Collected Essays (93 page)

He encouraged me to ask questions, and, feeling like Aladdin in the treasure cave, I asked him as many as I could think of. His mind was unbelievably fast and experienced. It seemed that, over the years, he had already thought every possible philosophical problem through to the very end.

Despite his vast knowledge, he still could discuss ideas with the zest and openness of a young man. If I happened to say something particularly stupid or naive, his response was not mockery, but rather an amused astonishment that anyone could think such a thing. It was as if during his years of isolated thought he had forgotten that the rest of the human race was not advancing along with him.

The question of why Gödel chose to live most of his life in splendid isolation is a difficult one. Although he was not Jewish, the Second World War forced him to flee Europe, and this may have soured him somewhat on humanity. Yet, he loved life in America, the comfortable position at the Institute, the chance to meet Einstein, the great social freedom. But he spent his later years in an ever-deepening silence.

The first time I saw Gödel, he invited me; the second two times, I invited myself. This was not easy. I wrote him several times, insisting that we should meet again to talk. Finally I phoned him to say this again.

“Talk about what?” Gödel said warily. When I finally got to his office for my second visit, he looked up at me with an expression of real dislike. But annoyance gave way to interest, and, after I’d asked a few questions, the conversation turned as friendly and spirited as the first. Still, toward the end of a conversation, when he was tired, Gödel would sometimes look at a visitor with an eerie mixture of fear and suspicion, as if to say, What is this stranger doing in my retreat?

Gödel was, first and foremost, a great thinker. The essence of the man is not to be found in his physical description, but rather in his ideas. I would like to describe now some of our discussions on mathematics, physics, and philosophy.

One of Gödel’s less well-known papers is a 1949 article called, “A Remark on the Relationship Between Relativity Theory and Idealistic Philosophy.” In this paper, probably influenced by his conversations with Einstein as well as by his interest in Kant, Gödel attempts to show that the passage of time is an illusion. The past, present and future of the universe are just different regions of a single vast spacetime. Time is part of space-time, but space-time is a higher reality existing outside of time.

In order to destroy the time-bound notion of the universe as a series of evanescent frames on some cosmic movie screen, Gödel actually constructed a mathematical description of a possible universe in which one can travel back through time. His motivation was that if one can conceive of time-travelling to last year, then one is pretty well forced to admit the existence of something besides the immediate present.

I was disturbed by the traditional paradoxes inherent in time travel. What if I were to travel back in time and kill my past self? If my past self died, then there would be no I to travel back in time, so I wouldn’t kill my past self after all. So then the time-trip would take place, and I would kill my past self. And so on. I was also disturbed by the fact that if the future is already there, then there is some sense in which our free will is an illusion.

Gödel seemed to believe that not only is the future already there, but worse, that it is, in principle, possible to predict completely the actions of some given person.

I objected that if there were a completely accurate theory predicting my actions, then I could prove the theory false—by learning the theory and then doing the opposite of what it predicted. According to my notes, Gödel’s response went as follows: “It should be possible to form a complete theory of human behavior, i.e., to predict from the hereditary and environmental givens what a person will do. However, if a mischievous person learns of this theory, he can act in a way so as to negate it. Hence I conclude that such a theory exists, but that no mischievous person will learn of it. In the same way, time-travel is possible, but no person will ever manage to kill his past self.” Gödel laughed his laugh then, and concluded, “The a priori is greatly neglected. Logic is very powerful.”

Apropos of the free will question, on another occasion he said:

“There is no contradiction between free will and knowing in advance precisely what one will do. If one knows oneself completely then this is the situation. One does not deliberately do the opposite of what one wants.”

As well as questions, I also brought in for Gödel’s enjoyment some offbeat theories of physics I had come up with recently. I was quite satisfied when, after hearing one of my half-baked theories, he shook his head and said, “This is a very strange idea. A bizarre idea.”

There is one idea truly central to Gödel’s thought that we discussed at some length. This is the philosophy underlying Gödel’s credo, “I do objective mathematics.” By this, Gödel meant that mathematical entities exist independently of the activities of mathematicians, in much the same way that the stars would be there even if there were no astronomers to look at them. For Gödel, mathematics, even the mathematics of the infinite, was an essentially empirical science.

According to this standpoint, which mathematicians call Platonism, we do not create the mental objects we talk about. Instead, we find them, on some higher plane that the mind sees into, by a process not unlike sense perception.

The philosophy of mathematics antithetical to Platonism is formalism, allied to positivism. According to formalism, mathematics is really just an elaborate set of rules for manipulating symbols. By applying the rules to certain “axiomatic” strings of symbols, mathematicians go about “proving” certain other strings of symbols to be “theorems.”

The game of mathematics is, for some obscure reason, a useful game. Some strings of symbols seem to reflect certain patterns of the physical world. Not only is “2 + 2 = 4” a theorem, but two apples taken with two more apples make four apples.

It is when one begins talking about infinite numbers that the trouble really begins. Cantor’s Continuum Problem is undecidable on the basis of our present-day theories of mathematics. For the formalists this means that the continuum question has no definite answer. But for a Platonist like Gödel, this means only that we have not yet “looked” at the continuum hard enough to see what the answer is.

In one of our conversations I pressed Gödel to explain what he meant by the “other relation to reality” by which he said one could directly see mathematical objects. He made the point that the same possibilities of thought are open to everyone, so that we can take the world of possible forms as objective and absolute. Possibility is observer-independent, and therefore real, because it is not subject to our will.

There is a hidden analogy here. Everyone believes that the Empire State Building is real, because it is possible for almost anyone to go and see it for himself. By the same token, anyone who takes the trouble to learn some mathematics can “see” the set of natural numbers for himself. So, Gödel reasoned, it must be that the set of natural numbers has an independent existence, an existence as a certain abstract possibility of thought.

