Three limericks.

Here are three limericks, which belong together. The second would be bizarre without the first: The third is parasitic on the other two, and is one of the most unusual limericks I know.

1. There was a young lady from Crewe,
   Whose limericks stopped at line 2.
   
   
2. There was a young man from Verdun,
   
   
3. 
   

Getting the bends.

This is a rather pretty piece of mathematics that I've always wanted to write up. It's a proof of the generalised Descartes circle theorem, one of very few mathematical theorems to have been immortalised in verse:

For pairs of lips to kiss maybe
Involves no trigonometry.
'Tis not so when four circles kiss
Each one the other three.
To bring this off the four must be
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.
If three in one, then is that one
Thrice kissed internally.

Four circles to the kissing come.
The smaller are the benter.
The bend is just the inverse of
The distance from the center.
Though their intrigue left Euclid dumb
There's now no need for rule of thumb.
Since zero bend's a dead straight line
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum.

To spy out spherical affairs
An oscular surveyor
Might find the task laborious,
The sphere is much the gayer,
And now besides the pair of pairs
A fifth sphere in the kissing shares.
Yet, signs and zero as before,
For each to kiss the other four
The square of the sum of all five bends
Is thrice the sum of their squares.

And let us not confine our cares
To simple circles, planes and spheres,
But rise to hyper flats and bends
Where kissing multiple appears,
In \small \rule[-1.5]{0.1}{0.1} n-ic space the kissing pairs
Are hyperspheres, and Truth declares -
As \small \rule[-1.5]{0.1}{0.1} n + 2 such osculate
Each with an \small \rule[-1.5]{0.1}{0.1} n + 1 fold mate
The square of the sum of all the bends
Is n times the sum of their squares.

Why should this rather pretty fact be true? Well, here's one way to think about it. If the \small \rule[-1.5]{0.1}{0.1} m^th sphere has centre at \small \rule[-1.5]{0.1}{0.1} O_m and radius \small \rule[-1.5]{0.1}{0.1} r_m then the distance from \small \rule[-1.5]{0.1}{0.1} O_m to \small \rule[-1.5]{0.1}{0.1} O_1 is \small \rule[-1.5]{0.1}{0.1} r_m + r_1. To put it another way, the points \small \rule[-1.5]{0.1}{0.1} O_m lie at the vertices of an \small \rule[-1.5]{0.1}{0.1} n+1-simplex, whose side lengths are determined by the various radii. This \small \rule[-1.5]{0.1}{0.1} n+1-simplex has to fit somehow into \small \rule[-1.5]{0.1}{0.1} n-dimensional space, so it must have hypervolume \small \rule[-1.5]{0.1}{0.1} 0. The first step will be to study how the hypervolume of a simplex depends on the lengths of its sides.

It won't be necessary to actually derive a formula. Surprisingly enough, all we'll need is a qualitative description of the kind of formula involved. To make the derivation easier, we can suppose that one of the vertices is at the origin. Then the hypervolume is proportional to the determinant of the \small \rule[-1.5]{0.1}{0.1} n+1 by \small \rule[-1.5]{0.1}{0.1} n + 1 matrix \small \rule[-1.5]{0.1}{0.1} A whose columns are the vectors corresponding to the remaining vertices. What do we know about these vectors? Well, we certainly know their lengths. Also, for any pair \small \rule[-1.5]{0.1}{0.1} v and \small \rule[-1.5]{0.1}{0.1} w, we know the length of \small \rule[-1.5]{0.1}{0.1} v-w. But \small \rule[-1.5]{0.1}{0.1} |v-w|^2 = |v|^2 + |w|^2 - 2 v \cdot w, so we know the dot product of any pair.

These dot products are the entries of the matrix you get when you multiply \small \rule[-1.5]{0.1}{0.1} A by its transpose. So we can work out the determinant of that matrix, which is the square of the determinant of \small \rule[-1.5]{0.1}{0.1} A. What kind of expression does this give for the hypervolume? Well, it's \small \rule[-1.5]{0.1}{0.1} \sqrt{P}, where \small \rule[-1.5]{0.1}{0.1} P is a polynomial in the lengths of the sides. As the determinant is calculated as a homogeneous polynomial of degree \small \rule[-1.5]{0.1}{0.1} n + 1, and the dot products are expressions of degree \small \rule[-1.5]{0.1}{0.1} 2 in the lengths of the sides, we get that \small \rule[-1.5]{0.1}{0.1} P is homogeneous of degree \small \rule[-1.5]{0.1}{0.1} 2n + 2.

Now, in the case we're considering, the sides are of the form \small \rule[-1.5]{0.1}{0.1} r_l + r_m. So the hypervolume is the square root of some homogeneous polynomial \small \rule[-1.5]{0.1}{0.1} Q of degree \small \rule[-1.5]{0.1}{0.1} 2n + 2 in the radii of the spheres. What can we say about \small \rule[-1.5]{0.1}{0.1} Q? Well, it must be symmetric, since it defines a symmetric quantity. We can say more. Let's fix all of the radii except one, \small \rule[-1.5]{0.1}{0.1} r_m, and let \small \rule[-1.5]{0.1}{0.1} r_m grow very large. The simplex will now no longer fit into \small \rule[-1.5]{0.1}{0.1} n-space: It will have positive hypervolume \small \rule[-1.5]{0.1}{0.1} V which will also grow very large. How does the growth of \small \rule[-1.5]{0.1}{0.1} V depend on \small \rule[-1.5]{0.1}{0.1} r_m? Pick another vertex \small \rule[-1.5]{0.1}{0.1} O_l. We can find some fixed \small \rule[-1.5]{0.1}{0.1} R such that all of the vertices except \small \rule[-1.5]{0.1}{0.1} O_m are always in the sphere \small \rule[-1.5]{0.1}{0.1} S centred at \small \rule[-1.5]{0.1}{0.1} O_l with radius \small \rule[-1.5]{0.1}{0.1} R. Taking another sphere \small \rule[-1.5]{0.1}{0.1} T, of the same radius, about \small \rule[-1.5]{0.1}{0.1} O_m it is clear that the whole simplex is contained in the convex hull of these two spheres. This hull consists of a hypercylinder of height \small \rule[-1.5]{0.1}{0.1} r_l + r_m and radius \small \rule[-1.5]{0.1}{0.1} R, with two hemihyperspherical caps. Its volume is a linear function of \small \rule[-1.5]{0.1}{0.1} r_m.

That is, the hypervolume of the simplex grows no faster than some linear function of \small \rule[-1.5]{0.1}{0.1} r_m. So \small \rule[-1.5]{0.1}{0.1} Q, the square of that hypervolume, grows no faster than some quadratic function of \small \rule[-1.5]{0.1}{0.1} r_m. That is, \small \rule[-1.5]{0.1}{0.1} Q contains no terms of degree greater than two in any of the variables \small \rule[-1.5]{0.1}{0.1} r_m. Let \small \rule[-1.5]{0.1}{0.1} b_m be the bend of the \small \rule[-1.5]{0.1}{0.1} m^th hypersphere, that is, the reciprocal of \small \rule[-1.5]{0.1}{0.1} r_m. Then \small \rule[-1.5]{0.1}{0.1} Q times the product of the squares of the bends of all the hyperspheres must be a polynomial, \small \rule[-1.5]{0.1}{0.1} R, in those bends. What's more, \small \rule[-1.5]{0.1}{0.1} R is symmetric and homogeneous of degree \small \rule[-1.5]{0.1}{0.1} 2(n+2)-(2n+2) = 2.

