KATY:So you think that my example demonstrates the subjectivity of categorization just as much as the subjectivity of analogy-making?
ANNA:Absolutely! And it’s precisely its subjective nature that lends categorization so much interest. Our categorizations are influenced by many factors: the place we happen to be in; our current goals; our knowledge; our culture; our emotional state; our obsessions; and who knows what else!
Categorization is reliable; analogy is suspect
KATY:Touché! I can see now that I was a bit off base in focusing on the distinction between objective and subjective, although I was getting close to the target. Now though, thanks to your help, I have at last located the bull’s-eye, the true distinction between analogy-making and categorization. Indeed, it’s obvious, after the fact! It all comes down to the question of certainty versus riskiness. Comparing a categorization’s reliability to that of an analogy is like comparing day to night. When I make a categorization, there is no chance of error, because I am just connecting something in my environment with a category that it matches; in such a cognitive act, there is no risk-taking. When I recognize a table, a chair, a piano, a melody, or what-have-you, I’m not blue-skying it; I’m just perceiving something as it is, end of story. I call a spade a spade; that’s categorization for you! Doubt is not on my radar screen when I categorize. However, making an analogy is always a gamble. You’re taking two situations and hoping that they will match, but there is no guarantee whatsoever that your guess will be right. The activity of analogical guesswork is a minefield of uncertainty. To make an analogy is to make a risky bet, knowing clearly that one may well lose one’s shirt. Take, for instance, the political analogies discussed in Chapter 6 — the ones that guided the course of the Vietnam war, at least on the American side. While some were on target, others were way off course, and nobody at the time could tell the wheat from the chaff. That’s the nature of analogy; to make an analogy is to put one’s good money on one’s very unreliable intuitions. What say you, my friend?
ANNA:Good try, but once again no soap. To start with, you’re fooling yourself if you think that categorization is always reliable and certain. Didn’t we just agree a moment ago that categorization is not objective? Well, for similar reasons, categorization often leads one into error, which means it cannot be relied on.
KATY:Can you give me some examples?
ANNA:Very gladly! You might put salt into your coffee, having taken one white powder for another. You’re thus the victim of a mistaken categorization! Then again, having been instructed to “take the second left”, you might turn down a driveway instead of a road. Wrong category once again! Or you might rub shaving cream in your hair, having taken it for hair lotion. Or you might drink rubbing alcohol, if it has been poured into a bottle labeled “cream soda”. Or you might happily chomp into a hot pepper, thinking it is merely a red pepper. Or you might mistake a flower bulb for an onion and chop it up in your salad. Or you might look up into the sky and think you see a bird when actually it’s a plane. Or you might take Mars to be a star instead of a planet. All of these events are mistaken categorizations. And think of stereotypes based on such things as sex or race or nationality or age or profession or religion — they too are a kind of categorization, made overhastily or unreflectively. Most people buy wholly into their stereotypes without realizing that such coarse-grained judgments of other people are often way off the mark. In short, stereotypes are a frequent source of deeply erroneous categorizations.
KATY:Your examples are quite convincing, Anna, I agree. All of the categorizations that you’ve just listed do amount to errors, and I accordingly acknowledge my mistake. A cognitive process that often leads to errors can certainly not be claimed to be always reliable! One point more for you, Anna, but I nonetheless must slightly take the edge off your glee by pointing out that in all the cases that you cited, the deluded individual is persuaded of their categorization’s correctness until some event in the world reveals that it was wrong. Thus we are always persuaded that our current categorizations are correct, even if later we come to realize that we were mistaken. For this reason, I stand behind my claim that a feeling of certainty — even if it’s just a subjective, fallible feeling — constitutes the dividing line between categorization and analogy-making, because one always believes lock, stock, and sinker — er, lock, stock, and barrel, that is! — in one’s categorizations, whereas one is always distrustful, as well one should be, of one’s own analogies.
