More on Math

As I have been meaning to do for several years now, I finally looked up a copy of Euclid’s Elements and got to reading.  A complete text – and the one I have been reading – can be found here, in Greek and in a translation a former professor would refer to as a “Bad English Version” done by a physics professor at the University of Texas, Dr. Richard Fitzpatrick.

Compared to the modern text I use at school, I was immediately struck by one thing in particular: Euclid’s proofs rely on a lot of actual drawing, marking off, and otherwise constructing bits and adding pieces to the diagram he starts with.  In contrast, the proofs in my current textbook rely mainly on learning a bunch of principles, and then applying them to other diagrams.

The problem with this second approach, I have found, is that it frustrates most students, who are “proving” things which are perfectly obvious – hey, look, the computer-drawn graphic of a rectangle with some lines on in results in some congruent lines!  Euclid’s approach says, “Fine, smart-aleck, you draw those equal lines.  No, you don’t get a ruler, that’s not how the game works, and anyway how did that ruler get so precise in the first place?”

Off-hand, I can’t remember how my own geometry instruction in high school compared, though vague memories suggest it was somewhere between the two.  I am the sort of person who is perfectly content to play with any sort of logical system for any good or bad reason (up to and including general time-wasting and neglect of chores), so I’m not sure I would have attached any importance to the difference if I did not have to teach the subject now.

Of course, the one thing worse than completely abstracted proofs has to be the ridiculously contrived “real life” problems.  My own conclusion so far is that for most practical applications – not that I have to deal with any often, so I may be way off the mark here – a few essential facts and a basic understanding of trigonometry will get you much further much faster than messing around trying to understand the “application” of the proof of some triangle theorem to a particular scenario.  This is not to write off geometry – the “essential facts” have to come from somewhere.

I am not sure what to make of this – or how exactly to apply it a month and a half into the school year, anyway.  To summarize my current thoughts on the subject: geometry should be a hands-on course, essentially a puzzle game, and stick to obvious applications if any – maybe looking at Kepler’s laws, or something like that.

A Bit on the Shutdown

On the one hand, you have a bill that was passed by Congress and signed into law, and has survived a Supreme Court challenge to one of its key provisions.  The will of the people having been declared, its opponents should, in a reasonable society, just get on with life and let the political process run its course.

On the other hand, this same law has been so divisive that in the three years since its passage, the political strategy of the American Right has been consumed by the effort to repeal, nullify, or otherwise pull the teeth of this measure, culminating most recently in a much-questioned decision on the part of Republicans in Congress to defund its operation as part of the budget (or what passes for one these days).

Other scandals, any one of which might have brought down another administration, have come and gone.  Terrorists sacked an American embassy in Libya, and nothing much has happened.  We found out that the government has been collecting data it has no business with, and nothing much has happened.  The administration passed a questionable education bill, encouraging use of a hotly-debated Common Core program, and there was barely any controversy because the Affordable Care Act, more commonly known as Obamacare, was drawing all the fire.  The IRS admitted to targeting political groups opposed to the President, and after five minutes of infamy the corruption seems to be forgotten by most.  Political chicanery of the worst sort – you would think – has arrived in the public eye and passed on with barely any notice, but the conservative war drums against Obamacare beat on.

I bring up these other scandals to debunk a common liberal complaint, that the Right only objects to Obamacare for selfish political ends, because it is a measure proposed and supported by a President of the opposite party.  On objective grounds, any of the incidents above would make better political ammunition than an (admittedly entirely partisan) healthcare bill which at the very least appears to have been passed with good intentions.  If the only goal were to disgrace the current administration, there are more effective talking points than, “We object to a bill which he thinks will help the country”.

I have said in other places that I do not agree with the strategy of the shutdown – but merely because I have nope that political exchanges might remain civil.  (Not that this is actually a feature common in the American political system.)  But with liberal leadership refusing to so much as consider the compromise of passing the bits of the budget that are not controversial, it seems that perhaps the defunders had it right after all: they have demonstrated, for anybody to see who cares to, that American liberals are not prepared to go anywhere without this bill – and the control of American health care it hands over to the government.

Further, even assuming that control were admitted in general to be both Constitutional and wise, the authority as exercised so far has shown no respect for the liberties, especially religious liberties, of American citizens.  I am not, on reflection, sure a Christian congressman could in good conscience vote to fund a measure which forces people to pay for others’ contraception and abortion (via the mechanism of an insurance plan, to be sure – but even if this absolves the person paying for the insurance, it merely shifts the guilt to the insurer).

And one more thing: the showdown, which turned into a shutdown, was surely not a step taken without contemplation.  This is a fight that someone, at least, on the Right thought would be won – either make enough of an impression to at least force a compromise, or to serve as ammunition come elections next year.

Yet the American Left’s reaction might best be classified as disbelief.  “Why would they pick a fight over this?”  “Don’t they understand we’re going to help people?”  “Why don’t they just recognize the law was passed, and get on with life?”  The Left seems, on the whole, to be amused by, oblivious to, or straight up shocked by the fact that anyone – let alone half a modern country – would think that the best thing a central government can do is get out of the way.  That half the people who involve themselves in politics still take Reagan’s dictum seriously and find the words, “We’re from the government and we’re here to help,” frightening – that seems beyond the Left’s institutional comprehension, even though individual liberals are quite familiar with the existence of their everyday conservative counterparts.

And yet the government is shut down, the drums against Obamacare beat on, and the Democratic-controlled Senate and the President himself are unwilling to budge even so much as to pass the rest of the budget.  A new era of non-partisanship, indeed.