Shades Of Reality
Chapter Two
Categorization
One of the hallmarks of human intelligence is the ability to form generalized concepts from specific facts. Pick up a rock. Let go of it. It falls down. Pick up a piece of wood. Let go of it. It falls down. Pick up an apple. We could go on listing things one after another. Or we could simply make the generalized statement that if you pick up any object and let go, it will fall down. Without the ability to generalize like this, we would have no recourse but to continue listing every specific object and explicitly stating what happens when you pick it up and then let go.
Generalization stems from the mind 's ability to detect and extract patterns of similarity from otherwise unrelated entities. (A rock, a piece of wood, and an apple each have very little in common. But they all respond similarly to the operation of "pick up" followed by the operation of "let go.")
The existence of similarities provides the mind with a powerful (and often misused) tool for understanding and coping with what would otherwise be an unmanageable array of separate attributes. The process of recognizing and grouping distinct patterns of similarity is called "categorization," and the corresponding groups that result are called "categories."
Example One:
To illustrate the concept of categorization, let's imagine that you have the collection of marbles shown in Figure 2.1, and you are told that you must divide them into two separate categories: "white" and "black." Which marbles belong to which categories?
(Figure 2.1)
Answer: The marbles which belong to the "white" category are: A, B, D, H, and I, while the marbles in the "black" category are: C, E, F, and G.
Example Two:
If you thought that the previous example was trivial and foolish, then consider trying to perform exactly the same task on the collection of marbles shown in Figure 2.2:
(Figure 2.2)
Now the problem is not so trivial! Clearly, marble A belongs in the "white" category and marble I belongs in the "black" category. But what about all of the other marbles, the ones that are neither black nor white, but different shades of gray?
The usual solution to this problem is to put all of the light gray marbles into the "white" category, and all of the dark gray marbles into the "black" category. But then you are still faced with the problem of deciding at which marble you should "draw the line."
Another approach to solving the problem of categorizing the marbles in Figure 2.2 might be to introduce a third new category called "gray." But instead of solving the problem, this approach only makes matters even worse. Now two "draw-the-line" points must be established, one that separates white from almost white, and one that separates black from almost black! (After all, marbles in the real world are never perfectly white or perfectly black.)
The most radical solution to the problem is to simply create nine separate categories, each containing a single marble! This solves the 'draw-the-line" problem. But now you've merely traded nine marbles for nine categories, and so you're no better off than when you started!
Pseudocategories
There is a good reason why you are experiencing difficulties in trying to sort the marbles in Figure 2.2 into categories: You are trying to categorize items on the basis of an attribute whose values lie on a continuum.
The marbles in Figure 2.1 have only two discrete color values (black and white) with no marble having an intermediate value. Therefore, the separation into two discrete categories is intrinsic. Furthermore, the color value of every marble in a given category is exactly the same (either black or white) as every other marble in that same category.
In the case of Figure 2.2, there are no intrinsic categories. If you artificially try to force the marbles into a small number of categories (of alleged "sameness"), you end up producing categories whose constituent members are not all the same. In fact it might even be possible to find two different marbles in the same category that have significantly different colors. To make matters even worse, it might even be possible to find a "white" marble and a "black" marble that are more nearly equal to each other than either one is to some of the other marbles in its own category!
This consequence is illustrated in Figure 2.3. Look at marble E in the "white category" and compare it to marble F in the "black category." There is very little difference in their colors. Their shades of gray are almost identical -- and yet they are in two completely separate categories! But now compare marble A in the "white category" with marble E. Both marbles are in the very same category, and yet the two marbles have profoundly different colors!
(Figure 2.3)
At this point you must recall the definition of categorization: The proccess of grouping distict patterns of similarity. If it's possible for two members of the same category to be significantly different, and if in the very same grouping it's possible for members of two different categories to be almost the same, then the attempt at categorization has obviously failed in concept.