We are now very close; there just aren't many polynomials that \small \rule[-1.5]{0.1}{0.1} R could be. \small \rule[-1.5]{0.1}{0.1} R has to be a linear combination of the sum of the squares of the bends and the square of their sum. What's more, the above argument shows that there is a collection of mutually tangent hyperspheres with a given collection of bends if and only if \small \rule[-1.5]{0.1}{0.1} R of that collection is \small \rule[-1.5]{0.1}{0.1} 0. We can tell what linear combination \small \rule[-1.5]{0.1}{0.1} R has to be by considering the degenerate situation with \small \rule[-1.5]{0.1}{0.1} b_m = 1 for \small \rule[-1.5]{0.1}{0.1} m at most \small \rule[-1.5]{0.1}{0.1} n, and \small \rule[-1.5]{0.1}{0.1} b_{n+1} = b_{n+2} = 0. Here we have a collection of spheres of equal radius wedged between two parallel hyperplanes. The sum of the squares of the bends is \small \rule[-1.5]{0.1}{0.1} n, and the square of their sum is \small \rule[-1.5]{0.1}{0.1} n^2, \small \rule[-1.5]{0.1}{0.1} n times as big. So \small \rule[-1.5]{0.1}{0.1} R must be proportional to the difference of the square of the sum of the bends and \small \rule[-1.5]{0.1}{0.1} n times the sum of their squares, completing the proof.

Before looking at some consequences, there is another result which drops out of the proof. In the case \small \rule[-1.5]{0.1}{0.1} n=1, \small \rule[-1.5]{0.1}{0.1} R is proportional to \small \rule[-1.5]{0.1}{0.1} (b_1 + b_2 + b_3)^2 - b_1^2 - b_2^2 - b_3^2 = 2(b_1 b_2 + b_2 b_3 + b_3 b_1), so \small \rule[-1.5]{0.1}{0.1} Q is proportional to \small \rule[-1.5]{0.1}{0.1} r_1 r_2 r_3 (r_1 + r_2 + r_3). Now, given any triangle with sides \small \rule[-1.5]{0.1}{0.1} a, \small \rule[-1.5]{0.1}{0.1} b and \small \rule[-1.5]{0.1}{0.1} c, and semiperimeter \small \rule[-1.5]{0.1}{0.1} s, circles centred at the vertices with radii \small \rule[-1.5]{0.1}{0.1} s-a, \small \rule[-1.5]{0.1}{0.1} s-b and \small \rule[-1.5]{0.1}{0.1} s-c will be mutually tangent. So by the above work, the square of the area of the triangle should be proportional to \small \rule[-1.5]{0.1}{0.1} (s-a)(s-b)(s-c)(s-a + s-b + s-c) = s(s-a)(s-b)(s-c). To find out what the scale factor is, consider the right-angled triangle with sides \small \rule[-1.5]{0.1}{0.1} 3, \small \rule[-1.5]{0.1}{0.1} 4 and \small \rule[-1.5]{0.1}{0.1} 5 units. Then \small \rule[-1.5]{0.1}{0.1} s = 6, \small \rule[-1.5]{0.1}{0.1} s-a = 3, \small \rule[-1.5]{0.1}{0.1} s-b = 2, \small \rule[-1.5]{0.1}{0.1} s-c = 1, \small \rule[-1.5]{0.1}{0.1} s(s-a)(s-b)(s-c) = 36, and the square of the area is \small \rule[-1.5]{0.1}{0.1} 6^2, which is also \small \rule[-1.5]{0.1}{0.1} 36. It follows that for any triangle the area is \small \rule[-1.5]{0.1}{0.1} \sqrt{s(s-a)(s-b)(s-c)}. This is Heron's formula.

Suppose we have a plane with three mutually tangent spheres, each of radius \small \rule[-1.5]{0.1}{0.1} 1, on it. Sitting nestled between the spheres and the plane is a sphere of radius \small \rule[-1.5]{0.1}{0.1} r. We can find \small \rule[-1.5]{0.1}{0.1} r using the method outlined above. Let the bend of the smaller sphere be \small \rule[-1.5]{0.1}{0.1} b. Then \small \rule[-1.5]{0.1}{0.1} (b + 3)^2 = 3(b^2 + 3), so \small \rule[-1.5]{0.1}{0.1} 2b^2 - 6b = 0, so \small \rule[-1.5]{0.1}{0.1} b = 3, and the inner sphere has radius \small \rule[-1.5]{0.1}{0.1} \frac{1}{3}. Although we had to solve a quadratic equation, we ended up with a rational value. But that isn't surprising in this case: We already knew one of the roots would be \small \rule[-1.5]{0.1}{0.1} 0, which is rational, as there is clearly another plane tangent to all three spheres and parallel (that is, tangent at infinity) to the other plane. The rationality of the other root follows from the fact that the sum of the roots of a quadratic equation in \small \rule[-1.5]{0.1}{0.1} x is minus the coefficient of \small \rule[-1.5]{0.1}{0.1} x.

Applying this in the rather well behaved \small \rule[-1.5]{0.1}{0.1} 3-dimensional setting gives another pleasant result in the same spirit as the Descartes theorem. Fix three mutually tangent spheres of bends \small \rule[-1.5]{0.1}{0.1} u, \small \rule[-1.5]{0.1}{0.1} v and \small \rule[-1.5]{0.1}{0.1} w. Take some sphere \small \rule[-1.5]{0.1}{0.1} S_0 tangent to all three. What happens when we construct a sequence of spheres, taking each \small \rule[-1.5]{0.1}{0.1} S_{k+1} tangent to \small \rule[-1.5]{0.1}{0.1} S_k and to the original three spheres, but with \small \rule[-1.5]{0.1}{0.1} S_{k+1} different from \small \rule[-1.5]{0.1}{0.1} S_{k-1}? Well, the bends \small \rule[-1.5]{0.1}{0.1} b_{k-1} and \small \rule[-1.5]{0.1}{0.1} b_{k+1} of \small \rule[-1.5]{0.1}{0.1} S_{k-1} and \small \rule[-1.5]{0.1}{0.1} S_{k+1} are the roots of the quadratic \small \rule[-1.5]{0.1}{0.1} (x + b_k + u + v + w)^2 = 3(x^2 + b_k^2 + u^2 + v^2 + w^2), so they must sum to minus the coefficient of \small \rule[-1.5]{0.1}{0.1} x, namely \small \rule[-1.5]{0.1}{0.1} u + v + w + b_k. This gives \small \rule[-1.5]{0.1}{0.1} b_{k+1} = u + v + w + b_k - b_{k-1}. Letting \small \rule[-1.5]{0.1}{0.1} K = u + v + w, this gives the successive equations:

 \small
\begin{array}{lcccl}
b_2 &=& K + b_1 - b_0 && \\
b_3 &=& K + b_2 - b_1 &=& 2K - b_0 \\
b_4 &=& K + b_3 - b_2 &=& 2K - b_1 \\
b_5 &=& K + b_4 - b_3 &=& K + b_0 - b_1 \\
b_6 &=& K + b_5 - b_4 &=& b_0 \\
b_7 &=& K + b_6 - b_5 &=& b_1
\end{array}

and so on, with period \small \rule[-1.5]{0.1}{0.1} 6. So we get a chain of \small \rule[-1.5]{0.1}{0.1} 6 spheres threading around the original three. What's more, the spheres which are opposite in this chain have bends summing to \small \rule[-1.5]{0.1}{0.1} 2K, twice the sum of the bends of the original three. This result has also been cast in verse: 'The mean of the bends of each opposite pair is the sum of the three through the thoroughfare'. This couplet is taken from a longer poem in a somewhat obscure book that I don't have to hand at the moment. There's a neat illustration, together with a hint at another proof that you always get a chain of length \small \rule[-1.5]{0.1}{0.1} 6, here.

Flashback #4.

A mystery wrapped in an enigma wrapped in a puzzle:

YE wons woS a skUlPu. ne.mT dan
hU insist,ntli. we.FT hiS ra.t hanT
cUS mi. s; cUS mi.
woS hiS impDt,nt pli.
sat ,lDn in a pit ful oF sanT

Talking to myself.

My last post was a lengthy quote of something I wrote when I was still a Christian. I made a couple of arguments then that I no longer find compelling. Since they also correspond to arguments that some of my friends give for the truth of Christianity, I think I should respond to them here.

The Bible, in particular the old testament ... clearly explains the character of God and His relation to us. Furthermore, it gives evidence that it is the word of God. The different books were written by different people in different places and cultures, but they all agree with one another and with the state of the world and have the same ultimate purpose; to give glory to God. They clearly explain the human condition in a way that resonates with the heart, though at the same time it offends. They contain many prophecies. Some of these prophecies were claims about what would happen in the future but before the end of the world. These prophecies have all come true, and not in such a way as to have been planned. Read any of the gospels, and you will see frequent references to predictions in the old testament of the events that occurred. For example, a description of the suffering of crucifiction was given before it had been invented.

