Thursday, November 23, 2006

Infinite Symbols

I must apologise, first, for delaying my Thursday post to Friday, and stepping in front of our blogg-mistress as I lumber along the paths of the e-cosmos. Second, I apologise to any percussionists who got envious thinking that this posting was going to be about really loud cymbals. It's partly in response to a question from the comment-box, regarding the representations of "infinity" on computers. Third, I apologise because this posting will be long, but Nancy has given me the ritual for making a "read more". And though it is long, you will be glad because most of it is by Chesterton, and not by me.

But in order to get to Chesterton, you will have to endure a little more from me first. So, let us consider that phrase about representations of "infinity". At this splendid truth, all Chestertonians will sit back and sigh gratefully, murmuring "solvitur ambulando". (That is the witty and complete reply to the old line that "motion is impossible because it requires an infinite number of steps, each crossing half of the distance to be covered". The Latin reply means "Let it be solved (by, or while) walking.")

But I am not going to show you the meanings of such interesting infinite things as ¥ or À0 or tan(p/2) or A* or

lim x-1
or even (to resort to an archaism):
10 GOTO 10
which are all ways of representing infinity.

Rather, I will let our Uncle Gilbert address the far more complex problem of the representation of abstractions which are not mathematical, and which are infinite if anything is. Moreover, unlike limits, or challenges to the Halting Problem, or such mathematical oddities, the infinite topic is one which is much more relevant to our daily life as poor weak human beings, who "wait with joyful hope for the coming of our Savior Jesus Christ".

(And now, a double pun for any oboe or bassoon players out there.)


Good - you took the risk, and the reed indeed broke. Ahem. Now, here's GKC:

Every religion and every philosophy must, of course, be based on the assumption of the authority or the accuracy of something. But it may well be questioned whether it is not saner and more satisfactory to ground our faith on the infallibility of the Pope, or the infallibility of the Book of Mormon, than on this astounding modern dogma of the infallibility of human speech. Every time one man says to another, "Tell us plainly what you mean?" he is assuming the infallibility of language: that is to say, he is assuming that there is a perfect scheme of verbal expression for all the internal moods and meanings of men. Whenever a man says to another, "Prove your case; defend your faith," he is assuming the infallibility of language: that is to say, he is assuming that a man has a word for every reality in earth, or heaven, or hell. He knows that there are in the soul tints more bewildering, more numberless, and more nameless than the colours of an autumn forest; he knows that there are abroad in the world and doing strange and terrible service in it crimes that have never been condemned and virtues that have never been christened. Yet he seriously believes that these things can every one of them, in all their tones and semi-tones, in all their blends and unions, be accurately represented by an arbitrary system of grunts and squeals. He believes that an ordinary civilized stockbroker can really produce out of his own inside noises which denote all the mysteries of memory and all the agonies of desire. Whenever, on the other hand, a man rebels faintly or vaguely against this way of speaking, whenever a man says that he cannot explain what he means, and that he hates argument, that his enemy is misrepresenting him, but he cannot explain how; that man is a true sage, and has seen into the heart of the real nature of language. Whenever a man refuses to be caught by some dilemma about reason and passion, or about reason and faith, or about fate and free-will, he has seen the truth. Whenever a man declines to be cornered as an egotist, or an altruist, or any such modern monster, he has seen the truth. For the truth is that language is not a scientific thing at all, but wholly an artistic thing, a thing invented by hunters, and killers, and such artists long before science was dreamed of. The truth is simply that - that the tongue is not a reliable instrument, like a theodolite or a camera. The tongue is most truly an unruly member, as the wise saint has called it, a thing poetic and dangerous, like music or fire.

