43
The composer Paul Hindemith was once conducting a rehearsal of one of his more dissonant orchestral compositions. At one point, he rapped his baton and said, “No, no, gentlemen; even though it sounds wrong, it’s still not right!”
44
The pianist Leopold Godowsky once visited a composer-friend and found him composing merrily away with operatic scores all over the piano. Godowsky said, “Oh, I thought you composed from memory!”
45
I once heard a radio interview with the pianist Artur Rubinstein. On the whole, the interviewer struck me as incredibly trite and stupid. Out of the blue, he asked, “Mr. Rubinstein, do you believe in God?” There was a tense pause. “No,” replied Rubinstein, quite definitely. “You see, what I believe in is something much greater!”
46
This anecdote reminds me of a riddle: What is it that’s greater than God; the dead eat it, and if the living eat it, they die? (See §53, for answer.)
47
When Mark Twain was asked what he thought of the music of Richard Wagner, he replied, “Oh, it’s probably not as bad as it sounds!”
48
Music and Mathematics. The mathematician Felix Klein was once at a party where the company was discussing the correlation between mathematics and music with respect to both tastes and aptitudes. Klein looked more and more puzzled and finally said, “But I don’t understand; mathematics is beautiful!”
49
Pitch and Color. Our visual spectrum happens to be less than one octave, that is, the highest frequency of light that we humans can perceive is not quite twice the lowest frequency. (By contrast, our auditory spectrum encompasses several octaves.) If our visual spectrum were a little more than an octave, I wonder whether two colors an octave apart would have the same psychological similarity as two notes an octave apart.
I once put this question to a rather famous Italian physiologist. He replied, “Ah, that’s a beautiful question!” I also put it to an equally famous musicologist. He answered in an irritated tone, “That’s obviously unverifiable!” So you see, the musicologist was really a logical positivist at heart (in the bad sense), whereas the physiologist was not.
But is the question really unverifiable in principle? Is it inconceivable that science might one day find a means of extending our visual spectrum? Perhaps it is. But isn’t it possible that we might one day meet intelligent beings from another planet whose visual spectrum is more than an octave and simply ask them?
50
Absolute Pitch. I was once riding in the front seat of a car driven by the computer scientist Dr. Marvin Minsky. In the backseat were two scientists from Bell Telephone Laboratories. The conversation turned to the subject of absolute pitch. Marvin said to them, “You know, Ray here has absolute pitch.” One of the two asked me, “How accurate is your sense of absolute pitch?” For some odd reason or other, I didn’t hear the question, so he said somewhat louder, “I say, how accurate is your sense of absolute pitch?” Upon which, Marvin turned around and said to them, “Oh, I forgot to tell you—he’s also deaf!”
51
Once at a mathematics conference, one of the speakers gave me an account of a paper he was about to deliver. I found the account incomprehensible. As he was talking, Marvin Minsky walked by and said to him, “No, no; your trouble is that you’re confusing a thing with itself!”
52
I love Marvin Minsky’s quote on the jacket cover of the book The Mind’s I. 4
This great collection of reflections provides you with your own quite special ways to understand things such as why, if you don’t read this book, you’ll never be the same again.
53
The answer to the riddle of §46 is nothing.
54
I have told several philosophers that despite my great love for the Taoist philosophers Laotse, Chuangtse, and Liehtse, perhaps my favorite philosopher of all is Ferdinand the Bull. One of them took this seriously and earnestly tried to convince me that Ferdinand couldn’t be a philosopher. “A philosopher is necessarily human,” he said. I can’t see why this must be true! Didn’t Ferdinand have a pacifist philosophy?
55
1 read in some philosophy book or other that perhaps the one true philosopher was the little girl of nine who was looking out a window and suddenly turned to her mother and said, “But what puzzles me is why there is anything at all!” The following comments, made by children I have known, have definite philosophical overtones.
Vincent (aged 3). When Vincent was about to go up in an airplane for the first time, he asked his father, “When we go up, will we also get small?”
Barry (aged 5 or 6). Barry once said, “I hope I never get to be ninety-nine!”
“Why?” I asked.
“Because when you get that old, you could die!”
Miriam (aged 8). Miriam is the daughter of a mathematical logician. She has either inherited or acquired many of her father’s characteristics. At one point during dinner, her father said, “That’s no way to eat, Miriam!”
She replied, “I’m not eating Miriam.”
Jennifer (aged 6). Jennifer is the daughter of a philosopher. One morning, her brother Jon (aged 8) came down to breakfast and played one April Fools’ joke after another on the parents. Then Jennifer came down, and Jonny tried an April Fools’ joke on her.
“What’s the matter with you, Jonny,” she said. “Today’s not April Fool!”
“It isn’t?” he cried in astonishment.
“April Fool!”
