Outhouse |

The best quote from the matrix has to be the kid breaking out, "Do not try to bend the spoon — that's impossible. Instead, only try to realize the truth: there is no spoon." Once mentioning the matrix, we cannot continue without reviewing Carl Sagan's apple pie. (and of course the musical rap cover.)

This post offers some notes on Ontology.

Here, we are going deep into Meinong's Jungle, wherein prowl pegasii and unicorns, which must subsist in

*some*sense since we can talk about them. This is Meinong's Gegenstandstheorie. Apparently, there are many levels of "real," if you think about it hard enough.

Oh my God, WTF is happening here? This blog has gone crazy! Well, yes, sort of, but there is some scrap of coherence here, I think... It all started when a student asked, "what is i, anyway?" I was stumped, and I pretty much still am, but here is my answer anyway. Obviously it's a complicated one, but, here goes!

First, let's get rid of the obvious canonical answer: i = sqrt(-1), but you knew that.

Now, what do we make of this? Well, there is at least some major utility to it, eh? Never minding whether or not i is an actual part of reality, I mean. It's a nice tool, like a number line, or a bubble level: helps you figure things out. Sort of like saying "my very educated mother just served us nine *pies," it's a construct you can use whether or not it has any meaning grounded in reality.

(* Sorry, Pluto's not a planet. Don't blame me, take it up with this guy.)

Let's look at some of the utility we get from i. For one, consider helical antennas! They make circularly polarized radiation and that is really neat. I once worked on a satellite that used those. That tubular structure in the picture is a multi-element quadrifilar helix. The polar coordinate representation of exp(i*theta) is a visual model for all kinds of oscillatory phenomena, including the motion of airplanes.

Here's another example of something you can do with i. You can take the cube root of 1! What, you're not impressed? How many cube roots do you think there are? (Hint: thew sqrt of 4 = +/-2, right? Why should it make sense to have two square roots and only one cube root?) Actually plain old +1 does have three cube roots (as it should) and they are 1 and -1/2 +/- sqrt(3)i/2. You don't want i just to get the root of -1, you want it even for mundane cube roots of plain old positive numbers!

Renee Descartes disagreed. He found i a lamentable necessity and saddled it with the pejorative "imaginary" label, but that, we'll see isn't really fair.

** ToDo: Put something intelligent here in the middle... ***

So the answer takes us full circle, back to the spoon only rephrased like this: "Do not ask if there really are imaginary numbers. That's impossible. Instead, only try to realize the truth: the real numbers are

**all**imaginary."

Cool! And all this time, I thought that you had disagreed with me.

ReplyDeleteThis turned into the best lecture I ever gave.

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