Archive for June, 2011

Solution to cricket balls problem

Last week, I set a little brain-teaser. Out of 12 balls, eleven are normal, and the other is either heavier or lighter than the others. Using a simple set of balances no more than three times, find out which ball is the odd one out, and whether it’s heavier or lighter than the other balls.

(And may I add that it is while preparing this post that I realised that the functionality in Word that allows one to use mathematical symbols appears not to be recognised by WordPress!)

First, a bit of notation.

Define set N as the set of all balls known to be normal. (Before the first weighing, this set is clearly empty.)

Define set X the set of all balls not yet known to be normal.  (In other words, balls in this set could be light, or they could be heavy, or they could be normal … we just don’t know. Set X is the set of all unknowns – i.e. those balls about which we as yet know nothing.)

Define set H the set of all balls not yet definitely known to be normal, but known not to be light (i.e. any single ball in this set could be normal, or it could be heavy).

Define set L as the set of all balls not yet definitely known to be normal, but known not to be heavy (i.e. any single ball in this set could be normal, or it could be light).

It is also useful to define #S as the number of elements in any set S. So, if, say, #H= 3, then there are 3 balls in set H.

At each weighing, there are a number of possible outcomes. For each of these outcomes, different strategies must be applied. So, to ease understanding, any given point in the procedure will be described by a sequence of numbers indicating the outcomes at previous weighings. So, for instance, if we are considering point 1.3.2, then we have had 3 weighings; at the 1st weighing, we had Outcome Number 1;  at the 2nd weighing, we had Outcome Number 3; and at the 3rd (and final) weighing, we had Outcome Number 2. All possible combinations of outcomes will be considered, and the odd ball identified as heavy or as light by the end of the third weighing.

Right. Let us begin.

To begin with, all balls are in the set X . So  #X = 12.

Weighing 1:

Weigh 4 balls against 4 other balls. There are 2 possible outcomes:

  1. Both sides are equally balanced
  2. One side is heavier than the other.

Let us first consider Outcome1. (We will consider Outcome 2 later.) If the sides are equally balanced then the 8 balls we have weighed are now definitely known to be normal. The other 4 , that weren’t used in the weighing, aren’t yet known to be normal. So we have:

 #N = 8

 #X = 4.

Now, let us consider Outcome 2, where one side is heavier than the other. Clearly, the 4 balls that weren’t placed in the scales are all normal; none of the 4 balls on the heavier side can be light; and none of the 4 balls on the lighter side can be heavy. So we have:

#N = 4

#H = 4

#L = 4

Let us first follow the branch down from Outcome 1, where we have #N = 8 and #X = 4. The second weighing can now be as follows:

Weighing 2:

On one side of the scales, place 3 balls from set N, and on the other, place 3 balls from set X  .

                                    N N N     vs.     X X X  

There are 3 possible outcomes:

1.1  The side with the balls from set X  is heavier

1.2  The side with the balls from set  X  is lighter

1.3  The two sides are equal

If we have 1.1, then it is clear that the odd ball is one of the 3 balls from set  X that were used in the weighing. And it is also obvious that these balls cannot be light – i.e. they belong to set H. And all the other balls apart from these three are now known to be normal. So we have:

#N = 9

#H = 3

We may now solve the problem at the 3rd weighing:

Weighing 3:

Weigh one ball from set  H against another ball from set H.

H    vs.    H

There are two possible outcomes:

1.1.1      One side is heavier than the other – in which case, the ball on the heavier side is the odd ball, and it is heavier than the other 11.

1.1.2      The two sides are equally balanced – in which case, the ball from set  H that hadn’t been used in this weighing is the odd ball, and, once again, it is heavier than the other eleven.

Now, let us go back to Point 1.2, i.e. we have had two weighings, and at the second weighing,  the side with the balls from set  X  is lighter.

It is clear that the odd ball is one of the three balls from set  X  that were used in the weighting. And it is also obvious that these balls cannot be heavy – i.e. they’re all in set L. And all the other balls are now known to be normal. So we have:

#N = 9

#L = 3

We may now solve the problem at the 3rd weighing:

Weighing 3:

Weigh one ball from set against another ball from set  L.

L    vs.    L

There are two possible outcomes:

1.1.3      One side is heavier than the other – in which case, the ball on the lighter side is the odd ball, and it is lighter than the other 11.

1.1.4      The two sides are equally balanced – in which case, the ball from set L that hadn’t been used in this weighing is the odd ball, and, once again, it is lighter than the other eleven.

Now let us go back to consider Point 1.3, i.e. to the point where we have had 2 weighings, had Outcome 1 at the first weighing (i.e. the two sides were equally balanced), and Outcome 3 at the 2nd weighing,  the two sides are, again, equally balanced.

It follows that the odd ball out must be that one ball that had been in set X   after the 1st weighing, but which didn’t take part in the 2nd weighing; and also that all the other balls must be normal, i.e. we have

#N = 11

 # X  = 1

So we may now solve the problem with the 3rd weighing.

Weighing 3:

Put the ball in set X  on one side, and any ball from set N in the other.

