Thursday, September 23, 2010

4294967295, or, the Last Number

Today is the Equinox, the beginning of Autumn. As I am sure you know by now (as Chesterton points out) since the God of Christianity is the real God of the universe, Christianity has something say about everything from pork to pyrotechnics, from pigs to the binomial theorem - so too, Christianity has something to say about Autumn. I apologise for the pun, but I cannot resist...
Christianity spoke again and said: "I have always maintained that men were naturally backsliders; that human virtue tended of its own nature to rust or to rot; I have always said that human beings as such go wrong, especially happy human beings, especially proud and prosperous human beings. This eternal revolution, this suspicion sustained through centuries, you (being a vague modern) call the doctrine of progress. If you were a philosopher you would call it, as I do, the doctrine of original sin. You may call it the cosmic advance as much as you like; I call it what it is - the Fall."
[GKC Orthodoxy CW1:321]
Yes, the Fall. (Which means that business with the snake and the fruit in the Garden happened in late summer, hee hee.) With the arrival of the autumnal equinox, Autumn (or the Fall, if you call it what it is) comes to our Northern Hemisphere, and our thoughts are naturally urged to contemplate the mystery of time.

Now, everyone will expect some sort of stiff physics or perhaps speculative cosmology - but that is not suitable for our column, even though GKC certainly mentioned Einstein nearly a hundred times in his various writings. I especially like the bit about how people ought to be fined if they mention his name without knowing what they are talking about. (That's in ILN May 23 1931 CW35:526) But we are not going into that sort of thing here - at least not today. If you're moving near the speed of light, however, perhaps we'll go into it yesterday, hee hee.

No; actually I was thinking more about numbers - which sometimes happens for me, since I play with numbers a lot - but also because of my recent writing about the chirality of letters. Now, the common Man (as GKC loved to say) does not normally think there is any difference between a NUMBER and a DIGIT. In fact, the use of that word "difference" is a hilarious pun, and gets into one of those things that - uh. As usual, Chesterton has gone into this, and he says it far better than I can:
They differed from the reality not in what they looked like but in what they were. A picture may look like a landscape; it may look in every detail exactly like a landscape. The only detail in which it differs is that it is not a landscape. The difference is only that which divides a portrait of Queen Elizabeth from Queen Elizabeth.
[GKC TEM CW2:245]
"They differ from the reality not in what they looked like but in what they were." There is a huge difference between a number and a representation of that number - and there is no pun at all in that context, since the "difference" isn't the mathematical kind of subtraction. One does not subtract paintings from English sovereigns; it is that old line about mixing apples with oranges... the possibility of fruit salads notwithstanding. Indeed, it is this grand conundrum which has misled several otherwise wise writers into distortions which ought not be dignified by the term "paradox", even though that is the word which is usually applied. But I am wandering and you are lost too. (Whew!) Let us return to numbers, or rather words about numbers, and about digits.

You see, I was thinking about time, and starts and ends - about measuring (as God states was His design in fashioning the various lamps in the sky, Gen 1:14) and about giving order to things. As soon as we examine this truth, we find a paradox, and we don't have to be a Chesterton to appreciate it. The paradox is that the sun and the moon do just one thing: the sun appears to travel, from east to west, over and over again. The moon does the same, while it slowly gets bigger and then smaller, and moves slowly from west to east against the background of the stars. (Yes there are some other wobbles but we'll let those details aside for today.) The sun and moon appear to make the same motions - but they make them in seeming endlessness. We have that grand insight into science given by GKC in his "Ethics of Elfland" chapter of Orthodoxy about how God says "Do it again!" to these heavenly bodies - and even if you have already read that chapter I urge you to Do It Again.

Ah, the intoxicating grandness of astronomy, where everyone on earth "has access at all times to the original objects of his study ... the masterworks of the heavens belong to him as much as to the great observatories of the world." [Robert Burnham Jr., Burnham's Celestial Handbook 5] Did you know that of the seven so-called "Liberal" Arts, there are four (or five) which are properly in the technical or scientific realm, and not in the non-science side? Oh yes. All four of the Quadrivium are about numbers in one sense or another: this is nothing new, but has been understood in this sense as far back as the twelfth century - for example, see the Didascalicon of Hugh of St. Victor:
Now, multitude which stands in itself is the concern of arithmetic, while that which stands in relation to another multitude is the concern of music. Geometry holds forth knowledge of immobile magnitude, while astronomy claims knowledge of the mobile.
[The Didascalicon of Hugh of St. Victor translated and annotated by Jerome Taylor; Book 2 chapter 6]
Or, to put it as computer scientists do, in the form of a tree (which is sometimes called an "outline"):
a. The discrete
1) the absolute = Arithmetic
2) the relative = Music
b. The continued
1) the stable = Geometry
2) the moving = Astronomy

[Above from The World of Mathematics 85]

We know a bit more about mathematics now, and might adjust this tree and add to it significantly, but it is a good start, and does indeed suggest that there is a lot more to the intellectual life than a stream of dull plot-schemes, or the dates kings died on, or philosophical dreams (or nightmares) aimed at dethroning God... hee hee! The fact that the typical philosopher is not set on understanding Reality but on dethroning God is stated in Chesterton's grand epigram about modern thought: "With a long and sustained tug we have attempted to pull the mitre off pontifical man; and his head has come off with it." [GKC Orthodoxy CW1:237] However! I have digressed, since it is quite tiresome to continue to read the whine of the liberal arts crowd here on the INTERNET. I wonder where they think computers come from! Plants, maybe. Ahem. The point here is not to make a digression on liberal arts - or even on liberal sciences - but to suggest an important truth about numbers - and about time.