I asked him how best to perceive pure abstract possibility. He said three things, i) First one must close off the other senses, for instance, by lying down in a quiet place. It is not enough, however, to perform this negative action, one must actively seek with the mind, ii) It is a mistake to let everyday reality condition possibility, and only to imagine the combinings and permutations of physical objects—the mind is capable of directly perceiving infinite sets, iii) The ultimate goal of such thought, and of all philosophy, is the perception of the Absolute. Gödel rounded off these comments with a remark on Plato: “When Plautus could fully perceive the Good, his philosophy ended.”

Gödel shared with Einstein a certain mystical turn of thought. The word “mystic” is almost pejorative these days. But mysticism does not really have anything to do with incense or encounter groups or demoniac possession. There is a difference between mysticism and occultism.

A pure strand of classical mysticism runs from Plato to Plotinus and Eckhart to such great modern thinkers as Aldous Huxley and D. T. Suzuki. The central teaching of mysticism is this: Reality is One. The practice of mysticism consists in finding ways to experience this higher unity directly.

The One has variously been called the Good, God, the Cosmos, the Mind, the Void, or (perhaps most neutrally) the Absolute. No door in the labyrinthine castle of science opens directly onto the Absolute. But if one understands the maze well enough, it is possible to jump out of the system and experience the Absolute for oneself.

The last time I spoke with Kurt Gödel was on the telephone, in March 1977. I had been studying the problem of whether machines can think, and I had become interested in the distinction between a system’s behavior and the underlying mind or consciousness, if any.

What had struck me was that if a machine could mimic all of our behavior, both internal and external, then it would seem that there is nothing left to be added. Body and brain fall under the heading of hardware. Habits, knowledge, self-image and the like can all be classed as software. All that is necessary for the resulting system to be alive is that it actually exist.

In short, I had begun to think that consciousness is really nothing more than simple existence. By way of leading up to this, I asked Gödel if he believed there is a single Mind behind all the various appearances and activities of the world.

He replied that, yes, the Mind is the thing that is structured, but that the Mind exists independently of its individual properties.

I then asked if he believed that the Mind is everywhere, as opposed to being localized in the brains of people.

Gödel replied, “Of course. This is the basic mystic teaching.”

We talked a little set theory, and then I asked him my last question: “What causes the illusion of the passage of time?”

Gödel spoke not directly to this question, but to the question of what my question meant—that is, why anyone would even believe that there is a perceived passage of time at all.

He went on to relate the getting rid of belief in the passage of time to the struggle to experience the One Mind of mysticism. Finally he said this: “The illusion of the passage of time arises from the confusing of the given with the real. Passage of time arises because we think of occupying different realities. In fact, we occupy only different givens. There is only one reality.”

I wanted to visit Gödel again, but he told me that he was too ill. In the middle of January 1978, I dreamed I was at his bedside.

There was a chessboard on the covers in front of him. Gödel reached his hand out and knocked the board over, tipping the men onto the floor. The chessboard expanded to an infinite mathematical plane. And then that, too, vanished. There was a brief play of symbols, and then emptiness—an emptiness flooded with even white light.

The next day I learned that Kurt Gödel was dead.

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Note on “Kurt Gödel”

Written 1981.

Appeared in
Science 82
, 1982.

My meetings with Kurt Gödel in 1971 and 1972 were among the great events in my life. I was very much in the position of the seeker meeting the great sage. A wonderful stroke of good fortune. I wrote up my recollections to use as a section in my book
Infinity and the Mind
(Birkhäuser, 1982,) and I also published this bit in
Science 82,
a popular magazine that insisted on their name every year, always staying in synch.

Martin Gardner

On the table is a drinking glass with a rubber membrane stretched across the top, taut as a drumhead. Resting on the membrane is a quarter from Martin Gardner’s pocket.

“Watch,” he says and pushes on the quarter with one finger.

POP! The quarter passes through the membrane and rattles in the bottom of the glass. I peer at the rubber sheet, looking for a hole, but there is none.

“It went through the fourth dimension,” Gardner says with a smile. “I have another fourth dimension trick. It’s called Card Warp. I make a little hyperspace tunnel that turns a card inside out.”

Martin Gardner knows a lot about magic. He knows a lot about other things, too. He has published books on logic, relativity, and particle physics. He has written books on magic, and books on mathematics. He is perhaps the world’s leading expert on
Alice in Wonderland, Through the Looking Glass,
and other works by Lewis Carroll. He has annotated the classic American baseball saga, “Casey at the Bat.” A few years ago he published an engaging and realistic novel about the intricacies of Protestant theology.

But among scientists, Martin Gardner is best known and loved as the dean of puzzledom, the ringmaster of mathematical games. Each month for the last 24 of his 66 years, Gardner has taken some abstruse branch of mathematics and transformed it into a series of profoundly entertaining puzzles and games in a column for
Scientific American.

His column is a place where the most serious of mathematicians come to play. He serves as impresario, often presenting problems as part of a larger discussion of a mathematical concept, adding twists and special effects to puzzles devised by others and creating many of his own.

Gardner’s topics have ranged from numerology to knot theory, from probability to plane geometry. Recently he wrote a pair of columns called “Nothing” and “Everything.” The collected columns fill nine volumes now, all but one volume in print. Meticulously researched and brilliantly written, each column stands as a permanent royal road into some region of mathematics.

Although he has earned the respect of professional mathematicians, Gardner is the quintessential amateur, casting his intelligence and wit toward whatever catches his eye, learning (and inviting his readers to learn) as he proceeds.

One would suppose that he has a strong background in mathematics—at the very least a master’s degree and probably a doctorate. Not so. He has never taken a college level course in mathematics.