Dr. Thursday's Post

Praise of Simple Addition

When I walked along the pier at Ostend; and I heard some sailors uttering a measured shout as they laboured, and I remembered that sailors still sing in chorus while they work, and even sing different songs according to what part of their work they are doing. And a little while afterwards, when my sea journey was over, the sight of men working in the English fields reminded me again that there are still songs for harvest and for many agricultural routines. And I suddenly wondered why if this were so it should be quite unknown, for any modern trade to have a ritual poetry. How did people come to chant rude poems while pulling certain ropes or gathering certain fruit, and why did nobody do anything of the kind while producing any of the modern things? Why is a modern newspaper never printed by people singing in chorus? Why do shopmen seldom, if ever, sing?

If reapers sing while reaping, why should not auditors sing while auditing and bankers while banking? If there are songs for all the separate things that have to be done in a boat, why are there not songs for all the separate things that have to be done in a bank? As the train from Dover flew through the Kentish gardens, I tried to write a few songs suitable for commercial gentlemen. Thus, the work of bank clerk when casting up columns might begin with a thundering chorus in praise of Simple Addition.

"Up my lads and lift the ledgers, sleep and ease are o'er.
Hear the Stars of Morning shouting: 'Two and Two are four.'
Though the creeds and realms are reeling, though the sophists roar,
Though we weep and pawn our watches, Two and Two are Four."

[GKC, "The Little Birds Who Won't Sing" in Tremendous Trifles]

Addition. Sums. Adding. To be precise, a closed, associative, operation with an identity and an inverse, defined over sets both finite and infinite... While most mathematics is thought "hard" (if you are a doll) or "difficult" (if you are not Newton or Cauchy or Euler or Gauss) it is nearly a truism of language to speak of things being "as easy as addition" or "as simple as two plus two equals four" - even for such a non-math guy as our Uncle Gilbert:

Mr. Blatchford, with colossal simplicity, explained to millions of clerks and workingmen that the mother is like a bottle of blue beads and the father is like a bottle of yellow beads; and so the child is like a bottle of mixed blue beads and yellow. He might just as well have said that if the father has two legs and the mother has two legs, the child will have four legs. Obviously it is not a question of simple addition or simple division of a number of hard detached "qualities," like beads. It is an organic crisis and transformation of the most mysterious sort; so that even if the result is unavoidable, it will still be unexpected. It is not like blue beads mixed with yellow beads; it is like blue mixed with yellow; the result of which is green, a totally novel and unique experience, a new emotion.
[GKC, What's Wrong With the World, CW4:155]

Here already we find something transcendent about addition - I do not mean the underlying argument GKC is making - I mean the curious fact that there are some kinds of "addition" which do not work like the numbers. In the world of paint pigments, blue plus yellow equals green. But let us look a little at numbers and find out what is going on, and perhaps we shall also have a "novel and unique experience": the "new emotion" which provokes praise of addition.
Read more.

In order to talk about addition we have to talk about numbers. One of the first problems one encounters in talking about numbers is the same as in other fields: distinguishing between the thing-in-itself and its representation in print. Perhaps the simplest way of clearing up this puzzle is to show you some different forms of a "simple" addition problem. We'll use the same one GKC quoted.

Written English:
Two plus two equals four.

Algebraic notation:
2 + 2 = 4

Latin:
II et II est IV.

Typical personal computer (x86-based):
Given that EAX contains 0000 0000 0000 0000 0000 0000 0000 0010
and EBX contains 0000 0000 0000 0000 0000 0000 0000 0010
after performing ADD EAX,EBX
EAX contains 0000 0000 0000 0000 0000 0000 0000 0100.

Real addition:
**
(stuck together with)
**
(is the same as)
****

The last one is about as close as I can come (here in the e-cosmos) to the "thing-in-itself"; the others forms are "representations". They are ways of indicating - even suggesting - the true details contained only in the last display. You see, there is a real thing, addition, which has a real meaning in the real world. Get out some coins (I don't care what value they are) - or some paper clips - or a few somethings. Put two of them in front of you, on your left side, and two of them on your right side. You see there are two here, and also two there. (Of course you do. It's simple!) Now slide them together into a little heap. Now there are four, even though you might be able to tell which two were on the left and which two were on the right - but that doesn't matter - there are four together, which are somehow the same as two and another two: though they were apart, now they are joined.

It is simple because there are just a few, and we can handle (that is USE OUR HANDS) to manipulate them (yes, I know that's just using a Latin term for the same thing...)

What if there were more than we would care to shove around? Or we were "adding" things that are not "shove-able"?

Ah. That's where "addition" (the symbols, or the representations) come in.

Taxonomists throw their hands up in the air over human beings - the species homo sapiens (Man-the-wise). Someone, searching for a way of expressing the unique catholicity of our species, and thereby demonstrating the perfection of the species sapiens, is always making a suggestion of another gerund to go here. Some have said faber (the maker, because we make tools - things for making - ah, a recursive thought!); others ludens (the player, because we play games); still others ridens (the laugher, for obvious reasons, hee hee). Tolkien, with his deep penetration, and his true love of words, named his reasoning beings the Quenta - the Speakers - and if he played the taxonomy game, he might have proposed homo loquens - man, the speaker.

