[-1.5]{0.1}{0.1} R of the type described above to the power set of the power set of \small \rule[-1.5]{0.1}{0.1} X. It is here, on this bleak mountaintop of abstraction, that there is a surprise. The sets of subsets of \small \rule[-1.5]{0.1}{0.1} X which are closed with respect to this connection are precisely the topologies on \small \rule[-1.5]{0.1}{0.1} X.

Proof: Let \small \rule[-1.5]{0.1}{0.1} {\cal T} be a set of subsets of \small \rule[-1.5]{0.1}{0.1} X closed with respect to the connection. Then there is a relation \small \rule[-1.5]{0.1}{0.1} R which is taken to \small \rule[-1.5]{0.1}{0.1} {\cal T} by the connection. That is, \small \rule[-1.5]{0.1}{0.1} {\cal T} is the set of subsets of \small \rule[-1.5]{0.1}{0.1} X consistent with \small \rule[-1.5]{0.1}{0.1} R; \small \rule[-1.5]{0.1}{0.1} {\cal T} is the set of sets \small \rule[-1.5]{0.1}{0.1} O such that, for all \small \rule[-1.5]{0.1}{0.1} x and \small \rule[-1.5]{0.1}{0.1} {\cal U} with \small \rule[-1.5]{0.1}{0.1} xR{\cal U}, if \small \rule[-1.5]{0.1}{0.1} x \in O then \small \rule[-1.5]{0.1}{0.1} O \in {\cal U}. In particular, as the empty set \small \rule[-1.5]{0.1}{0.1} \emptyset contains no points, it is in \small \rule[-1.5]{0.1}{0.1} {\cal T}. As \small \rule[-1.5]{0.1}{0.1} X is in every ultrafilter, \small \rule[-1.5]{0.1}{0.1} X \in {\cal T}. If \small \rule[-1.5]{0.1}{0.1} A and \small \rule[-1.5]{0.1}{0.1} B are in \small \rule[-1.5]{0.1}{0.1} {\cal T}, \small \rule[-1.5]{0.1}{0.1} xR{\cal U}, and \small \rule[-1.5]{0.1}{0.1} x \in A \cap B, then \small \rule[-1.5]{0.1}{0.1} x is in both, so both are in \small \rule[-1.5]{0.1}{0.1} {\cal U}. But then \small \rule[-1.5]{0.1}{0.1} A \cap B \in {\cal U}, as \small \rule[-1.5]{0.1}{0.1} {\cal U} is an ultrafilter. That is, \small \rule[-1.5]{0.1}{0.1} A \cap B \in {\cal T}. If each set in a family \small \rule[-1.5]{0.1}{0.1} ({\cal U}_i)_{i \in I} is in \small \rule[-1.5]{0.1}{0.1} {\cal T}, and \small \rule[-1.5]{0.1}{0.1} x \in \bigcup_{i \in I}{\cal U}_i, then \small \rule[-1.5]{0.1}{0.1} x is in one of them, so one of them (and hence their union) is in \small \rule[-1.5]{0.1}{0.1} {\cal U}. That is, \small \rule[-1.5]{0.1}{0.1} {\cal T} is closed under arbitrary unions. Putting it all together, \small \rule[-1.5]{0.1}{0.1} {\cal T} is a topology on \small \rule[-1.5]{0.1}{0.1} X.