ANNA:You’ve put it very well, Katy, but now I must cast cold water on your idea. All categories have their zones of uncertainty, and exactly the same thing can be said for analogies. If you take a category — any category — and you try to trace its borders, then gray zones will immediately start appearing. Take the category of clothing, for instance. You wouldn’t hesitate for a split second in saying that a coat, a jacket, a pair of pants, a skirt, and a sweater belong to this category, while a blanket, a pistol, and a cell phone do not.
KATY:I don’t see the slightest uncertainty about membership in the category of items of clothing.
ANNA:If you’ll let me finish, perhaps you will… Just focus for a moment on the fringes of this category and you’ll see that your sense of security starts to wobble. Is a hat an article of clothing? What about a belt or a scarf? What about gloves or socks or a headband? What about ski goggles or flippers for swimming? Or consider the category of furniture. Is a piano a member of the category? What about a toy piano? A beanbag chair? How about a laundry basket? A chandelier? A toy chest? A coatrack?
KATY:Well, you’ve raised extreme examples. It would take me a while to make up mind about some of these cases.
ANNA:Of course! Things are far from black and white! There’s no doubt that you will go back and forth in some of these cases, and even after making up your mind about them all, if you ask your friends, it’s guaranteed that you’ll get back all sorts of diverging opinions, everyone having their arguments ready at hand. At some point your head will start spinning and you won’t be sure any longer what you think about these marginal cases. And we can also recall some cases we mentioned earlier, such as whether Pluto is a planet or not (on this topic, it would seem wise to keep an open mind), or whether an accused person is guilty or innocent (indeed, it’s precisely because of marginal cases of category membership like these that the professions of lawyer and judge exist).
KATY:I must admit that I’m coming to feel that I’m less and less sure about what I thought I was sure about, alas. I now concede that uncertainties exist in categorization, but they always lie out at the fringes of categories, while most members of categories are close to the core and far from the margins, and thus are safely removed from the battle zones of doubt. So in categorization, uncertainty, although possible, is rare. But in analogy-making, uncertainty is pretty much the rule, not the exception. Are you going to disagree with me once more?
ANNA:Well, I’m afraid that once again you’ve fallen for a stereotype. Just go back and think of all the automatic, unconscious, run-of-the-mill analogies that I brought up earlier, such as the doorknob that easily turns, rather like hundreds of others that you’ve already turned, or Joanie, who, always true to herself, is perennially late to appointments, or this French fry, crunchy and warm, just like its plate-mates, or this elevator, which can be relied on to work, just like so many similar ones in similar buildings. These are certainly all analogies, but you’ll grant also that they are very certain and that one does not tremble in one’s boots in relying on them.
KATY (laughing): What can I say? Once again I find I’m in agreement with you, Anna. You’ve knocked down my pins. You’ve rolled a strike! At last we’re in harmony.
The telephone rings.
KATYANNA (groggily): Hello… Ah, uh… who is it?
A male voice replies: Good morning, Katyanna! It’s Dounuel. I’m so relieved to have reached you! I just woke up from a very strange and upsetting dream, and if you have a moment, I’d appreciate it
if you were willing to talk about it a little with me.
KATYANNA:You woke me up, you know? But it’s all right — I’m coming to my senses, and that’s good. So go ahead and tell me about your troubling dream.
DOUNUEL:Well, in my dream I was divided into two people who were both extremely stubborn, and they were having a huge argument. Being split in two in that way was a very weird feeling, I tell you.
KATYANNA:What a peculiar dream! And what were these opponents arguing about?
DOUNUEL:It was most unusual. Although intellectually they were bitterly opposed to each other, they were actually very good friends, and they had decided that they would write a book together, but they hadn’t yet started it. One of them was vehemently insisting that the book would have to be written in French, while the other one was equally ardently demanding that it be written in English. It was a ferocious battle, and yet they were both using the most polite and friendly language with each other.
KATYANNA:My goodness! What a peculiar nightmare! But this imaginary book that they were hoping to write together — what was it going to be about?
DOUNUEL:Oh, you know, my standard old hobbyhorse — the unity of analogy-making and categorization. No surprise there!
