Demons and Democracy: the Strange Color of the Nearby

(If you are reading along: Orthodoxy Chapter IV "The Ethics of Elfland" first five paragraphs CW1:249-252)

I told you last week there would be no demonism on our adventure into Elfland... and there won't be. Except in one punning way, which is one of the really beloved lines spoken by Father Brown. You will perhaps disagree. I think it reveals his humility, and the too-easily forgotten fact that priests must themselves confess their sins:

"I am a man," answered Father Brown gravely; "and therefore have all devils in my heart."
[GKC, "The Hammer Of God" in The Innocence of Father Brown]

One of the prayers the priest says at Holy Mass hints at this authentic view of the priesthood:

"Accept O Holy Father, Almighty and Eternal God, this spotless host, which I, your unworthy servant, offer to You, my living and true God, to atone for my numberless sins, offenses, and negligences; on behalf of all here present and likewise for all faithful Christians living and dead, that it may profit me and them as a means of salvation to life everlasting. Amen.

But we are not here to examine liturgy... but rather demon... ahem, excuse me. I mean democracy.

No, not as in the opposite of "Republican"! (And I thought I was skating on thin ice with the reference to magic. Hee hee.)

But - as I warned you last time, you must pay very close attention to the words here. There will be some words you know, and perhaps have some strong feelings on - like "democracy" and "liberal" and "tradition". Most importantly you need to know that these terms are being used in their full, classical, "rich" or perhaps "proper" sense (in heraldry, something "proper" means it is shown in its true colors), and emphatically not as political terms.

You are puzzled. We have crossed a bridge, into a world which I have rather carefully avoided naming (though you are all writing "May 22 - entering Elfland" down in your hiker log books). You thought we would be meeting strange, remarkable, unexpected, surprising creatures - hobbits and elves or dwarves (dwarfs, if you are not a Tolkien person) or maybe Milo or Bastian or that Poppins woman - or at least Spock or E.T. or even Mr. Potter (it is a wonderful life, you know?) And instead I bring up GKC's drudgery of what sounds like government or maybe education. Oh, how surprised you'll be... we shall indeed meet some remarkable creatures. You must be bold... and you will be surprised!
Wave wand here if you feel bold.

Now, having mentioned Milo, the bored hero of The Phantom Tollbooth, you will be surprised, as I was, in what we find in the very first sentence from our chapter:

When the business man rebukes the idealism of his office-boy, it is commonly in some such speech as this: "Ah, yes, when one is young, one has these ideals in the abstract and these castles in the air; but in middle age they all break up like clouds, and one comes down to a belief in practical politics, to using the machinery one has and getting on with the world as it is."
[CW1:249, emphasis added]

If you've never read that book, you must know that "The Castle in the Air" is the prison to which the princesses of Sweet Rhyme and Pure Reason were banished, and the demons guard the path to it... Ah, so you are starting to feel something? Good, good. It's starting, then.

The start is a bit slow. As usual, GKC uses a rather complex analogy, drawn from his own late Victorian youth, and the world he knows, to try to explain something even more complex. It is a kind of parable, laden as usual with verbal fireworks, and a confusing term or two, but it contains something striking, which is like chrome. For "the strong chromium" as a friend of mine says, "has the strange color of the nearby". Yes, chrome and so many metals act as mirrors when polished. And this is the first kind of magic GKC reveals to us.

We must understand what GKC is doing. He now has "to trace the roots of my personal speculation" - that is, explain how he started to get to HIS view of things, which is so different from what we saw in the previous chapters. He believes in "Liberalism" - but not in "Liberals". Note! I have no time to give you the grand explanation of what these words meant in England of 1908 - but that precision hardly matters to us. You will see it clearly, as in a mirror (!) very shortly, as GKC proceeds. But... if you find this too confusing, read these lines:

They said that I should lose my ideals and begin to believe in the methods of practical politicians. Now, I have not lost my ideals in the least; my faith in fundamentals is exactly what it always was. What I have lost is my old childlike faith in practical politics. ...

I was brought up a Liberal, and have always believed in democracy, in the elementary liberal doctrine of a self-governing humanity. If any one finds the phrase vague or threadbare, I can only pause for a moment to explain that the principle of democracy, as I mean it, can be stated in two propositions. The first is this: that the things common to all men are more important than the things peculiar to any men. Ordinary things are more valuable than extraordinary things; nay, they are more extraordinary. Man is something more awful than men; something more strange. The sense of the miracle of humanity itself should be always more vivid to us than any marvels of power, intellect, art, or civilization. The mere man on two legs, as such, should be felt as something more heartbreaking than any music and more startling than any caricature.
[CW1:249-250]

(Yes, there's a ref. to Manalive where a telegram reads "Man Found Alive With Two Legs" - what you haven't read it yet? Get busy.)