Suppose that \small \rule[-1.5]{0.1}{0.1} {\cal T} is a topology on \small \rule[-1.5]{0.1}{0.1} X, and let \small \rule[-1.5]{0.1}{0.1} R be the relation \small \rule[-1.5]{0.1}{0.1} {\cal T} is taken to by the Galois connection, and suppose that the connection takes \small \rule[-1.5]{0.1}{0.1} R to \small \rule[-1.5]{0.1}{0.1} {\cal T}'. It is enough to show that \small \rule[-1.5]{0.1}{0.1} {\cal T}' = {\cal T}. Evidently \small \rule[-1.5]{0.1}{0.1} {\cal T} \subseteq {\cal T}', so it's enough to show that, for any set \small \rule[-1.5]{0.1}{0.1} C \not\in {\cal T}, we have \small \rule[-1.5]{0.1}{0.1} C \not\in {\cal T}'. Let \small \rule[-1.5]{0.1}{0.1} C be such a set, and let \small \rule[-1.5]{0.1}{0.1} O be the interior of \small \rule[-1.5]{0.1}{0.1} C. As \small \rule[-1.5]{0.1}{0.1} C isn't open, there is some \small \rule[-1.5]{0.1}{0.1} x \in C \setminus O. Let \small \rule[-1.5]{0.1}{0.1} {\cal F} be the set of all open neighbourhoods of \small \rule[-1.5]{0.1}{0.1} x, together with the complement of \small \rule[-1.5]{0.1}{0.1} C. Any finite intersection of sets in \small \rule[-1.5]{0.1}{0.1} {\cal F} is nonempty, so \small \rule[-1.5]{0.1}{0.1} {\cal F} can be extended to an ultrafilter \small \rule[-1.5]{0.1}{0.1} {\cal U}. Any neighbourhood of \small \rule[-1.5]{0.1}{0.1} x is in \small \rule[-1.5]{0.1}{0.1} {\cal U}, so \small \rule[-1.5]{0.1}{0.1} xR{\cal U}. But \small \rule[-1.5]{0.1}{0.1} C \not\in {\cal U}, so \small \rule[-1.5]{0.1}{0.1} C isn't in \small \rule[-1.5]{0.1}{0.1} {\cal T}', as required.

This result is remarkable enough, but there's more. It turns out that compactness and Hausdorffness correspond closely with similar properties of relations. Say a relation \small \rule[-1.5]{0.1}{0.1} R is surjective if, for every \small \rule[-1.5]{0.1}{0.1} {\cal U} there is at least one \small \rule[-1.5]{0.1}{0.1} x with \small \rule[-1.5]{0.1}{0.1} xR{\cal U}. Say \small \rule[-1.5]{0.1}{0.1} R is injective if, for any \small \rule[-1.5]{0.1}{0.1} {\cal U}, there is at most one \small \rule[-1.5]{0.1}{0.1} x with \small \rule[-1.5]{0.1}{0.1} xR{\cal U}. If \small \rule[-1.5]{0.1}{0.1} R is a function, these definitions are exactly the usual definitions of injectivity and surjectivity.

Claim \small \rule[-1.5]{0.1}{0.1} 1: Let \small \rule[-1.5]{0.1}{0.1} R be a relation, as above, and let \small \rule[-1.5]{0.1}{0.1} {\cal T} be the topology that \small \rule[-1.5]{0.1}{0.1} R is taken to by the Galois connection. Then \small \rule[-1.5]{0.1}{0.1} {\cal T} is compact if \small \rule[-1.5]{0.1}{0.1} R is surjective.
Proof: By contradiction. Pick any open cover of \small \rule[-1.5]{0.1}{0.1} X with no finite subcover, and let \small \rule[-1.5]{0.1}{0.1} {\cal F} be the set of complements of the sets in the cover. Then any finite intersection of sets in \small \rule[-1.5]{0.1}{0.1} {\cal F} is nonempty, so \small \rule[-1.5]{0.1}{0.1} {\cal F} may be extended to an ultrafilter \small \rule[-1.5]{0.1}{0.1} {\cal U} on \small \rule[-1.5]{0.1}{0.1} X. By surjectivity, there is some point \small \rule[-1.5]{0.1}{0.1} x with \small \rule[-1.5]{0.1}{0.1} xR{\cal U}. \small \rule[-1.5]{0.1}{0.1} x must lie in some set \small \rule[-1.5]{0.1}{0.1} O of the original cover. But \small \rule[-1.5]{0.1}{0.1} O can't be in \small \rule[-1.5]{0.1}{0.1} {\cal U} (its complement is), contradicting the definition of \small \rule[-1.5]{0.1}{0.1} {\cal T}.

