Solving the Problem

I was speaking with a friend recently about how some time ago (when I was doing my PhD) there was a big to-do about one of those talking dolls that said "Math is hard". (No, there is no hidden joke here because today is the feast of St. Matthias.) In one of those impromptu seminars grad students have in the hall or in the lab, we discussed the topic and I volunteered to try to find the official answer. Of course, I found it in Aquinas, but the answer came as a surprise to some.

((click here to find out the answer!))


The answer is that math is easy. Oh, yes, I know this provides just another reason to detest Aquinas and Thomism and medieval philosophy and the Catholic Church - or at the least, Chestertonian computer scientists who write about such things. I know you were expecting a discussion of that question, and you can find a short one here, but I have another topic today - though a related one. Hee hee.

Instead of dolls, I want to talk about another aspect of education, which may help me make some progress in my study of pedagogy... Incidentally, did I ever tell you that I used dolls to help teach computer science? Oh, yes, I did, I had forgotten. (See here for details!) What was I teaching? The curious and very general tool called "recursion" which also happens to be related to today's topic.

So what is today's topic again?

It's the very strange term which one hears in the Media from time to time, usually referring to how some grade-school teacher doesn't want to teach addition or such things, because she wants the children to learn "problem-solving skills". I've heard that a few too many times now, and think it is silly. You see, I am a computer scientist, and have been in the field for over 30 years, having faced a bizarre range of wacky questions both in academics and in industry - and except for a very tiny handful of tools, I cannot even imagine what they mean by a "problem-solving skill". One of the most powerful I have is finite state machines - I would undertake to teach an attentive fifth-grade class this, and have them reasonably fluent in a few weeks - but I doubt that anyone teaches that. It's too hard. Another tool very powerful, is recursion - which I taught to college students with those wonderful nesting dolls. But I am sure there would be restrictions to that pedagogical method, not to say a series of odd comments about the sanity and stability of the professor.

Well, I do not know what skills they mean, since the few useful methods we have to actually solve problems are not the ones they teach! And when I mention something that is traditionally taught in grade school - I guess I ought to say had been traditionally taught - that is, long division - these people throw things at me, whining about how we have calculators now. But I am not going to talk about long division either - maybe another day. Oddly enough if you cannot do long division, certain parts of mathematics and computer science will be forever closed to you as being beyond your understanding. How sad. (And I thought they say how they want the children to learn problem-solving skills???)

Let us instead look at another branch of mathematics, which you may have heard about indirectly without knowing there is such a thing. It is a very splendid and interesting branch, with very easy and fun parts, as well as very hard and exciting and difficult parts, rather like hikes or sporting events or even cooking recipes... I was motivated by a posting by our friend "Old Fashioned Liberal" who alluded to "degrees of separation". I have never seen the movie, but I seem to recall that the number "six" is used.

Now, I might spend a lot of energy - and blogg-space - writing about six. It is a very interesting number. It is called "perfect" because its factors other than itself sum to itself. (Two others like this are 28 and 496.) Or we might bring up "psephy": the very curious old trick from ancient days, wherein numbers are written as letters, and discuss the "number of the beast" (Rv 13:18) which has three sixes. But I won't.

Instead, I will tell you just two things about six, which perhaps may augment the topic of the movie - which I seem to recall hearing was the suggestion that each human is "related" by no more than "six degrees of separation" to each other human. Demonstrating such a thing would require a "problem solving skill" in spades - and I don't know if they proved it in the movie or just assumed it. But I happen to have a small acquaintance with one "problem solving skill" - that interesting branch of mathematics called "graph theory" - and it might play a role in solving the problem.

However, I don't feel up to tackling that issue today, so I will tell you about two others which are much easier and more fun. But first, since you may be thinking you got onto some math guy's blogg, and not the Chesterton blogg, I will give you the relevant Chesterton quote, which is really wonderful and may give you chuckles...

An infinite number of years ago, when I was the chief weakness of a publisher's office, I remember that there was issued from that establishment a book of highly modern philosophy: a work of elaborate evolutionary explanation of everything and nothing; a work of the New Theology. It was called "The Great Problem Solved" or some such title. When this book had been out for a few days it began to promise an entirely unexpected success. Booksellers sent to ask about it, travellers came in and asked for it, even the ordinary public stood in a sort of knot outside the door, and sent in their bolder spirits to make inquiries.

Even to the publisher this popularity seemed remarkable; to me (who had dipped into the work, when I should have been otherwise employed) it appeared utterly incredible.

After some little time, however, when they had examined "The Great Problem Solved", the lesser problem was also solved. We found that people were buying it under the impression that it was a detective story. I do not blame them for their desire, and most certainly I do not blame them for their disappointment. It must have exasperated them, it would certainly infuriate me, to open a book expecting to find a cosy, kindly, human story about a murdered man found in a cupboard, and find instead a lot of dull, bad philosophy about the upward progress and the purer morality. I would rather read any detective book than that book. I would rather spend my time in finding out why a dead man was dead than in slowly comprehending why a certain philosopher had never been alive.
[GKC "Reading the Riddle" in The Common Man 60]

Now, I promised two interesting things about six which will shed some light on OFL's phrase about "degrees of separation".