X       vs.      N

If the side containing the ball in set X  is heavier, then the ball on that side is the odd ball, and is heavier than the others; if the side containing the ball in set  X  is lighter, then the ball on that side is the odd ball, and is lighter than the others.

We have now considered all possibilities from Outcome 1 at first weighing, and in each case, have solved the problem within three weighings, as required.

Now let us return to the first weighing, and consider how we should proceed if we had Outcome 2, i.e. if, when we weighed 4 balls against 4, one side was heavier than the other. As we said, we would then have:

#N = 4

#H = 4

#L = 4

In this case, for the 2nd weighing is as follows:

Weighing 2:

Place on each side of the scales 2 balls from set H, and one from set L.

H H L     vs.    H H L

There are two possible outcomes:

2.1  The two sides are equally balanced

2.2  One side is heavier than the other

Let us first consider 2.1.

If the two sides are equally balanced, then it is clear that the 6 balls used in the last weighing are now known to be normal, and that the odd ball is one of the two balls in set L that weren’t used in the last weighing. This gives us:

 #N = 10

#L = 2

So we may now solve the problem with the 3rd weighing.

Weighing 3:

Weigh one of the balls in set  L against the other ball in set L .

L    vs.    L

Obviously, the ball on the lighter side is the odd ball, and it is lighter than the other 11.

Now, let us return to Point 2.2, where, at the second weighing, one side was heavier than the other.

There can be two possible reasons for this:

(i)            One of the 2 balls in set  H on the heavier side is the odd ball, and is heavier than the others

(ii)           The ball in set  L on the lighter side is the odd ball, and is lighter than the others

In either case, the 2 balls in set L  that weren’t used in Weighing 2 are now known to be normal. Also, the ball in set  L on the heavier side, and the 2 balls in set  H on the lighter side, are all now known to be normal.

This gives us:

#N = 9

#H = 2

#L = 1

We may now solve the problem with the 3rd and final weighing.

Weighing 3:

Weigh one of the balls in set  H against the other ball in set H .

H    vs.    H

There are 2 possible outcomes:

2.2.1      The two sides are equally balanced

2.2.2      One side is heavier than the other

If we have 2.2.1, then, clearly, the two balls that were used in the 3rd weighing are now known to be normal. This means that the ball in set  L is the odd ball out, and is lighter than the others.

If we have 2.2.2, then clearly, the ball on the heavier side is the odd ball, and is heavier than the others.

We have now considered every single possibility, and have solved the problem, as required, within 3 uses of the scales.

(I often take this problem as an illustration of the importance of defining one’s terms well: if all terms are well defined from the start, everything tends to fall into place.)

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When the awkward becomes adequate: “An Awkward Echo” by Mark Dietz

Back in the 90s, when I first became acquainted with this strange thing called the net, I became a regular contributor to the now sadly defunct discussion board on the Penguin Classics site. We soon developed a small core of regular contributors – though others came and went – and, after a while, we started, in effect, exchanging essays. There were three of us particularly who indulged in this sort of thing: one, I subsequently found out, lived quite close to me, and we became friends in the real world as well as in the cyber-world (after initially having crossed swords on the board); and the other was a very genial chap who lived across the Pond named Mark Dietz – and who, I am pleased to see, has recently discovered this blog. I had thought I had lost contact with Mark, until he traced me down quite recently on Facebook. And he told me that some of the ideas he was considering while conversing with us on the Penguin website he had subsequently developed into a doctoral thesis, and that the thesis has now been published under the title An Awkward Echo: Matthew Arnold and John Dewey. And, what’s more, both I and my other fellow essayist on the old Penguin site had been named in the Preface to the work. 

All of this is a preamble explaining why I have been reading an academic thesis on education and the nature of communication. Academic thesis it may be, but it has none of the dryness normally associated with academic theses. I found it fascinating to see the thoughts Mark had previously sketched out on that Penguin site now being expounded in such intricate detail. 

So what is this book about? Mark himself admits towards the end of the book that he finds it difficult to express in a few words the theme of the book. One may say it is about education, but that seems an inadequate description of the book’s contents: the perspective of the book seems wider than that. If I were to try to do what the author himself says he finds difficult – i.e. state in a few words what the book is about – I’d say it is about communication, and about the problems and difficulties involved in transmitting something from one mind into another. Can a thought, an idea, be transmitted precisely? The obvious answer to this is “no”. Leaving aside the possibility of telepathy, the thought has to go from one mind to another via some sort of medium – in the form of spoken words, perhaps, or a written text. And this text can only be an approximation of what had been in the author’s mind – maybe a good approximation, maybe a poor one, depending upon the skill of the author. But approximation it is – an awkward echo. And when the reader reads the text, the reader’s interpretation introduces another level of uncertainty. The echo that the reader takes in can be a very awkward echo indeed. But how awkward? Can these awkwardnesses (which is about as awkward a word as has ever appeared on this blog) be negotiated? If so, how? 