Verbal fireworks? Or words on fire doing work? You will hunt very hard to find a real politician, in either England of 1908 or America of 2008, who acts on these terms, who sees the miracle of humanity.

But (you say) I thought you said it wasn't going to be about politics?

GKC dragged it in, not me. But proceed, and be surprised:

This is the first principle of democracy: that the essential things in men are the things they hold in common, not the things they hold separately. And the second principle is merely this: that the political instinct or desire is one of these things which they hold in common. Falling in love is more poetical than dropping into poetry. The democratic contention is that government (helping to rule the tribe) is a thing like falling in love, and not a thing like dropping into poetry. It is not something analogous to playing the church organ, painting on vellum, discovering the North Pole (that insidious habit), looping the loop, being Astronomer Royal, and so on. For these things we do not wish a man to do at all unless he does them well. It is, on the contrary, a thing analogous to writing one's own love-letters or blowing one's own nose. These things we want a man to do for himself, even if he does them badly. I am not here arguing the truth of any of these conceptions; I know that some moderns are asking to have their wives chosen by scientists, and they may soon be asking, for all I know, to have their noses blown by nurses. I merely say that mankind does recognize these universal human functions, and that democracy classes government among them. In short, the democratic faith is this: that the most terribly important things must be left to ordinary men themselves - the mating of the sexes, the rearing of the young, the laws of the state. This is democracy; and in this I have always believed.
[CW1:250]

Now, the well-read Chestertonian will immediately hear the echo of a very famous quote: "if a thing is worth doing, it is worth doing badly." [GKC, speaking of Woman in What's Wrong With the World CW4:199] And, for completeness, this idea of "blowing your own nose" also appears in Heretics, and I leave it for anyone who wants a nice workout side hike; you can find it in CW1:203.

Again, you are wondering: where is the magic? Where are the strange beings?

YOU ARE SEEING THEM. Elvish zoologists call them "humans". (You'll find out about the magic soon enough.) GKC supposes that there is some fundamental law which underlies all sorts of things - which he sums up in the idea that we ought to do certain things for ourselves, as much as we possibly can. (In another context, this idea is a part of the design method called "Subsidiarity" - an ancient idea, and part of Catholic Social Teaching, and about which I have written elsewhere.) But as a broadly used term in government, not delving into the depths or distinctions of some governing mechanism, "democracy" means "rule by the people" - the people choose the arrangements. Like Athens of long ago... Ah, so nice. The splendid happy life of ancient Greece and all that. Houses, Senates, meetings, elections... Ahem. But GKC does not sit and bask in the Mediterranean warmth. He reveals some real depths to this supposedly well-known term, and gives us some singularly rich insights:

But there is one thing that I have never from my youth up been able to understand. I have never been able to understand where people got the idea that democracy was in some way opposed to tradition. It is obvious that tradition is only democracy extended through time. It is trusting to a consensus of common human voices rather than to some isolated or arbitrary record. The man who quotes some German historian against the tradition of the Catholic Church, for instance, is strictly appealing to aristocracy. He is appealing to the superiority of one expert against the awful authority of a mob. It is quite easy to see why a legend is treated, and ought to be treated, more respectfully than a book of history. The legend is generally made by the majority of people in the village, who are sane. The book is generally written by the one man in the village who is mad. Those who urge against tradition that men in the past were ignorant may go and urge it at the Carlton Club, along with the statement that voters in the slums are ignorant. It will not do for us. If we attach great importance to the opinion of ordinary men in great unanimity when we are dealing with daily matters, there is no reason why we should disregard it when we are dealing with history or fable. Tradition may be defined as an extension of the franchise. Tradition means giving votes to the most obscure of all classes, our ancestors. It is the democracy of the dead. Tradition refuses to submit to the small and arrogant oligarchy of those who merely happen to be walking about. All democrats object to men being disqualified by the accident of birth; tradition objects to their being disqualified by the accident of death. Democracy tells us not to neglect a good man's opinion, even if he is our groom; tradition asks us not to neglect a good man's opinion, even if he is our father. I, at any rate, cannot separate the two ideas of democracy and tradition; it seems evident to me that they are the same idea. We will have the dead at our councils. The ancient Greeks voted by stones; these shall vote by tombstones. It is all quite regular and official, for most tombstones, like most ballot papers, are marked with a cross.
[CW1:250-251]

If you don't already know it, "tradition" comes from a Latin verb, trado which means "I hand over, hand on, pass on" - and "I entrust". So many things you have, and think of as your own, have actually been handed on to you, often with great expense and at great effort on the part of others who were holding them in trust... you are so used to them you have forgotten where they came from - and (alas) you are most often utterly unaware of the power, the intense and extraordinary thing (for which "magic" might be the only good term) that you are capable of wielding with them.

What, for example? Do you really need me to tell you? OK: by what power are you reading this? (I don't mean the computer; I mean your ability to read, and to understand what I have written.) Your language is a tradition. No revolutionary, no "liberal" - in our modern sense - can escape it. Every science, every field of study presupposes language as a basis for its work; in that sense, even the hardest of the hard sciences is just another Liberal Art.

