Entropy or Progress?

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I recently had to spend a lot of time with an elderly uncle and aunt (and they would probably resent being characterized as ‘elderly’ if they saw this post).  And, while they have more marbles than most citizens have, despite their combined ages of 181 (and you can go right ahead and be insulted, if you want to), they are increasingly resentful about almost every aspect of life.  They were very unhappy about “Gay Marriage,” they’re unhappy with Donald Trump —that actually goes for most of my friends and relatives— and scornful of Bernie Sanders, and critical about the curricula of most schools (they were both teachers in their day), and on and on.

Listening to their rants, night in and night out, I realized that what they’re upset about is that things are being done differently than they were in the seventies.  More drugs in the streets, more teens getting pregnant, more misbehavior in Washington (DC, that is), the News readers on TV aren’t as well informed, stores don’t carry the food items they used to, and so on.

Of course, they’re older, so you expect grumbling about things.  But they grumble about good things, too.  They have a cell phone, but they grumble about features on it that they do not use.  In other words, they grumble about service packages, which are different from the very small selection of service packages that were available in, say, 1980.

The funny thing is that they’re ultra liberal, and their only beef about Bernie Sanders is that he goes off the deep end.  (They think he should be President, and Hillary C. should be Vice President, never mind the political implications of that setup.)  Having a female president is probably one of the newfangled, unproven ideas that they have absolutely no faith in.

But, you see, that is the essence of Conservatism.  Change is bad.  The most intelligent (OK, I’m being judgmental here) conservatives would say that too rapid change is bad, but change is inevitable.

I stumbled on the following insight when trying to endure the endless grumbling of this pair of relatives.  They equate the constant onslaught of newfangled-ness with disorder.

If we only did things the way we’ve got it completely figured out, we wouldn’t have all these snafus.  If we only didn’t have Obamacare, the Health Insurance companies would know exactly how to painlessly rip off their consumers.  (Right now, the ripoffs are painful, and Health Insurance companies are playing a dangerous game.  The lack of transparency of the old insurance system guaranteed a certain amount of immunity from public scrutiny, which is no longer present.  Utility boards are increasingly aware of the profit margins of insurance companies, and one of these fine days, they will see legislation controlling it.)

It was the disorderliness of public life that was at the heart of which initiatives conservatives would support, and which they would not.  The new wave of T-Party conservatives uses the liberals themselves as their compass: if the Democrats want it, it must be bad, which is a lazy way of analyzing political initiatives.  Instead of looking at the problems of society with a view to ameliorating them, conservatives assess legislation based on the merits of whether it will help business (Big Business, because small business cannot afford lobbyists) and whether it will help vilify Democrats.  They have moved away from the principle of does it increase disorder?

Everyone is familiar with the Second Law of Thermodynamics:

In any isolated system, the total Entropy increases.

Entropy is a mysterious quantity to most people.  I only know to explain it by example.  For instance, if you have a container with every gas molecule in it bouncing about with the exact same speed, its entropy would be zero.  If one molecule goes at a different speed, the entropy increases a little bit.  If all the molecules are zipping around at different speeds, the entropy is huge.

Someone (James Clerk Maxwell) proposed an experiment as follows.

Get a sample of gas into a container divided into two parts, with a door in the middle.  There is a little demon at the door, and he (or she) lets fast molecules pass from one side into the other side (the "fast" side), and lets slow molecules in the fast side into the slow side.  Gradually, over time, the slow side will accumulate the slower molecules, and the fast side will accumulate the faster molecules.

The resulting total entropy will be lower than it was before the demon got to work, because the spread of the speeds in the two halves is less than they used to be.  The gas molecules were disorderly before, and a little more orderly now.  So, in principle, entropy can be lowered, but it involves knowing the speeds of individual molecules, so that the disorderliness can be systematically reduced.  Incidentally, this experiment demonstrates that information is the opposite of entropy, so entropy is a measure of lack of information.

According to Wikipedia, entropy is also a measure of how much heat (or other energy) is present which cannot be made to do work.  As you can imagine, low-temperature heat can't do very much work, even if there's oodles of it lying around, such as in our office-rooms, for instance.  In trying to make the air in my office do some work, I have to do even more work to extract the heat in the first place.