Yes, in the process of wisdom, we first reduce reality to the spoken word (So we say "two plus two equals four.") Then we move to writing:

(that is, "two added-to two amounts-to four")
and all the other variations which have been encoded over the millennia.

I should, however, point out that the form using a computer is even more of a cheat than the others. And it will come as a real surprise to some, because it is not technical, but philosophical. Language, be it symbolic representations of sounds (Remember that "to" represents the same sound as "two"!) or symbolic representations of words ("2" represents "two" but also duo or zwei<>deux or dos!) possesses a strange characteristic, deriving directly from something supernatural. Language, the spoken word (and therefore the written word too) has an infinite or eternal dimension - one-sided, yes, but infinite in the sense that it is "unbounded".

My gosh, haven't you felt that these "Dr. Thursday" posts are going to run on and on? Hee hee.

We know full well that we'll get tired, have to go home from work or school, go to bed, or whatever. But at a given moment, we "feel" that the words might go on and on, as long as is needed. And so, we can think of numbers that perhaps we might never really say - numbers that no computer, no collection of computers, could ever store - numbers that might not even mean anything in the "real" world - but really big numbers... googol-plex, and such - and we can, by that power residing in the part of us which is NOT physical, immediately think of adding one to it.

An aside: If you need more on this, I direct you to The Phantom Tollbooth by Norton Juster, in which the Mathemagician presents a good commentary on infinity. His brother, King Azaz, makes the same comment, though in somewhat more veiled language. As you may know, our hero Milo receives infinite gifts from each of them, but then I must not reveal too much. Go and read it yourself.

In other words, our grasp of the abstraction implicit in numbers and addition is somehow derived from the supernatural trait called the imperishability of the soul: since WE can imagine talking on-and-on, long enough to finish the addition, we can grasp the general notion of addition. Now for the shock.

Computers cannot do this.

The addition which is "native" to computers is NOT that kind of addition. It looks like regular addition, and will work as long as one keeps things "small enough" - but.

First, this addition is a representation, just as much as "2" is a representation for "two" (and so on) - it has to be, because the "numbers" in the computer are themselves representations! We tech guys write zeros and ones (well actually most of the time we use the regular numbers, and the machinery fixes them during compilation), but actually the "numbers" are just a higher voltage and a lower voltage on separate wires, or tiny thin strands of metal on a wafer of quartz, or regions of magnetized iron particles on a spinning plate.

Second, this addition is what math guys call "modular addition" - or the grade school teachers might call clock math. There is a wrapping-around that happens, sooner or later, and if you count high enough, you have to start over again at zero. Ever notice how if you call a friend at 11 AM and say you'll meet for lunch in two hours, you add 11+2 and get one? Yes, that's right... on a typical computer (either x86 or 68000), if you add two to 4294967295 you will also get one. It's true, and for the same reason. On clocks, the wrapping happens at zero, which is also called twelve. On most computers, the wrapping happens at the number called 2³² or 4294967296, which is also called zero, because 4294967296 requires thirty-three bits to write, and these computers can only add 32 bits at once. (Sure, if you are a tech, you can think of tricks - I know several, but we are not going into that today.) Now, most of the time we don't need to actually count up to 4 billion, so we don't have a problem with this wrap-around. But we have to know that it's there. Why is it there? Why will something like that always be there? Simply because one has to build the machinery to hold the data. Either a "register" (the thing that holds the 32 bits, and wraps around) or memory, or hard drives, or whatever it may be - all such things are finite.

But the human mind is not.

Thus, there arises, even here, in the dull simplicity of a very technical (and perhaps very boring) little matter - the matter of addition - we are faced with ETERNITY - with one of the greatest thoughts possible. And that brings us squarely face to face with God and religion. Which is as it should be:

...very uneducated rich men who loudly demanded education. And among the marks of their ignorance and stupidity was the particular mark that they regarded letters and figures as dead things, quite separate from each other and from a general view of life. They thought of a boy learning his letters as something quite cut off, for instance, from what is meant by a man of letters. They thought a calculating boy could be made like a calculating machine. When somebody said to them, therefore, "These things must be taught in a spiritual atmosphere", they thought it was nonsense; they had a vague idea that it meant that a child could only do a simple addition sum when surrounded with the smell of incense. But they thought simple addition much more simple than it is. When the Catholic controversialist said to them, "Even the alphabet can be learnt in a Catholic way", they thought he was a raving bigot, they thought he meant that nobody must ever read anything but a Latin missal.
But he meant what he said, and what he said is thoroughly sound psychology. There is a Catholic view of learning the alphabet; for instance, it prevents you from thinking that the only thing that matters is learning the alphabet; or from despising better people than yourself, if they do not happen to have learnt the alphabet.
[GKC, The Common Man 166-7]