Five Thousand B.C. and Other Philosophical Fantasies Page 5
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Solution to the Gödelian Machine Puzzle. The sentence NPR*NPR* says that the repeat of NPR* is not printable (i.e., the sentence is true if and only if the repeat of NPR* is not printable). But the repeat of NPR* is the very sentence NPR*NPR*! So this sentence is true if and only if it is not printable. If the sentence is false, then it is printable (since the sentence says that it isn’t), which would mean that the machine is capable of printing a false sentence. But we are given that the machine is accurate and never prints false sentences. So it must be that the sentence is true, hence what it says is really the case, which means that it is not printable. And so the sentence NPR*NPR* must be true, but the machine cannot print it.
The philosopher J. Michael Dunn once showed the above problem to his son, Jon. After the boy understood it, he said to his father, “One thing I would like to know. Why would anybody want to construct such a machine?” The father thought for a moment and replied, “Well, it would be nice if we could have an accurate machine that could print out all true facts about the world. But it seems that such a machine is not possible.”
The whole point, of course, is that no accurate machine can possibly print a sentence that says that the machine cannot print it. In the very process of printing it the machine would falsify it! The situation is reminiscent of the scene in Romeo and Juliet in which the nurse comes running to Juliet and says, “I have no breath.” Juliet replies, “How can you have no breath when you have breath left to say ‘I have no breath’?”
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I started this chapter with self-annihilating sentences, and it seems only fitting to end with what might aptly be called a self-annihilating conversation. The dialogue is not original. It was devised by Goodwin Sammel, a musician I first met in my University of Chicago days who has always taken an interest in mathematical matters. When he first heard about Gödel’s theorem he came up with this exchange:
A: It’s true!
B: It’s not!
A: Yes, it is!
B: It couldn’t be!
A: It is true!
B: Prove it!
A: Oh, it can’t be proved, but nevertheless it’s true.
B: Now, just a minute: How can you say it’s true if it can’t be proved?
A: Oh, there are certain things that are true even though they can’t be proved.
B: That’s not true!
A: Yes, it is; Gödel proved that there are certain things that are true but that cannot be proved.
B: That’s not true!
A: It certainly is!
B: It couldn’t be, and even if it were true, it could never be proved!
Notes
1
Raymond Smullyan, What Is the Name of This Book? (Englewood Cliffs, N. J.: Prentice-Hall, 1978), p. 208.
2
Sigmund Freud, The Future of an Illusion (Third edition, The International Psycho-Analytical Library # 15, translated by W. D. Robson-Scott, The Hogarth Press, 37, Mecklenburgh Square, London, and the Institute of Psycho-Analysis, 1943).
3
Ambrose Bierce, The Devil’s Dictionary (New York: Hill & Wang, 1957).
4
Douglas Hofstadter and Daniel Dennett, The Mind’s I (New York: Basic Books, 1981).
4
A Query
I have one Catholic friend with whom I have had many a lively discussion. I once asked him which of the following two types, both of which act extremely well by Christian standards, he considered the better: The first type is sympathetic and kind by nature and acts lovingly toward his neighbor simply because he feels like it; he does not have to force himself to do so. He does not act the way he docs because of any moral principle nor out of obedience to any commandments; he simply feels like acting as he does. By contrast, the second type, though he in fact behaves as well as the first, does not do so spontaneously but forces himself to do so because he believes it is the right thing to do. I asked my friend which type was the better. After thinking for a moment, he replied, “The first one sounds to me as if he is more in a state of grace, but the second one’s actions have more salvation value.”