Claim \small \rule[-1.5]{0.1}{0.1} 2: Let \small \rule[-1.5]{0.1}{0.1} {\cal T} be any topology on \small \rule[-1.5]{0.1}{0.1} X, and let \small \rule[-1.5]{0.1}{0.1} R be the relation that \small \rule[-1.5]{0.1}{0.1} {\cal T} is taken to by the Galois connection. Then \small \rule[-1.5]{0.1}{0.1} {\cal T} is compact iff \small \rule[-1.5]{0.1}{0.1} R is surjective.
Proof: The 'if' follows from Claim \small \rule[-1.5]{0.1}{0.1} 1 and the fact that \small \rule[-1.5]{0.1}{0.1} {\cal T} is closed with respect to the Galois connection. To prove the 'only if', suppose that \small \rule[-1.5]{0.1}{0.1} {\cal T} is compact, and let \small \rule[-1.5]{0.1}{0.1} {\cal U} be any ultrafilter on \small \rule[-1.5]{0.1}{0.1} X. Suppose for a contradiction that every \small \rule[-1.5]{0.1}{0.1} x \in X has an open neighbourhood not in \small \rule[-1.5]{0.1}{0.1} {\cal U. These neighbourhoods form an open cover, which therefore has a finite subcover. The complements of the sets in this subcover are in \small \rule[-1.5]{0.1}{0.1} {\cal U}, and their intersection is empty, contradicting the fact that \small \rule[-1.5]{0.1}{0.1} {\cal U} is an ultrafilter. So there is an \small \rule[-1.5]{0.1}{0.1} x such that every open neighbourhood of \small \rule[-1.5]{0.1}{0.1} x is in \small \rule[-1.5]{0.1}{0.1} {\cal U}, so that \small \rule[-1.5]{0.1}{0.1} xR{\cal U}. As \small \rule[-1.5]{0.1}{0.1} {\cal U} was arbitrary, \small \rule[-1.5]{0.1}{0.1} R is surjective.

Claim \small \rule[-1.5]{0.1}{0.1} 3: Let \small \rule[-1.5]{0.1}{0.1} R be a relation, as above, and let \small \rule[-1.5]{0.1}{0.1} {\cal T} be the topology that \small \rule[-1.5]{0.1}{0.1} R is taken to by the Galois connection. Then \small \rule[-1.5]{0.1}{0.1} R is injective if \small \rule[-1.5]{0.1}{0.1} {\cal T} is Hausdorff.
Proof: By contradiction. Let \small \rule[-1.5]{0.1}{0.1} {\cal U} be an ultrafilter, and let \small \rule[-1.5]{0.1}{0.1} x \neq y \in X be such that \small \rule[-1.5]{0.1}{0.1} xR{\cal U} and \small \rule[-1.5]{0.1}{0.1} yR{\cal U}. Then we can find disjoint open sets \small \rule[-1.5]{0.1}{0.1} O and \small \rule[-1.5]{0.1}{0.1} P with \small \rule[-1.5]{0.1}{0.1} x \in O and \small \rule[-1.5]{0.1}{0.1} y \in P. Then \small \rule[-1.5]{0.1}{0.1} xR{\cal U} implies that \small \rule[-1.5]{0.1}{0.1} O \in {\cal U}, and \small \rule[-1.5]{0.1}{0.1} yR{\cal U} implies that \small \rule[-1.5]{0.1}{0.1} P \in {\cal U}. But \small \rule[-1.5]{0.1}{0.1} O \cap P = \emptyset \not\in {\cal U}, contradicting the fact that \small \rule[-1.5]{0.1}{0.1} {\cal U} is an ultrafilter.

Claim \small \rule[-1.5]{0.1}{0.1} 4: Let \small \rule[-1.5]{0.1}{0.1} {\cal T} be any topology on \small \rule[-1.5]{0.1}{0.1} X, and let \small \rule[-1.5]{0.1}{0.1} R be the relation that \small \rule[-1.5]{0.1}{0.1} {\cal T} is taken to by the Galois connection. Then \small \rule[-1.5]{0.1}{0.1} R is injective iff \small \rule[-1.5]{0.1}{0.1} {\cal T} is Hausdorff.
Proof: The 'if' part follows from Claim \small \rule[-1.5]{0.1}{0.1} 3 and the fact that \small \rule[-1.5]{0.1}{0.1} {\cal T} is closed with respect to the Galois connection. To prove the 'only if', suppose \small \rule[-1.5]{0.1}{0.1} {\cal T} isn't Hausdorff, and let \small \rule[-1.5]{0.1}{0.1} x and \small \rule[-1.5]{0.1}{0.1} y in \small \rule[-1.5]{0.1}{0.1} {\cal T} be distinct but not separated by any pair of open sets. Let \small \rule[-1.5]{0.1}{0.1} {\cal F} be the set of open sets containing either \small \rule[-1.5]{0.1}{0.1} x or \small \rule[-1.5]{0.1}{0.1} y. Any finite intersection of sets in \small \rule[-1.5]{0.1}{0.1} {\cal F} is an intersection of an open set containing \small \rule[-1.5]{0.1}{0.1} x with one containing \small \rule[-1.5]{0.1}{0.1} y, so is nonempty. Hence \small \rule[-1.5]{0.1}{0.1} {\cal F} can be extended to some ultrafilter \small \rule[-1.5]{0.1}{0.1} {\cal U}. Then \small \rule[-1.5]{0.1}{0.1} xR{\cal U} and \small \rule[-1.5]{0.1}{0.1} yR{\cal U}, so \small \rule[-1.5]{0.1}{0.1} R isn't injective.