I rather overstated the case here. The different books clearly demonstrate that they were written in different places and cultures by the differences in what they say. It is a stretch to say that, for example, the book of Esther's purpose is to give glory to God, when it doesn't even mention Him. There are huge numbers of prophecies, some of which are recorded as having been fulfilled. There is no unified explanation for this; several factors must be considered:

• There are a lot of potential prophecies, most of which don't obviously refer to anything concrete.
• The fulfilment of some of the prophecies was clearly planned and deliberate.
• Some of the events recorded as having been fulfilments of prophecy may not have happened.
• It is often necessary to be generous in the interpretation of the original language to make the prophecy stick.
• Some of the prophecies (of the exile, for example) may have been written after the fact.
• Treating biblical prophecy as predictive has repeatedly been shown to lead to disappointment.
  

The first, second and fourth points in particular apply to the prophecies of the crucifixion.

Evidence for this is to be found in documents of the new testament. We have manuscripts of these documents, some of which were written by eyewitnesses, going back to within a generation of the originals. It is possible to argue the truth of the resurrection from the internal textual evidence of these documents, but I do not have space to do so here, nor am I an expert in textual criticism. The other historians of the time mention the crucifiction, and they also document the spread of Christianity. The early witnesses claimed that Jesus had risen from the dead. At the time they could have easily been proved wrong by simply producing Jesus' body: Nobody did so, though the Jewish establishment had strong reasons to. They claimed that the body had been stolen, by those same witnesses. But the witnesses clearly believed what they were saying; they were prepared to die in painful ways rather than to deny it. This would not have been worth it if they had made it up. Nor were there just a few witnesses. Look at 1 Corinthians 15:3-7. Here Paul makes a big claim; that huge numbers of people saw Jesus after his resurrection, and that they were still around; that you could check by going to ask them. Again, he could have been easily discredited if he was wrong, unless large numbers of people were so convinced of a lie that they were prepared to die for it.

We have a great deal of evidence, in the form of the gospels and other contemporary documents. We wish to distinguish between the following two scenarios:

Scenario 1: Jesus' life, death and resurrection happen pretty much as recorded in the gospels, and as a result the early church grows rapidly. In this scenario the disciples behave in a rational kind of way, and the main problem is the miraculous nature of the physical events.

Scenario 2: The key miracles recorded in the gospels (for example the resurrection) do not occur. Nevertheless, the disciples very quickly begin to tell everybody that Jesus was raised from the dead, and are prepared to stick to this story to the point of death. This teaching spreads rapidly and the church grows at an enormous rate. In this scenario there is nothing to confuse a physicist. The problem is the rather odd behaviour of the disciples.

Now, since scientists began to look seriously at the world, they have been surprised by just how regular it's behaviour is. It is now extremely difficult to convey just how accurate the predictions of, for example, quantum electrodynamics are. On the other hand, the behaviour of people is normally observed to be far less rigid. However much it may damage our pride to think it, humans very often do not behave in a rational (or even rationally explicable) manner. It is well documented that we often make (and indeed believe) extraordinary but false claims. The phenomenon of cult suicide shows that, in the right circumstances, people will even die for beliefs for which they do not have sufficient grounds.

It is therefore to be expected that, at least scientifically, there is no explanation of the events of scenario 1. Any such explanation must be supernatural. What is slightly more surprising is that it is possible to find reasonable explanations for the events of scenario 2. That is, the odd behaviour of people is of such a kind that it is perfectly possible that they would behave as in scenario 2. I shall deal with this aspect of the particular points raised in the quote above later.

Odd behaviour, even of people, is still odd, and so the events of scenario 2 are still improbable. However, even this improbability is greatly reduced by the following consideration. There were quite a few itinerant miracle workers and teachers in the time of Jesus. Almost all of them are now lost to history; their followers did not consistently claim that they had risen from the dead or, if they did, they failed to ignite a revolution on the scale of the early church. This failure is, in fact, why we have not heard of most of them. The fact that there were so many, however, makes it more plausible that one of them would have such an effect that we would still be deeply influenced by them today. In order for that to happen, though, it would be necessary for people to behave in a way that, in retrospect, looks odd in just the way that we see.

To illustrate this last point, consider the national lottery. For any particular person, it is extremely improbable that they should win. But when we see a news story about somebody who has won, we are not particularly surprised. The reason for this is that there are so many people competing, and we would not expect to hear about all the others who did not win.

This kind of analysis is one reason why not all historians are Christians.

Of course, given the presupposition of a benevolent and omnipotent God, scenario 1 is very easily explicable. Accordingly this line of thought in no way disproves either that God exists or that Jesus was God and was raised from the dead. But these things cannot be presupposed in an argument for the existence of such a God. To see this point more vividly, consider that the presupposition of a deceptive and quite potent God makes scenario 2 extremely easily explicable. But it would not be valid to therefore conclude the existence of such a God from the available evidence.

Evidence for this is to be found in documents of the new testament. We have manuscripts of these documents, some of which were written by eyewitnesses, going back to within a generation of the originals.

It is disputed whether the authors were eyewitnesses. The documents were written, at best, 30 years after the events they record. We must not expect them to be accurate. This is particularly true of their accounts of the appearances of Jesus after the resurrection, since the earliest does not mention any resurrection appearances at all, and the other 3 narratives diverge widely at this point.

The other historians of the time mention the crucifiction, and they also document the spread of Christianity.

There is little doubt that the crucifixion occured, or that Christianity spread widely. Many religions spread rapidly despite major setbacks; look at the Jehovah's witnesses. I don't claim to understand how this happens, but it appears to be a part of normal human behaviour rather than a supernatural phenomenon.

The early witnesses claimed that Jesus had risen from the dead. At the time they could have easily been proved wrong by simply producing Jesus' body: Nobody did so, though the Jewish establishment had strong reasons to.

These claims were not made in public until at least 7 weeks after the crucifixion. By that time, in that climate, the corpse would have been unrecognisable.

The witnesses clearly believed what they were saying; they were prepared to die in painful ways rather than to deny it. This would not have been worth it if they had made it up.

The apostles did not die explicitly for a claim, 'We saw Jesus after His death,' but for a movement in the spread of which they had invested their lives. That people are prepared to die does not even mean that they have sufficient reason to believe in their cause, as martyrdoms of people with conflicting views throughout history show.

Look at 1 Corinthians 15:3-7. Here Paul makes a big claim; that huge numbers of people saw Jesus after his resurrection, and that they were still around; that you could check by going to ask them. Again, he could have been easily discredited if he was wrong, unless large numbers of people were so convinced of a lie that they were prepared to die for it.

If we include verse 8 too, we see that Paul listed the damascus road experience as one of the appearances. This indicates that the appearance to the 500 people he mentions may have fallen similarly short of the detailed tactile experiences in the Gospels. It's certainly a leap to suggest that, if Jesus wasn't raised, all 500 had made up lies for which they were prepared to die. What's more, it wouldn't have been all that easy to check up on Paul. The world wasn't as well connected then as it is now, and the claim is difficult to falsify: You may just be speaking to the wrong people. The letter was written to Christians, who almost certainly didn't check the claim out thoroughly.

Having dealt with some individual details, I should mention that I don't know exactly what happened. This is unsatisfying (in that I would very much like to know), but I do not think it violates intellectual integrity. Since we are dealing with people, and since the events were thousands of years ago in a different cultural context which we do not fully understand, it is not possible to give a definitive timeline. Nor is it necessary. The fact that I do not have a simple, coherent, and definitive storyline worries me as little as the fact that I do not have a simple, coherent and definitive storyline explaining every claim of alien abduction that has been made.

Flashback #3

I've managed to dig up an email I sent to a friend when I was still a Christian. It shows how I would have answered some of the main objections at the time, and it supports my view that, irrespective of whether it is true, Christianity (like many other religions) is coherent. I now disagree with some of what I said: I'll deal with that in a separate post.

