Learning a Lesson with a Looking Glass

Happy Ascension! Today might be said to be the feast day of mathematics, since it was the day Jesus told His apostles to go out and matheteusate all nations... The Greek was translated to the Latin discipuli which usually is left as "disciples" - but it has another English translation: "the Learners".

Yes, as I mentioned recently, math is easy - and today being its feast day, I thought we'd look at another odd but common truth which you would never guess is mathematical, yet is one of its most important stones in its foundation: the truth we call Right and Left...

((click here to proceed))

Of course you want to hear some Chesterton before you suffer through such an odd affair, so I will supply it:

[Syme the policeman poet said] "...So it is in mere curiosit! y that I make my queries. First of all, what is it really all about? What is it you object to? You want to abolish government?"
"To abolish God!" said Gregory [the anarchist] opening the eyes of a fanatic. "We do not only want to upset a few despotisms and police regulations; that sort of anarchism does exist, but it is a mere branch of the Nonconformists. We dig deeper and we blow you higher. We wish to deny all those arbitrary distinctions of vice and virtue, honour and treachery, upon which mere rebels base themselves. The silly sentimentalists of the French Revolution talked of the Rights of Man! We hate Rights as we hate Wrongs. We have abolished Right and Wrong."
"And Right and Left," said Syme with a simple eagerness, "I hope you will abolish them too. They are much more troublesome to me."
[GKC The Man Who Was Thursday CW6:490]

In order to set my stage, I shall invoke another text, sometimes very Chestertonian in form, and one of my! favourite sources of delight: the famous "Calvin and Hobbes" ! comics o f Bill Watterson. If you have the collection called Attack of the Deranged Mutant Killer Monster Snow Goons you will find them on pages 35 and 37. Yes, perhaps they are on-line somewhere, but I cannot take the time to find them, so I will merely give you the dialog. You must bear in mind that "Calvin" is a six-year-old boy and "Hobbes" is his stuffed tiger who often speaks with Calvin when they are alone...

Calvin: (to Hobbes) Help me with this homework, OK? What 6 + 3?
Hobbes: 6+3, eh? Well, this one is a bit tricky. First we call the answer "Y" as in "Y do we care?" Now Y may be a square number, so we'll draw a square and make this side 6 and that side 3. Then we'll measure the diagonal. (He draws the square and labels it.)
Calvin: (staring at the drawing) I don't remember the teacher explaining it like this.
Hobbes. (waving the pencil knowingly) She probably doesn't know higher math. When you deal with high numbers, you need higher m! ath.
Calvin: (measures the diagonal) But this diagonal is just a little under two.
Hobbes: OK. Here, I'll draw a bigger square.

Now, assuming you've stopped laughing, you are wondering what this has to do with Chesterton. Oh, ye of little faith! How soon you have forgotten this important line:

Men can construct a science with very few instruments, or with very plain instruments; but no one on earth could construct a science with unreliable instruments. A man might work out the whole of mathematics with a handful of pebbles, but not with a handful of clay which was always falling apart into new fragments, and falling together into new combinations. A man might measure heaven and earth with a reed, but not with a growing reed.
[GKC Heretics CW1:117]

Obviously, Hobbes is proposing to do just that. In the intervening pages, Calvin's mom goes to meet with his teacher, producing some very funny allusions much in the s! tyle of A. C. Doyle's references in the "Sherlock Holmes" stor! ies to m ysterious cases about which we never hear anything more... (All C&H readers recall the "Noodle Incident" and all those sirens at noon...) Be that as it may, Calvin's father takes a hand and tries to instruct his son in the far greater mystery of simple addition:

Dad: Here, maybe this will make more sense. (pun in the text) I have eight pennies. I ask you for four more.
Calvin: I say forget it. You're the one with a steady paycheck.
Dad: Just give me four pennies. Good. How much money do I have now?
Calvin: Investments and all?
Dad: (exasperatedly) No. Just here on the table.
Calvin: Eight cents.
Dad: No, eight plus four is twelve, see? Count them up.
Calvin: But those four are mine!

Ah... yes, please finish laughing. And now attend.