Ah. But consider this sentence again: "Tradition is the democracy of the dead." Are you starting to feel something powerfully magical here? Not just the idea of fairy tale, of some fanciful story told in the nursery. No; the idea that something - ah, like a magic wand - something almost unimaginably powerful was given to you long ago - given as your very own inheritance - and you've always had it. But you never really paid any attention to it, never thought of it as important: that odd wooden stick you've carried, dangling from your belt, or the sparks that flew out when your hand is near it... You don't like that hint of the "m" word? Then think of a pencil in your pocket, and its graphite smears on your hand. (Milo is given such a wand by the Mathemagician.) Not personal enough? Then how about speech? You do not even understand the power of your tongue? Why are so many powerful things attributed to speech? (Read the Epistle of St. James for details!) You have indeed inherited rich, oh, so rich, gifts. You abandon or neglect them at your peril.

One more paragraph brings us to the end of today's leg of our journey, and it is a very elegant summary of today's topic:

I have first to say, therefore, that if I have had a bias, it was always a bias in favour of democracy, and therefore of tradition. Before we come to any theoretic or logical beginnings I am content to allow for that personal equation; I have always been more inclined to believe the ruck of hard-working people than to believe that special and troublesome literary class to which I belong. I prefer even the fancies and prejudices of the people who see life from the inside to the clearest demonstrations of the people who see life from the outside. I would always trust the old wives' fables against the old maids' facts. As long as wit is mother wit it can be as wild as it pleases.
[CW1:251-2]

That word "ruck" is not a typo. (Yes, I had to look it up; I like to look things up, it's fun, you can find so many other things while you are looking!) "Ruck" means "the undistinguished multitude, the crowd of ordinary persons or things". It is one of GKC's favourite topics - you can find all kinds of references throughout GKC's writing to "The Common Man" - there's even a book called that.

GKC wants us, just to begin with, to see ourselves in a mirror (the magic thing with the strange color of the nearby!) It is magic, not because of the mirror, but because of us. We are unusual, we are marvels, we are extraordinary - because we are simply ordinary. And we have not even ventured into preternature (the land of fairy and magic), much less supernature (the Land of the Living, where He dwells in Whose image we are made.)

So - the next time you see a mirror, stop and behold the image and the likeness of God. A broken, sad, weak, confused, often whiney, nasty, mean and rude image, perhaps insanely preferring a demon as a model - but still His image: "For religion all men are equal, as all pennies are equal, because the only value in any of them is that they bear the image of the King." [GKC, Charles Dickens CW15:44] Nothing in the kosmos, except the Most Blessed Sacrament, is as holy as a human being. (Today, for much of the world, is the feast of Corpus Christi, which is transferred to this coming Sunday in the U.S.) Father Brown hints at this great credal mystery:

"It's the first effect of not believing in God that you lose your common sense, and can't see things as they are. Anything that anybody talks about, and says there's a good deal in it, extends itself indefinitely like a vista in a nightmare. ... all because you are frightened of four words: 'He was made Man.' "
[GKC, "The Oracle of the Dog" in The Incredulity of Father Brown]

Yes - one of the effects of reading GKC is we start seeing things as they are. Even in mirrors.

--Dr. Thursday

PS After writing this - and I did it AFTER lunch this time: I find that I have not been of much help at all on this leg. I shall ask you to read GKC's five paragraphs by yourself, and think about them. They are worth it. You will find yourself in awe of such company as we have, both living, and traditional.

Thursday's Dr. Thursday Post: Infinity

It may be a stretch of the imagination to connect last Sunday's gospel (the woman at the well) with our discussion of last Thursday - or perhaps not. The woman's "madness" was shattered - as if a spell was broken - by the Voice of Authority who told her "Go get your husband". So deep was her restoration that she was able to bring others to that same fountain... Ah. But for today I shall resist plunging into the deep waters this imagery brings up.

In thinking of insanity, and Lent, I must bring to your attention one of the most unusual and perhaps most insightful views of a gospel event I have ever read. The event is the "Good Thief" hanging in crucifixion next to Jesus - an apologist defending Christ even on Calvary! "We are but suffering as we deserve - but This One has done nothing wrong... Lord, remember me when You come into Your Kingdom."

The insightful view is not mine. It is contained in the rich notes and the amazing play-sequence, "The Man Born To Be King" of Dorothy L. Sayers (DLS), a series of radio plays she wrote on the life of Christ. I don't have the text here to transcribe, so I shall merely give you a hint of her argument. She claims that the Good Thief perhaps took Jesus to be a harmless nut-case - a crazy man - YET - the thief still treats Him kindly, and "plays along" - only to receive a most unexpected reply. The scene DLS only hints at is the one I love to ponder: for behold, later that day, the Lord would tell the thief, "Nope, I wasn't nuts, but it was kind of you to think so. The charity you showed to the harmless lunatic You showed unto Me!" A strange, yet somehow most dramatic view. Read it for yourself.