Unfortunately, as the population grows, and other processes take place that cannot be helped, the size of the same old problems we’ve always faced become larger, exacerbated by the extremely wealthy manipulating the laws to keep more of their wealth.  So the government has to feed, clothe and shelter a larger population under the poverty line (some of whom have incredibly large TV sets, to the indignation of conservatives), with lower tax revenues.  Why can’t we feed and clothe them as we did before?

This constant howl about why can’t we do it as we did before?  which the conservatives bring up is disingenuous, because a lot of what conservatives did before was accomplished by fooling with deficits, so that expenses were simply put off until the Democrats were once more in office, and needed to clean up the mess the Republicans had left behind.  The present mess is that Republicans did not put through any major maintenance of public highways and airports and harbors and research facilities, which must now be undertaken with lower tax revenues.  They were hoping that it would take exactly four years for Democrats to confess that they were not up to the job of mucking out the Aegean Stables.

Republicans are not going to be much better at mucking out anything, except their perceived handicaps of Democrats in the Washington bureaucracy.  They can save a few pennies by firing hundreds of career bureaucrats in Washington, but their own young bloods will demand an enormous salary for doing the same jobs, and will do them badly.

What Republicans are best at is creating imaginary crises which they can proceed to solve, such as the fictitious Social Security Crisis, which will probably get solved, if some idiot of the GOP is elected president, by borrowing money from the Veterans Administration, or from some useful part of the budget, and giving it to some conservative think-tank.

Liberals and Democrats do not change things just for the heck of it.  It is necessary to solve ever increasing problems creatively, with ever-decreasing tax revenues.  And, while nobody is watching, even with an inflation rate that is miniscule, businesses everywhere are raising prices, just because they can.  So things change, all in favor of Business, which are the darlings of the GOP!  I will blog very soon about Business.  And it will not be favorable.

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Paris, and a Sane Response--Paul Krugman

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Many analyses by Paul Krugman seem to be right on the mark, and this one is no exception.  People in the US and the UK and France and Germany, and Ireland, in short, all the countries that sometimes feel like making an overwhelming response to terrorism, should do some serious, rational thinking, to have their thoughts ready for such events as those of Paris.

Paul Krugman, a Left-Of-Center columnist and political writer, gave a very succinct analysis of the events of Paris, and all such acts of terrorism, those that have made it to our awareness, and the dozens that have not.

The intention of the terrorists is to provoke a more outrageous response from the highly-armed Western nations (US, UK, Russia, etc) than the random violence they caused.  They get more bang for their buck from the response, than from the senseless murders themselves.

For instance Donald Rumsfeld (Paul Krugman writes) capitalized on the indignation following 9/11 to do what he considered a clean sweep of the Middle East, and attacked Iraq, which was remarkable for being a nation in which there was a delicate balance between the Shia and Sunni sects of Islam, not to mention a small minority of Christians as well.  Now what do we have?  Not only have we killed thousands of innocent Iraqi civilians, we have thrown the entire region out of balance.

Vast multitudes in the US find it tiresome to dwell on the subtleties of International politics.  Is the answer to simply blunder in and make grand gestures?  Those who find it unthinkable that direct action cannot solve the problem of terrorism must be sat upon.  Direct action can only put our children in even greater danger.  Only by no military action can the efforts of terrorists be eroded.  Political action: certainly.  Humanitarian action: yes.  Education.  Persistence.  Resolve.  But counterattack: not a good idea, says Paul Krugman, and after reading his article, I tend to agree.

Meanwhile, my immediate sympathetic response to the horror of the City of Paris seems to have only resulted in those from other parts of the world that suffered attacks both before and after Paris feeling slighted.  There is a constant undercurrent of terrorism out there, and there is a growing multitude of friends and families and compatriots of the victims of terrorism crying out for retaliation, feeding into the intentions of the terrorists for a grand conflagration.  A grand conflagration only helps the terrorists.

The Deal with Bernie Sanders

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I just read the most amazing article about what one journalist thinks is going on with Bernie Sanders.

This piece by Matt Taibbi was linked to by someone on my Fb page, and I think it articulates almost all of the thoughts that I have been trying to verbalize over the last year or so.  While most political journalists and bloggers are carrying on about how "un-electable" Bernie Sanders is, Mr. Taibbi writes that Bernie Sanders is so focused on his somewhat unpopular message simply because he's one of the few politicians who believes the system is not broken.