Now we can easily imagine an alternative state of things, roughly similar to that produced in Watts' allegories, a system, that is to say, whereby the moods or facts of the human spirit were conveyed by something other than speech, by shapes or colours or some such things. As a matter of fact, of course, there are a great many other languages besides the verbal. Descriptions of spiritual states and mental purposes are conveyed by a variety of things, by hats, by bells, by guns, by fires on a headland, or by jerks of the head. In fact there does exist an example which is singularly analogous to decorative and symbolic painting. This is a scheme of Esthetic signs or emblems, simple indeed and consisting only of a few elemental colours, which is actually employed to convey great lessons in human safety and great necessities of the commonwealth. It need hardly be said that I allude to the railway signals. They are as much a language, and surely as solemn a language, as the colour sequence of ecclesiastical vestments, which sets us red for martyrdom, and white for resurrection. For the green and red of the night-signals depict the two most fundamental things of all, which lie at the back of all language. Yes and no, good and bad, safe and unsafe, life and death. It is perfectly conceivable that a degree of flexibility or subtlety might be introduced into these colours so as to suggest other and more complex meanings. We might (under the influence of some large poetic station-masters) reach a state of things in which a certain rich tinge of purple in the crimson light would mean "Travel for a few seconds at a slightly more lingering pace, that a romantic old lady in a first-class carriage may admire the scenery of the forest." A tendency towards peacock blue in the green might mean "An old gentleman with a black necktie has just drunk a glass of sherry at the station restaurant." But however much we modified or varied this colour sequence or colour language, there would remain one thing which it would be quite ridiculous and untrue to say about it. It would be quite ridiculous and untrue to say that this colour sequence was simply a symbol representing language. It would be another language: it would convey its meaning to aliens who had another word for forest, and another word for sherry, and another word for old lady. It would not be a symbol of language, a symbol of a symbol; it would be one symbol of the reality, and language would be another. That is precisely the true position touching allegorical art in general, and, above all, the allegorical art of Watts.

So long as we conceive that it is, fundamentally, the symbolizing of literature in paint, we shall certainly misunderstand it and the rare and peculiar merits, both technical and philosophical, which really characterize it. If the ordinary spectator at the art galleries finds himself, let us say, opposite a picture of a dancing flower-crowned figure in a rose-coloured robe, he feels a definite curiosity to know the title, looks it up in the catalogue, and finds that it is called, let us say, "Hope." He is immediately satisfied, as he would have been if the title had run "Portrait of Lady Warwick," a "View of Kilchurn Castle." It represents a certain definite thing, the word "hope." But what does the word "hope" represent? It represents only a broken instantaneous glimpse of something that is immeasurably older and wilder than language, that is immeasurably older and wilder than man; a mystery to saints and a reality to wolves. To suppose that such a thing is dealt with by the word "hope," any more than America is represented by a distant view of Cape Horn, would indeed be ridiculous. It is not merely true that the word itself is, like any other word, arbitrary; that it might as well be "pig" or "parasol"; but it is true that the philosophical meaning of the word, in the conscious mind of man, is merely a part of something immensely larger in the unconscious mind, that the gusty light of language only falls for a moment on a fragment, and that obviously a semi-detached, unfinished fragment of a certain definite pattern on the dark tapestries of reality. It is vain and worse than vain to declaim against the allegoric, for the very word "hope" is an allegory, and the very word "allegory" is an allegory.

Now let us suppose that instead of coming before that hypothetical picture of Hope in conventional flowers and conventional pink robes, the spectator came before another picture. ...

[See here for the ARC entry; it is truly amazing.]

... Suppose that he found himself in the presence of a dim canvas with a bowed and stricken and secretive figure cowering over a broken lyre in the twilight. What would he think? His first thought, of course, would be that the picture was called Despair; his second (when he discovered his error in the catalogue), that it has been entered under the wrong number; his third, that the painter was mad. But if we imagine that he overcame these preliminary feelings and that as he stared at that queer twilight picture a dim and powerful sense of meaning began to grow upon him - what would he see? He would see something for which there is neither speech nor language, which has been too vast for any eye to see and too secret for any religion to utter, even as an esoteric doctrine. Standing before that picture, he finds himself in the presence of a great truth. He perceives that there is something in man which is always apparently on the eve of disappearing, but never disappears, an assurance which is always apparently saying farewell and yet illimitably lingers, a string which Is always stretched to snapping and yet never snaps. He perceives that the queerest and most delicate thing in us, the most fragile, the most fantastic, is in truth the backbone and indestructible. He knows a great moral fact: that there never was an age of assurance, that there never was an age of faith. Faith is always at a disadvantage; it is a perpetually defeated thing which survives all its conquerors. The desperate modern talk about dark days and reeling altars, and the end of Gods and angels, is the oldest talk in the world: lamentations over the growth of agnosticism can be found in the monkish sermons of the dark ages; horror at youthful impiety can be found in the Iliad. This is the thing that never deserts men and yet always, with daring diplomacy, threatens to desert them. It has indeed dwelt among and controlled all the kings and crowds, but only with the air of a pilgrim passing by. It has indeed warmed and lit men from the beginning of Eden with an unending glow, but it was the glow of an eternal sunset.