KATYANNA:Now there’s a theme that I know like the back of my hand! At least you were pretty much yourself in your dream.
DOUNUEL:Yes, luckily. But it was awful to feel myself split into two pieces that were fighting so intensely over the choice of language. It was such a relief when I finally woke up and realized I was just one ordinary person, not plagued by schizophrenia. So thank you, Katyanna, for having let me let off some steam about my disturbing dream. And now, thanks to you, I’m feeling fresh as a daisy. It’s time for me to get back to work on my book, which, I’m glad to report, is nearly done (I’m just putting the finishing touches on a dialogue that comes at the very end, like icing on a cake), and since today is an odd-numbered day, I’ll write in French, as is my custom — and then tomorrow I’ll write in English, and so forth and so on.
KATYANNA:What a coincidence, my dear brother! Can you believe that exactly the same thing just happened to me?
DOUNUEL:No! Tell me about it! I’m all ears!
KATYANNA:Yes, indeed, dear brother, I had a parallel dream, a similar dream, a comparable dream, an analogous dream…
DOUNUEL:You wouldn’t mean a dream belonging to the very same category?
KATYANNA:Ah, yes — just the phrase I was looking for! You couldn’t have known it, but your phone call woke me up from my own very upsetting nightmare. In it, I too was split into two people who were having at each other like two angry little gremlins, although always using extremely polite words. And something of your hobbyhorse must have rubbed off on me after all these years, because one of the angry gremlins was insisting that analogy-making is the core of cognition, while the other one maintained with equal vehemence that categorization played that role. It was all nonsense, of course, as I now realize clearly, but at the time it really seemed as if it made perfect sense for these two strange gremlins to be arguing that these are two mental processes that differ from each other.
DOUNUEL:How droll it is to see ourselves taken over at night by wild fantasies, making us believe in notions that by light of day are clearly sheer nonsense! But luckily you’re wide awake now, just as I am, and as they say, all’s well that ends well! And so before hanging up, let me just wish you a lovely day, my dear sister. Good-bye!
KATYANNA:And likewise to you, my dear brother! Good-bye!
And thus, just as the sun was rising, after this troubled night during which she had imagined herself split into two subselves plunged in a bitter struggle, Katyanna arose with a feeling of serenity, happiness, and inner peace, thanks to her recovered unity.
NOTES
Prologue
Page 8There is a famous Russian poem… Selvinsky (1920).
Page 16One day… the idea came to me… Poincaré (1908), p. 52.
Page 21likeness is a most slippery tribe… Plato (1977), p. 231a.
Page 21a mobile army of metaphors… Nietzsche (1873), p. 46.
Page 22The light of human minds, is perspicuous words… Hobbes (1651), Chapter V, 36.
Page 22Expressing oneself with metaphors has the quality… Alberic (1973), pp. 146–147.
Page 22A science that accepts images is… Bachelard (1934), p. 47.
Chapter 1
Page 38Censorship is the mother of metaphor… Jorge Luis Borges, quoted in Manea (1992), p. 30.
Page 38Leisure is the mother of philosophy… Hobbes (1651), IV, 46.
Page 38Death is the mother of beauty… From the poem “Sunday Morning”, in Stevens (1923).
Page 60I’ll tell you something… Robert Pond, “Fun in metals”, John Hopkins Magazine, April 1987, pp. 60–68, quoted in Murphy (2002), p. 18.
Page 61A language is a dialect with an army… Max Weinreich (1945). “The YIVO and the problems of our time”. Lecture delivered at the Annual YIVO (Yiddish Scientific Institute) Conference, New York, 5 January 1945.
Chapter 2
Page 97Our goose is cooked… Chiflet (1985) and Whistle (2000).
Page 112A famished fox observed some grapes… Morvan de Bellegarde (1802).
Page 112Driven by hunger, a fox was lusting… Phædrus (1864), Book 4, Fable 3.
Page 112We can’t have all we seek, alas… Benserade (1678), p. 108.