Having read over the last paragraph, it seems to me to be an inadequate – indeed, awkward – summary of the book, but since I don’t think I could do much better, let us move on. Mark considers two directions of thought – that of the author attempting to communicate with the reader (i.e. rhetoric, the art of persuasion), and, in a reverse direction, that of the reader trying to understand the author (i.e. hermeneutics, the art of interpretation). And in particular, he considers two major poles: Matthew Arnold, that apostle of authority and of established cultural values, and the American educationalist John Dewey, whose emphasis was on personal experience. It is hard to imagine two more sharply contrasted figures, both in terms of their personalities and their ideas, but for all their apparent differences, Mark finds surprising common ground between the two, and argues that they were approaching the same issue from different directions. 

This finding a middle ground is very typical of Mark’s approach. In our times, the very phrase “middle ground” reeks of anodyne compromise, of intellectual and moral pusillanimity, of lack of passion. But there is nothing anodyne about the search for the middle ground here: indeed, Mark is a passionate searcher for this ground, arguing that only in this middle ground can ideas from different sources meet, and interact fruitfully: 

One of the great peculiarities of our age is that we have taught ourselves that great thought takes the form of extremes, and only those thoughts that challenge the past and move us quickly and forcefully into the future are intellectually credible. The diminished middle ground with its nuance and hard-won sense is all too often missing in our arguments today. But, in most ages, I suspect the middle ground is that to which we must turn for an adequate solution to our needs, be they educational, or social, or spiritual, or intellectual.

 This search for the middle ground takes us on a fascinating intellectual journey. Mark is not prepared either to accept wholeheartedly or to jettison any ideology, even those with which, one suspects, he may not personally sympathise. Whatever the ideology, his aim is always to see if elements of it, in conjunction with elements of other ideologies, can produce something worthwhile. Or, if we cannot help the various echoes being awkward, let us, instead of chafing at the fact, consider at which point the echoes may become, at least, adequate

Having first given us sketches of the ideas and the personalities of Arnold and of Dewey, and having defined and proposed what he terms a “educational pluralism” – which, Mark contends, is what both Arnold and Dewey were approaching from their very different perspectives, Mark goes on to examine the nature of our perceptions. He starts with Hume, who argued that the relationship of causality, i.e. our understanding of how the world around us works, is not something that we perceive, but rather a concept we formulate in our own minds. This mode of thought (simplified to the point of absurdity though it may be in my summary above) has, says Mark, “arguably … dominated philosophy from the German idealists to today’s post-structuralists”. But Dewey, the pragmatist, wasn’t impressed: he refused to allow that any thought could exist independently of experience. He refused to go down the path to a dead end in which each human mind is trapped within its own confines, and introduced the image of the “tentacled mind”, which has the ability, born of experience (which, as Mark reminds us, is derived from the Latin ex peri – i.e. “from trial”), to leap across boundaries. The individual human mind, in Dewey’s world, is by no means an isolated thing in a sea of other isolated things, and imagining it to be such leads us only into dead ends. 

In the next chapter, “The Critical Mind”, Mark then gives us a potted history of the art of literary criticism, of interpretation. To simplify (as I must here, even with the attendant danger of introducing echoes so awkward that Mark may well be gritting his teeth in frustration as he reads this), traditional criticism privileged the author: here, the best reading was that which came closest to understanding the author’s mind. The “New Criticism” that emerged in the earlier decades of the 20th century abolished the author’s mind as irrelevant, since we have no way of knowing for sure what the authorial intentions were: instead, it privileged the text. And in our own times, postmodernist ideology privileges the reader: all that matters is the reader’s interpretation. True to form, Mark isn’t willing to throw any of these away; instead, he tries to see how these three approaches can be combined, can be made to interact each with the other, to produce something that cannot be produced in any ideological extreme – a whole that is greater than the sum of its parts. In the process of doing this, Mark provides wonderfully lucid accounts of all these different ideologies, not allowing his own personal biases to deflect him from fairness. 

Here, I must confess that I have long had difficulty with much of postmodernism. The first of these difficulties is to do with the seeming inability of proponents of postmodernism to write clearly. Now, I do appreciate that difficult ideas require difficult vocabularies; however, I do know from my personal reading experience that it is possible to explain to varying degrees of lucidity all sorts of extremely complex ideas; and it does, I admit, make me suspicious that I have never encountered any degree of lucidity at all in any ideological exposition I have come across written by the postmodernistas. However, I am content to lay the blame on myself on this matter: no doubt it’s just me being thick. The more important issue is that they appear to me either to be saying too much, or too little. If they are saying that all responses are necessarily subjective, and that any subjective response is as valid as any other subjective response, then that is clearly nonsense: I doubt very much whether anyone proposing so preposterous an idea would carry their paranoia of authority so far as to take equally seriously, were they to be taken gravely ill, the diagnoses of an experienced doctor and of an unpractised amateur. However, if all postmodernistas are really saying is that different readers can legitimately differ in their interpretations of a single text, then, as Shakespeare’s Horatio might have said, there needs no “-ism”, my lord, come from academia to tell us this. And in practice, the ideas of postmodernism have led us, I believe, to doubt the importance of what is termed “High Culture” to such an extent that we seem no longer to believe it worth propagating: if differing subjective responses are all we have to go by, then, if we are to develop communal values (as we must if we are to live together in a community), the only criterion we can go by is that of popularity – i.e. by those subjective responses that occur most frequently to the greatest numbers of people. Under such circumstances, that which is valuable but difficult stands little chance, and it is deeply ironic, though, sadly, not surprising given the prevalence in our time of so much of postmodernist thought, that democracy is sidelining those very cultural values that Arnold thought would prepare us for it. 