Thanks for your questions about Christianity. Before I dive into answering them, I should probably make a few remarks. First, as I mentioned, my beliefs are not those of all Christians. There are some who would answer your questions differently, and some who have not thought about them. Second, I have reordered your questions for convenience; the answers to some of them depend on ideas raised in the answers to others. Third, I have tried to reproduce your questions as accurately as possible, but if I have oversimplified or misrepresented them then please clarify them. Fourth, the foundation for many of the answers will lie in God; that is, I believe that the focus of Christianity is God, and not Christians. Many of the questions you have asked reflect the impression (too often given by Christians) that God's ultimate motivation is to please/save/convince us. Of course, the question of God's ultimate motivation is difficult; we as creatures must expect it to be a struggle to understand the Creator. Nevertheless, He has shown enough of Himself to answer the questions you have asked. Finally, I have a habit of using words like 'good', 'sin', 'forgive' etc. to mean what the Bible means by them rather than what they mean in common usage. I have tried to point out all the places where I have done this. Without further ado:

>What does it mean to say that 'God is good'? That is, what is a sensible notion of goodness?
Here the Biblical usage is close to the original English meaning of the word good. That is, 'good' means 'In accordance with the character of God' as neatly summed up by the phrase 'what Jesus would do', at least for the actions of people. Similarly a thing is good if God is pleased by it. The definintion of goodness is the first example of something in my reply with its foundation in God. It follows that the statement 'God is good' is tautologous, at least if God is consistent within Himself.

However, as so often happens in mathematics, having defined a word in a technical way, it is worth looking to see why it corresponds closely with how you use it in natural language. First, obviously, when God has explained how He would like us to behave, he has by this definition explained the notion of goodness. Second, God has made us 'in His image'; we have some idea of good within ourselves, in our conscience. This is a weak reflection of true goodness, as defined by God, but we can often see that we 'ought to' behave in a certain way. We can see that some things (like murder, to pick an obvious example) are not good. We can also see that we ourselves are not good, even by the weak standards we can see for ourselves. But we can also be confused by many things in the world that we can see are not good, and ask why they are allowed by a good God:

>Why are there natural disasters and suffering in the world if God is both omnipotent and good?
First, from the answer to the last question, it is clear that what God does is good, even if we don't think so. However, this answer is not very satisfying, since the goodness of God does correspond somehow to our own notions and it is hard for us to see a good reason for the suffering in the world. It turns out that the answer lies in the character of God. Namely, God is just. That is, things that are not good make God angry, and God punishes people who are not good (that is, who are sinful). As I mentioned earlier, even we can see that we are not good. But often we do not realise the extent of our sin.

For example, God has created us and has given us many things that are good, such as life, food, the beauty of nature and in particular the beauty of mathematics. The good response to this is to thank God, and to devote our time and energy to Him. We do not do this. Indeed, we add insult to injury by devoting our time and energy to the things that have been created, without reference to the God who created them. We devote ourselves to mathematics without acknowledging the God who created it. Or we seek security or extended life or comfort for their own sakes, and ignore God who gives them. I mention those things that I do; you may devote yourself to other things. But all people insult God in this way.

Again, we often mistake our own notions of goodness for true goodness, and try to live by reference to them. But our own standards of goodness are considerably weaker than that of God. Take a look at Matthew chapters 5-7, for a small sample of God's standard. The Bible likens the comparison between our own standards and God's to that between the ground and the sky. So by setting up our own standards we demean goodness, and so insult God.

All this is aggravated by the fact that these sins are against God himself. Even we can see that a crime is aggravated by the greatness of the being against whom it is committed. For example, it is far worse to kill a human being than it is to kill a chicken. If you go to an art gallery and damage one of the paintings, that is worse than tearing up a photograph of the same painting, as could be seen by the response of the gallery attendants to these two actions. But we have sinned against God, whose greatness I cannot even adequately describe.

God's punishment is just and proportionate, and consists of suffering in this world, death, and eternal suffering after death. The infinity of the punishment corresponds to the infinity of the greatness of God, against whom we have sinned. To sum up; suffering in this world is punishment for the fact that we do stuff displeasing to God. This raises an obvious question:

>Why were we created able to do stuff displeasing to God?
Again the answer lies in the character of God. Take a look at Romans chapter 9 verses 19-23. This section of Romans is answering a closely related question; 'If God is in control then why does He blame us for the fact that we are sinful?' (v19), and the answer given also answers the question you have asked. First, note that the question is about the motivation of God: 'Why does God ...?'. So we should expect to have some difficulty answering it; we are vastly inferior to God and we can only get a small and limited inkling of the goodness of God's motivation (v20). Indeed, the following analogy is drawn: We make clay, which is vastly inferior to ourselves, into whatever we want without worrying about whether the clay is convinced that what we are doing is right. It would be strange if there was a protest at a pottery factory seeking 'rights for clay' because the clay was being made for a variety of purposes without consulting the clay itself (v21). Given all this, it is surprising that we are able to understand anything of God's motivation at all. But we are. God is righteous, just, and angry at sin. He is powerfully able to punish sin. He wants to express these things. But He cannot do this without there being actual sin to be justly condemned. So he made us capable of sin (v22). Note that God's ultimate motivation is not for us, but for Himself. Nor is it that He should have pleasure, but rather that He should express who He is. I shall return to verse 23 later.

>Will people in heaven have free will; that is, could they freely choose to disobey God?
Yes.
>Will they?
No.
>So what was the point of Earth: If God can create beings with free will who do not disobey him, why didn't He?
See the answer to the previous question. Heaven is different from Earth, in that in heaven God will be praised for who He is, as He has revealed Himself to be through what happened on Earth.

>What does it mean to be 'forgiven' by God?
>What was the point of Jesus dying on the cross; couldn't God have just forgiven us?
The normal sense of the word 'forgiven' is that one person, who has been wronged by another, forgets the guilt of that other. This occurs usually in a situation in which the person who did the wrong is truly sorry. God, being just, cannot forgive us in this sense. It is against His character that guilt should go unpunished when the guilty party not only isn't sorry, but continues do the same wrong. But as I have explained above, we all (including myself) are in that position. Yet God will not punish me in Hell for my sin; He has forgiven me in a different sense.

God's character contains not only absolute justice but also absolute love. This absolute love is also expressed in this world. God loved me so much that he arranged for somebody else to take my punishment. This was not a trivial matter. You could not take my punishment, because you have your own to deal with. Nor could anbody else sinful, for the same reason. But innocent things like rocks and trees also can't take my punishment; they cannot suffer as men. So an innocent man was needed. What God did was this: He Himself became a human being, lived a good life, was killed, and was punished on my behalf. This was not a result of anything good in me that caused God to love me, but was pure mercy (v23). In this God gave full expression to His love. God has mercifully allowed me to rely on this and this alone for my righteousness (as judged by God), and to come to a relationship with him by these means.

>If God wants a relationship with me, why hasn't He made Himself more obvious to me?
Well, firstly, as mentioned a few of questions ago, God's ultimate desire isn't for you to have a relationship with Him. If it was, you would have a relationship with Him. However, I guess this doesn't really get at the idea underlying your question, which I shall now try explore. There certainly is a sense in which God wants you to have a relationship with Him; He loves you and it would please Him to have such a relationship. Therefore it would be good for you to have a relationship with Him. But how can you be expected to when He hasn't made Himself more obvious? Wouldn't it be a good idea for Him to convince you by showing up in your room and doing a few miracles?

Be warned; you may find the answer to this question offensive. Take a look at what Jesus said in Luke chapter 16 verses 19-31. Two men die; one (Lazarus) goes to heaven and the other to hell. The one in hell asks for Lazarus to come and alleviate his suffering, but it is explained that this is impossible. So he says 'I've got 5 brothers; send Lazarus to warn them'. The idea being that, if they had the greater evidence of Lazarus rising from the dead and warning them, then they would repent. The response is that this is simply not true. If they have not been convinced by 'The law and the prophets' then they won't be convinced even if they do see such a miracle.

'The law and the prophets' refers to the Bible, in particular the old testament. This clearly explains the character of God and His relation to us. Furthermore, it gives evidence that it is the word of God. The different books were written by different people in different places and cultures, but they all agree with one another and with the state of the world and have the same ultimate purpose; to give glory to God. They clearly explain the human condition in a way that resonates with the heart, though at the same time it offends. They contain many prophecies. Some of these prophecies were claims about what would happen in the future but before the end of the world. These prophecies have all come true, and not in such a way as to have been planned. Read any of the gospels, and you will see frequent references to predictions in the old testament of the events that occurred. For example, a description of the suffering of crucifiction was given before it had been invented.