In order to help reveal more of this, I have to go back to something I mentioned previously, and about which I recounted an interesting story almost a year ago: the fact that computers cannot add. Perhaps I here need to stress an even more surprising truth, and one which will distress the public educrats who read this blogg: the fact that computers do not use numbers. Yes, computer do not deal with numbers at all. What they deal with is something else - and the closest I can come to describing it without formalisms is to say, like I did previously, that they deal with things which have properties like railroad cars, that can be manipulated. (The Latin here is rather exaggerated; we do not use our hands (Latin manus) to manipulate railroad cars unless they are toy railroad cars!) But yes, those things which we call BYTES are arranged, and linked together and "manipulated" - they are physical in a way that no number is physical.

Remember how (long ago) we learned about counting? We had three sheep or three cherries or three toy engines... we found out the mystery that this "three" thing was like "red" or "soft": it did not depend on the things, but on the three-ness of those things. There has been much written about this mystery of number - but it is not a mystery. It is caused by some refusing to learn the first Learning, which is why the Greeks called it "Mathematics" = "That Which is Learned". Again, Chesterton says it far better than I can:
It is as easy to be logical about things that do not exist as about things that do exist. If twice three is six, it is certain that three men with two legs each will have six legs between them. And if twice three is six, it is equally certain that three men with two heads each will have six heads between them. That there never were three men with two heads each does not invalidate the logic in the least. It makes the deduction impossible, but it does not make it illogical. Twice three is still six, whether you reckon it in pigs or in flaming dragons, whether you reckon it in cottages or in castles-in-the-air.
[GKC ILN Nov 11 1905 CW27:59]
However, we are not concerned today with these things that do not exist, with flaming dragons or with castles-in-the-air, but with something far more mysterious: the LAST NUMBER.

Yes. (Oh boy, exciting!) Now, you will laugh about this, since you probably think it is a joke. It's not a joke. You may have followed Milo through The Phantom Tollbooth and asked the Mathemagician about the Biggest number, or the Longest number - and he has sent you up the infinite staircase, or out along the infinite line - both of which take you to the same place. But that is another sort of thing. (People worry about infinity, and get confused; they must have been absent that day, or haven't gotten that far in their coursework. Oh well; another time for that.) But I do not mean THAT sort of Last Number.

If you have owned a car long enough - or have bought a used one which someone else has owned long enough - you will know what I am talking about. There comes a point when the odometer "goes back to all zeros" - suddenly you have a New Car again. (hee hee) It sounds very Christian, let's see: "unless your car turn and be built again, it shall by no means..." Ahem. But we know that the car is not new. It is still the same old car. But the odometer says something astoundingly small!

Yes. But there is a better example, and one which we are even more familiar with, and will tell my point far better. It is called the Clock.

Now we all know how to add - or we think we do until we go to balance our checkbooks, though that usually means subtraction - and usually we are pretty good at adding. And I am sure that if I gave you a number, you feel fairly certain that you would be able to add two to that number and give me the correct answer. Let's try a little quiz:

I ask and... you answer:

Five plus two is... Seven.
Seven plus two is... Nine.
Nine plus two is... Eleven.
Eleven plus two is... Thirteen.

Good. Now, let's just try it with some other words put alongside, shall we? And we'll see what happens.

It's eleven o'clock in the morning now - let's meet in two hours - will you be free?

And you respond: At one this afternoon? Yes, I'm free.

Ah ha! But you just added two to eleven and got one!

Is that oops or sure? Well, well. What is going on?

It's the same as the car odometer - and the same as the computer. The clock provides us with just one "wheel" and when it runs out, we start over. We do as Chesterton says God tells the sun: "Do It Again." The repetition after noon is our homage, not (oddly enough) to the non-Copernican terms of Sunrise and Sunset, which is the way the Romans named the hours, but to the far more mysterious zenith and nadir of the globe, to midnight and noon.

In the computer, there is something rather like an odometer - there is a fixed number of places, sort of like those little dials of digits, and when they fill up, it starts over again at zero. However, because the computer's "numbers" (which are NOT numbers) represent values using a base-two scheme, the Last Number of a computer isn't all nines. That was the Great Media Fear some ten years ago, remember? They called it Eetook, or Y2K - the comet which was supposed to hit on New Year's of 2000 (or maybe 2001 depending on when you believe the millennium began, hee hee) No; in a thirty-two bit computer, which is what most personal and business machines are these days, the Last Number looks like this in our common tongue:
or if you prefer it in words:
four billion, two hundred ninety-four million, nine hundred sixty-seven thousand, two hundred ninety-five.
It's the last number, because after it comes zero again. (Remember, as I said, the things in the computer are NOT numbers, but that is another topic for another day. Huh? Are you looking at a computer? Do you see numbers here? I didn't think so. Hee hee!)