The converses to claims \small \rule[-1.5]{0.1}{0.1} 1 and \small \rule[-1.5]{0.1}{0.1} 3 are false. This remarkable pattern is a shadow of a pair of adjoint functors, which I hope to say a little more about soon.

Survival of the thickest?

I recently came across a novel argument by Alvin Plantinga, which aims to show that it is irrational for a human to believe both in naturalism and the theory of evolution by natural selection. Unfortunately, Plantinga never managed to write it up properly, but a draft may be found here. It is only worth reading the first part; the rest consists of a turgid and incomplete response to some objections which differ greatly in character from that which I give below. Essentially, Plantinga argues that evolution is unlikely to produce rational creatures whose reasoning is reliable: This gives those who believe that they are the result of unguided evolution a reason to doubt the reliability of their own reasoning, and so to doubt all of their conclusions; in particular they should doubt their belief in evolution.

Plantinga does not query the idea that natural selection can produce rational creatures. Instead, he aims to show that rational creatures arising in this way are unlikely to reason reliably; that their beliefs are likely, on the whole, to be false. It is necessary to introduce the idea of rationality in this way in order to make the argument apply to us. Evolution produces a great diversity of creatures, but we are the only rational creatures on Earth.

Since selection occurs on the basis of behaviour, Plantinga considers four possible ways in which the beliefs of a rational creature could be related to its behaviour.

1. Not at all. In this possibility, beliefs have no effect at all on behaviour. So true beliefs are no more likely to be selected for than false ones. Natural selection is unlikely to produce creatures with such useless baggage. Also, we would have no way to recognise that creatures of this kind were rational. We have no way of knowing that, for example, trees are not rational in this way. Since a key starting point of the argument was that we are the only rational creatures on Earth, the word rational must be being used in some sense that excludes this possibility. Humans are recognisably rational; it is ridiculous to think that our beliefs do not affect our behaviour.
   
2. Beliefs do affect behaviour, but in a way unrelated to their content. The same comments apply to this possibility as to the first.
3. Beliefs are causally efficacious: The creatures act on the basis of their beliefs. However, the behaviour this produces is maladaptive. Such creatures would die out. The probability of rational creatures produced by natural selection being of this kind is therefore tiny. Equally, our own continued survival as a species shows that we are not of this type.
   
4. Beliefs are causally efficacious: The creatures act on the basis of their beliefs. What's more, the behaviour that this produces is adaptive, so that the creatures survive. By a process of elimination, we are of this type. This is a good sign, since any rational creature produced by natural selection is overwhelmingly likely to be of this type.
   

Plantinga claims that the first two types of rationality would be fairly probable, a claim which the above analysis shows to be incorrect. He agrees that the third case is improbable. We can therefore focus in on the fourth case, where he claims that the probability that the beliefs of the creatures are, on the whole, reliable is not much more than 1 in 2. After all, it is possible to conjure up scenarios in which a creature survives by acting on the basis of false beliefs. To quote him:

Perhaps Paul very much likes the idea of being eaten, but when he sees a tiger, always runs off looking for a better prospect, because he thinks it unlikely that the tiger he sees will eat him. This will get his body parts in the right place so far as survival is concerned, without involving much by way of true belief. Or perhaps he thinks the tiger is a large, friendly, cuddly pussycat and wants to pet it; but he also believes that the best way to pet it is to run away from it. . . . or perhaps he thinks the tiger is a regularly recurring illusion, and, hoping to keep his weight down, has formed the resolution to run a mile at top speed whenever presented with such an illusion; or perhaps he thinks he is about to take part in a 1600 meter race, wants to win, and believes the appearance of the tiger is the starting signal; or perhaps. . . . Clearly there are any number of belief-cum-desire systems that equally fit a given bit of behavior.

Plantinga acknowledges that most such scenarios involve creatures whose belief-desire structures are, on the whole, maladaptive; they are only appropriate in very special circumstances. He therefore introduces a more subtle scenario. In this scenario, the creatures believe that everything is conscious, so that for example they have no word for 'tree', but rather one for 'conscious tree'. In particuar, almost all beliefs of the creature are false. This is, of course, a simplification of what early humans believed.