What was Calvin's mistake? What was his father's mistake? What is going on here? Are we talking about money - investments and all - or something else? Is this the b! eginning of a sermon on ownership and property, a commentary on the Commandments, or an elaboration of Leo XIII's Rerum Novarum with reference to Distributivism, Distributism, or something similar?

No. (Though it could be... but not this time.) It is my attempt at trying to awaken you to a very mysterious property, one of the most well-known relations of all the relations of things that we know of: the property of adjacency, of "things-being-next-to-each-other" - that is, of the mystery of Right and Left. Calvin's father is a lawyer and so he never studied the special branch of mathematics called "Set Theory" - which can go by other names like "Discrete Math" or "Number Theory". I am not going to scare you, but I want you to read on, and try very hard to start noticing something you are seeing constantly - and then realize that there was once a time that you did not yet grasp it, as Calvin did not grasp it... and finally you learned it. Alas! , you learned it too well, for it became a commonplace, and yo! u forgot about it... until a lunatic like me comes along and reminds you that it is still there.

One of the things we are taught - I mean we computer people, and we scientists, and we mathematicians - is to pay attention to detail, and try to see things as they are. That is very Chestertonian, and you no doubt have heard me quote such things before:

I would insist that people should have so much simplicity as
would enable them to see things suddenly and to see things as they are.
[GKC ILN July 13 1907 CW27:506]
...the object of my school is to show how many extraordinary things even a lazy and ordinary man may see if he can spur himself to the single activity of seeing.
[GKC Tremendous Trifles 6]

Yes. Now here is the idea I want you to see: What do we mean by addition? Can there be various kinds of such a basic and obvious thing?

It is so dreadfully obvious to us in computing, since we have to work with! a MACHINE, which does NOT understand, and will not EVER understand... yes, it may be hard for you to realize, but we can do NOTHING AT ALL with computers unless we use the very small collection of tools which are part of the machine - these are symbols, actually, which stand for certain simple kinds of actions or tasks. Most people walk around assuming computers can do "math" - by which they mean adding and stuff like that.

Here I must be Scholastic, and shout, with a blow on the table, distinguo! (I distinguish!)

Of course they do NOT add. No more than a smaller river "adds" when it pours its water into a larger one at the confluence of the waters. (I thank my good friend lUkE for this analogy; you may also recall a Tom Petty song about the river kissing the ocean...)

But as interesting (or dull) as this question is, we are not going to talk about that today and here. It would take a while, too, and though it is interesting, we n! eed to get at something far more interesting and hard to expla! in - the idea that Calvin's father was trying to teach.

What do we mean by addition? What do we mean - I use this word "mean" here and I do not know what other word to use - what do we mean when we place something next to something else and try to consider them in the joint sense, as something which is now single, though paradoxically it remains composite and multiple?

Is this sounding fantastic? Or confusing? Oh, why am I struggling to write about this as I wish?

Pardon me while I invoke my muse...

Oh dear little Alice, whose name is Truth:
Oh thou who dwellest in the Wonderland of the Real World,
Call upon the Holy Spirit, Who pours out fire in sevenfold stream on those who beseech His aid,
and give me the words that will bring true light. Amen.

Ah... I could try all day to explain this, and yet fail - even if I tried with both hands...

AHA.

"I'm sure I didn't ! mean" Alice was beginning, but the Red Queen interrupted her impatiently.
"That's just what I complain of! You should have meant! What do you suppose is the use of a child without any meaning? Even a joke should have some meaning - and a child's more important than a joke, I hope. You couldn't deny that, even if you tried with both hands."
"I don't deny things with my hands," Alice objected.
"Nobody said you did," said the Red Queen. "I said you couldn't if you tried."
[Lewis Carroll, Through the Looking Glass, chapter 9 "Queen Alice"]

Ah. Now we have something. Let us try with both hands.

(I must control myself here - the hand is a wonderful thing, and there are books about it, not counting those on anatomy, but I must not write another just now.)

In order to proceed, let us talk about another kind of number: numbe! r as used the grammarians, which means whether a word is "! singular " (that is, just one) or "plural" (that is, more than one). This idea is so dramatic and so penetrating of thought that many languages embed the "number" in a word as an ending or suffix. (Some, like Vietnamese, use another method.)