I had previously thought I would write up a "proof" about GKC's interesting mathematical bit about the circles, but there will be more of this philosophical geometry before you know it, and I don't feel like making such a long detour today. So let us proceed. We have finished GKC's comments on lunacy and madness - which he expresses using the mystery of the circles: infinite in one sense (for it has no end) yet still not so very large (for it is no bigger than it is drawn). We have seen an omnibus labelled "Hanwell" and thought about those unfortunates who believe themselves to be chickens, or glass, or Kings of England, or Jesus. We have heard of the limits of literature, the risks of reason - and been challenged to cut off our own head if it offends us. What is all this? Why are we seriously contemplating insanity? GKC has a reason, and not merely a poetic one.

Click here to continue the adventure.

GKC tells us himself what he is up to:

I have described at length my vision of the maniac for this reason: that just as I am affected by the maniac, so I am affected by most modern thinkers. That unmistakable mood or note that I hear from Hanwell, I hear also from half the chairs of science and seats of learning to-day; and most of the mad doctors are mad doctors in more senses than one. They all have exactly that combination we have noted: the combination of an expansive and exhaustive reason with a contracted common sense. They are universal only in the sense that they take one thin explanation and carry it very far. [CW1:225]

We might take this as the bridge-passage, the musical riff that brings us from Heretics to Orthodoxy. Recall that in Heretics we saw a long line of men - writers, thinkers, philosophers - men whom GKC respects, even admires - some of whom he would readily claim as friends - and yet men with whom he is in bitter and utter disagreement: "a Heretic - that is to say, a man whose philosophy is quite solid, quite coherent, and quite wrong." [Heretics CW1:46]

Those men are the men LIKE the lunatics. Note he does NOT say they ARE lunatics! He is not pulling an ad hominem argument. He is talking about a general idea, dealing with the IDEAS of those men. What does he tell us about them? He says those are the men with the SMALL PATTERNS, even though they are "infinite":

But a pattern can stretch for ever and still be a small pattern. They see a chess-board white on black, and if the universe is paved with it, it is still white on black. Like the lunatic, they cannot alter their standpoint; they cannot make a mental effort and suddenly see it black on white.[CW1:225]

He proceeds to give an example (about materialism) but almost immediately points out that he is NOT making an argument about the detail, but about the generality. He links the flaw in the materialist view of the kosmos back to the flaw in the man in the asylum. It may be true enough. But it is so much smaller a truth than can be found elsewhere.

I hope you are reading along with me - and so you will readily note that it is futile for me to try to skip the example. GKC himself tried to do that. In one of his amazing leaps, he goes from that example to a stark generality of epistemology (the study of knowledge itself): "In one sense, of course, all intelligent ideas are narrow. They cannot be broader than themselves." [CW1:226] It is the paradox of words, the strangeness of a homework assignment like "Define 'infinity' and use it in a sentence." It hints at another mysterious line of GKC's which he put in another mystery: "Alone on earth, the Church affirms that God himself is bound by reason." ["The Blue Cross" in The Innocence of Father Brown]

Whew, let's stop for a bit. Do you feel stuck in a swamp of ideas? You are wrong. It's the brisk fresh air. You are at a peak of a mountain, and seeing a vista. It's at these points where you feel most congested, you are actually most free, and actually presented with a greater wideness of vision than elsewhere. So let us pick this matter apart so we can grasp where we were and better handle where we're going next. I can't do all the epistemology, I didn't bring that in my knapsack today. Let's see if we can deal with it directly. Let's read it again:

"In one sense, of course, all intelligent ideas are narrow. They cannot be broader than themselves."

The point of the paradox is we can handle things far bigger than our hands - because we have words which can reduce infinity to eight letters. (Count them: I, N, F, I, N, I, T, Y.) The strict philosophers will now throw eggs at me, saying I have committed the "Fallacy of Equivocation", confusing the word "infinite" and the idea "infinite". But I catch the eggs, and scramble them to make our lunch. They are not reading along. (Recall "poetry floats on the infinite sea"...) It would be just as adequate for me to cite the Summa of Aquinas (I Q10 A1) to help them out, since they like that kind of citation, it shows I do read those kinds of things. Ahem! But for us, this "paradox" is as simple as this mountain-peak. We're stopped here - and need to choose a path. But we can choose ANY direction - as long as it's down. (We are walking, you know; remember we said last week, "let it be solved by walking".)

The "fallacy of equivocation" is a kind of error in logic, in the use of words. How about an example? Here's one: saying "God is limited because he is only three letters long". But GKC is telling us there is the same kind of error in saying "we cannot hold the idea of 'infinity' since it is INFINITELY BIG". It is a paradox in reason itself, not merely written by GKC, to state without further quibble, that Infinity is narrow, and God is limited. This is not because of the things-in-themselves, but because of our equipment. (We are on a journey, we are NOT going EVERYWHERE AT ONCE. We are walking, and so are SOMEWHERE.)