You are in a strongly priveleged position with respect to those 5 brothers. You also have the new testament, of which the same things can be said. That new testament, and other historical documents of the time, provide convincing historical evidence of the life, death and resurrection of Jesus, who was God as well as a Man. The claim of the story I mentioned is that the evidence you have is more than convincing, but that it is your sin that prevents you being convinced; and even more shockingly that your claim that if God were to appear to you and do a miracle then you would believe is false; that your sin would lead you to explain away even this. Worse, your sin might lead you to worship the miracle and ignore the God who performed it.

>Can we trust the Bible, given that it makes such strong claims (for example, that miracles took place)?
I am afraid I cannot give a complete answer to this question here, as the amount of evidence available is so great. I reccomend that you have a look at the Bible itself, and check some of the things I mentioned earlier. I also recommend that you come to the talk at 1:10pm on Monday 12th at the guildhall, titled 'Can I trust the Bible?', which will explain some of the key evidence. I shall take a little space here to argue for the truth of just 1 key miracle; the resurrection.

Evidence for this is to be found in documents of the new testament. We have manuscripts of these documents, some of which were written by eyewitnesses, going back to within a generation of the originals. It is possible to argue the truth of the resurrection from the internal textual evidence of these documents, but I do not have space to do so here, nor am I an expert in textual criticism. The other historians of the time mention the crucifiction, and they also document the spread of Christianity. The early witnesses claimed that Jesus had risen from the dead. At the time they could have easily been proved wrong by simply producing Jesus' body: Nobody did so, though the Jewish establishment had strong reasons to. They claimed that the body had been stolen, by those same witnesses. But the witnesses clearly believed what they were saying; they were prepared to die in painful ways rather than to deny it. This would not have been worth it if they had made it up. Nor were there just a few witnesses. Look at 1 Corinthians 15:3-7. Here Paul makes a big claim; that huge numbers of people saw Jesus after his resurrection, and that they were still around; that you could check by going to ask them. Again, he could have been easily discredited if he was wrong, unless large numbers of people were so convinced of a lie that they were prepared to die for it.

The available evidence was strong enough to convince me. But you should look at it for yourself. You mentioned Hume's argument that, since miracles are so improbable, you should believe almost any alternative (such as, they were all strongly motivated liars). I need to look up the argument again to check I have it right, but it seems to implicitly assume that miracles are very highly improbable; in particular, that they do not have divine cause. So it assumes the conclusion.

I apologise for any poor explanations I have given (with hindsight, I should have gone and slept for a bit and finished this in the morning).

Getting away with murder.

Though what I will describe in this post is probably irrelevant for most people, you may find it interesting as a worked example of my foolishness. Just to back that up, in case anybody had been duped by the fact that I do discuss philosophical issues into thinking that I'm any good at it, I'm not. You should be able to see that by the end of this post.

After I lost my faith, I remained confident that evangelical Christianity (the brand I had happened to hold) was still coherent, I just no longer believed that it was true. I knew there were objections, and was sure they could be worked around. A week and a bit ago I went to a talk and realised that there were two objections for which, although I'd thought I knew how to deal with them, I didn't. I'd just assumed that I'd already worked it out sometime, and not thought about it much.

Both objections are to the idea that the Bible is in any sense wholely true. The first refers to God's recorded act of genocide. Again and again the Bible says that God is good, so how could He do such a morally reprehensible thing? The second refers to the promised return of Jesus. Jesus Himself was pretty sure that it would be within his generation, as was Paul. The author of the book of revelation referred to one of Jesus' parables to give an impression of even greater urgency. But it hasn't happened.

I discussed the first objection with the speaker and with the folk at a Bible study group I attend, and I think I can now see a way around it. There is no getting around the fact that the Bible claims God committed genocide; the question is whether He was morally in the wrong. A crude application of the idea that moral principles apply to all persons equally would say that since it's wrong for Adolf Hitler to commit genocide, it's also wrong for God. But that misses an important idea; that the context and not simply the action must be taken into account. The fact that it is morally OK for a policeman to handcuff a prisoner doesn't make it morally OK for the prisoner to handcuff the policeman. The equality only says that if the whoever is now a prisoner were at some point to become a policeman, and vice versa, then it would be OK for them to do the handcuffing. To apply the principle, you need a sufficient similarity of context. This opens up a grey area, but most moral situations are so far into the black or the white that in practice this isn't a huge problem.

Now, the context here, according to the Bible, is that God is the creator and people are the creation. This huge asymmetry in the context means that we cannot say that an action is immoral for God just because it would be immoral for anybody else. Of course, we now face a different problem. The Bible claims that God is recognisably good. The principles I mentioned a few days ago only tell us that an action of God is good if it is in His own ultimate interest. However, that God shows His goodness by acting in His own interest is exactly what the Bible seems to claim. So how is genocide in God's interest? Nowhere is it claimed that God is a racist or a sadist. Instead the given reasons are tied to God's practical purposes and self-expression. I don't claim to be able to see that massacre was what was best for God, but then I don't have a good enough understanding of the world to see that it wasn't.

I thought that all of this could be circumvented by looking at the recorded personality of Jesus. After all, Jesus is supposed to be God, and most people who wore WWJD wristbands didn't think indiscriminate slaughter was high up on the list. But though he never did anything of the kind himself whilst he was on Earth, Jesus seems to have been convinced that God had wiped out nations. So Jesus' behaviour on Earth was at a higher standard. But it's reasonable to think of this as one of the things that goes along with the incarnation. That is, in becoming human, Jesus gave up his right to treat other humans as property, and acted as if He were as morally accountable as anybody else. This is a special case; we can't expect God in general to behave like that.

I still have no idea how to resolve the second objection. But since I also had no idea how to resolve the first one for a while, I have good reason to keep looking.

Flashback #2

For a bit of a change, here's a fairy tale I once wrote. I'm not a big fan of popular fairy tales; children are obviously able to cope with the more glaring untruths they contain, but are probably influenced by the more subtle lies which are hinted at. For example, fairy tales often suggest that people can be neatly divided into two categories, the good and the bad, and that social class, nationality and physical appearance are good ways to tell who is in which category. An idea which often filters down into later life is magical thinking, including the idea that, when you believe something strongly enough, it will happen. This is true to a certain extent of things within our own heads, so I decided to stick to that context. Here is the story:

Once upon a time, in a very familiar place where the sky was almost always blue, there lived a strong and handsome prince. He lived in a grand castle on a hill with a dense forest all around. He was not only strong and handsome, but he was also courageous. Once, he had climbed all the way to the top of the highest tree in the land. He had opened his eyes, looked around, and seen his very own castle below him. He had thought how much like a ribbon the road that ran from his door looked as it wound into the distance. He had been astonished by how far away the forest went in all directions, and especially downwards. And he had hardly even been scared a bit.

But although he was so strong, and so brave, and so handsome, and he owned a grand castle and was prince of the whole land, he was still very sad. Every day he would look around at all the things that he had, and would say: 'I would be happier without all of these things if only I had somebody else to talk to'. For, you see, he was the only living person in the whole of the land. The only face he saw was his own, reflected in the gleaming treasure that filled his rooms. The reason for this was that the prince was a character in a story, and that the person who had written the story did not have enough imagination to think of anybody else at all to put in the story with him.

One day, the prince was feeling very sad indeed. He thought to himself 'Maybe, if I was more sure of how strong and how brave I am, I would be happier'. So he decided that he would not only climb to the top of the tallest tree in all the land, but that he would hang from the highest branch by only one hand. But when he had climbed all the way to the top, and was hanging there by one hand, he was still sad. He said to himself 'I would be happier even if I was not so strong and so brave as I am, if I just had somebody I could be friends with'.

Just as he had finished saying this, he noticed that there was a small cloud of dust above the road in the far distance. He had never seen such a thing before, so he climbed down out of the tree as fast as he could, and ran along the road to see what it could be. After he had been running for almost an hour, he saw what the cloud was. It was dust being kicked up by the hooves of a mighty horse. Riding on that horse was a knight in armour that shone as brightly as the brightest of his treasures, and the knight was holding a small teddy bear.