Suppose we were to meet such a creature, who spoke a different language to us. Let us suppose that their word for 'conscious tree' is 'shrubbery'. We, after talking to them for a while, would conclude that their word 'shrubbery' meant tree, and that their beliefs about trees were on the whole correct, but that they also had a false belief; namely, that trees are conscious. But any account of beliefs that accords with the way we use the word (to account for something which humans express by their language) must be such that in this case the beliefs of the beings are on the whole true, as we see, but that they include the false belief that everything is conscious.

This shows that Plantinga is simply working with an inadequate notion of what a belief is. That is fair enough; after all, we don't (as far as I know) have a good working definition to which he could refer. Nor can I give one. This unfortunately leads to incorrect arguments being put forward from time to time, of which Plantinga's is an example.

We may, however, say that we would expect some of the beliefs of the beings to be false. For example, it seems to be better to believe that there are causal links in situations where there are not than to risk missing actual causal links. So we would expect evolved rational beings to believe in causal links where there are none. We can easily observe that this is true of ourselves; it is called superstition. The field of studying what errors evolved beings might be expected to make is a growing one.

On the whole, we have found that we make errors of the expected kinds, and so this endeavour allows us to clear up our thinking. For example, those who understand some of this often actively guard against their tendency towards superstitious thinking. A sensible study of the limitations we would expect to find in ourselves given that we have evolved leads to humility, and to hope that understanding the truth about ourselves may yet set us free, but it does not swamp us in the kind of radical doubt Plantinga envisages.

Discussing Dawkins.

Very few of the responses to 'the God delusion' that I have come across deal with the main argument it presents. So I was very pleased to see that the Faraday institute had arranged a discussion on that very issue. If you're not familiar with the argument, you should probably have a look at it before reading on.

Unfortunately, the discussion was unsatisfying for a couple of reasons. First, the format led to the speakers talking past, rather than to, each other. Second, nobody who spoke accepted that Dawkins' argument was valid. I don't think it is valid either, but I would have liked to know how those who do accept it defend it. I had hoped that a discussion of the argument would involve a speaker who accepted it. In any case, it is worth exploring some of the ideas that were introduced.

The discussion began with each of the two main speakers giving a 20 minute presentation. Richard Swinburne, a prominent opponent of Dawkins, was up first. Swinburne mentioned that he did not accept Dawkins' argument, but did not explain where the problem is. He spoke on a related theme, reiterating an argument he had proposed before 'The God Delusion' was written. The main link, apart from the similarity of form, seems to be that this argument was criticised in the chapter of TGD that contains the key argument.

The argument may be summarised by the phrase 'Theism is a simple explanation'. After a digression on the nature of scientific explanations (which I will not go into; it isn't relevant to the argument, and better accounts of scientific explanation exist), Swinburne introduced the following criteria for accepting that a hypothesis is true, given some evidence:

1. The evidence should be a likely outcome, given the truth of the hypothesis.
2. The evidence should be an unlikely outcome, given the falsehood of the hypothesis.
3. The hypothesis should be simple.
4. The hypothesis should fit in with our background knowledge.

He then mentioned that if we're looking for a theory of everything, the fourth criterion is irrelevant. He introduced some evidence; there are beings to whom the universe is rationally intelligible. He also introduced a hypothesis: 'There is a God, omnipotent, omniscient and perfectly free.' He checked the criteria:

1. Being free, God will not be limited by irrational inclinations: He will act rationally. Being omnipotent, God will have perfect moral knowledge. Being rational, God will wish to do what is good. Being omnipotent, God will do what is good. It is good to create other beings, in order to do them good. This involves creating a universe for them to live in. It is good to make that universe rationally intelligible to them. So God will do so.
2. The world is a complicated place. It is unlikely to have come into existence by chance.
3. Only one being, with 3 qualities, is hypothesized.
   

It is worth noting that the argument relies on the simplicity only of a hypothesis about God, whereas Dawkins' claim was that God Himself, rather than any hypothesis about Him, was necessarily complex.

Next in line was Colin Howson, an atheist philosopher. He had prepared a powerpoint presentation beforehand. He reviewed Dawkins' argument, and rejected it on the grounds that there is no need to consider how complex God is to obtain a probability of His existing by chance; he claimed that this probability is automatically 0. Thankfully, he had anticipated the argument Swinburne presented, and he argued against it.