Mark, however, with a fairness that I must admit would have been well beyond me, smoothly and elegantly brings together such apparent opposites as Arnoldian reverence for High Culture (which, he points out, was not intended to prepare us for democracy rather than act as a bulwark against it), and reader response theories of our own time. He proposes a “pluralism” – a middle ground that is no mere compromise, but which incorporates elements from all the different approaches, and in which the authority of the author does not compromise the independence of the reader, nor vice versa; in which the reader is by no means merely a passive element in the process; and in which the reader’s response, the text, and indeed, even the author’s intentions, are all equally privileged. One may ask, as practitioners of New Criticism did, “Can we really know the author’s intentions?” To which Mark will no doubt answer: “Up to a significant point, yes.” The echo may be awkward, but it is not completely unrelated to that which it is echoing. The echo may indeed, in many cases, be adequate. Or even, perhaps, more than adequate. For why should we not see it as a positive feature that the authority of the author, while not to be rejected, is not absolute? Why is it not a positive feature that the reader is not merely a passive component in the procedure, but takes an active part in the creation and apprehension of meaning? 

*** 

Any summary of complex ideas is bound to be a simplification, and I fear I have, in the above, over-simplified, and, hence, distorted, many of the ideas that Mark expresses in his book with such elegance. I suppose the only remedy for that is to recommend the book itself, which, though a thesis, is as far from academic dryness as I can imagine. 

One final word: it is a temptation on reading works of non-fiction to skip, or at least merely to skim through, the notes. That would be unadvisable here, as Mark frequently allows himself a far more personal expression in his notes than he does in the main body of the text. These notes, ranging in reference from Moby-Dick to Star Trek, are frequently a delight in themselves. My favourite, I think, is the little riff Mark gives us on the theme of sentimentality – an issue we used to discuss often back in the days of the Penguin Classics discussion board. Most people, I think, would see sentimentality as necessarily a Bad Thing, but Mark, keen to find elements of worth in everything, no matter how obscurely hidden, thinks otherwise: 

Now, according to a rather curious definition, sentimentality is false emotion; however, if we take a closer look at sentimental emotions what bothers us is not their falseness, but a tendency to represent emotions that are too large, too close to the surface, too awkward, too cloying, too freely expressive. In none of this is falsehood really at blame; were falsehood present, we would have another complaint altogether – chicanery, deceit, subterfuge – all of which suggest a second level to the emotion, a something underneath the surface, an irony, perhaps. Sentimentality is thus, to my mind, if it is true sentimentality, to be valued as sentimentality (and frankly, I do think we ought to learn how to value sentimentality, for without it the world is missing a rather common and surprisingly varied ingredient), an emotion that resides on the surface of life – untempered, unalloyed, unprofound – large, voluble, broad, impersonal, awkward, and vast. 

Such scrupulous regard for fairness, such determination to examine everything for any possible nugget of value, is entirely typical of the author, and is among the many features that make this academic treatise so enjoyable to read.

Brain teaser: 12 cricket balls

This one’s an old one, but a good one. Some of you may have seen it before. 

You are given twelve cricket balls. 

(For readers on the other side of the Pond, cricket is the finest of all sports, and you really don’t know what you’re missing. But let us not digress.) 

You are given 12 cricket balls, which look identical, but one of them is a bit different from the other eleven. Although it looks like the other eleven, it doesn’t weigh the same. It could be lighter, or it could be heavier. You don’t know. 

The other eleven balls all weigh the same as each other. 

And you also have one of those old-fashioned scales. You can put as many balls as you like on either side of the scales, and you will know whether the two sides are equal in weight, or whether one side is heavier than the other. And that is all.

You are allowed to use these scales three times, and three times only. And from that, you must identify the ball that doesn’t weigh the same as the others. And you must be able to say whether it is heavier or lighter than the others.

Over to you. (Answer some time next week.)

(Hint: don’t start with weighing 6 balls on one side against 6 on the other: all that will tell you is that either one of the balls on the lighter side is light, or that one of the balls on the heavier side is heavy. And that doesn’t really get you too far.)

Literature and religion

Well, it had to be done. I have, this week, started to read Dante. Not in the original, of course – I am a very poor linguist – but in Robin Kirkpatrick’s translations, which come with detailed introductions and copious commentaries and notes, invaluable for someone like myself, shamefully ignorant as I am both of the historical and the cultural context of the Commedia. And it is a dual language edition, so I can, at least from time to time, glance across to the Italian text to get at least a feel of what the original sounds like. 

So why Dante? Well, life is too short for anything but the best. And how do I know this ranks among the best? Well, I don’t, of course. But given that one cannot check everything out for one’s own self, one has to rely to a great extent on the judgement of others – one has to go by reputation. Which is not to say that one abnegates one’s own judgement, of course; but one’s judgement can only be applied to what one has already read, and, except in the obvious cases of twaddle that are recognisable from a mere extract or two, this judgement doesn’t take one very far in deciding what to read next from that vast and as yet unknown ocean of literature. One has little choice but to rely on reputation. 