Here, in this dim picture, its trick is almost betrayed. No one can name this picture properly, but Watts, who painted it, has named it Hope. But the point is that this title is not (as those think who call it "literary") the reality behind the symbol, but another symbol for the same thing, or, to speak yet more strictly, another symbol describing another part or aspect of the same complex reality. Two men felt a swift, violent, invisible thing in the world: one said the word "hope," the other painted a picture in blue and green paint. The picture is inadequate; the word "hope" is inadequate; but between them, like two angles in the calculation of a distance, they almost locate a mystery, a mystery that for hundreds of ages has been hunted by men and evaded them. And the title is therefore not so much the substance of one of Watts' pictures, it is rather an epigram upon it. It is merely an approximate attempt to convey, by snatching up the tool of another craftsman, the direction attempted in the painter's own craft. He calls it Hope, and that is perhaps the best title. It reminds us among other things of a fact which is too little remembered, that faith, hope, and charity, the three mystical virtues of Christianity, are also the gayest of the virtues. Paganism, as I have suggested, is not gay, but rather nobly sad; the spirit of Watts, which is as a rule nobly sad also, here comes nearer perhaps than anywhere else to mysticism in the strict sense, the mysticism which is full of secret passion and belief, like that of Fra Angelico or Blake. But though Watts calls his tremendous reality Hope, we may call it many other things. Call it faith, call it vitality, call it the will to live, call it the religion of to-morrow morning, call it the immortality of man, call it self-love and vanity; it is the thing that explains why man survives all things and why there is no such thing as a pessimist. It cannot be found in any dictionary or rewarded in any commonwealth: there is only one way in which it can even be noticed and recognized. If there be anywhere a man who has really lost it, his face out of a whole crowd of men will strike us like a blow. He may hang himself or become Prime Minister; it matters nothing. The man is dead.

[GKC, G. F. Watts]

So. If this does not express something quite a bit greater than the various infinities I mentioned above, then I will turn in my wand and go back to building pipe organs. But I hope you will read it carefully, then go and bang on some cymbals...

A postscript. Just in case someone feels compelled to quote that line in Orthodoxy that "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] you might consider this first:
It's true we know the infinite
Because we put our bounds on it:
The end of endless lines we're finding
Reduced by logic's truthful binding
And see the All within one glance:
A start point; a means to advance.
[by me, Dr. Thursday]
You see, GKC can use the term "infinite" and remain reasonable because it is not crossing the infinite sea; but he does understand that it is uncrossable. That's what all those symbols are sounding! Yes, exactly. It's very Chestertonian, too. In mathematics, infinity is just a form of limit. Hee hee.


  1. "All language is used like that; you never get a question answered literally, even when you get it answered truly." ... Father Brown, "The Invisible Man"

    Methinks you answered the question about infinity truly, but not literally. Though your answer was nearly infinite in length.

    Chesterton is astonishing, is he not? A casual reader might think of this selection on Watts as being in the tradition of the nominalists. But on the contrary, Chesterton is incredibly pro-reality. This selection you quoted, which on the surface is about language and symbols, is really perhaps the most solidly stated defence of Reality our man ever penned. And the greatest wordsmith of the 20th Century deftly explains to us here the weakness of words!

  2. Dr. Thursday,

    Infinity is an interesting subject, with a rich history dating back to the ancient Greeks who really opened infinity to all sorts of speculation and investigations. (Referring to our previous discussion about calculus, Greeks started to experiment with mathematical methods that eventually lead to calculus.)

    One could analyze it from many viewpoints, but to start with the most basic thing, its generaal definition, infinity itself is NOT a limit, it is contradictory to the very basic definition of infinity -- Latin “in” means “not” and “finis” means “end, boundary, or limit,” thus infinity means “without end, boundary or limit.” By definition, infinity is therefore a negation of anything finite.

    I don’t want to wade into a discussion on poetry, so I will only say that a lot of poetry that pretends to deal with issues rationally will not float easily, and a lot of such poetic waxing quickly sinks like a boat made from Swiss cheese. (That is why I avoid writing poetry. If I ever try it, it would be something silly, nonsensical in Chestertonian sense and funny, something that would float easily, and if it sank, I wouldn’t care a bit. :-)

    Here is the full quote 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. The result is mental exhaustion, like the physical exhaustion of Mr. Holbein. To accept everything is an exercise, to understand everything a strain. The poet only desires exaltation and expansion, a world to stretch himself in. The poet only asks to get his head into the heavens. It is the logician who seeks to get the heavens into his head. And it is his head that splits."