First, I will give you a very curious fact from graph theory, but translated. (I suspect this is the fact that lurks behind that movie.)

In any group of six people, there must be either three mutual acquaintances or three mutual strangers.

And now you prove it, Doctor?

Proof? You want proof? Hee hee. I'd rather ask another question, oh "skilled problem-solver": How many kinds of groups of six people can be formed, considering just the relation of being acquainted? How can you find out?

Please don't grab a sheet of paper, it will take you quite a little bit of time, since there are 156. (See here if you wish to see what they look like!) These groups run from six total strangers to six mutual acquaintances. If you thought there were 32768, you get partial credit, since that's how many distinct groups of "six-humans-related-by-acquaintance" there are, but not the kinds of groups. Here's what I mean. If one group has all strangers except for Andy and Bill, and another all strangers except for Edith and Francesca, these are the same "kind" since both have one pair of acquaintances.

(In graph theory I have asked the number of graphs having six vertices, which are unique up to isomorphism.)

The fun thing is to show it... but as you have pointed out, this is not a mathematics blogg.

Now, let us take something a bit more profound - and a bit more human - which also has six in it, though it requires seven people.

Huh? you say. How can that be?

That is the famous "fencepost problem", sometimes called "the error of plus or minus one" - a certain organist I know has a musical way of putting it: "Consonance is always just a half-step away." (hee hee) But you are distracting me, and you will understand in just a moment!

Here's the statement of the fact:

Every Catholic is related through no more than six links to every other Catholic.

Yes, and here is how the chain works:

Joe Catholic

His parish priest

His bishop

The Pope

Another bishop

Another priest

Another Catholic

Now, this particular kind of graph is called a "tree" - er - but I have no time to give you all the details now. (You may recall the words in St. John's Gospel about "I am the vine and you are the branches", which suggests that our Lord also knew about this.)

As you can see, there are seven individuals, and six links between them. In another class during grad school our professor called attention to this hierarchical structure, pointing out there are just four levels to the tree... he noted that the comparable structure for academics is far deeper. And yes, in case you hadn't caught on, the very common file technique for Unix and Windows and all comparable systems - the hierarchical file system of directories and files - parallels the structure of the Catholic Church! (Oh, yes, the "root" directory is just the computer's analogy for the Pope. How elegant that computing offers a demostration of the distinction between clergy and laity... alas, but that is a topic for another time and place.)

Lest you think I have gone completely askew into the depths of my tech world and forgotten Chesterton, I shall present Chesterton's own explanation, which keeps things linked, yes, whether pork or pyrotechnics, pigs or the binomial theorem...

...a philosophical connection there always is between any two items imaginable. This must be so, so long as we allow any harmony or unity in the cosmos at all. There must be a philosophical connection between any two things in the universe; if it is not so, we can only say that there is no universe, and can be no philosophy. A possible connection of thought there is, then, between any two newspaper paragraphs, though we may not always happen to think of it. And, as I say, it is my mental malady that I almost always do happen to think of it.
[GKC ILN Feb 17 1906 CW27:127]

And if that does not help, I will give you the real stinger, which goes back to my original matter of pedagogy and the far larger topic of the purpose of things...

I am sure that, in so far as there is any sort of social breakdown, it is not so much a moral breakdown as a mental breakdown. It is much more like a softening of the brain than a hardening of the heart. What does seem to me to have slackened or weakened is not so much the connection between conscience and conduct clearly approved by conscience, as the connection between any two ideas that could enable anybody to see anything clearly at all. It is not a question of free thought but of free thoughtlessness. The difficulty is not so much to get people to follow a commandment as to get them even to follow an argument. It seems to tire their heads like a game of chess when they are in the mood for a game of tennis. And in truth their philosophy does seem to be rather like a game of tennis, with the motto of "Love all." But, it will be noticed that the rules of tennis are really rather more arbitrary than the rules of chess; only, while they claim the same obedience, they are easier to obey. It seems to me that this modern mood does not mind anything being arbitrary so long as it is also easy. It does not inquire into the authority or even the origin of any order which it has come to regard as ordinary. It only asks to move smoothly along the grooves that have been graven for it by unknown and nameless powers - such as the powers that organise the tubes or the trams. It does not object to ruts if they are also rails. It does, indeed, wish to be comfortable, and will sometimes abandon convention for the sake of comfort. But it seems to me that this generation has rather less than its fathers and grandfathers of the special sort of discomfort that used to be called divine discontent. Divine discontent, of the older sort, was disposed to drive its questions backwards against the movement of existence and discover the causes of things. The old abstract revolutionist would have had the star-defying audacity to ask who it is who really runs the trams or controls the tubes. Most of the young rebels of to-day are content to ask whether they will not soon be made a little bigger or a little quicker or a little more convenient. In other words, the individual has indeed a certain kind of independence but I am not sure that it is the kind of independence which requires most intelligence.
[GKC ILN March 13 1926 CW34:58]

Maybe we need to start teaching graph theory in grade school... along with long division. It might just solve the problem! Math is easy, you know...