On another occasion, Jennifer had just come home from a movie. She said to her mother, “Mommy, what is the best movie ever made? And I don’t want you to tell me what you think is the best movie; I want you to tell me what is the best movie.”
David (aged 10). My wife and I were once with David’s family at a drive-in theater. The first feature was excellent, but the second feature looked as if it were going to be terrible. One of the adults suggested that we leave. David of course wanted to stay, and so an argument began.
“Why don’t we take a vote?” I suggested.
“No!” said David. “That’s not fair because the majority will win!”
Natalie (aged 8). Natalie is the daughter of a mathematical logician. The family was visiting us for a weekend, and one evening we all had a lively philosophical discussion about time. For some perverse reason or other, I took the position that time is unreal.
Next morning at breakfast, someone asked of two acquaintances which was the older?
“Bill is older by two years,” I remarked.
“How could he be?” asked Natalie. “Didn’t you say that time was unreal?”
56
Natalie’s remark reminds me of G. E. Moore’s famous proof of the existence of an external world. He held up a hand and said, “Here is a hand.” Then he held up his other hand and said, “Here is another hand. Hands are objects, hence objects exist.”
I am also reminded of a conversation I once had with the logical positivist O. Bowsma. I took an extreme view, holding that minds were essentially independent of bodies.
“I can easily imagine myself in another body,” I said. “I am fully prepared for the possibility that next week I might find myself in a totally different body, say, one with three arms.”
“You are really prepared?” asked Bowsma.
“Absolutely!” I replied.
“Tell me,” said Bowsma, “have you bought yourself another glove?”
57
As the conversation continued, I became more and more wildly idealistic. Bowsma had an objection to just about every statement I made.
“Tell me,” I finally asked, “do you believe I am being inconsistent?”
“No,” he replied.
Another philosopher present said, “What you are saying is too vague to be inconsistent!”
58
There is a curious thing about inconsistency. In the formal mathematical systems mainly in use today, consiste
ncy is absolutely essential, for without it the whole system breaks down and everything can be proved. It has therefore been argued that if a person is inconsistent, he will end up believing everything. But is this really so?
I have known many inconsistent people, and they don’t appear to believe everything. First, it is difficult to live long enough to believe everything. Second, even if we were immortal and inconsistent, we would not necessarily believe everything. I say this for the following reason: If we were consistent in our inconsistency, then we might end up believing everything, but it is more likely that an inconsistent person would be just as inconsistent in the way he carried out his inconsistency as he is about other things, and this would be the very thing that would save him from believing everything.
The inconsistent people I have known have not seemed to have a higher ratio of false beliefs to true ones than those who make a superhuman effort to maintain consistency at all costs. True, people who are compulsively consistent will probably save themselves certain false beliefs, but I’m afraid they will also miss many true ones!
59
Here is a little paradox:
YOU HAVE NO REASON TO BELIEVE THIS SENTENCE.
Do you have any reason to believe the above sentence or don’t you?
60
Have you heard the business executive’s paradox? It was invented by the literary agent Lisa Collier of Collier Associates. The president of a firm offered a reward of $100 to any employee who could provide a suggestion that would save the company money. One employee suggested, “Eliminate the reward!”
61
My favorite paradox of all is known as hypergame. It is due to the mathematician William Zwicker.
A game is called normal if it has to terminate in a finite number of moves. An obvious example of a normal game is tic-tac-toe. Chess is also a normal game, assuming tournament regulations. Now, the first move of hypergame is to state which normal game is to be played. For example, if you and I were playing hypergame and I had the first move, I might say, “Let’s play chess.” Then you make the first move in chess, and we continue playing chess until the termination of the game. Another possibility is that on my first move in hypergame, I might say, “Let’s play tic-tac-toe,” or “Let’s play casino,” or any other normal game I like. But the game I choose must be normal; I am not allowed to choose a game that is not normal.
The problem is, Is hypergame itself normal or not? Suppose it is normal. Since on the first move of hypergame I can choose any normal game, I can say, “Let’s play hypergame.” We are then in the state of hypergame, and it is your move. You can respond, “Let’s play hypergame.” I can repeat, “Let’s play hypergame,” and the process can go on indefinitely, contrary to the assumption that hypergame is normal. Therefore, hypergame is not a normal game. But since hypergame is not normal, on my first move in hypergame I cannot choose hypergame; I must choose a normal game. But having chosen a normal game, the game must finally terminate, contrary to the proven fact that hypergame is not normal.
An amazing paradox indeed!
62
A Moral Paradox. The philosopher Jaako Hintikka makes the delightful argument that one is morally obligated not to do anything impossible. The argument, which ultimately rests on the fact that a false proposition implies any proposition, is this: Suppose Act A is such that it is impossible to perform without destroying the human race. Then surely one is morally obligated not to perform that act. Well, if Act A is an impossible act, then it is indeed impossible to perform it without destroying the human race (since it is impossible to perform it at all!), and therefore one is morally obligated not to perform the act.
