| I want to use the simplest example possible, and I want to be as pictorial as possible, So I'm going to talk about the simplest geometrical entities there are: points and lines. A point is a zero-dimensional object, a line is one-dimensional, and the way we get from one to the other is the subject of this discussion. Thinking from small to big, that question becomes, "How do we get from a point to a line?" And the simple answer turns out to be, "You don't. It's impossible." The proof of this comes from Georg Cantor, one of the greatest mathematicians of all time. On the other hand, thinking from big to small, the question becomes, "How do we get from a line to a point?" And that question turns out to be relatively easy. So easy, in fact, that I'll give two different demonstrations, the second of which derives from another late 19th Century mathematician, Richard Dedekind, who was a friend and admirer of Cantor. We'll tackle this problem first, then I'll talk about what it means more generally.
First, let's get precise, so we understand exactly what these two questions mean. Then we'll look at how they can be solved-or if not, why not. To go from a point to a line means, in effect, how can we manipulate a point (including movement, endless duplication, rearrangement, etc.) to come up with a line-or, to make thinking about it simple, a line segment, meaning part of a line of finite length. To go from a line to a point means much the same thing: how can we manipulate a line to come up with a point?
Because going from a point to a line is impossible, we'll discuss it last. First, we'll discuss what is possible, so you can understand what it means. The easiest way to make a point from a line takes two steps.
First, you duplicate it:
Second, you take the duplicate and make it intersect the original:
The intersection is a point.
This is very simple, and part of its simplicity comes from the fact that we've used a second dimension when we take the second line and rotate it so that it intersects with the original. This is another indication of the superiority of thinking big, from more dimensions to less, rather than the other way around. But some might consider it cheating, if we're trying to strictly compare turning zero-dimensions into one versus the reverse. So let's take another approach. This one derives in spirit from the Dedekind cut, a mathematical construction on the number line that resolves the seeming contradiction between the continuous nature of the line, and the discrete nature of any specific number.
First take a line and cut it twice, to make a line segment: Cutting a line once is the simplest sort of manipulation you can perform in one dimension, so it's clearly an allowable manipulation. Once you have a line segment, the rest is easy: you just shrink it. For simplicity's sake, we'll say you shrink it in half. How does that help us get a single point? Simple: you repeat the process an infinite number of times. Now some folks argue that you can't do anything an infinite number of times-and physically, they're absolutely right. But conceptually, you can, and we're dealing with concepts here. And when you do that, what you end up with is a single point.
In fact, this is precisely what a decimal representation of a real number is all about-only shrinking the line segment by 1/10 each time, instead of ½. Think of the number pi, the ratio of the diameter of a circle to its circumference, a decimal that goes on forever, but begins 3.1415.... We can think of it in terms of line locations on a line segment, each one a smaller subsegment than the one before. For each step, the numeral stands for a segment whose lower bound is the numeral, and whose upper bound is the next numeral.
We begin with step zero (mathematicians often like to start with zero), with 3 on a line segment of length 10:
Next, step one: we have 3.1: We take the segment from 3 to 4, and take the sub-segment from 3.1 to 3.2:
Now, step two: We do the same to get the segment for 3.14:
And step three, for the segment for 3.141:
And so on, and so on, to infinity.
This process gives us an infinite number of numbers, each corresponding to a single point. For the number 3, for example, we simply take a line segment starting at 3 for every step from step one onward.
Nerd Alert! All others please ignore!
The Dedekind Cut is a concept that makes all of the above quite simple and obvious. As Wikipedia explains, The cut is:
a partition of the rational numbers into two non-empty parts A and B, such that all elements of A are less than all elements of B and A contains no greatest element. The cut itself is, conceptually, the "gap" defined between A and B. In other words, A is every number between the cut and any number lower than the cut, and B is every number between the cut and a number greater than the cut. The cut itself is in neither set.
The concept is very important because:
The Dedekind cut resolves the contradiction between the continuous nature of the number line continuum and the discrete nature of the numbers themselves. Wherever a cut occurs and it is not on a real rational number, an irrational number (which is also a real number) is created by the mathematician. Through the use of this device, there is considered to be a real number, either rational or irrational, at every point on the number line continuum, with no discontinuity. |
End Nerd Alert! We now return you to your regularly scheduled diary.