I will try once more. (This one is great, and will shock any computer scientists in the audience.) Watch carefully, and I will use YOUR computer to represent BIG integers, including for example, the number of electrons required to fill the sphere bounded by the diameter of the most distant galaxies. Or, even bigger: the factorial of that number. Or even bigger: that number raised to its own power... that many times. Big numbers. BIG big numbers. HUGE numbers. (Even more than GKC weighed.) I can even use the computer to deal with transfinite numbers, the mysterious "aleph-one", which is the cardinality of the real system of numbers. And there are even others... Wow, look: before your very eyes, all those things are being communicated by what I have just written! HAVE I NOT COMMUNICATED THEM TO YOU? Of course I have. They are formally represented - as ideas. No, not directly as tick-marks on a sheet of paper or tokens in a box. (Please. Don't be silly. When was the last time you saw 1000 of ANYTHING? We gave that up about 5000 years ago, when the Egyptians began to write Ç to stand for "ten".) You cannot represent such gigantic numbers by that means. It is like asking how much God weighs. It does not have meaning. But you can communicate the idea of such numbers - which means you have communicated the number. The idea of such vast quantities has a meaning, and so we can accomplish the communication of that idea. And if we failed to say it in symbols of mathematics, we would resort to the symbols of poetry: I think of "Tonight" in "West Side Story":

Today the minutes seem like hours,
The hours go so slowly...
And still the sky is light...

That is what GKC is saying. In order to talk about anything, and reason about anything, we use something narrow. We do not have the infinite time or an infinite box of tokens to play around with the real thing, so we use what we can.

OK. Maybe it was futile - I ought to stick to my own toys - so let's resume with GKC, and I will let the high-tech philosophy for others to play with. Perhaps this next sentence will tell you the same thing, which is just GKC's own version of the very important Principle of Contradiction: "Nothing can be, and not be, at the same time."

A Christian is only restricted in the same sense that an atheist is restricted. He cannot think Christianity false and continue to be a Christian; and the atheist cannot think atheism false and continue to be an atheist.
[CW1:226]

Again, please read this carefully. You need to think about the simple sentences, not about some deep quippy insult or brag. You can put in any partisan or sectarian words you like, and it has JUST the SAME meaning and power. YOU ARE ON A PEAK of FREEDOM, my friend, not stuck in a swamp! Try it again. Then we'll proceed.

Now that you have a NEW tool, then, we shall actually approach this example of materialism - and its opponent, spiritualism. (We are using the terms rather generically here; materialism means there is nothing but material: nothing spiritual at all. Whereas spiritualism means there also exists an unseen realm.)

I shall quote at length again, because you are surely tired of reading MY words, and also because the "verbal fireworks" here are SO good:

...there is a very special sense in which materialism has more restrictions than spiritualism. Mr. McCabe thinks me a slave because I am not allowed to believe in determinism. I think Mr. McCabe a slave because he is not allowed to believe in fairies. But if we examine the two vetoes we shall see that his is really much more of a pure veto than mine. The Christian is quite free to believe that there is a considerable amount of settled order and inevitable development in the universe. But the materialist is not allowed to admit into his spotless machine the slightest speck of spiritualism or miracle. Poor Mr. McCabe is not allowed to retain even the tiniest imp, though it might be hiding in a pimpernel. The Christian admits that the universe is manifold and even miscellaneous, just as a sane man knows that he is complex. The sane man knows that he has a touch of the beast, a touch of the devil, a touch of the saint, a touch of the citizen. Nay, the really sane man knows that he has a touch of the madman. But the materialist's world is quite simple and solid, just as the madman is quite sure he is sane. The materialist is sure that history has been simply and solely a chain of causation, just as the interesting person before mentioned is quite sure that he is simply and solely a chicken. Materialists and madmen never have doubts.
[CW1:226-7]

(An aside: if you are wondering who "McCabe" is, you can read his chapter in Heretics CW1:157 et seq. Joseph McCabe (1867-1955) was a Roman Catholic priest turned rationalist.)

And while I greatly doubt that you can possibly be satisfied with my writing, here I must leave you for today. Please try to think a little about these things. Not about the math, or about the epistemology, the knowledge OF meanings of words and ideas - but ABOUT the meanings, and the ideas.

Still lost? When we think about our mother (let us say) we do not think about her picture, but about HER. But when we talk about her, we may show the picture, or use that six-letter word - but everyone knows who it is we are talking about, even if they have never met her or seen her. IN THE SAME WAY: when we think about infinity, we do not think about that splendid and funny little proof of the math dudes about "increasing without bound" or about a bottomless box of tokens - nor simply about that eight-letter word - but we use that word to talk to others about, as I have just done with you.