In a recent post, I suggested that the four most significant pillars of Western literature are Homer, Dante, Cervantes and Shakespeare. This is contentious, of course – not for what has been included, but for what has been left out: what about the Greek tragedians, one may ask? Or the Romans? And why stop at the early 18th century? – why not move on to Goethe and Heine and Pushkin, or to Tolstoy and Dostoyevsky, or on into modernism? Indeed. This is why trying to compile lists of these things, or attempting to define the criteria of inclusion into a literary canon, is such a fruitless activity. But however we define the literary canon, there is no-one whose place in it is more secure than Dante’s. To anyone with an interest in literature, this is a peak that just has to be scaled. 

But approaching something such as the Commedia leads to a somewhat awkward question: how should we, in our secular and disbelieving age, approach religious literature? And religious art as well, for that matter. And religious music. Of course, I do not mean to suggest that one has to have religious belief to appreciate or even love works with religious content: it should come as no surprise, for instance, to see Richard Dawkins choose Bach’s St Matthew Passion amongst his Desert Island Discs. But the question still remains: do we secularists appreciate such works despite their religious content? Or could it be that, despite our professed disbelief, we love these works because of it? 

I find this an endlessly fascinating question, and I fear there is no way to discuss this question without at least touching on that awkward and potentially embarrassing issue of one’s own religious beliefs, or lack of them. I would prefer not to go into this too deeply here, firstly because I am not at all learned in matters of theology, and secondly because it is an extremely contentious area, and whatever personal view I put forward, I am bound to anger and even alienate at least some of my readers. But if I am to write about literature (which is after all the primary purpose of this blog), then it’s hard to see how I can escape writing about religious literature; and if I am to write about religious literature, my own religious (or irreligious) outlook is far from irrelevant. So, with apologies beforehand to anyone whom I might offend, I might as well take a deep breath and dive in: how do I, who profess no particular religious belief and who usually write “None” when asked to specify my religion on an official form, approach a work as the Commedia which is entirely imbued with Christian thought? Is it possible to take anything at all from the Commedia without, at the very least, a leap of the imagination into this world? And can one be capable of such a leap without having, at the very least, some sort of sympathy with religious belief? 

Before we dive into discussion of such matters, we should, I think, acknowledge that the extreme poles of devout belief and of unconditional disbelief are but that – poles; and that, between these poles, there are almost infinite shades of grey, almost infinite nuances. The very presence of these nuances is often denied by the Dawkinsites, who appear to insist that either one believes or one doesn’t, and that there can be no room for “I don’t believe but…”.  I think this is the first of many points where I part company with the Dawkinsites: far from dismissing this but as intellectual pusillanimity, as they appear to do, it is in this region of but that we find depth and richness: but for this but, our lives, beset as they would be with immutable certainties, would, to my mind, be arid indeed. It is in this region of but that the human imagination, without which there can be no art or music or literature – or even, for that matter, science – flourishes. Far from dismissing this but, let us welcome it, and look into it a bit deeper: but what

My own first tentative steps into this region of but are motivated by certain feelings – feelings of yearning and of awe – that I know I often have, but which I cannot articulate to any degree of precision. Vaguely defined though these feelings may be, I know I experience them when I listen to certain music, or when I read certain lines of poetry. And not just then: there are all sorts of occasions when I know I feel certain things that, for want of a better word, may be described as “transcendent”, since they appear to transcend the mere physicality of the world I inhabit. Now, it may well be that “appear” is the key word here, and that these feelings of transcendence are no more than my synapses synapsing and my neurons doing whatever it is neurons are supposed to do. Yes, perhaps. But do I want to see myself as no more than a collection of neurons and synapses? The answer, for all my professed rationality, is “No, I don’t”. It may well be that I am no more than the sum of my constituent physical parts, but there is a very big part of me – no doubt deeply irrational – that rebels at such an idea. The Dawkinsites will tell me to amputate away that irrational part of myself , but I cannot. And neither, frankly, do I want to. Those who demand that, out of respect for rationality, we deny ourselves these feelings, might as well demand that we deny ourselves feelings of sexual desire, which, after all, are equally irrational. Of course, some have exhorted us to do just that, but they have been no more successful than I think Dawkins & co will be in their exhortations. Irrational or not, there are some aspects of our humanity that we cannot reasonably be expected to part with; and insisting, like some modern Pentheus, that we do so , seems to me unrealistic, and possibly, as Pentheus himself found, dangerous. 

But at this stage, the Dawkinsite may ask: “What about the truth? Does that count for nothing?” But, in keeping with their oft-stated dictum that everything must be challenged, I think it is worth challenging, at least as an intellectual exercise, the belief (for it is no more than that) that the importance of truth overrides everything else. Can we be sure of this? What if the truth makes us unhappy? What if the truth diminishes the richness of our lives? This is not to deny the importance of the truth, but if everything should be questioned, then let us, as Ibsen did in play after play, question also the nature of truth itself, and of the human capacity to apprehend it, and also its importance when balanced against other claims. And let us not assume that we are in full possession of the truth in the first place: such  an assumption is, to say the least, highly questionable. 

