Discrete / continuous

Imagine you're a biologist studying an isolated species of birds on an island, and every bird is either black or white or red. There are no gray birds, and no pink birds, no intermediate colors. You might say that color in these birds is a qualitative trait. The birds come in discrete colors.

On the next island over, there is a different bird species. It also contains some black birds, and some white ones, and some red ones. But most of them are some shade of gray or pink. In other words, their colors fall somewhere between the extremes of black, white, and red. So in this species, color is quantitative. The birds come in continuous colors.

Gradients

When I was a kid, no one had any idea how a fertilized egg “knew” how to develop into an embryo. We knew that one cell divided into two, two into four, etc., but then what? At some point, the embryo would have to somehow “figure out” which end was the head (and which the tail), and which side was up (and which down).

It turns out this is accomplished by using chemical gradients. A gradient is where something (in this case the concentration of a substance) varies from a low value at one end to a high value at the other end. The value varies smoothly from end to end, in fact we can say it varies continuously. So, for instance, there is a chemical that is stronger at the head end of the embryo, and weaker at the tail end (or vice versa, it's not important which). Each cell along the gradient uses the concentration at its location to “know” which part of the body it's in, and develops accordingly. Other gradients are used to create limbs and other body parts.

Line-Drawing

Lots of issues come down to drawing a line. In some cases it's easy. Consider the first species of birds above. Nature has conveniently separated them into obvious groups. Drawing lines is easy.

But what about the other group? A much harder problem. There's no obvious place to draw lines, or even to know how many should be drawn. And yet some of the birds are very different from each other (remember there are a few that are pure white, black, or red).

Sometimes it doesn't really matter exactly where the line is drawn. We just need to draw it somewhere in a general area, and then everyone can use it. For instance, in the US people are considered adults at age 18. But it would not be the end of the world if we drew this particular line at 17 or 19. The same with speed limits. If the speed limit on a particular highway is 50 miles per hour, it probably wouldn't matter much if were changed to 45 or 55. In cases like this, we're imposing a sort of discrete system on a continuous substrate.

Misc, to-do

  • sometimes people offer the following types of argument against proposed new or changed policies:
    • in response to arguments against raising the minimum wage to $15, they ask: "if $15 is good, why not $20, why not $100?"
    • in response to an (imaginary) proposal to reduce a speed limit from 75 to 65, they might ask: "if 65 is good, why not 55, why not 5 miles per hour?"
    • on possible response might be "why not 85, why not 200?"
    • (not sure if this one qualifies) in response to recognition that US slavery (or our treatment of the Indiajs) was bad and had bad consequences, they ask: "why don't we waste our time also condemning the [ancient and far-away] crimes of [some ancient and far-away people]? People have been rotten bastards forever, so why are you singling out certain crimes?
    • tangential?: moral disappointment
    • one possible response to that might be: why are you singling out the accomplishments of your favored country? After all, other countries have done good things, too.
    • two main points I'm trying to make
    • sometimes we have to draw lines, even though it's not obvious how
    • there is such a thing as diminishing returns
    • I need a story to show that not everything is correlated in a linear way. how about this?
    • Alice buys new untested household robot Bob. Bob has lots of useful features but he's not very smart, but he is programmed to learn. Alice is relaxing on the patio with drink in hand, chatting with friends and enjoying the day. Alice thinks her drink would be better if it were a little colder, and she wants to show off her new toy to her friends. She calls out to the robot: "Bob, bring me ice." Bob asks "Is ice good?" and Alice replies, "yes, ice is good." Bob instantly creates (in his internal fabrication unit) one ice cube, and puts in Alice's drink. Alice looks into the drink and looks at Bob. She says "More ice, please, Bob." Bob says "more ice is better?" Alice says "yes, more ice is better." Bob instantly creates a three-ton block of ice and dumps it on Alice, killing her. The point is that pretty much everybody recognizes that it's often the case that just because a little of something is good, it's not the case that a lot would be better.
    • In the real world no one thinks we should have a minimum wage of $100 an hour. And even the stupidest conservative knows that no one thinks that. So why do they bother asking? Well, clearly it's rhetorical. The point seems to be, if it's not immediately, blindingly obvious where to draw a line, that must mean no line can or should be drawn. Therefore there should be no minimum wage at all.
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