The World as I Found It (11 page)

Mary Queen of Scots!

The Headless Horseman!

Cronus eats his young! Or was it Rhea?

Eats his young? gasped Eddie, catching his chest. Eats his —

And with that, Eddie cut loose with another shriek.

B
UT WHY
would you concern yourself for three years with a paradox about a Cretan who said all Cretans were liars?

Oh, more than three years, said Russell, delighted with Ottoline's incomprehension. Five years I spent—God, more, once I started wrestling with such nonsense as the round square and the present king of France. The problem with these last two, the king and the square, is how denoting phrases like these can describe, with seeming verisimilitude, nonexistent squares and monarchs. To the layman, it sounds silly, wasting one's time on puzzles like this, but you see such absurdities are the experiments of the logician. A theory can't hold if it works only part of the time or works only in certain isolated cases. The absurdity, the exception, no matter how trivial, belies the crack in the theory that sends the good ship
Ars Logica
to the bottom …

It was a warm, overcast day, and they were sitting on the sand, talking as the waves coughed and sloughed to their feet. Things between them were better now that the two last guests, Lytton and Eddie, were gone. That morning, Russell had left on the same train with them, traveling as Eddie's guest in his first-class compartment. Ottoline's house staff were on the same train, third class. Russell didn't travel far. When they reached Winchester he got off, saying he was stopping there to visit an aunt. It was an indifferent story — Lytton certainly didn't swallow it — but Russell didn't care. It was enough for decorum's sake.

Before noon he was back at Studland, and now, except for Ottoline's personal maid, Brindy, they were alone. Better yet, they were feeling that sense of repose that comes after having made love — once urgently in his cottage, and then again outside, this time against a tree, his trousers round his ankles as he struggled to roll a condom over his plumb bob before she changed her mind or thought she heard someone coming. Russell was still picking bark from his palm.

They both agreed it was better, infinitely better. With less than a day left, they were now working hard at putting a gloss on the weekend, both eager to forget the tension and remember instead this
one perfect day
. Sitting there, Ottoline felt she saw with fresh eyes this brilliant, difficult man whom, for all her doubts, she still loved. With marked detachment, and some pleasure, she had watched him suffer at dinner the night before, but now, having purged herself of resentment, she was nursing him so he could withstand those barren weeks without her in his Cambridge bachelor rooms. Yet even here, her impulse was not entirely selfless: as she well knew, the happier he was when they parted, the less demanding he would be when she turned her attentions once more to Lamb.

As for Russell, he was determined to be cheerful and optimistic. He would not dwell on his imminent departure, nor on thoughts of Lamb or other petty jealousies. He very much wanted to be diverted, and he was flattered when Ottoline asked for her first lesson in logic so she might better understand his mind and work. Even if it was a bit of a sop to assuage his bruised ego, she was genuinely curious. He, on the other hand, was anxious to improve her mind, and he wasn't starting from nothing. Some years back in Edinburgh, much to her brother Arthur's annoyance, Ottoline had spent a year in college, where she had taken a general course in logic. Unfortunately, Ottoline had done badly in the course, further diminishing her already precarious sense of her abstract mental abilities. But here Russell was quick to put her at ease, promising to be simple and clear and to start at the beginning. So saying, he began by telling her that the logic she had studied was of the old, Aristotelian kind, no doubt employing syllogisms of the sort called Barbara:

Russell was saying: This kind of reasoning dominated Western logic for two thousand years, and in some quarters, especially church quarters and the schools, it still dominates logic and severely hampers it. Aristotle employed many other types of syllogism besides Barbara, and if all logic were syllogistic, this would be splendid. But real progress came only with the modern recognition of asyllogistic processes, which of course confound syllogistic reasoning. Leibnitz made some progress, but even he had too much respect for Aristotle to break his hold. So the real modern period of logic dates from the publication of Boole's
Laws of Thought
in 1854. But Peano and Frege, working independently, were the ones who made the biggest contribution to modern logic.