It is at this point I realise how far I am from being that sceptical, rational creature I like to imagine myself. I’m not. But it doesn’t follow that I must therefore be a credulous, superstitious fool, for, contrary to Dawkinsite doctrine, it’s not merely a case of adhering either to one pole or the other: there are many shades of grey in between, and these shades must be acknowledged. True, I’m not entirely sure on what shade of grey in this in-between I stand on: indeed, I am not even sure I stand at all – our lives are not static, and neither are our frames of mind, even from one moment to the next. It is this constant flux in the unfathomable depths of our minds that gives our lives whatever richness they have: in this sense, uncertainty is not merely to be rejected – it is to be welcomed. 

It is with all this in mind that I approach religious literature. For I know I am not alone in feeling as I do: all human societies, in all parts of the world and in all times, no matter how primitive or how sophisticated, have had a belief in what may be termed– once again, very vaguely, for we have no vocabulary to express these things precisely – “spiritual”, i.e. something other than the physical. This inclines me to suspect that our yearnings towards spirituality, towards transcendence, towards whatever vaguely defined term we care to use, are innate, and cannot, for all the Dawkinsite exhortations, be wished away. And neither, I think, should we wish to wish them away. 

Beyond this, I don’t know that I am prepared to speculate. If I were asked if I believe in a personal God, I’d answer I probably don’t – “probably”, because I don’t know what shade of grey I may be on at any given moment. But despite my agnosticism, possibly leaning towards (but no more than possibly leaning towards) atheism, I do often find in religious writing – or in religious music, or in religious art – an expression of something which I cannot adequately describe, but which I know is far from alien to my sensibilities. That towards which we yearn may well be illusory for all I know, but the yearning itself isn’t. So, when I find myself in a state of rapture when listening, say, to Beethoven’s Missa Solemnis; or when I find myself responding to the poetry of Gerard Manley Hopkins describing a world charged with the grandeur of God; or when I am moved by Rubens’ Descent From the Cross; I rather suspect I am affected because of, and not despite, the religious content of these works. 

So, onwards and upwards, as they say: Dante. Given its reputation, it’s a bit daunting, but there’s no point merely sticking in an apprehensive toe into the water: one just has to dive in, and hope for the best. And of course, one cannot read something like the Commedia without knowing the Bible. My own knowledge of the Bible has mostly been absorbed from secondary sources: until, say, a couple or so years ago, I had read very little of the Bible.  Oh, of course, I paid lip service to it: it’s a pillar of Western civilisation; our entire culture, the norms of the society we live in, all flow from the book; the King James version is among the great literary miracles of the world; and so on, and so forth. It was far easier paying this lip service than actually reading the thing. So, after I finished my reading of all the Shakespeare plays about a couple of years ago, I decided that I really had to get to know the Bible at first hand. Since then, I have been reading through it, book by book (it’s not a volume that should be rushed through) – for, of course, the Bible is not a single book, but an entire library. And soon, I shall be tackling the Book of Psalms, which, a friend of mine (who I hope is reading this, and from whom I’ve nicked without permission quite a bit of the contents of this post – I’m sure he knows what I’m talking about) once described to me as “food for the soul”.  Well, in for a penny, in for a pound, as they say – not only will I be diving into Dante, I plan also to dive into the Book of Psalms, with the aid only of the renowned Tyndale Old Testament commentaries by Derek Kidner. And, depending on how my reading progresses, and how much I can take in, I hope to be reporting on those one hundred and fifty psalms, one by one, on this blog. For whoever may be interested. 

No more boasting like a fool – this deed I’ll do before the purpose cool.

I’m famous!

Dear Reader,

This blog you are now reading is, I gather, ranked 24,389,427th in the world.

I feel so proud!

The finest decade: 1601-1610

Looking through the history of Western culture, has there ever, I wonder, been a decade quite as rich in achievement as the first ten years of the 17th century?

Shakespeare was, in this decade, at his very peak. Exact dating is difficult, but it appears that by 1601, the start of the decade, Shakespeare already had behind him Hamlet and Twelfth Night. What followed was every bit as remarkable – Othello, King Lear, Macbeth, Antony and Cleopatra. In comparison with works of such order, even plays such as Measure for Measure, Troilus and Cressida or Coriolanus – works that would be considered towering works of genius from any other writer – are often, in this context, regarded as “relatively minor”. By the time the decade ended, Shakespeare was engaged on his final trio of dramatic masterpieces – Cymbeline, The Winter’s Tale and The Tempest. And amongst all this was the publication in 1609 of one hundred and fifty four sonnets – works that ensured that even if Shakespeare had never written a single play, we’d still be thinking of him as our greatest writer.

And also in this decade, over in Spain, Cervantes published the first of the two parts that make up Don Quixote, and began work on the second. If, as is said – I can’t remember by whom: indeed, I think I just made it up – that the four principal pillars of Western literature are Homer, Dante, Shakespeare and Cervantes, it is surely remarkable that two of them were at their peak at the same time.