And this limited limitlessness applies even to the matter of God, which we do not narrow to a mere word of three letters, and Who has even more meaning and even more intimacy to us than our very mothers...

Onward to the next the peak, dudes!

--Dr. Thursday

A GKC Debate about Math

Back in October of 2006, our Dr. Thursday elicited some interesting discussion when he brought up Math, GKC and an Ignatian Asylum, after which a spirited debate ensued.

"Wild Goose", citing the One-Should-Always-Have-A-Healthy-
Skepticism-For-An-Encyclopedia-Where-Anyone-Can-Claim-Expertise
Wikipedia, stated:

It is not quite true that “When somebody [Newton] discovered the Differential Calculus there was only one Differential Calculus he could discover.”

Leibniz also discovered Differential Calculus, in a different form, arguably, a more durable and suitable calculus:

“The infinitesimal calculus can be expressed either in the notation of fluxions or in that of differentials. “

To which Dr. Thursday responded:

I should not have added [Newton] to GKC's words. It was in the 1660s (or so) that both Newton and Leibniz "discovered" calculus; yes, almost simultaneously, though Newton seems to have priority.

Alas, "a quarrel arose between the followers of Newton and the followers of Leibniz, and unhappily it grew into a quarrel between the great men themselves..." [The World of Mathematics 143, 286 et seq; Purcell, Calculus 156, 278]

And, if you're still following this (for full arguments, please see the comments section of the above referenced posting), "Wild Goose" continued:

You have brought up an excellent topic. The way I see it, the main point of your post was that math and science have their own fixed rules, while poetry has its own, due to the free will of the author or the creator. (“But when Shakespeare killed Romeo he might have married him to Juliet's old nurse if he had felt inclined.”) You are saying that there is only one calculus, while there may be a virtually infinite number of plays or plots along the lines of Romeo and Juliet, limited only by the author’s imagination. But I think that would be like comparing apples and oranges.

After which "DavyMax", a new commenter as far as one can tell that sort of thing, just today responded:

To simplify things a bit. Godel proved the essential intuitive nature of mathematics. But Wild Goose, you seem to be implying that intuition and imagination are one in the same. They are in fact quite different. Without seeing a proof I may intuitively think that there are infinitely many primes or that the Reimann hypothesis is true or the Axiom of Choice. However, this is because I would think, for instance, as I do, that the Axiom of Choice is in fact the truth. Clearly, this is quite different from writing a different ending of Romeo and Juliet (or preferably some other story I rather like the ending of Romeo and Juliet). Mathematicians do not intuitively think something because that's the way it sounds nice or because they think it's "cool." It's because they think it is TRUE. That is the ultimate goal. They may and often do choose to explore an idea because it is pretty or beautiful, but not because of those goals in mind but because they know from experience that the truth most often is pretty and beautiful.

This reminds me also of the line from V for Vendetta that says something of Artists telling the truth with lies. This is true in the sense that they are trying to express some inner truth through any means neccesary and we all understand what they're doing. Mathematicians are seeking rather than expressing truth when using intuition. Whereas, artists are expressing a truth already experienced when using imagination.

I thought I'd bring the whole thing to the fore because with a post that old, it's hard for people to jump back into the conversation. But I wanted to thank DavyMax for finding us and joining in the conversation by attempting to revisited this topic, if others are interested. Anyone want to respond to DavyMax? I thought he brought up an excellent point about Truth, and the difference between intuition and imagination.

Dr. Thursday's Thursday Post

Life in Three Dimensions: Part II - The Doctor and the Dolls (or) Pronouns, Beer, Pyramids, and the "abbr."

Whatever was the man who built the pyramids, one feels that he must (to put it mildly) have been a clever man.
[GKC, William Blake 101]

I shall now resume my discussion of dimensions, but because you are almost certainly a "lit'ry" person I am going to have to explain the next step with quite a bit of care. Do not worry! Do not fear! It is not very complex, but it is very philosophical, and you will have some fun on the journey.

Our next step in our study is the very useful and fundamental trick of mathematics and computing called recursion. Recursion is nothing more than a very fancy use of the idea of a pronoun, but a sophisticated, nay, a magical and powerful pronoun - one which has lost its fear of crossing over into the realms of numbers, sets, functions and functionals, and all the complexities of (drum-roll) THE LEARNING which we call "Mathematics".

But perhaps you would rather find out about why computer scientists play with dolls. No, not the ones that say "Math is hard"! These kind.
You may have seen this beautiful Russian doll - it stars in a wonderful children's book called The Doll in the Window by Pamela Bianco. As you may know, this is made of wood, and there is a seam around the middle. If you twist carefully, it opens, and inside, you will find
another doll.