And not yourself? asked Ottoline cattily.

Well, he said with an embarrassed smile, I'm getting to my place in things. But to return to Peano and Frege: one great contribution they made was to show that propositions that traditional logic thought to be of the same form were in fact quite different. Take the propositions “Socrates is mortal” and “All men are mortal.” Aristotelian logic would say they are of the same form. But consider: “Socrates is mortal” has Socrates — a single man — for its subject, whereas “All men are mortal” takes as its subject a universal class consisting of all men. The persistent failure to grasp fundamental logical distinctions like these made for all manner of bad metaphysics and generally bad philosophy. Modern logic has finally uncovered many of these problems, and one way it achieved this was through the development of logical notation.

So saying, Russell took a stick and drew on the wave-smoothed sand:

stands for
not

   stands for
or

    stands for
and

  stands for
therefore

Hence, he said, you might write in signs:

Meaning: either
p
or
q
; and not
p
; therefore
q
.

Ottoline rolled her eyes. And for you this is as simple as one, two, three.

One gets used to it, he demurred. I can't use it to order supper.

Mmmm … Or to mind your p's and q's.

In any event, he resumed, silently drawing other signs in the sand, as you can see, there are other symbols and more complex propositions. The advantage here is that the signs are more easily taken in at a single sweep. Avoiding the connotations of words, they isolate the sheer logic — the bones — of a statement, showing something that is at once simple and highly abstract.

I see, I see, said Ottoline, rubbing her arms. It
is
getting abstract, isn't it.

No, not so abstract, soothed Russell. Just listen, please … Now, where was I? Right! Frege and Peano came to logic through mathematics. I also came to logic through mathematics, but with a more philosophical bent. You see, mathematics is most philosophical in its beginnings, when we ask such general questions as how we can deduce one thing from another, or what logic even is.

Anyway, I had long wanted to systematically reduce mathematics to logic. Even as a boy I can remember asking my brother why a mathematical axiom was so, only to hear him reply, Because it is so. In college, I found this lack of a foundation in mathematics even more bothersome. Hegel's
Greater Logic
I thought was muddled nonsense. I found Kant's contention that mathematics and logic are independent of experience equally unsatisfactory. Why was I to believe that arithmetic consists of empirical generalizations that somehow work? I couldn't tolerate it. What I wanted to establish was a way of deducing mathematics that was rigorous, defensible and scientific. At the Paris conference in 1900, Peano and his students very much impressed me — their discussions were so extraordinarily precise! Well, part of the reason was the logical notation that Peano used. So taking Peano's notation, I invented my own, more extensive notation for logical relations. It was like a microscope. Suddenly, I was able to see to the root of questions that hitherto had eluded me. Russell sighed. What I didn't know then was that Frege had already covered much of the same ground.

Oh, how awful! said Ottoline.

Well, said Russell, betraying a slight smirk of satisfaction. As things turned out, it was rather worse for Frege. You see, in 1901 Cantor made a proof that there was no greatest cardinal number —

Cardinal number?

Don't concern yourself with that. All you need to know is that while working with Cantor's proof, I discovered a contradiction having to do with classes. This posed no small problem — in my mathematics, I had defined number in terms of class, with classes of classes, and classes of classes of classes, and so forth. So it wasn't a silly little logician's conundrum at all. It was really quite fundamental. And the question was, if we had a class — say, the class of odd numbers — could that class be a member of itself?

Wait, protested Ottoline dizzily. Please, you mustn't run on like this. You must go more slowly.

No, no — Already he was gesturing. Really, it's not that hard to see. It goes back to our Cretan — the one who called all Cretans liars. Consider: if what the Cretan says is true, then he's a liar and his statement is false. If what he says is a lie, on the other hand, then he's telling the truth while at the same time lying. The same principle applies to the contradiction I found in Frege's mathematics.