And let us not forget the King James Bible. It was published in 1611 – this year is the four hundredth anniversary of that momentous event – but much of the work on it would, I imagine, have been done in this decade. (And yes, I know, much of it is a rehash of Tyndale’s Bible, but still…)

In addition, Ben Jonson wrote in this decade some of his finest work – most notably, The Alchemist and Volpone. And towards the end of the decade, John Donne started work on his Holy Sonnets. (While his other poems are often hard to date, it is most likely that many of them were the products of these years.)

There was other fine writing as well – the Jacobean era was, after all, the most glorious period of English drama; and inSpain, there was the prolific Lope de Vega. It is an indication of the sheer stature of Shakespeare and Cervantes that literature even of this quality seems to be in their shadow.

Rubens: Descent from the Cross (Antwerp Cathedral)

Other areas of the arts were flourishing as well. The young Rubens was already well-established, and in the final years of this decade, was engaged on those two massive works now in Antwerp Cathedral – The Raising of the Cross, and the almost unbearably moving Descent from the Cross, in which the body of Christ seems to be falling and soaring at the same time. Meanwhile, inSpain, El Greco was putting on to canvas his very personal and idiosyncratic mystical visions. Equally idiosyncratic and personal were the late paintings of Caravaggio: having killed a man in 1606, Caravaggio spent the last few years of his brief life on the run, and found himself in Naples, Sicily, Malta, unable to settle anywhere for long. Before his untimely and mysterious death, his strange and violent death-haunted imagination put on canvas some of the most intensely tragic and powerful of visions.

Meanwhile, if we turn to music, Monteverdi composed Orfeo, among the earliest and greatest of all operas, and, in 1610, the monumental Vespers. In England,  William Byrd was composing at the height of his powers: his two books of Gradualia were published within these years.

Science was not left behind in this either. Just before the start of this decade, in 1610, William Gilbert published his astonishing discovery that the earth itself was a huge magnet; and the decade ended with Galileo observing with his new-fangled telescope the moons of Jupiter, and the phases of Venus. In between, Johannes Kepler formulated and published his ground-breaking Laws of Planetary Motion.

It seems unimaginable that all this could happen within the space of just a few years. Suddenly, the world seemed to cast off its old clothes, and enter a new phase. Was there something in the air? One tries to find connections between all these achievements, even though one knows that such connections are, at best, fanciful. Shakespeare and Jonson knew each other, and, one imagines, they would have known the other dramatists and poets in London at the time; but mostly, all these great men of genius were working independently. But fanciful or not, it’s hard not to see some of Shakespeare’s tragic spirit in those late paintings of Caravaggio. And, although the sun-baked plains of La Mancha are as far as can be imagined from the deep claustrophobic gloom of Glamis Castle, one can imagine Don Quixote agreeing with Macbeth that “Nothing is but what is not”.

At much the same time as Shakespeare was writing King Lear, El Greco was

El Greco: View of Toledo (Metropolitan Museum New York)

painting Toledo during an apocalyptic storm. (Dating of this painting varies: some date it to the late 1590s rather than to the 1600s, but let’s not allow such little details to get in the way!) Storms of such elemental violence could, no doubt, occur inEngland as well: every time I read those storm scenes of Shakespeare’s play, I find it difficult to banish from my mind El Greco’s tempestuous vision.

And then I look at Carvaggio’s painting of the boy David holding Goliath’s head. The living David seems rather bland – a smug, self-satisfied lad, lacking much in the way of character. All the character is in Goliath’s head, which, though severed, seems

Caravaggio: David with the head of Goliath (Galleria Borghese, Rome)

still a living, sentient being. This decapitated head is Caravaggio’s self-portrait: here, he pictures himself conquered by a mere boy. (“To the boy Caesar send this grizzled head…”) And, though defeated, though decapitated, I can almost hear Caravaggio agreeing with Shakespeare’s Cleopatra, that “’tis paltry to be Caesar”.

But of course, this is all mere fancy. The air of that decade was nothing special: it wasn’t so different from the air we breathe now. But when I think back on what was achieved in those few years, and I cannot help but be struck with wonder. It’s almost as if all the finest aspects of human genius were concentrated into that brief space of time.

Is general knowledge trivial?

Back in the 1980s, a board game called “Trivial Pursuit” became all the rage. The trivia that was pursued in that game was general knowledge. What is the capital of Colombia? Which is the longest running soap on British television? Who sculpted the Medici Tombs in Florence? What is measured in units of Amperes? Which team won the first World Cup in football? Who was the Greek goddess of love and beauty? And so on. The sciences and the arts, history and geography, sports and entertainment, the erudite and the populist – they were all indiscriminately rolled into one long series of questions and answers, into questions that had answers, into questions that could easily be asked and answers that could easily be expressed. It was all great fun. It still is. But nonetheless, it is, the creators of the game claimed, trivial. Sports or sciences, rivers or mountains, soaps or Shakespeare – it’s all ultimately trivia: none of it really matters