To summarize, the process of categorization is valid only to the extent that the attribute being categorized is not represented by a continuum (or nearly a continuum).
Attempting to partition a continuum
into "categories" will generally pro-
duce groupings of questionable validity.
I therefore refer to such attempted groupings of a partitioned continuum as forming pseudocategories.
However, returning once again to Figure 2.2 (or Figure 2.3), suppose we were to simply throw away some of the "middle" gray marbles (such as D, E and F) and, in their place, substitute new marbles that look more like the very light ones (such as A, B, and C) and/or the very dark ones (such as G, H, and I). Then the attempt at grouping the marbles into "black" and "white" categories would become more meaningful. The less disparity that exists among the individual members of a pseudocategory, the more valid that grouping becomes (i.e., the more the pseudocategory becomes a true category). In the extreme limit, when each of the marbles exactly resembles either marble A or marble I, then we once again have a case similar to the one shown in Figure 2.1 where the process of categorization is completely meaningful. (We will see in the next chapter that categories are merely examples of something known as "crisp sets," and pseudocategories are examples of something known as "fuzzy sets.")
Example Three:
As another example of the kinds of inconsistencies that can arise when attempting to partition a continuum into categories, consider the following advertisement that an automobile repair shop recently ran in a local newspaper:
The repair shop has, in effect, tried to define "categories" of expensiveness -- those bills which total less than $200 (the inexpensive category), those bills that total less than $300 (the somewhat more expensive category), etc. But now consider two different customers, one whose total bill is $199, and the other whose total bill is $201. After their respective rebates, the first customer ends up paying $189, while the second customer's final bill is only $181. The customer whose initial bill was in the inexpensive "category" ends up having to pay more than the customer whose initial bill was more expensive!
Lists
Take a sheet of paper and a pencil, and make a list of all your favorite foods. You don't necessarily have to list the foods in order of preference, but the final list must be complete and accurate. In other words, you must include every food that you like, and only those foods that you like.
Initially, the task is very easy. You would probably start out by listing your most favorite foods. My list would start out with:
Next, after you've gone through all of your very favorite foods, you would start to include those foods that you like pretty much. In my case, they might be:
Eventually, you're going to run out of the easy items and start reaching foods like (in my case) lettuce, pickles, and onions (or whatever items are appropriate in your case). You eat these things, but is it appropriate to include them as some of your favorite foods? In fact, look at the very last item at the bottom of your list. Are you sure that even that item belongs on the list? Where do you "draw the line" between those items that must be included on the list, and those that must not? (Remember, the list must be both accurate and complete.)
Whenever you make a list of items, you are attempting to group the universe into two distinct categories: those items that belong on the list, and those items that don't belong. If every item on a list "belongs" on the list just as much as every other item on the list, and if every item not on the list "doesn't belong" on the list just as much as every other item not on the list, then the process of listing works well. For example, if I were to ask you to make a list of every state in the USA that begins with the letter A, you might write:
Similarly, if I were to ask you to list the names of all the former presidents of the United States, you would write: George Washington, John Adams, Thomas Jefferson, etc. Once again it would be possible to create a list that was both accurate and complete (since each of those names, and only those names, belong equally well on the list).
But not all lists can be created so "crisply." When attempting to create a list of items for which such clear-cut distinctions do not exist (such as in the case of trying to list your "favorite foods"), then the concept of listing starts to become "fuzzy." Eventually, you must deal with items that kind of belong on the list and which, at the same time, also kind of don't belong.
As another example, if someone were to try to make a list of all the "endangered species" on our planet, where would they "draw the line" between endangered and not endangered? And even if such a list were made, would the species at the very bottom of the list be significantly more endangered than the next species in line that just missed getting on the list? Would it be meaningful to say that the one species was truly endangered while the other one wasn't endangered at all, simply because it failed to make it onto the list?