An apparently similar viewpoint appears in Meister Eckhart’s ninth talk of instructions, titled “How the Inclination to Sin Is Always Beneficial.”1
Know that the impulse to wrong is never without use and benefit to the just person. Let us notice that there are two sorts of people involved. One is so constituted that he has little or no impulse to do wrong, whereas the other is often strongly tempted. His outward self is easily swayed by whatever is at hand—swayed to anger, pride, sensuality or whatever, but his better nature, his higher self, remains unmoved and will do no wrong, or be angry, or sin in any way. He therefore fights hard against whichever vice is most natural to him, as people must who are by nature choleric, proud, or otherwise weak and who will not commit the sin to which they are liable. These people are more to be praised than the first kind. Their reward is also greater and their virtue of much higher rank. For the perfection of virtue comes of struggle, or, as St. Paul says, “Virtue is made perfect in weakness.”
The impulse to sin is not sin but to consent to sin, to give way to anger, is indeed sin. Surely, if a person could wish such a thing, he would not wish to be rid of the impulse to sin, for without it he would be uncertain of everything he did, doubtful about what to do, and he would miss the honor and reward of struggle and victory. Because of the impulse to evil and the excitement of it, both virtue and its rewards are in travail born. The impulse to wrong makes us the more diligent in the exercise of virtue, driving us to it with a strong hand, like a hard taskmaster, forcing us to take shelter in doing well. The weaker one is, the more he is warned to strength and self-conquest; for virtue, like vice, is a matter of the will.
Despite my general fondness for mystic writers, there are several things about the above passage that I find extremely disturbing. For one thing, certain parts are downright illogical! Is it really true that if a person had no impulse to sin he would be uncertain of everything he did and doubtful about what to do? Are saints and angels uncertain and doubtful about what to do? Aren’t there other more useful and interesting things with which to occupy one’s time than “exerting noble efforts to overcome one’s sinful impulses”?
I think that Benjamin Franklin really hit the nail on the head in his astute essay titled “Self-Denial Not the Essence of Virtue.”2 He begins the essay as follows:
It is commonly asserted, that without self-denial there is no virtue, and that the greater the self-denial the greater the virtue.
If it were said, that he who cannot deny himself any thing he inclines to, though he knows it will be to his hurt, has not the virtue of resolution or fortitude, it would be intelligible enough; but, as it stands, it seems obscure or erroneous.
Let us consider some of the virtues singly.
If a man has no inclination to wrong people in his dealings, if he feels no temptation to it, and therefore never does it, can it be said that he is not a just man? If he is a just man, has he not the virtue of justice?
If to a certain man idle diversions have nothing in them that is tempting, and therefore he never relaxes his application to business for their sake, is he not an industrious man? Or has he not the virtue of industry?
I might in like manner instance in all the rest of the virtues; but, to make the thing short, as it is certain that the more we strive against the temptation to any vice, and practice the contrary virtue, the weaker will that temptation be, and the stronger will be that habit, till at length the temptation has no force, or entirely vanishes; does it follow from thence, that in our endeavors to overcome vice we grow continually less virtuous, till at length we have no virtue at all?
Franklin then further elaborates his point. He ends the essay with the following paragraph:
The truth is, that temperance, justice, charity &c. are virtues, whether practiced with, or against our inclinations, and the man, who practices them, merits our love and es
teem; and self-denial is neither good nor bad, but as it is applied. He that denies a vicious inclination, is virtuous in proportion to his resolution; but the most perfect virtue is above all temptation [emphasis added]; such as the virtue of the saints in heaven; and he, who does a foolish, indecent, or wicked thing, merely because it is contrary to his inclination (like some mad enthusiasts I have read of, who ran about naked, under the notion of taking up the cross), is not practicing the reasonable science of virtue, but is a lunatic.
Notes
1
Meister Eckhart, “How the Inclination to Sin Is Always Beneficial.” In Meister Eckhart, A Modern Translation, Raymond Bernard Blakney (New York and London: Harper and Brothers, 1941), p. 12.
2
Benjamin Franklin, “Self Denial Not the Essence of Virtue.” In Pennsylvania Gazette (February 18, 1734).
3
Three Fantasies
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Simplicus and the Tree—An Open Air Symposium
SIMPLICUS: I am enjoying this tree.