The knight got off the horse, and said 'Hi. Who are you?'. Now the prince did not think that this was a very respectful way to address the prince of so large a kingdom, but he did very much want to talk. So he said 'I am the prince of this land and of all the surrounding country, from the sea to the great river. Before addressing such a great person, you should remove your helmet and grovel.' The knight took off her helmet, and said 'What's a grovel?' But the sight of her face left the prince too surprised to reply; she was just a little girl.

'I'm Anne. What's your name?' said the little girl. The prince thought about this for a while, and eventually gave up. He did not have a name. This upset him, for he thought that he really ought to have such a simple thing as a name. When she saw how sad he was, and he told her that he did not have a name, Anne decided that she would give him one. Now, she had named something before with only a little bit of help, and she decided to do just the same again. So she said 'I shall call you Spot'.

The prince was delighted with his new name, and was thrilled to have Anne to talk to. They talked for what seemed like days. He told her all about his castle and his treasure and his forest, and she told him about her house and her family and her dog. He wanted to know how she had come to be in his forest, and she explained that she had been listening to a story about a handsome prince and had begun to drift towards sleep, and had been dreaming about riding through the forest when she met him.

But she explained that soon she would have to go because every morning she had to go to school. This made the prince very sad, for he knew that without her he would soon become lonely again. When Anne heard this, she felt sorry for the lonely prince, and thought seriously about what she could do to help him. Soon, she had an idea. She said to the prince 'You can come and live in my imagination.' The prince was not sure how to get into her imagination, and asked what he ought to do. 'I'll imagine you, silly!' said Anne, and closed her eyes.

She imagined the prince just as hard as she could and soon he found himself in her imagination, which was a busy, exciting place, full of interesting people. There was McDougall, the only dinosour who had stayed behind when the others had left for Saturn in their spaceship (because, you see, he was terribly afraid of heights). There was a fat man in a big red coat trimmed with white fur. Usually he was very jolly, but at the moment he was not; he was suffering from a strange disease called 'existential angst'. He gave the prince a bottle of a dark fizzy liquid called 'the real thing'. There was Anne's imaginary friend Bill, who somehow managed to get people to blame Anne for all the naughty things he did. And there were hundreds of others. With so much company, the prince was very happy.

Now as the years went by, and Anne grew up, she began to thing more and more about the sorts of things that adults worry about, like spelling and tea. Indeed, if you spoke to her, you would probably not be able to find out about the prince. But every night she would fall asleep and the fat man and McDougall the dinosaur and prince Spot and all the others would come out to romp and play.

Flashback #1.

One of the things I plan to do with this blog is to put up some of the things I have written in the past, which are currently distributed as notepad files across at least 3 computers, so as to have them all at my fingertips wherever I go. Since one such item is linked to my loss of faith, this seems a sensible time to post it. It's a slightly modified version of an email I sent to a friend last Christmas explaining some of the aspects of Christianity that I have trouble with.

The first issue is one of evidence. I don't mean to deny that there is evidence for Christianity, but to claim that what evidence there is is extremely weak. A major theme of this weakness is the existence of reasonable alternative explanations.

A characteristic example is personal spiritual experience. It is an interesting fact that most people at some point have a compelling sense that there is something greater than themselves, and that this often happens in a religious context. The powerful and mind-enveloping nature of these experiences means that not only their inherent interest but also the experiences themselves demand (and often suggest) an explanation. Such experiences are often taken to be evidence of Christianity: They may be described, for example, as 'a sense of the presence of God'. Typically, it is not the fact that many people have had such experiences which is used as evidence. Instead, many people are persuaded of the truth of Christianity partly on the basis of their own personal experiences.

The weak point of such evidence is the availability of alternative explanations. The physiology of such experiences has been studied, and purely physical mechanisms have been suggested. The idea is not too complicated: Not all parts or states of the brain have the function of directly determining the nature of the world. For example, although we dream when we are asleep, and the dreams themselves are mind-enveloping, we do not afterwards think that the reality suggested by dreams is in any sense true. It is reasonable to suppose that mystical experiences are similar; they are a state of the brain produced by factors in the real world other than the ones that they immediately suggest. This is backed up by the fact that they can be produced by epilepsy, by certain drugs, and (to a lesser extent) by particular frequencies of infrasound and certain kinds of architecture.

Another typical feature of this kind of evidence is that it is necessary even for those who use it to admit that there must be alternative explanations. Mystical experiences are not limited to Christians, or even to theists. A Christian and an agnostic have the same difficulty in explaining the deep spiritual experiences of Muslims or Buddhists. The explanation that this is God revealing Himself to them is open to the objection that he is doing so in a distinctly deceptive manner. On the other hand, the more typical (physiological or diabolical) explanations remove the value of the original evidence for Christianity: My own spiritual experiences are as likely to have a physiological or diabolical cause as those of other people.

The point of this is not just to undermine the argument from spiritual experience, but more generally to illustrate the kind of weakness posessed by many of the usual things cited as evidence for Christianity, such as the Bible, other reports of miracles, the existence and rapid initial growth of the Church etc. This argument is in no way intended to demonstrate the falsity of Christianity: Indeed, no argument of this kind could ever do more than undermine particular arguments for Christianity. But if all arguments and evidences for Christianity may be countered in this way, that is a real issue.

To make the point more vividly, imagine that the claim 'We don't have conclusive proof of the existence of aliens because they have the technology to carefully observe us without revealing themselves' is true. (Warning: In what follows I don't intend to draw an analogy between aliens and God. That is a different thought-experiment). Suppose further that the alien observers have only recently arrived, that they are extremely intelligent, and that they are able to gather physical information about us at any level of detail they wish. In particular, they have access to all current documents including the Bible, the Qur'an and so on. Would they come to the conclusion that Christianity is true? If so, on what grounds might they do so?

This issue can also be approached from a more philosophical angle. Some statements are generally accepted as either true or false. Others, such as 'if when scampi green', are exempted from this distinction, and declared meaningless. Some examples are less obvious. For example, in the middle of the last century two different interpretations of quantum mechanics were devised. The first (the 'many worlds' interpretation) states that all the time, all over the place, the world is 'branching' into different worlds, and that everything that could happen, actually happens in some possible branch. According to this theory, the reason why we don't experience the world as branching in this way is that we ourselves do the same thing. Any particular branch of you only has memories and experiences within their own branch; the different events in different branches are experienced by different versions of you. This is an extremely strange theory, but it has experimental backing. The second (the Copenhagen interpretation) says that there is no such thing as a real world made up of particles in particular places moving at particular speed. Instead, the world is a 'superposition' (a kind of funny mixture) of such worlds. However, if you make any measurement at all, then the world mysteriously changes and becomes a mixture in which the experiment you performed (and no other) has a definite outcome. So no experiment can produce a funny mixture of results reflecting the funny mixture that the world 'really' is. This is also an extremely strange theory, and this also has experimental backing.

The odd thing is that the experimental backing of the two theories is exactly the same. Indeed, if you work through the mathematics you find that these two totally different theories make precisely the same predictions. That is, any experiment that supports either must support both, and vice-versa. So it is not clear what to do with statements like 'it is the copenhagen interpretation, and not the many worlds interpretation, which is correct'. The universe behaves just the same way whichever way you do the mathematics, and it seems odd to pick out one way of making the same predictions as true, and the the other as false. This is an area in which objective notions of truth are on shaky ground. But if there is no way to distinguish Christianity from the alternative explanations given for the evidence, then statements like 'The claims of Christianity are true' end up on similar ground. If there is no experiment we can perform to demonstrate, for example, that God answers prayer (and there is no experiment we can perform to show that He does not), then statements like 'God answers prayer' are in this sense meaningless.

One of the most obvious objections to this line of thinking is that there is an experiment that I can perform to determine whether such statements are true. All I have to do is wait. Eventually, I'll die, and then, if the claims of Christianity are true, I'll know it. This objection could be caricatured as 'Evidence? You want evidence? Well, you'll get evidence. Plenty of evidence. But by then it will be too late.'