But did you know that Greek and Egyptian and Hebrew have a third form of this grammatical number? Yes - besides "one" or "many" they had a special word for "two" of a thing. Even Latin and English have traces of this powerful urge to call attention to those very special things that come in pairs, like hands and gloves, feet and shoes - twins of various kinds.

There is something very interesting about our hands, and I might talk about the important idea of "chirality" (which comes from the Greek word for "hand") in chemistry, and note how most of the molecules of living things are "handed" - the amino acids are all "left-handed", and there are sugars named from the Latin words for right (dextrose) and left (levulose! ) - and then there are things in physics like the "right-hand rule" for electric current and magnetism, or the "spins" of subatomic particles... what a world. No wonder Gabriel Syme found them troublesome!

But there is another aspect of our hands which provides the foundation for addition, the one which Calvin's father was trying to teach his son... From an incredibly young age, we get used to the idea that we have "sides" to our bodies, and that we can place things on our left or on our right, and that we bring things together by reaching out our hands and then pulling inwards - we carry or bring back those two separate things, and by our own hands we make them ONE.

Hence, by our "trying with both hands" we take two things and "bring them back" to be one... In Latin, they said relatio. That is what a relation is, the bringing back, of two things, to be one thing.

Now, immediately there springs up a quirk, which is what neithe! r Calvin nor his father grasped, and why I have needed to writ! e about this at such length. Dad had eight pennies, Calvin had four. The pennies are put together, but in what kind of "together" were they put? Calvin saw ownership persisting even in the collection - perhaps those were his four rare Indian-head pennies, and did not want to lose sight of them. Calvin's dad saw the dreary summation he kept a calculator to help him with, and merely counted the collection as "twelve".

There are other views. If you look at digits rather than words, and try adding - no let us say placing together - Dad's 8 with Calvin's 4, you might say the result is 84 or maybe 48. Here you show a mystic wisdom, for this is "addition" of another form, know to computer scientists as "concatenation" or more formally "addition of the free monoid over a finite alphabet"... but I am scaring you with these terms.

Do not be scared. This is a good insight, and points out Syme's struggle with right and left. You most likely know that Hebrew and Arabic a! re written from right to left. Perhaps you have heard that there was a form of ancient Greek that was written in "boustrophedon" = as the cow plows, or as we might say these days, as the ink-jet printer prints: on the first line print from left-to-right, and then go down and print from right-to-left. But, you scream, there is a big difference between "dog" and "god"! Correct. And from the gloomy halls that you just ran from, the theoretician responds:
"That is because the free monoid is not commutative."

You wanted an answer, now try with both hands. But Chesterton also has answered for us:

I remember a man ... who told me he was on a spiritual plane ("we are on different planes") on which yes and no, black and white, right and wrong, right and left, were all equal. I regarded him as I should any boastful aviator who told me that from the height to which he had risen all London looked like an exact chess-board, with all the squares and streets th! e same size. In short, I regarded him as a liar. London street! s are no t equally long, seen from a flying-ship or from anywhere else. And human sins or sorrows are not equally serious, seen in a vision or anywhere else.
[GKC ILN Aug 15 1914 CW30:145]

Yes, Alice, there is a difference between left and right. The amino acids in living things are "left-handed". If someone brewed a batch of right-handed amino acids in a lab, they would be chemically identical, and contain nothing extra, and have nothing left out - yet no earthly thing could use them to build proteins, any more than a right shoe can go onto a left foot.

And that is the great mystery I am trying to reveal. It is not only the very living parts of our bodies which possess handedness, but also the powerful and splendid human invention of letters and numbers.

We need to see this, as hard as it may be. And if you need a further demonstration,

!rorrim a tuohtiw siht daer ot (sdnah htob htiw) yrT

Even the very letters and number! s use so casually have their right and left hands - a separation as final as the Last Judgement:

And he shall set the sheep on his right hand, but
the goats on his left. [Matt 25:33]

Therefore what God has separated, let no calculator join...

discrete math symbols