Logic

Mr. Charles Ormsby said that he was in danger of being ‘sent down.’ More specifically, “I’ve tried almost everything, and failed. The last thing, now, is mathematics. You’ll soon find me hopeless, in which case my father will have to take me into his firm.”

Vic asked, “Will you also fail there?”

“Most certainly. My mistakes will cost him many thousands of pounds. I expect that he’ll be noticeably angry. When I point out that a gentleman shouldn’t be in business in the first instance, he may become apoplectic.”

Ormsby was tall and thin with a scrambled face and large glasses. He looked hopelessly unathletic, and also unlikely to be very successful with the ladies. However, he spoke too well to be as stupid as he claimed, or, indeed, to be stupid at all. Vic didn’t have much expertise in judging social class in England, but, given what Vicki had told him, he would have put Ormsby down as an upper-class person interested in literature. It was surprising only that he should have failed in that area. Anyhow, to work!

Ormsby was in a special program that was designed for people with no natural aptitude for mathematics, and it had its own instructional materials. The first object was to introduce the student to mathematical symbolism with a series of examples. The first one began with the sentence,

S1a. Every positive real number has a unique positive square root.

Ormsby was a little put off by the term, ’real number’, but, not wanting to get into imaginary numbers such as the square root of minus 1, Vic explained that it was any number that could be represented by a decimal. Then came the mathematical version of S1a:

S1b. Let x > 0. There exists a
unique y > 0 such that y^{2}= x.

That provoked: “I don’t agree at all with letting x be things. I can let the charwoman in to clean my rooms, and I can let my dog kill a neighboring cat, but I do not let inanimate objects, much less abstract ones, do things, thank you very much.”

This was something Vic hadn’t expected. And, then, there was that English way of saying, as it sounded, ‘thenk you very much.’ The intonation, particularly with the ‘much’, gave it the reverse meaning. Before Vic could react, the other continued, “And then, do numbers really exist? As soon as one names a number, there it is. You can’t then deny it, and there’s no point in saying that it exists unless it’s also possible to deny its existence.”

There was a lot going on in Ormsby’s head, and it occurred to Vic that Frank Collins would find him fascinating. Following Collins’ example, Vic decided to be as rigorous as possible.

Introducing the variable as a symbol
that could be replaced by the names of numbers, if done consistently,
Ormsby reluctantly agreed. Using logic and explaining the function of
parentheses, Vic introduced:

(x)(x + 0 =x)

with the reading, ‘For any x, x plus 0 is equal to x’. After grimacing and making funny gestures, Ormsby allowed that it made sense, and was, in fact, true. Vic then wrote down,

(x)[(Nx & x>0) => (Ey)(Ny
& y> 0 & y^{2} = x)]

He then read it out as:

For any x, if x is a real number and is positive (greater than 0), then there is at least one y which is a real number, is positive, and has a square is equal to x.

Vic was delighted when Ormsby replied, “That allows that there might be more than one such number, whereas the original said that there’s only one.”

“Excellent. We now have to remove that possibility, so we’ll add to the formula. We can simplify it if we assume that we’re only dealing with real numbers.”

(x){ x > 0 => (Ey)[( y >0
& y^{2} =x) & (z)[ z > 0 => (z^{2} = x
=> z = y)]]}

That took some explaining: “The second half says that, for any positive number, z, if it’s the square root of x, than it’s identical with the previously mentioned y. So there can only be one.”

“Well, I do see that this is an improvement.”

“The only trouble is that, using this kind of symbolism, it would take forever to get anywhere. That’s why mathematicians don’t like logic.”

“They must be an impatient lot.”

It gradually became clear that Ormsby didn’t want to study mathematics, but was happy to learn some logic. Vic was happy to oblige, but wasn’t sure what requirements Ormsby might have to meet. Not wanting him to be ‘sent down’, Vic went to see Collins the next day.

Collins was in Magdalen College, pronounced ‘Maudlin’, and it was even more forbidding than Harvard’s Widener Library, particularly for Americans.

When the first American Rhodes Scholars had come over, one Oxonian was said to have remarked, “I suppose that Americans do have a right to exist. But I don’t see why they have to exercise that right at Oxford.”

Of course, such comments were never serious. Moreover, virtually everyone recognized that Britain couldn’t have defeated Nazi Germany without the aid of America and Russia. Since almost everyone had also preferred Franklin Delano Roosevelt to Stalin, Americans, other than military people, were fairly well accepted at Oxford in their considerable numbers. But there were still jokes:

Magdalen was known for its high proportion of foreigners, particularly Americans. At a movie shown recently in a local theatre, there was a scene of cannibals paddling a large canoe rapidly down a river. Someone had stood up and shouted, “Well paddled, Magdalen!”

However, it was better to be ridiculed than hated, and Vic found the porters at the gate quite polite, and quite willing to direct him to Collins’ rooms.

Collins, unengaged, asked Vic in for tea. When asked about Ormsby, he laughed and replied, “He’s in college here, and I’ve met him occasionally. That young man is so well connected that he can do anything he wants short of assaulting the vice-chancellor. He won’t be sent down, and, if he wants to learn some logic, we have no reason to object. It’s not as if he wants to study alchemy and mix noxious chemicals in his rooms.”

“No. He’s certainly no danger to anyone. I think he’s actually quite intelligent, but very eccentric.”

“England is the world’s best place for eccentrics, and Oxford and Cambridge are the best places in England. How about you, Vic? You could join the people who stand in punts and joust with poles trying to knock one another into the water. Or you could invent something all of your own.”

“What do you do?”

“I’m an Australian. We’re descended from criminals, and we’re a practical people. We don’t engage in foolishness.”

“Vicki and I once did think of chasing butterflies with nets.”

“Very British. Particularly if done in quaint costumes.”

“I think we’ll have to wait for spring. It wouldn’t be a credible activity on a cold rainy day.”

Raising his teacup, Collins replied, “You aren’t going to let that stop you, are you?”

The next day, when Vic happened to point out to Ormsby the many connections between logic and computer theory and practice, he found him somewhat interested. He would have expected Ormsby to dismiss angrily any sort of technology, but he asked some questions. It turned out that he hated most people, and hoped that they would be replaced by computers. Vic pointed out, “Computers deal only with ones and zeros, and it can take some sophisticated programming even to get one to add up a column of figures.”

“How can that be?”

“The sum may overflow the register in use, and arrangements have to be made.”

“I read something about computers doing all sorts of things.”

“As for the future, no one really knows.”

“But we’re young. There’s a deal of time.”

“What people do you want to replace with them.”

“All those snivelly little people who create endless difficulties when one wishes to do the simplest things.”

What had at first sounded like Fascism on a grand scale came down to something quite limited and personal, the replacement of a rude post office official who said that a package wasn’t wrapped properly by a computer which would send it on its way without any backtalk. Vic had to laugh. But, then, he had on his hands a man who, for almost the first time, really wanted to learn. He could deal with that.

After his next mathematical session with Collins, Collins asked Vic about Ormsby. After filling him in, Vic added, “I might have him meet Vicki. Even though she’s an enthusiast, she might put to rest some of his wilder ideas about computers.”

Nodding his approval, Collins went on to say that he would like to meet Vicki himself, and asked when she would next be in Oxford. Vic was aware that Collins, a bachelor living in college in the traditional Oxford way, was hardly the monastic type. He seemed not to have a girl friend, but he looked at women in pubs, and liked joking with them. There weren’t many mathematical young women, and he would surely like Vicki.