The main problem with this objection is that it relies fundamentally on the person who faces God after death being me; that is, being the same person who is currently writing this email. This brings me to the second issue, which has to do with the Christian account of the self.

It is necessary at this point to indulge in a little pedantry. When I used the word 'me' to refer to 'the person who is currently writing this email', I was referring not to myself over all time, but only over a very short period of time, in which it is obvious what it means to say that I remain the same person. That is, I was referring to an instance of myself. There doesn't seem to be a word that summarises the idea of a person-instance in this sense, and I shall want to refer to this concept quite a bit. So, from now on I shall use the word 'being' to mean what I would otherwise have to summarise by the phrase 'person-instance'.

The point is that the objection above relies fundamentally on the idea that two beings (me and a particular being who God ultimately judges) correspond. The normal account of this (indeed, it is so commonly assumed that it is fundamental to the language; hence the pedantic paragraph above) is that the way these two beings correspond is via a person, or self, of which both are instances. More generally, it is assumed that we have a way to assign each being a person (of which that being is said to be an instance) in such a way that two beings correspond if and only if they are instances of the same person. Picking out a correspondence of beings over a short period of time is usually quite easy; after all, over such short periods each being is associated to a body, and the bodies remain quite stable over these short periods. By extending the correspondences this gives us, we get a way to identify beings which, in most normal circumstances, really does allow us to associate lots of beings together as instances of a person. Thus, for example, I can be identified as an instance of the same person as a particular baby born 23 and a bit years ago.

But this simple way to find a correspondence between beings cannot be all that we mean when we talk about selves. After all, hundreds of millions of people believe in reincarnation: That is, that particular beings alive today are instances of the self-same people as other beings who died many years ago. This identification is not made by the process described above. Similarly, as mentioned above, Christian doctrine relies on a similarly extrapolation of the notion of a person. But the very notion of people breaks down when faced with a simple thought-experiment.

Imagine yourself in a sci-fi world where there is a civilisation on the moon. Imagine further that the normal way for people to get to the moon is via the port-o-matic teleportation device. You go into a small room, close the door, and press the magic button. But when you open the door, hoping to greet your smiling lunar friends, you find that you are still on Earth. The explanation is obvious: The teleporter has malfunctioned. But now suppose that the malfunction was not in failing to produce a copy of you on the moon, but in failing to destroy the copy left on earth. That is, there are now two beings who might sensibly be described as being you; call them Matt1 (on Earth) and Matt2 (on the moon). Actually, this may not be the right way to put it: We want to say that the beings Matt1 and Matt2 are both the instances of the same person as the being Matt, who walked into the teleporter. But equally, we want to say that Matt1 and Matt2 are now instances of different people. There is no way to assign people to Matt, Matt1 and Matt2 that meets these conditions.

I'm sure you have seen thought-experiments of this type, and that you could concoct more complicated or striking examples in which, perhaps, Matt1 remains a Christian but Matt2 apostatises. But there is no need to picture such a fantasy world to get at this phenomenon. It is already with us, in split-brain patients. There is no consistent way to assign these beings as people. The Christian account of the world relies strongly on the identification of beings consistently into people: Each person is supposed to persist eternally in heaven or hell, for example. The beings in heaven and hell are very different to those they are associated to on Earth. Those in heaven are perfectly good, and those in hell are perfectly evil. On the other hand, some of the beings on Earth are pretty mixed. So it is not clear which beings on Earth are to correspond to which after death. That those after death may have memories of particular lives on Earth is one possibility, but memories are flimsy things and consideration of people with total memory loss suggests that memory should not be essential to the self. The issue is particularly confused by people whose personalities are totally changed by severe brain injuries, and by those with severe dementia.

This issue may well become more clear-cut within our lifetimes. Scenarios such as those in the thought-experiment may become practical. Artificial intelligence may finally be accomplished. Or possibly scientists will discover that these scenarios are impossible, because something inside each normal human brain does not obey the laws of physics. This would correspond to the discovery that we have non-physical souls, would resolve all of these issues in favour of the religious view, and would be hugely surprising for scientists everywhere.

The third and final issue relates to the Big Questions, those which have no generally accepted answers today, despite having been studied by brilliant thinkers for thousands of years.

Traditionally, it has been claimed that religions such as Christianity provide answers to these Questions, but it is not clear that they do. Very often, the answers beg the question, by postulating some aspect of God to which the same Question still applies. A classic example is the Question 'Why is there anything at all, rather than nothing?'. Initially, it looks like this Question is answered by Genesis 1:1; The reason why there is anything at all is that it was created by God. But this begs the question; after all, God is Himself supposed to be something, and the Question 'why is there anything at all, rather than nothing?' applies equally to God as to the created universe.

Similar remarks apply to Questions about the nature of things, such as 'Why does what there is behave in a regular fashion?', or 'What is the purpose of everything?'. Another interesting example is the Question 'Why are there so many simple Questions that we don't know the answers to?' Each such Question corresponds to an attribute of God. The fact that we do not know the answers to the Questions applied to those attributes is called the ineffability of those attributes of God. So this Question corresponds to the idea that the ineffability of God is itself an ineffable attribute.

C.S. Lewis said 'I believe in Christianity as I believe that the sun has risen: not only because I see it, but because by it I see everything else'. Here he is referring to the explanatory power of Christianity. This explanatory power is usually taken to lie in the light shed on the Big Questions by Christian doctrine (after all, smaller questions may be left to science, history, geography, mathematics, common sense etc.). But all that the standard answers to these Questions do is to pull them back to a different level; the same Question, asked at the new level, is answered by the statement that God is ineffable. But this is no answer at all, and certainly isn't a satisfying explanation. In this way, much of the supposed explanatory power of Christianity is nullified.

What should I do?

I mentioned in my first post that I'd talk a bit more about my moral beliefs here. I'm not going to be writing about the details of what I think is right and wrong, but rather about why I'm not too worried by the fact that I do think some things are right and others wrong. After all, in my last post I mentioned that I don't believe in God. Nor do I believe in anything else supernatural. If you claim to derive your morality from some supernatural source, you may think I'm being inconsistent here. After all, isn't any moral system automatically supernatural?

One variation of this argument, which was given at a talk I attended yesterday, begins with the idea that we think that some things are right, and that others are wrong. We make such judgements about things by comparing them to an absolute moral standard. If you are sceptical about this claim, consider the fact that we feel so strongly that some things are evil that we are not prepared to accept the possibility of argument on the matter: We would say that they are absolutely wrong. But it is impossible to produce such a moral standard from examination of the natural world, however carefully you do it. Careful investigation may reveal that a person's actions caused extreme suffering, but it cannot reveal that that suffering itself is a bad thing. So it is impossible that the world we live in, which includes this moral standard, can be explained in purely naturalistic terms.

I'm afraid that in order to bring out the flaw in this argument, I'm going to have to indulge in a little formalism. The argument is trying to show that a particular limitation (to naturalistic explanations) leaves us unable to account for the world that we observe. Some evidence (our strong moral feelings) is given for a proposition (the existence of a moral standard). It is demonstrated that the truth of this proposition cannot be explained given the limitation. But it is the evidence for the proposition which is part of the world that we observe, not the proposition itself. So it is necessary to show that, given the limitation, we are unable to account for this evidence. But that is not true.

To put it another way, it is possible to explain how there could be conscious beings like us who feel extremely strongly on moral matters even in a purely natural world in which there is no moral standard. Evolutionary theorists have had a good stab at this, and have made enough progress to show that it is possible in principle, though they are very far from a complete theory. I do not mean to imply by this that evolutionary theory can answer moral questions; to do so would be to justify, rather than simply explain, our moral feelings and beliefs. When I discussed this with the speaker after the talk, he concluded that although it is possible to account for the qualities of morality which he mentioned from within a naturalistic framework, it is not possible to justify any particular moral statement. That, I think, is fair enough.