Similarly, the second doll has a seam, within which you will find...

a third doll. Not to prolong the tension, here is the whole family:

Now, these are my own - yes, now you can tell your friends that there really is a computer scientist who plays with dolls! But hang on for the punch line. It's not what you will expect. When I taught at the Unnamed School, they had a set in the department office, and once I sent a student there to borrow them to explain the SAME thing I am about to explain to you.

You see, there are several interesting things about these nesting dolls. First, the overall idea of a thing and another thing related to it, the same but somehow smaller - something defined in terms of itself - in the abstract, we call this recursion, and that is the topic for today. Second, there is something curious one may discover somewhere about the fifth or sixth doll - will the next one have a seam? Will there be yet another doll inside? Does it ever stop? And what would happen if it didn't?

You think this is NOT Chestertonian? Not only is it in a Chesterton book, but it is used in a rather horrifying lesson in theology by Father Brown: "there is this about such evil, that it opens door after door in hell, and always into smaller and smaller chambers." ["The Sign of the Broken Sword" in The Innocence of Father Brown] Hopefully, we won't go in the opposite direction today, which is the very same thing: "One went into larger and larger windowless rooms, rooms big with Babylonian perspective; but one never found the smallest window or a whisper of outer air." [GKC, Orthodoxy CW1:266] Indeed, I have a point in all this, if you are patient: "...for me all good things come to a point, swords for instance." [ibid.]

Some weeks back, a posting, or a comment, began to explore this line from Orthodoxy: "Poetry is sane because it floats easily in an infinite sea; reason seeks to cross the infinite sea, and so make it finite." [CW1:220] We are going to see two things here - one of them quite finite, like my eight dolls; the other is infinite, and yet limited, so we need not cross any infinite seas today. We'll do that next week, hee hee.

So! If you wish to find out more about dolls, beer, and pyramids, press here.
The doll, of course, is just an attractive starting point. I am not proposing to go into "recursion" as we do in computer science, but only to give you a rough, poetic idea about it, in order that I can show you something curious, and perhaps "artistic". Well - some may call it art, others just a pattern, but in order to talk about it, we need to grasp the idea behind (not inside) the dolls. So, let us begin with a pronoun.

The Apostles' Creed begins with the pronoun "I"; but it goes on to rather more important nouns and names.
[GKC, The Well and the Shallows CW3:350]

The pronoun, you may recall, is just a name for something or someone, which has the remarkable property of both linking and unlinking from person to person, and thing to thing - a magical kind of verbal mirror, which by reflection can show any chosen individual or individuals. It is a real time-saver - it permits real simplifications of what would be very complex ideas. (Though, as any magic, it has its risks. It is easy to forget what the REAL thing is to which the current reflection is to correspond!) But we are not going to play that kind of game - the kind where one has to provide hints in parentheses. Heaven knows I use them a lot - but then they are another kind of magic, for another kind of simplification - or simpleton.

Now, pronouns in math are nothing new - except perhaps for the term. Us math guys usually call such things "variables" - grade-school teachers, for some strange reason, call them "placeholders" which sounds like a thing at a restaurant. But after all, "x" is just as valid a pronoun in a statement like

x + y = 7

as "he" is in a statement like

He is reading a book by Chesterton.

In both cases, we are told something and we are also left wondering about something. (hee hee: which book?)

However, unless one has played with some of the "advanced" kinds of mathematics, one rarely encounters the very special form of "pronoun" I am about to describe. I have had to make up an example which I can explain quickly, but will tell you what you need to know, so if you DO know about functions and recursion, don't be too shocked I did not use the factorial! (hee hee) But! its! levity! is! just! a! little! too! much! for! the! present! (Sorry - "x!" is read "x factorial" in math.)

OK. Now that I have told you all this, let us talk about beer. Ah! In our day, unlike GKC's, some of us buy beer in cans. I am not going to argue about that; let us just take it as a given. Some of us have more money, and can buy it in bottles or kegs, or more time, and can brew our own. But unlike these other containers the beer can has the wonderful property of being fun to stack into pyramids. Just to clarify for those few of you who may have never done this, a beer can pyramid is made by placing two cans side by side, and one can centered above their point of contact. Usually, people that drink beer from cans drink one can at a time, and so they are forced to build their pyramids as the ancients did: starting at the bottom. (If you want to hear about the fools who build them starting at the top, you must wait for my book on Subsidiarity.)

When one builds a pyramid, one must calculate the requirements first - just as our Lord told us. (see Luke 14:28) Let us say we wish to have a nice small one, with just six cans across the bottom. That means we'll need six cans, and however many cans make a pyramid with five cans at the bottom. Remember we need to stagger the rows as we go up, so if we have six on the bottom, we can only have five in the next row up. Fine.

So to make the pyramid with five cans on the bottom, we need those five cans, and however many are needed to make a pyramid of four.

So to make the pyramid with four cans on the bottom, we need those four cans, and however many are needed to make a pyramid of three.

So to make the pyramid with three cans on the bottom, we need those three cans, and however many are needed to make a pyramid of two.