I'll modify the terms of the argument, and the order of presentation, (though hopefully not the ideas) to fit in more closely with the phraseology above. Howson explained that the three criteria introduced above can be made precise in the framework of Bayesian inference. The first two criteria correspond to features of the calculation, whilst the third criterion corresponds to the idea of assigning prior probabilities to hypotheses on the basis of simplicity. He then attacked the three criteria individually:

1. On the basis of the hypothesis, we would expect the world to be a good place. But when we observe it, we find that it doesn't look that way. This is the problem of evil. Also, there are many conceivable universes containing beings which rationally comprehend them; So the possibility of our universe, even given this explanation, is tiny.
   
2. It is impossible in principle to assign a probability here; we don't know what the alternative explanations are. They certainly aren't limited to 'chance'. Scientists have had to be more imaginative than science fiction writers, to explain the real world. We must accept our poverty of imagination. We don't know the alternatives; we may not even be able to.
   
3. We have observed that assigning prior probabilities on the basis of simplicity is a good idea for explaining things within the world; the history of science is full of examples. But it is the world itself, including the history of science, which is to be explained. So any recourse to that history would be circular. There doesn't seem to be a reason why simplicity is intrinsically more probable than complexity.

This approach also knocks down any attempt to give a similar objective justification for science. But there is no need for science to be justified by such an abstract philosophical foundation.

At this point, the three interlocutors were each allowed to make a brief speech. Though they were in agreement that all three of Dawkins, Swinburne and Howson were wrong, they did not have time to adequately explain why. Indeed, the two main speakers did not take long to point out that the objections they had managed to articulate were simply missing the point. I've tried to phrase the above summary in such a way as to avoid similar misunderstandings, in order not to have to present them and pull them apart here.

The two main speakers were then left with 3 minutes each to deal with the points that had been raised. Swinburne mentioned that he could resolve the problem of evil, though not within 3 minutes. Howson expressed his doubts. That, apart from explaining what the interlocuters had misunderstood, was all they had time for.

Then there was a question time. I raised an objection to Howson's attack on the second criterion. We may not know what the possible explanations there are, but we can surely put crude bounds on the number of explanations of any given simplicity. For example, an explanation that cannot be described in English in a space shorter than the encyclopædia brittanica can hardly be described as simple, and we can put a bound on the number of remaining explanations. It's possible that we can do this in such a way as to put a lower bound on the probability of the hypothesis of theism. His response, that a simple explanation may require a high-level language for its expression, ignores the fact that high-level languages may be introduced and used within English. The process is called giving definitions. Soon after this, I had to leave early to get to a bible study group. The speakers never had a chance to develop the discussion, but I think there are some more ideas to be explored here.

The argument about the first criterion can certainly be developed. Howson pointed out that the hypothesis of theism simply does not explain physical events as precisely as modern physics; the universe we see is so specific that it is still highly improbable under this hypothesis. Swinburne never explicitly claimed it explained anything more than the existence of beings capable of rationally understanding the universe: He just hinted at it.

Now that these hints have been laid to rest, the evidence being explained can be seen to be simpler than the proposed explanation. After all, the explanation includes the idea of a being who is capable of rationally understanding all things (that's a small part of omniscience). But the explanation also relies on the existence of absolute moral truth. Although the problem of evil can be resolved (I've outlined one attack in an earlier post), the difficulty of the resolution shows that this moral truth is not as simple as it might at first appear, and accordingly brings out some of the complexity of the proposed explanation.

As Howson pointed out, we don't really have justification for assigning prior probabilities on the basis of simplicity. Indeed, why should we be able to assign probabilities at all? The idea of probability makes sense in the context of tossing a coin. We have a reasonable understanding of coins, and of how they behave. This ultimately relies on some crude understanding of how the world works. It is on the basis of such an understanding that we can assign probabilities. So we can't assign probabilities to such understandings, except on the basis of better ones. We can't assign probabilities to ultimate understandings at all; it doesn't make sense.

For a test case, consider the probability that there is a physical world at all. We have masses of evidence, pushing this probability up to 1. But discounting this evidence, in the abstract, what is the probability? The question makes no sense. It stretches the notion of probability beyond its proper boundaries. In a similar way, the hypothesis of theism just doesn't have a probability in the abstract, apart from the evidence. Reasoning on the basis that it does is the shared flaw of the arguments given by Dawkins and Swinburne.