Of course, this is correct up to a point. When the entire spectrum of human knowledge is reduced to a series of simple questions and answers, what can it be other than trivial? We are told of those who excel in this sort of thing because they have spent a long time mindlessly memorising an awful lot of facts by rote – the dates of British kings and queens, the lengths of the world’s great rivers, the capitals of the American states, and so on. Such people are often derided as “anoraks” – usually by those who would themselves take great offence if anyone were to look down on their harmless pleasures. But be that as it may, we may concede, I think, that memorising vast amounts of facts merely to be able to reel them off in pub quizzes is trivial, no matter how enjoyable it may be for those who do it. But while reducing general knowledge to a series of questions and answers may quite rightly be considered trivial, I remain unconvinced that general knowledge itself is trivial. Far from it. For, leaving aside those who simply memorize facts merely for the sake of it, possession of a good general knowledge is generally indicative of an intellectual curiosity, a desire to know about the various aspects of this fascinating world that we inhabit. And, conversely, a lack of general knowledge indicates the opposite – a lack of intellectual curiosity, an indifference. And neither of these polar opposites is, it seems to me, trivial

Of course, as ever, there’s too much for any one person to know. My own general knowledge is not, I think, too bad, but what I may happen to know is of minuscule proportions when set against all that I really should know, but don’t. My academic background is science, but ask me about … oh, I don’t know … ask me about, say, the structure of Planet Earth, and my ignorance will be apparent within seconds. I know there’s this thing called the core – which is, unsurprisingly, at the centre; and the thin, upper layer at the top is called the crust; and that, in between, there’s the mantle; and then there are these things called  … what are they called again? Ah yes … these things called tectonic plates, which move around a bit and cause earthquakes; and … er … that’s about it, really. No, really – that’s about it: that’s as far as my knowledge extends on this matter. Now, I am by no means proud of my ignorance in these and in other subjects: quite the contrary. But if I were to be asked a question on this subject in a general knowledge quiz, I would be likely not to know the answer. But if someone were to be able to answer questions on this subject – perhaps, even, quite difficult questions on this subject – then that ability to answer such questions would indicate to me not mere mechanical memorising of facts, but a knowledge of these important matters born of an intellectual curiosity. And it is to do all of us an injustice to describe this knowledge, or this intellectual curiosity, as merely trivial. It seems to me, on the contrary, something to be admired. 

And yet, we seem to do everything we can to kill off any intellectual curiosity in our children. Small children are full of curiosity: What’s this? What’s that? What does this do? Why does it do that? How does it do that? And yet, far from encouraging this, far from feeding their curiosity, we seem to do our best to kill it off. There is a wonderful scene near the start of Terry Gilliam’s 1981 film Time Bandits, in which a boy is looking through a book, and finds himself fascinated by various things he is discovering about Ancient Sparta. But when he tries to communicate his enthusiasm to his parents, all he gets are indifferent grunts: his parents are busy – well, if not busy, at least occupied – sitting passively in front of the television set, watching some vapid game show. Sure, this scene is an exaggeration for the purposes of satire, but it doesn’t seem to me too great an exaggeration: in the thirty years or so since that film was released, with all our wonderful advancements in digital technology offering even more channels and even more means of communicating vapidity, we have become ever more efficient in killing off whatever spark children might once have had, and making sure that they end up much like ourselves at an early an age as is possible. 

But alas, as with all human endeavour, perfection is not something we can ever achieve: despite all the disposal at our means, a few do manage to escape childhood with their intellectual curiosity still more or less intact, and who are capable of displaying a breadth, and sometimes even a depth, of knowledge on all sorts of things – even on such matters as movements of tectonic plates. Rather than concede our failure with these people, we find it easier to brand them “anoraks”, to claim that their knowledge is merely superficial and acquired from memorising facts by rote; we find it easier, indeed, to dismiss their knowledge itself as merely “trivial”. And we do: it makes us all feel better.

And in the meantime, we can watch on television people displaying without the slightest hint of embarrassment quite prodigious ignorance on just about everything, safe in the assurance that viewers are more likely to sympathise rather than otherwise with their ignorance. Of course, one cannot know everything, and what seems basic to one may not seem basic to someone else. For me, say, with my obsession about Shakespeare, knowing which play by Shakespeare features a pound of flesh may seem pretty basic; but to someone else, something about which I am totally ignorant may appear equally basic, and in their eyes, I would be but an ignorant oaf for not knowing about it. Fair enough – although I do strongly believe that knowledge of Shakespeare should not be regarded as esoteric, and that any educated English-speaking person should possess at least a cursory knowldge of the major Shakespeare plays. But leaving Shakespeare aside for the moment, when people display ignorance on basic matters across the board – not merely on literature, but also on science, on history, on geography, on current affairs, you name it – then, after a while, it all adds up, and a picture begins to emerge that I dearly hope is not representative of humanity in general. Surely so many of us can’t find ourselves living in so endlessly fascinating a world, and take so little interest in any aspect of it?

Of course, one has to be careful in saying this. To think that the great treasure-house of culture and of knowledge that we have accumulated across the centuries should belong to everyone, and should not be the preserve merely of an elite, is, as everyone knows, an elitist position to take. Far better denigrate the concept of knowledge itself as essentially pointless – as trivial. That way, we all end up feeling better.