May I share some mathematical models of America with you? They won't be elaborate or complex -- more like intellectual doodling, but I hope they will be fun/serious. For what it's worth, I have a Doctorate in math, but I’d like to limit things to a couple of mathematical snapshots.
(1.) This is the concept of reductio ad absurdum which is intimately related to something called "proof by contradiction". Not all mathematicians like this proof, since its "indirect", but sometimes that's all you've got.
It works, briefly, like this. Let's say you're convinced that A, B, and C are true propositions, but you just can't seem to prove D in the same way you proved A, B, and C (i.e., "directly"). So sometimes you fall back on proof by contradiction which works like this (be warned, it's a weird way to prove things).
You say, well what if we assumed D was "false" and then rigorously derived what that would imply. And IF D was false (remember, you want to prove that it is true), and IF this implied that A, B, or C was false (which you "know" are true), then (voila!), you've just proved that D is true. Sounds a little like a slight of hand, doesn't it, but believe it or not, this kind of reasoning was the origin of non Euclidian geometry (another story).
OK, now to politics.
I'm leaving out some steps here (which students always appropriately resent), but the above is all of a piece with something called the "contrapositive". You will see the similarity as we proceed and also see that this analysis relates perfectly to the fascist right wing (dem & pug equally) "model" of human existence.
But first to the logic. The contrapositve of A implies B, is Not B implies Not A. Indeed these two conditional statements are logically equivalent. Here's how it works: If you're in Albuquerque, then you're in New Mexico (a conditional statement which is clearly true). Now for the contrapositive: If you're Not in New Mexico, then you're Not in Albuquerque.
These can (and frequently do) take the form of syllogisms (at least 2 premises and a conclusion) and some of these syllogisms go all the way back to Aristotle, who even though he was totally wrong about practically everything in the physical world (note Galileo Galilei -- who was punished terribly for sticking a pin into Aristotle's TOTAL misunderstanding of motion, etc.), he had it all together when it came to logic. Indeed, Euclid based his Elements on Aristotle's insightful logic.
Well, I hope you're still with me (and even enjoying things a little?), so we can now talk about dem/pug fascism (dem/pug, because after the 2006 Congress of dem Republican, we now know that right wing dems = right wing pugs).
Here's the conditional statement. IF the Bush/fascist/Republican/neocon agenda was truly in touch with the social reality of human existence, than our economy would be great, our children wouldn't be massacred in money cow oil wars for the elites, Mother Nature wouldn't be gang banged by Texas energy corporations, our retirements would be solid, out heath plans would be solid, and American science wouldn't be under funded into an intellectual embarrassment (for the first time in American history!), and Wall Street would be a raging bull.
Formally put, IF A, then B.
But, remember Not B implies Not A, and since the United States of America is now wall to wall "Not B", the logical/mathematical implication is clearly Not A, i.e., the Bush/fascist/Republican/neocon take on human existence is FALSE.
Said more poetically, give a fool (why does president pinhead come to mind?) enough rope and he will hang himself.
Jesus put it even more succinctly: "Ye shall judge them by their fruits". And the left over fruits of these last eight years are rotten with failure.
Hey, we gave A a chance (eight years of a chance!), but the thumb of deductive logic is pointing straight down.
(2.) Shall we try another one? I'm not sure how I'm going to bring this one off, but it has to do with one of the most powerful and subtle concepts in the whole of mathematics, and this concept (or truth, since it’s more than a mere concept) is that of the Limit of calculus (or more generally, analysis).