During the course of writing this book I received a lot of help from many of my friends -- some of them had carefully read the entire manuscript from cover to cover, while others participated to a lesser extent. When it came time to write the Acknowledgments page, I was faced with the very same problem that all makers of fuzzy lists must face: Where do you "draw the line" insofar as whose name gets mentioned and whose doesn't? How much of a contribution of effort does a person need to make before he or she warrants a mention? I wanted to acknowledge everyone who participated, but then it wouldn't have been fair to the major participants to simply give everybody "equal billing," regardless of how little or how much of a role they played. (To see how I finally decided to handle the problem, look at the Acknowledgments page near the front of this book.)
Books
Whenever you read any non-fiction book, you will notice that the author has usually organized the book into chapters, and that each chapter may in turn be further divided into subsections. Each chapter and subsection then attempts to deal with one particular aspect of whatever subject is being presented.
But concepts are fuzzy and, as such, it is often impossible to present them in a straight linear fashion. One piece of information must overlap other similar pieces of information presented elsewhere in the book. You may notice that the author repeatedly discusses many of the same topics over and over again in different locations with each presentation coming from a slightly different perspective or in a slightly different context.
Consider, for example, the problem of trying to write a book about the history of the United States. In principle it might seem like it should be a rather straightforeward task to write such a book since history, after all, occurs linearly in time. But if a book were merely to present history as a series of chronological events, then it would not be able to discuss parallel developments taking place in other parts of the country. Nor would it be able to follow a single topic (such as the life of Abraham Lincoln) and continue to focus on that particular aspect of American history for any length of time.
And just as in the case of lists, the author must also decide which topics he considers to be important enough to be included in the book, and which ones will "just miss" making it in. In most cases, it would simply be impossible (or impractical) to include every single topic or event. For example, a "complete" book of American history would have to include all biographies (even yours and mine) since we too are a part of ongoing history.
And even for those topics that do get included, the author must decide to what extent they will be discussed before moving on to a new topic.
Equification
As was already pointed out, the process of categorization is valid only to the extent that the attribute being categorized is not represented by a continuum. But despite the warning of trying to attach a meaningful interpretation to the elements of a partitioned continuum, society still goes ahead and does it anyway. Grocery stores, for example, like to sort their fruits and vegetables into different size categories -- "small" tomatoes vs. "large" tomatoes, "jumbo" artichokes vs. "medium" artichokes, and so on -- as if nature really grew them in only one size or the other! And in many cases, the method of partitioning is so extreme that the entire size continuum is actually reduced to only a single "category"! (In this case the item is simply sold "by the piece," regardless of any size differences between the items.)
But regardless of the number of partitions used, there seems to be one common "mind-set" that almost always arises about the individual items in each pseudocategory. Borrowing from a certain well-known document written by Thomas Jefferson ("We hold these truths to be self-evident, that all vegetables are created equal ...."), I define -
Equification:
The pretense that all of the individual items
in a given pseudocategory are exactly equal.
For example, equification declares that every "large" tomato is exactly the same as every other "large" tomato. If the price of one "large" tomato is 79 cents, then the price of any other "large" tomato is also 79 cents, even if one of the tomatoes is somewhat smaller than the other.
All-or-Nothingism
There is one particular form of equification that is so prevalent in our society that I have given it a name of its own. It occurs when a continuum is partitioned into exactly two pseudocategories, and when all of the members of one of the pseudocategories are equified to some maximum value, while all of the members of the other pseudocategory are equified to a value of zero. I call this form of equification -- All-or-nothingism.
Examples of All-or-nothingism are abundant in everyday life. To discuss them all thoroughly would require a separate book. But the following illustrate a few representative examples:
As silly and unrealistic as most of these equifications are, almost nobody seems to question them! They represent a mind-set so firmly rooted in our Aristotelian way of thinking that most of Western society actually seems to accept them as being perfectly logical. (After all, you 'gotta' draw the line somewhere, right?)
We will return to discussions of "drawing the line" and other such all-or-nothing concepts many times throughout the remainder of this book.