Now, how do we make a line from a point? Well, it would be simple if we could make a line segment. Then all we'd have to do is stretch it to infinity. Or, if you prefer, we could just double its length, and then repeat an infinite number of times. So the real trick is getting from a point to a line segment. And that's what's impossible. How do I know? Simple-since Cantor showed the way with his "diagonal proof" back in 1891.
For simplicity sake, let's assume the line segment is segment going from 0 to 1. And let's assume-for the sake of argument-that you've managed to manipulate one point to fill it all. I'm going to show that this can't be done: assuming it's so produces a contradiction. At each step, you can only make one point into a finite number of other ones. So for each step, you write down the decimal representation of the points that you've created. (The order you write them in isn't important.) Now, we allow you an infinite number of steps, so the list can be infinitely long. I'm going to show that there's a number that isn't on this list, which means you actually haven't created the whole line segment, even in an infinite number of steps. The actual numbers aren't important, so let's just assume it starts of looking like this:
0.124809...
0.743821...
0.297537...
0.447302...
Constructing a number not on the list is simple-and here's where I'm following Cantor exactly. For the first decimal place, we chose any numeral except "1", the first decimal of the first number on the list. for the second decimal place, number, we choose any numeral except "4", the second decimal of the second number on the list. For the third decimal place, we chose any numeral except "7", the third decimal of the third number on the list, and so on, for all the numbers on the list. The number we generate in this manner cannot be on the list, because it differs from every nth number at the nth decimal place.
There's a lot of sophisticated mathematics that's inherent in this demonstration. For one thing, it shows that there's a difference between a countable and an uncountable infinity. This is what Cantor's proof was all about. Cantor originally used it to show that there are more real numbers (all possible decimal representations) than there are rational numbers (those expressible as a ratio of two integers: a/b). The concept of one infinity being "more than" another in turn gave rise to transfinite arithmetic. The fact that there are an infinite number of rational numbers in any line segment, and yet there are uncountably more numbers that aren't rational also helps illustrate some of the most important concepts in topology and how they relate to one another. Furthermore, in calculus, the Lebesgue integral-a way of calculating the area under a curve-remains unchanged if one removes all the rational numbers.
It's not important to understand the mathematics involved. What is important is simply to realize that the lack of symmetry between the little-to-bit and the big-to-little approaches is the doorway to a good deal of rich and complex mathematics that one might never have suspected in advance. It's almost like Alice discovering Wonderland.
Generalizing Beyond Mathematics
"Okay," you might say, "So what? What does this sort of obscure, geekland mathematics have to do with anything else, especially politics and ideology?" Good question.
First, it shows us something very fundamental about reality and how we conceive it. It shows that conceiving something on one level that's completely consistent and self-contained does not guarantee that we can go ahead and simply rely on what we know to understand a greater level of complexity. If this is true about some of the simplest mathematical objects imaginable, then it is very hard to see how we could assume that it's not true about anything else. And, of course, "conceiving something on one level that's completely consistent and self-contained" is very good description of a successfulform of fundamentalist thinking-let alone the failed forms that aren't even internally consistent.
Second, this has direct applicability to Kegan's model of cognitive development and its applicability to politics. The relationship of point (zero dimensions) to line (one dimension) to is directly analogous to the relationship between one level of cognitive development and the next. The point is content relative to the context of the line. One understands the point in terms of the line-and the Dedekind Cut makes this not just explicit, but formally precise. At the same time, one cannot understand the line in terms of the point-even if one generates an infinite number of points on the line, there are an uncountable infinity more points that are not accounted for. In fact, for some purposes-such as Lebesgue integration-the infinity of points one can generate make no difference whatsoever.
Both separately and together, these two points support an argument generally favoring liberalism over conservatism, while at the same time warning against becoming too smug. As I have argued in terms of Kegan's model of cognitive development, liberalism is inherently more compatible with a more sophisticated, more broadminded level of cognition. Conservatism is more compatible with level 3, where the self is constructed in terms of the social roles and relationships of the surrounding society, while liberalism is more compatible with level 4, in which the self is author to such roles and relationships.
However, Kegan's model also informs us that liberalism and level 4 can also be limiting, and that there's a higher level at which their taken-for-granted truths become open for critical questioning. It's just as possible for liberals to be closed-minded towards this higher level of complexity as conservatives are to the complexity that liberals are more comfortable with. This is not to say that both are equally close-minded, however, much less to imply that so-called "moderates" are superior. Rather, it's to remind us that all breakthroughs lead to new challenges, new problems that require new breakthroughs in turn. |