It's not all that hard. Look at the clock. After 12 comes 1. Yeah, our "12" on the clock ought to be "zero" - and on computers, midnight is called "00:00". If your clock or watch is digital, you may (as I do) have it set to use the 24-hour clock, sometimes called military time, in which case the Last Number looks like "23:59" or "23:59:59".

I had to hunt a good bit to find whether GKC ever played with this puzzle, and I am not sure he didn't. But he has a lovely joke about time:
Mr. Birrell ... remarked that all the children appeared to be consumed with a desire to ask him the time. He appeared mystified as to why they asked him the time. I am unable to answer with accuracy (although I have studied the phenomenon many hundred times) beyond being quite certain that it was not because they wanted to know. A careful examination of the conduct of Battersea Park children shows quite clearly that the mention of no hour of the day (however sensational) makes any difference at all to their dignified and dilatory behaviour. Children live in an almost entirely timeless world (in which they resemble the Deity of Thomas Aquinas), and most of us who can remember our childhood can remember a certain sense of spaciousness in the hours, a sense that might be called a kind of happy emptiness. ... And I think very few children (certainly not the countless hordes that lie in wait for Mr. Birrell and me in Battersea Park) take any particular interest at all in the time of day. If you followed the disgraceful example of Policeman Peter Forth, and answered them "A quarter past thirteen," I think the information would be received with a refined indifference.
[GKC ILN May 5 1906 CW:178-9]
Perhaps all my lengthy discussion of the Last Number will also be received with a "refined indifference" - but it has some relevance. There is a last number assigned to each of us also... As a true priest and teacher, Father Jaki often reminded people of the Four Last Things: death, judgement, heaven and hell. In thinking of the waning of the year, we may do well to ponder the coming Harvest - our Lord actually mentioned such a thing - and consider how our odometer is counting off the seconds and minutes and hours. How are we using our time and our lives? Is it fruitful? But it is also important to remember that while the northern hemisphere begins autumn, the southern hemisphere begins spring. This is NOT a matter of dualism. This is not a suggestion that we believe, as the pagans, in the Cycle or "Great Year" - this topic is examined in all its bitter barrenness in Jaki's Science and Creation. We are Christian: we believe in the Strict Linearity of Time. This absolute linearity of Christian history is also found in Chesterton's The Everlasting Man: "It declares that really and even recently, or right in the middle of historic times, there did walk into the world this original invisible being; about whom the thinkers make theories and the mythologists hand down myths; the Man Who Made the World. [CW2:398-9, emphasis added] You cannot have a middle of a circle... or to put it another way, no circle can have its center on its edge.

There is A LAST NUMBER to the Cosmos, and it WILL end. (The mystery of "end" is also something we shall consider another day, since it also makes people itch, like "infinity".) But the fact that we humans CAN add (unlike machines) suggests something infinite at work. In fact, we do have something infinite, and it is the very fact (among others) that we CAN imagine adding correctly, regardless of how huge a number, that we are distinguished from machines. And if you think about this a little more, you will begin to see into both "infinity" and "end"... but let it go for today.

A good friend of mine, a brilliant engineer, once put it into a satirical "poem" (I think that is what he called it). I don't have his precise phrase to quote but it was something like "intelligent rivers perform addition." That is the whole point. A computer cannot add, or rather it adds only in the way in which a river adds its water to the ocean.

But there is a greater point to be made, and I bring in the term "poem" intentionally, so that I can conclude with a Chesterton quote:
In one of his poems, he says that abyss between the known and the unknown is bridged by "Pontifical death." There are about ten historical and theological puns in that one word. That a priest means a pontiff, that a pontiff means a bridge-maker, that death is certainly a bridge, that death may turn out after all to be a reconciling priest, that at least priests and bridges both attest to the fact that one thing can get separated from another thing - these ideas, and twenty more, are all actually concentrated in the word "pontifical." In Francis Thompson's poetry, as in the poetry of the universe, you can work infinitely out and out, but yet infinitely in and in. These two infinities are the mark of greatness; and he was a great poet.
[GKC ILN Dec 14 1907 CW27:603-4]

1 comment:

  1. I rather like Chesterton's proposal that people be fined for talking about Einstein without having taken the trouble to find out what Einstein actually said. (I would add Godel and Heisenberg to the list.)

    It has been claimed that the theory of relativity somehow endorses 'relativism' in ethics. Martin Gardner (an admirer of GKC) knew better -- see the preface to his book RELATIVITY FOR THE MILLIONS -- though Carl Sagan didn't.

    The theory of relativity says that even though observers in different frames of reference may disagree in their observations, they will deduce the same basic physical laws. So IF relativity has anything to say about ethics, it would endorse the existence of universal ethical norms.

    The misunderstanding might have been avoided if Einstein had called his theory 'the Theory of Invariance' -- which he actually considered doing.


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