63
But doesn’t the following argument (sic!) show that one is morally obligated to do everything that is impossible?
Suppose that Act B is such that if one performs it, then the human race will be saved from destruction. Isn’t one then morally obligated to perform the act? Now suppose that Act B is impossible to perform. Then it is the case that if one performs Act B, the human race will be saved, because it is false that one will perform this impossible act and a false proposition implies anything. One is therefore morally obligated to perform every impossible act.
64
For those who like logic puzzles, here are some nice ones.
Problem 1. There are three brothers named John, Jack, and William. John and Jack always lie (make only false statements) and William makes only true statements. The three are indistinguishable in appearance. One day you meet one of the three brothers on the street and wish to know if he is John (because John owes you money). You are allowed to ask him only one question, which has to be answered by yes or no, and the question may not contain more than three words. What question would you ask? (The solutions to this and the next four problems are given in §66.)
Problem 2. Suppose in the last problem we change the conditions and make John and Jack the truth tellers and William the liar. You still want to know whether the brother you meet is John. Now what three-word question will work?
Problem 3. This time we have only two brothers. One is named John, and the other is not. One of the two always lies, and the other always tells the truth, but we don’t know whether John is the liar or the truth teller. You meet both brothers together, and you wish to find out which one is John. You are allowed to ask either one of them a three-word question. What question will do the trick?
Problem 4. Suppose in the last problem you are not interested in which brother is John but only in whether John is the truthful brother or the brother who lies. What three-word question will enable you to find out?
Problem 5. I once met these two brothers on the street. I had two distinct questions in mind. I knew that if I asked the first question, I would then know the correct answer to the second question, whereas if I asked the second question, I would then know the correct answer to the first question. Can you supply two such questions?
65
A Godelian Machine. Of the many mathematical machines I have used to illustrate Godel’s famous proof, the following is the simplest.
The machine prints out various expressions composed of four symbols: P,N,R,*. An expression is called printable if the machine can print it. A sentence is any expression of one of the four forms: (1) P*X; (2) NP*X; (3) PR*X; and (4) NPR*X, where X is any expression built from the four symbols. Each sentence is interpreted as follows: (1) P*X is called true if and only if X is printable; (2) NP*X is called true if and only if X is not printable (N is an abbreviation of not, just as P is an abbreviation of printable); (3) PR*X is called true if and only if XX is printable (XX is called the repeat of X, hence the letter R); and (4) NPR*X is called true if and only if XX is not printable.
We are given that the machine is completely accurate, that is, every sentence printed by the machine is a true sentence. The problem is to find a true sentence that the machine cannot print! (The solution to this problem and a discussion are given in §67.)
66
Solutions to the Puzzles of ∫64. For Problem 1: A question that works is, “Are you Jack?” and it is the only three-word question I can think of that does work!
Jack and William would both answer no to that question (Jack because he lies and William because he is truthful). John would answer yes (because John lies). Therefore, if you get yes for an answer, you will know that he is John, and if you get no for an answer, you will know that he is not John.
As for Problem 2: The very same question works! Only now a yes answer indicates that he is not John, and a no answer indicates that he is John.
Now, the solution to Problem 3: This is more subtle. The question, “Are you John?” is useless. Whatever answer you get could be the truth or could be a lie. A question whose correct answer you already know (such as, “Is water wet?”) is no good. You will then know whether the one addressed is truthful or not, but you won’t know whether or not he is John. The question, “Are you truthful?” is no good. You will get the answer y
es from both brothers.
A question that does work is, “Is John truthful?” John would certainly claim that John is truthful (regardless of whether or not John is truthful). John’s brother would claim that John is not truthful (correctly if John’s brother is truthful and falsely if John’s brother lies). So, a yes answer indicates that the one addressed is John. A no answer indicates that he is not John.
Another three-word question that works is, “Does John lie?” A yes answer to that question indicates that the one addressed is not John. A no answer indicates that he is John.
As for Problem 4: To find out whether John is truthful, the question, “Are you John?” now works! Suppose you get yes for an answer. If the speaker is truthful, then he really is John, in which case John is truthful. If the speaker is lying, then he is not John, hence John must be the truthful brother. So, whether the speaker is lying or telling the truth, a yes answer indicates that John is truthful. We leave it to you to verify that a no answer indicates that John is not truthful, regardless of whether the speaker is lying or telling the truth.
There is a pretty symmetry between the last two problems. To find out whether the one addressed is John, you ask him whether John is truthful, whereas if you want to find out whether John is truthful, you ask him whether he is John. This provides a solution to Problem 5: One question is, “Is John truthful?” The other question is, “Are you John?” Asking either question will enable you to know the correct answer, not of the question you ask, but of the other question.
Five Thousand B.C. and Other Philosophical Fantasies Page 4