So, my beliefs about the way that the world is are unable to justify any moral statements. Yet I do have some moral beliefs. Isn't that a little problematic? This is a topic which I was discussing with some friends recently, one of whom suggested I should start this blog. So I believe I ought to have a go at answering it. Before I do so, though, I shall explain why I think the problem is one that we all face. We believe some moral statements, and some amoral statements. Sometimes it is possible to deduce a moral statement from some other statements, some of which are amoral. But it isn't possible to deduce a moral statement from other statements, all of which are amoral. So, nobody's amoral beliefs are able to justify any moral statements. Yet almost everybody does have some moral beliefs. This is a little problematic.

However, I don't find it to be a problem in actually making moral decisions. Instead, I find that they are often obvious. When I bring to mind the idea of killing or hurting a person, it is immediately clear to my mind that it is wrong. When I hedge the idea about with details, it usually, but not always, remains clear that it is wrong. For example, it is not obvious that capital punishment is wrong. I may need to examine the world to find out such details. But there is no need for me to examine some other bits of the world to find out if the action itself is right, and I do not do so.

Given the subject matter, it is quite possible that the above summary is misleading. I'll try to clear up a few possible misunderstandings.

Sometimes other people disagree with me on moral questions. It is easy to imagine that this could occur to the extent that any moral statement would be meaningless. But it does not. In fact, it's quite rare. That is not too surprising, given that we're all members of the same species. This general agreement suggests that whatever part of my brain it is that sees what is right or wrong is functioning properly. So if I do kill somebody, then I deserve to be punished. On the other hand, if a person murders another but is unable to see that it is wrong, that person should be treated; something in them is not properly functioning. Aside from the physiological evidence, the general concensus allows us to talk about proper functioning in this way. But it does not provide any justification for any moral belief.

So am I setting myself up as an infallible moral judge? Absolutely not. I can easily think of times when I have had moral beliefs which were wrong, and I conclude that I must also currently have some false moral beliefs. In a similar way, I can easily think of times when I have had geographical beliefs which were wrong, and I conclude that I must also currently have some false geographical beliefs. But I'm nevertheless quite confident in my moral and geographical beliefs and in the ways that I come to them. I'm just prepared to accept that I might be wrong.

In some cases, I do not know whether a thing is right or wrong. If there were an absolute standard of morality, this ignorance would show that it could not be completely determined in the way described above. However, I do not believe in any such absolute standard.

I have probably also given the (false) impression that when I make moral decisions it involves no thought. However, the moral beliefs which I see most clearly are those which are most abstract, so it is often necessary to throw some amoral beliefs in and do some thinking to come to a conclusion in any particular case. To illustrate this, here are a couple of areas where I have very clear moral beliefs, together with some details of how they might be used together with amoral ideas.

Firstly, I have clear moral beliefs related to my own benefit. When I feel pain, it is clear to me that this is bad, and when I feel pleasure, that it is good. When I think of my own death, I can see that it is a bad thing. Of course, these are just crude examples, and some of the things that I feel are good in myself are much more complex, like liberty or self-expression. Initially it may appear that no sensible morality may be derived from this selfishness, but by mixing in some amoral ideas you can get surprisingly far. Nevertheless, a clear-eyed look at history is enough to show that it is possible to act in your own interest without your actions being good.

Secondly, I believe that all persons are equal from a moral point of view: That in deciding whether an action is right or wrong, the identity of the actors is irrelevant. Although this is one of my strongest beliefs, it is extremely abstract, and it is necessary to combine it with both the first belief and some amoral ideas. For example, when I meet someone, I know that they are a person and that I am a person (amoral belief). I know that when I feel pain, it is bad. So, when they feel pain, it is bad. So it is right for me to try not to cause them pain.

Of course, this account is greatly oversimplified. In order to apply the above ideas in complicated situations it is necessary to work out what the consequences of various courses of action are, which can involve complicated amoral reasoning, and then to evaluate how good or bad those consequences are, according to my moral beliefs. It is also necessary to decide whether particular things are persons. For example, is an embryo a person? Is a chimpanzee? I must confess that I do not have a precise definition of a person, but that in most cases it is clear. If in doubt, I err on the side of supposing things are persons. This works because so many things are obviously not.

I will conclude with an extraordinary claim: What I have outlined above also describes how you justify your moral beliefs. You may claim that you do so in another way, such as looking in the Qur'an. But those who look in the Bible and those who look in neither book have similar moral beliefs, which closely reflect parts of those books and are hard to reconcile with other parts (such is the nature of religious texts). If you claim that without a particular guidebook you would be unable to see for yourself that murder is wrong, I can only conclude that whatever it is in your brain that sees what is right or wrong is not properly functioning, and that you should see a doctor at once.

Who am I?

Those who stumble across this blog as they meander around the Well Worn Walkways may, struck by a spirit of curiosity or procrastination, wish to find out more about me, its nominally humble resident. So I'll kick off with the old philosophical chestnut 'Who am I?'.

Any non-vegetative readers will already have guessed some of the most important things about me: That I am human, that the time in which I am alive is the early 21st century and that I live in such relative luxury that I have easy access to a computer. Those whose eyes have glanced at my profile will have noticed that I unflinchingly mention mathematics in public. I have loved mathematics for as long as I can remember. Indeed, one of my earliest childhood memories is of computing 16^2 on the way to the toilet on a camping trip. Mathematics continues to be the love of my life; as I write this I am in the first year of a mathematics PhD.

Your accurate deduction from this that I am in my early twenties gives me an excuse to mention some of the other more mundane facts about myself. I'm male, white and British. I was born in Knutsford and brought up there by my wonderful parents. This already marks me out as extremely fortunate. This good fortune has continued throughout my life so far: My brother, though only three years older than me, threw himself into helping to bring me up and delighted in trying to teach me everything as he learned it. I was taught at the local high school by almost universally excellent teachers. The head of mathematics in particular was an inspirational teacher who got me involved with some national mathematics organisations. As a result, I was able to compete for the United Kingdom in the International Mathematical Olympiad, which took me all over the world. I've been a maths student (or 'mathmo') at Cambridge University for the last few years, and have somehow fooled them into letting me stay on to do a PhD.

What am I like as a person? I'm bright, and good at understanding ideas: I'm not so good at understanding people. I have enough social skills to get by, and enjoy the company of others, but don't usually form deep relationships. This is partly because I'm not very emotional. If I do get worked up, it's normally over abstract ideas. Such things as I do believe, I tend to believe strongly and in sharp contrast. I am burdened by the annoying trait that often goes along with this: Great difficulty making important decisions. I have always been satisfied with my lot, though this has never been tested by serious hardship.

I was brought up as a Christian, and was a nominal believer until my late teens, when I began to question whether the religion that I had been brought up in was true: I came to the conclusion that it was. A few months later I swallowed my pride and asked God to save me; I became a Christian for real. Over the next few years my relationship with Jesus deepened, and my faith was a very important part of my life. This was supported by my family, who are strong Christians, and by the churches I attended at home and at university, and the CICCU.

At the end of my fourth year at university, during a church service, I realised that I no longer believed in God. It was very sudden, and utterly reasonless. For a couple of months I tried to convince myself that it was test of some kind, and that I was still a Christian, but I became gradually and unwillingly convinced that this wouldn't wash. It wasn't until two months after that that I finally told my parents, whose reaction has been a model of loving acceptance. Other people's reactions have been more mixed, though they have often given me useful and interesting things to think about.

I still don't believe in much. I don't believe in God's existence or nonexistence; I don't know whether He exists or not. I don't know the answer to any Question worthy of the capital Q. This includes the Question 'Is it possible to be sure of the answers to any of the Questions?', so I'm not even an agnostic. I'm just confused.

I do believe some things, though. I believe lots of ordinary things, such as that Paris is the capital of France. I believe that the world works in a very regular way, to the extent that my habit of describing it with words (like 'Paris is the capital of France') isn't totally misguided. I believe that the world works even more regularly on very small scales, to the extent that our habit of describing it with mathematics is unreasonably successful. I believe that some particular mathematical systems (described better than I ever could elsewhere) are accurate descriptions of how the world works on a small scale. I have some particular moral beliefs: That some things are right and good and that others are bad, or even evil. I'll discuss these a bit more in my next post. Similarly, I have some aesthetic beliefs: That some things (especially mathematical arguments) are beautiful and others are ugly. That's about it.

Being human, I will have been a little inaccurate in my description of myself. You can come to your own, more accurate conclusions, by looking at what I write.