Page 112A certain fox from Normandy… La Fontaine (1668), Book III, p. 11.
Page 132When a dog eats the flesh of a goose… Henri Poincaré, as quoted by Roger Apéri in Dieudonné, Loi, and Thom (1982), pp. 58–72.
Page 133There Is No Word… Tony Hoagland, Poetry Magazine, July–August, 2012.
Chapter 3
Page 138Items to save when one’s house is burning down… Barsalou (1991).
Page 154Phelps is pretty much my double… Mark Spitz, quoted in an article by Drew Van Esselstyn in the New Jersey Star-Ledger, 15 August, 2008.
Page 155Back in 1972, they didn’t have a 50-meter race… Ibid.
Page 160Karnak Caps… From Egypt Sweet by Kellie O. Gutman (privately issued, 2005), p. 9.
Chapter 4
Page 188Ireneo Funes, the main character in Jorge Luis Borges’ short story… Borges (1962), p. 114.
Page 208As both émigré and physicist, Dr. Teller was aware of the Nazis’ lengthening shadow… Mark Feeney, in the obituary “Bomb pioneer Edward Teller dies”, in The Boston Globe (“Nation” section), 10 September, 2003.
Page 221Of course, the historical figure of mathematical fame… Swetz and Kao (1977), p. 7.
Chapter 5
Page 274Genius is 1 percent inspiration… Thomas Edison in 1902, as reported in the September 1932 edition of Harper’s Monthly Magazine.
Page 301If the only tool you have is a hammer… Maslow (1966), p. 5.
Page 313That memory is knowledge, that knowledge is going to interfere… Jiddu Krishnamurti, “First conversation with Dr. Allen W. Anderson” in San Diego, California, 18 February, 1974.
Page 313“Freedom from the Known”, the title of one of his most famous works… Krishnamurti (1969).
Page 315There are magic links and chains… James Falen, “Odelet in Praise of Constraints”, in Hofstadter (1997), p. 272.
Chapter 6
Page 330In their book Mental Leaps… Holyoak and Thagard (1995), p. 139.
Page 334The 1930s is a composite analogy… Khong (1992), p. 59.
Page 334The analogy of Munich raised the stakes… Khong (1992), p. 184.
Page 337One of the most interesting fndings of researchers… Khong (1992), p. 217.
Page 338These findings may leave us feeling… Gentner, Rattermann, and Forbus (1993), p. 567.
Page 368When I look at an article in Russian… Personal letter from Warren Weaver to Norbert Wiener, quoted in Weaver (1955).
Page 363Parfois, le succès ne fut pas au rendez-vous… Bertrand Poirot-Delpech, in the obituary “Sagan, l’art d’être
soi”, in Le Monde, 26 September, 2004.
Chapter 7
Page 388All summer long, without a care… La Fontaine (1668), Book I, p. 1.
Page 400The real problem with the interface is… Norman (1990), p. 210.
Page 408I will sette as I doe often in woorke use, a pair of paralleles… Recorde (1557).
Page 413Multiplying one number by another is… Bezout (1833), p. 12.
Page 415As long as one considers numbers as abstract entities… Bezout (1833), p. 13.
Page 420To divide one number by another means… Bezout (1833), p. 21.
Page 420The number to be divided is the dividend… Bezout (1833), p. 21.
Page 420One’s goal in doing a division is not always to find out… Bezout (1833), p. 21.
Page 436Categories let people treat new things as if… Spalding and Murphy (1996), p. 525.
Page 436Analogy is what allows us to see the novel as familiar… Gick and Holyoak (1983), p. 1.
Page 436In an analogy, a familiar domain is used… Clement and Gentner (1991), p. 89.
Page 436If one establishes that a given object belongs to a certain category… Anderson (1991), p. 411.
Chapter 8
Page 438Could anyone think… that they have always marched forward… Poincaré (1911), p. 31.
Page 440“8–3” is easily understood; 3 can be taken from 8… De Morgan (1831), pp. 103–104.
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