FIRST PHILOSOPHER: No, it is not the tree you are enjoying but the light from the tree. It is not the tree that is directly influencing your sense organs but only its reflected light. Therefore, you are enjoying the light of the tree.
SECOND PHILOSOPHER: No, no, it is not the light he is enjoying but rather the image the light forms on his retina.
THIRD PHILOSOPHER: This is superficial physiology! The retinal image could not affect him if his optic nerves were dead, and his optic nerves are but part of his brain and nervous system. Therefore, what he is really enjoying is the neural activities of his entire brain and nervous system.
FOURTH PHILOSOPHER: I think that it is misleading to say that he is enjoying this physiological activity; I would instead say that his enjoyment of the tree is this physiological activity.
CARTESIAN: All of you are wrong! His physiological process is only the material counterpart of an inner mental or spiritual process; it is this spiritual soul-activity that he is enjoying.
IDEALIST: Except that the evidence for what you call the material counterpart of this mental process is, as I have demonstrated, wholly inconclusive. I don’t believe in the existence of this “tree.” The proper way to phrase it, therefore, is that the mind or soul of Simplicus is enjoying his idea of the tree.
IDEALISTIC MYSTIC: I deny the existence of individual minds. There is no such thing as the mind of Simplicus! There is only one universal mind, called the world soul, cosmic consciousness, God, the Absolute, or whatever, and it is this universal or absolute mind that is enjoying the tree, which exists as one of its ideas.
REALISTIC MYSTIC:1 The viewpoint of my friend the idealistic mystic is about the opposite end of the spectrum from mine, and yet it comes closer to mine—in the sense of abstract identity or isomorphism—than any other yet expressed.
I start from the premise that reality is purely material. All that exists is the material universe, which for certain purposes might be broken down into material particles and their motions. Simplicus’s enjoyment of the tree is therefore indeed an event or set of events in the nervous system of the body of Simplicus. This viewpoint, though correct, seems to me only partial. Simplicus is not a closed physical system. When Simplicus has a thought, the particles of the cerebrum of Simplicus move not only in relation to each other but also in relation to every particle of the entire universe. I therefore wish to look upon the thoughts of Simplicus as an activity of the universe as a whole. Thus, instead of saying that it is Simplicus enjoying the tree, I would say it is the whole physical universe that is enjoying the tree.
FIRST LOGICAL POSITIVIST: I wonder if the viewpoints of the idealistic and realistic mystics really differ in content or merely in terminology. How do I know that when the first says material and the second mental, or the first physical universe and the second universal mind, that they are not merely using different words to denote the same thing?
SECOND LOGICAL POSITIVIST: I doubt that this question itself has any cognitive content. How in principle could one verify whether they mean the same or different things?
PHYSICIST: This type of question is out of my domain. I would like to return to the viewpoint of the realistic mystic. Naturally, this viewpoint interests me in that it uses the terminology of science. It has, however, one serious weakness that borders on the downright ridiculous. All right, he may translate the statement, “Simplicus enjoys the tree,” to, “The universe enjoys the tree.” Now suppose someone else—say, Complicus—comes along and claims to enjoy the tree. Again the mystic translates the statement, “Complicus enjoys the tree,” to, “The universe enjoys the tree.” So when the realistic mystic says, “The universe is enjoying the tree,” how can I possibly know whether it is Simplicus, Complicus, or someone else—or for that matter some dog—who is enjoying the tree?
REALISTIC MYSTIC: I should like first to remark that by profession I am also a physicist. Now certainly, when I do physics or am engaged in daily life activities I would use the more descriptive and specific terminology, “Simplicus is enjoying the tree,” or “Complicus is enjoying the tree,” rather than, “The universe is enjoying the tree.” Just because I regard Simplicus’s enjoyment of the tree and Complicus’s enjoyment of the tree as special cases of the universe enjoying the tree does not mean that I regard them as identical events. So of course when it is necessary to be more specific (which is most of the time), then I am specific. But for other purposes—which might be termed spiritual, mystical, or religious—I believe it more fruitful to regard these particular events as an activity of the universe as a whole.
