Oops, I did it again! Well, most people do. Why is this? Because it is an evolutionary trait that enables most mammals --and who knows how other life forms think?-- to retain and understand a larger volume of organized information than non-mammalian species. Here's a rough idea why this is such a brilliant strategy:
Method 1: unorganized knowledge of a little kid
Woodpecker - bird, has wings, flies, eats bugs, is noisy
Robin - bird, has wings, eats worms, is noisy
Dog - mammal, quadruped, barks, wags tail, slobbers over people that it likes.
Cat - mammal, quadruped, meows, purrs when likes people
Chicken - has feathers, lays eggs, can be eaten if cooked, eats worms
alligator - likes to stay in water, lives in zoos
people - talk, laugh when amused, cry when sad, eat when hungry (or get grumpy), like dogs and cats, come in male and female types, females have high voices.
Method 2: organized knowledge, with aspects of generalization creeping in
Mammals - quadrupeds, like and are liked by people
- dogs: bark, wag tails, slobber
- cats: meow, purr
- alligator: probably growls or something, live in zoos, exception: not liked by people.
Birds - flying, noisy
- Woodpecker, eats bugs
- Robin, eats worms
Two-legged walking types - male and female types with different appearance and voices
- chickens: bird-like, can be eaten, lay eggs
- people: like mammals (except alligators, of course)
Other types
- snake: slithers, unknown type, bites people
As you can see, with this --very hypothetical, of course-- classification, there's a lot less information to remember for each individual type. Obviously the Alligator has been seriously misclassified, and the Chicken has been classified as a biped, rather than as a bird.
Back to the question: why generalize? Now, every type of animal in a classification has some things in common with the others. Is it not reasonable to imagine that each type of animal has more in common, until proven otherwise? As soon as this child hears about its mother's "eggs", this classification scheme would be confirmed. (Can people be eaten, if cooked? Oh dear.)
Generalization in Abstract Theories
Mathematics is all about generalization. For instance, one is familiar with the idea of the diameter of a circle. What about a sphere? Of course. What about a square? Well, let's see now. After all, the diameter of a circle is the furthest apart two points in the circle could be. So this same definition can be used for a square, too! Or a cube. (In fact, a cube of side L has a diameter of L times the square root of 3, which follows from Pythagoras's Theorem. The diagram at right is intended to depict a cube in glorious 3-d. Suppose all the blue lines, purple lines and the green line all have length L. Now the diameter we want is the red line, which is the hypotenuse of the red-yellow-green triangle. The green side has length L. The yellow line is itself the hypotenuse of a blue-blue-yellow triangle, so its length squared is 2 times L squared.)
A prime number is an integer that can't be factored into smaller integers. Factoring is an elementary skill, and it's not hard to get little kids factoring a number like 30 into little prime factors. Once they get to algebra, a kid probably gets annoyed when the process of converting x*x + 7x into x(x+7) is also called the same name: factoring. Hey, wait; are there prime polynomials? Sure. If you can't factor it, it's a prime polynomial, such as x*x + 5.
Readers who know about complex numbers and irrational numbers probably see the opportunity here to re-classify the last example as a composite. So, of course, it's all relative, whether or not something is considered prime. Classification becomes a matter of semantics very soon.
Generalization, too, becomes a matter of semantics, and the stretching of the meaning of the term that we want to generalize has to pass the test of whether it is more useful or less useful once it is generalized. For instance, the expression "The only good chicken is a dead chicken" is useful only if you're thinking about eating it. Dead chickens, as we all know, lay only a limited number of eggs (if "lay" is the word we want, here), so if you want the eggs, you gotta keep the chicken alive.
The tendency to over-generalize is commonly taken to be a sign of the most extreme foolishness. But generalization, as with most abstract operations, has to be practiced before it is perfect. As I read in a book somewhere, a child who has seen numerous examples of flying birds will assume that all flying things are birds. An encounter with a bat, for instance, will dispel this misapprehension, and that's a part of growing.
Consider the opposite case. A child who does not generalize at all, who does not indulge in any kind of speculative generalization is, to my mind, a cause for concern. So, while a tendency to make reckless generalization has to be moderated, gently, if possible, the child who does not generalize at all ought to be gently encouraged to make generalizations, and test them out.
Archimedes
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