    Chesterton is saying that an attempt to rationally explain anything in its fullness, of something which has some real infinity in it, (crossing the ocean of infinity), is not only futile, but may lead to insanity - it is the “logicians’s head” that splits. (Cantor’s insanity is a good example.)

    Infinity is not a limit also in the philosophical sense. Aristotle gave the most profound analysis of infinity in his Physics:

    “Everything is either a source or derived from a source. But there cannot be a source of the infinite or limiteless, for that would be a limit of it.” (203b6)

    So the infinite is not the same thing as a limit!

    In fact, Aristotle uses this argument when he gives his 4th consideration for the belief in the infinite. “Because the limited always finds its limit in something, so that there must be no limit, if everything is always limited by something different from itself.” (203b20) The “no limit” (or no-limit) means infinity and it is a different thing from the limit itself.

    And I think this applies also in the mathematical sense. A function approaching a fixed value as it approaches infinity, is a different thing from the function itself, and both are different from the fixed value toward which the function is approaching, which is the limit of this function. (See definition of mathematical limit in ) The fixed value or a point thus cannot be infinite. The infinity works only in our imagination, (almost in a poetical sense), as we imagine the function crossing or moving in the infinite ocean of the plane or space in which the function is being drawn.

    I was hoping you would be able to produce a more substantial form of computer represented infinity than a “mere” 10 GOTO 10, which is nothing more than a trivial infinite loop, that would be, hopefully, produced only by a very ignorant novice in computer programming, and even that only by a bad mistake or an bad oversight. Besides, this example of infinity is only “infinite” in a very limited trivial aspect of infinity - that of trying to mimic an infinite or eternal time period, and even that not from minus to plus infinity, but only an interval from a concrete physical point in time to, hopefully, infinity. Still this “infinite loop” is infinite only in our imagination, and I doubt it would last longer than our life span, or very likely much shorter than that - most likely until the first power bump or power failure that would shut the computer down. Only bad computer programmers, or bad programming environments, produce these sort of these infinite loops, otherwise known as bugs, they have no use or purpose, (as far as I know), and they are nothing to brag about.

    I was hoping you would mention Cantor’s ideas about infinity, because after the Middle Ages the infinity took on a very different meaning, perhaps starting with Giordano Bruno, who muddled the subtle distinctions about the infinity of God maintained by Thomas Aquinas and the Church, and this lead to his death - he was, of course, burned as a heretic. It was indeed a dangerous thing to blur the subtle and painfully acquired distinctions about infinity that prevent one from falling into many “infinitely huge” errors we have inherited today.

    Wild Goose

  3. An infinite Wild Goose chase.

    The infinite loops you mention are more restrictive than the most closed room. There is no way out of such infinity - it's stifling! It's the prison of a rationalism that keeps returning on itself.

  4. Re. enless loops:

    "A small circle is quite as infinite as a large circle; but, though it is quite as infinite, it is not so large. In the same way the insane explanation is quite as complete as the sane one, but it is not so large. ... The circle returns upon itself and is bound. The cross opens its arms to the four winds; it is a signpost for free travellers." - Orthodoxy

  5. Kevin,

    A good point about the the circle and the cross. But, there are two meanings with respect to the infinty of the circle. In the Greek and Byzantine culture the circle and the sphere was considered a symbol of God, perfection, and heaven.

    Nowenere is this more obvious than in a typical Byzantine church, but here the symbolism of the “ball and the cross” is reversed. The cross-in-the-square church symbolizes the world, the spherical dome symbolizes the Heaven, the Divine World, and the Heavenly Jerusalem.

    In his Resurrection of Rome Chesterton is somewhat critical of the Byzantine art, logic, mathematics and Byzantine symbolism, and he elevated the Western or “Roman” interpreration. I will re-read the important passage in the Resurrection of Rome and in the Ball and the Cross, but it seems to me Chesterton perhaps neglected this classical Greek interpretation, and what has become famous is his “lunatic” aspect of infinity you quoted.

    Wild Goose


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