CHRISTIAN THEOLOGIAN: Since you brought up the word religion, may I ask whether you honestly believe it possible to incorporate into your purely materialistic framework the fundamental ideas of religion like God, soul, divine purpose, and reward and punishment? If all that exists is matter, what could it possibly mean for my soul to be immortal, and how could I anticipate punishment or fear reward?
PSYCHIATRIST: I think you meant fear punishment and anticipate reward.
REALISTIC MYSTIC (amused): I certainly can incorporate all these ideas into a purely materialistic framework. By God, I of course mean the entire universe. The word soul or mind I happen to use most frequently. I am not a dualist in that I do not regard soul as a substance, as I do matter. Rather, soul for me is a combination of memories and dispositions. If I have a beautiful recording of a musical composition and the record should fall and break, it is no tragedy, providing I can get another copy. What is important about a particular record is not its particular atoms but rather the pattern that has been impressed upon it. It is this pattern that might well be called the soul of the record—its propensity to reembody the musical idea. Likewise, a man’s soul consists of his memories and behavioral propensities. In this sense, it seems perfectly natural to also regard the universe as having a soul, which is its pattern. If you would prefer me to use the word God to mean this soul or pattern rather than its concrete embodiment, I would have no objection. After all, suppose if by magic each atom in the universe were replaced by an identical particle, or if this is empirically meaningless, suppose that all basic particles of the universe were thoroughly reshuffled but end in a pattern identical with the present one. I would hardly say that the universe had undergone any significant change; it would still have the same pattern or soul. I do, however, disagree with the idealist or dualist who thinks of soul as a substance, unless (is it possible?) he would allow a pattern to be called a substance. In this case, our difference is not metaphysical at all but purely terminological. This suggests the following thoughts on dualism versus monism.
I can see some sense in distinguishing a particular body from its pattern if there exists at least one other body with the same pattern. But since there is only one universe, it is hard to understand the difference between the universe and its pattern. This would mean that we can distinguish the mind of a man from the body of a man, or the mind of a dog from the body of a
dog, but in the limiting case of God, the body of God might be the same as the mind of God. Stated in the language of the mathematician, matter and mind may be different locally but the same globally.
To return, however, to the theologian’s second question, I first wish to remark that I have always found it exceedingly odd that many scientists—even those in the computer science field—are perfectly willing to use terms like thinking, purpose, reward, and punishment for both humans and computing machines but absolutely balk at the idea of applying these so-called anthropomorphic terms to the universe as a whole. Of course the universe is mainly inorganic, but so is a computer! I greatly fear that this is a sad reflection of the continuing egocentricity of mankind. Descartes thought that humans think but dogs do not. (His dog, however, thought otherwise!) People today who believe that humans think usually believe that dogs think as well. With plant life, people are doubtful, and when it comes to inorganic matter, there most people really draw the line! As if there is some social hierarchy—stones, plants, dogs, people! One calls stones dead and inert. Of course stones are dead in the purely biological sense. The word inert is, however, misleading, considering the fantastically rich life and activities of a stone’s inner molecular structure. But to balk at applying anthropomorphic terms to the whole universe, whose structure is so vast and complex compared with any person or computer—indeed, it includes all people and computers—to refuse this terminology for the universe as a whole strikes me as totally unwarranted. No, I certainly have every whit as much right to apply such terms as thinking, feeling, and planning to the universe as a whole as I do to entities that are only parts of the universe. Let the tough-minded think of these terms as purely operational. My so-called mysticism consists not of any metaphysical meaning attached to this terminology but purely of my emotional responses that such terminology tends to engender. At any rate, in this terminology, it of course makes sense to refer to the universe as having a purpose or as punishing or rewarding us for our actions. For example, I would say that the universe punishes a baby—for its ultimate good—for sticking its hand into a fire.