So to make the pyramid with two cans on the bottom, we need those two cans, and however many are needed to make a pyramid of one.

BUT A PYRAMID OF ONE CAN IS ONE CAN.

So that means I need one, and two, and three, and four, and five, and six cans, or 21 cans total. (Great - we could do it with one case, and have three left over to enjoy later.)

Now, remember how I told you about how a PRONOUN can simplify complicated ideas? Wouldn't it be nice if I could have written this six-pyramid explanation?

Let's say that P(n) is the pronoun we want. P(n) means "How many cans it takes to build a pyramid with n cans on the bottom row". Shorthand. OK, now here is the fun thing. We know TWO THINGS about our pronoun - and one thing we don't know - just like real pronouns.

FIRST THING: P(1) is 1. A pyramid with one can on the bottom requires one can.

SECOND THING: For any number bigger than one - call it "SHE" (just to bend the rules) we need THAT MANY cans, and ALSO however many cans it takes to make a pyramid of ONE LESS than "SHE". But we can say it like this:

P(she) = she + P(she-1)

Now, why on earth did I mention "abbr." in my title? Well, "abbr." is the ABBREVIATION for the word "abbreviation". This sounds like a joke, but is really just an instance of what we call a self-referential object. That is, an object which somehow contains itself. Every so often you will see someone on TV standing in front of a TV monitor, which is showing that picture. So on the TV you see (somewhat smaller) a person standing by a TV, and on that TV you see (even smaller) a person standing by a TV, etc... Maybe you have some of those Russian dolls? The big doll opens along its middle, and inside is another doll, very similar, but smaller. It also opens, and inside is another doll...

NOW YOU HAVE BEGUN TO UNDERSTAND WHAT WE CALL RECURSION: a function (or rule, or idea) which somehow contains itself within itself - like the dolls. In real-world computing, we use recursion for certain kinds of work - but we must (repeat MUST) always check to see that we know TWO THINGS, as we saw with our beer-can formula. We must have a "stopping point" (we call it the "terminating condition") which is simply defined. Like the smallest doll without a seam, or the pyramid of one can. And we also have the "recurring" rule, which uses the idea itself on a PORTION of the problem - remember how each time our pyramid is smaller by one? A doll-with-a-seam will have another doll inside. But sooner or later we shall hit our terminating condition - and get our answer. No infinite sea, remember?

Yes, the Beer Can formula which we have given above has a simpler form, in which the recursion has been taken out. It looks like this P(she) = she * (she+1) / 2. You can check this for six, because 6 times 7 divided by 2 is 21.

Ah. Fine. Nice. If you need help, you ought to ask. Because now we're going to ask a very hard question.

What if there is no terminating condition?

In the real world, we hit the limits of physics. If we make a TV picture depend on itself, sooner or later we get down to the resolution of the screen - the "graininess" or the pixels by which the picture is formed. If we make a sound depend on a sound, we usually call that feedback, and end up blowing out the amp. If we apply the idea to any physical idea, sooner or later we get down to something which does not divide - a living thing ends abruptly with cells, a material (chemical) thing with atoms. (We'll defer discussion of the atomic particles for the time being, except to note that whatever THEY are, they are NOT like the gross structures of our world, simply comprised of atoms.)

"But Doctor" - you moan, thoroughly confused and disgusted by the high mathematical jargon I have been spewing - "Is there anything at all Chestertonian in this 'self-referential' idea anyway?"

There sure is - at least two places:

There was a debating-club in Bedford Park, on which I first tried my crude ideas with even cruder rhetoric. It deserved better treatment. It was frightful fun. It was called the "I.D.K."; and an awful seal of secrecy was supposed to attach to the true meaning of the initials. Perhaps the Theosophists did really believe that it meant India's Divine Karma. Possibly the Socialists did interpret it as "Individualists Deserve Kicking". But it was a strict rule of the club that its members should profess ignorance of the meaning of its name; in the manner of the Know-Nothing movement in American politics. The stranger, the mere intruder into the sacred village, would ask, "But what does I.D.K. mean?"; and the initiate was expected to shrug his shoulders and say, "I don't know," in an offhand manner; in the hope that it would not be realised that, in a seeming refusal to reply, he had in fact replied. [GKC, Autobiography CW16:148]

And in "The Loyal Traitor" contained in GKC's Four Faultless Felons, we find "the word is One." While I can point out this has nothing to do with - er - let us say the Prologue of St. John's Gospel, I must, unfortunately, refrain from commenting further, lest I reveal too much of the mystery puzzle of that particular story.

But in mathematics, in the realm of ideas, what happens when we proceed into this strange infinite realm of never-ending recursion? Some kind of computational black hole of never-ending number? Well...

Stay tuned, and next time I will show you. Please stick with me here - the next chapter will give you something fun to do, and a result you can use in your Christmas decorating...