The other liberal candidates, e.g. Hillary Clinton, appear to be more moderate and electable simply because (Matt Taibbi says) they're cynics, and they're bought up by Big Business, and at the end of the day, they will play the game of the interests that have bought up the political process lock stock and barrel, whereas Bernie Sanders is completely uninterested in playing any sort of game.

To understand the complexity of Matt Taibbi's argument, and to appreciate the evidence he puts forward in support of it, you just have to read the post in its entirety.

When I first read it, what caught my eye is the fact that the political scene has to be viewed in the context of how Citizens United remains standing, despite the considerable force that liberals within Government and without have brought to bear on it.  Others may pay lip service to wanting to bring it down, but Bernie Sanders is almost the only one who goes about every day as if Citizens United must fall, and life must go on after it does.  Most of us (Taibbi writes) can't bear to think about "the little old lady who is on the point of freezing to death," at least not for very long, because some utility has decided to shut off her heating.  But Bernie thinks of nothing but these things, which have no business to happen in an enlightened country such as we wish ours is.

That's Bernie Sanders explained.  I sure wish it is true.

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Donal Trump is showing us how not to be Stupid

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We have to take a good hard look at ourselves.

Just repeating the mantra that everyone other than ourselves is stupid clicks with (unfortunately) the sort of people with whom we have very little in common.  We want to connect with people who will help us solve the problems that we face now, and will face shortly.  We can't be choosy about who's going to help us.

Trump might not be an idiot, but this constant ranting does tend to suggest that his best days are behind him.  Unless he is brilliant, and is implementing a Machiavellian scheme to convince the nation that he is stupid and harmless, he really is stupid and harmless (there I go, doing exactly what Trump himself is doing).  So he keeps us guessing whether his silliness is a brilliant smokescreen, or genuine silliness from the essence of Trump.  I vote for the latter.

On the other side of the coin, it is time we stopped poking fun at the GOP, and took stock of how we should proceed.  Sanders and Clinton have set an excellent example in the debates: they declined to indulge in mudslinging (though they were not reticent about criticizing each other on substantive issues, even if the actual words they used were carefully calculated), and it is time we stopped simply grumbling, and made concrete suggestions which conservatives can support, even if unwillingly.  If there is one criticism of Bernie Sanders, it is that he seems to have turned his back on compromise.  This is understandable, because the GOP, at least, seems to be uninterested in compromise.  John Boehner's resignation seems to suggest that he had decided that non-compromise had been taken as far as it could.

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Sleeping Positions

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I saw this one on a Fb post, and I thought I should share.

It is funny, and pokes gentle fun at all these psychological reports about things that are highly speculative at best.

College Students These Days

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Gone are the days when information, facts and analysis could be given to students without a lot of decoration.

I just passed a classroom this afternoon where students were playing some sort of game.  On one hand, there are lots of highly educational games for middle-schoolers which our undergrads may never have seen, and it is just possible that one of these may make more of an impact on a jaded freshman than a simple lecture, even if the lecture is beautifully delivered with Powerpoint slides, a sprig of parsley and a cherry on top.

I myself occasionally use a game or two to draw in the attention of a student whose hands are sneaking towards their cell phones to stave off utter boredom.  But then, I'm teaching future teachers, and they need to have a trick or two to lay on their future students, who are even more likely to have their attentions wander.

Some faculty getting into the action

One usually assumes that each new generation learns more sophisticated information than the last.  But we've arrived at a point where the trend is reversed: some material actually has to be taken out, to make space for games and entertainment.  I don't mean my own institution, oh no.  We would never do that.  But other institutions do a lot of this.  Cut the curriculum in half, and put in lots of fun.  No fun, no work.  Jack* is not going to be a dull boy if he can possibly help it.

But good luck getting your fresh young graduate to do the work of a senior employee, if you're hiring!  Chances are he only learned about the first 15 letters in the alphabet, because the last--however many-- had to be jettisoned to make space for the occasional period of fun.

Unless you hire somebody from our school :)

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[* An obscure reference to the old adage that 

"All work and no play

Makes Jack a dull boy."

--just in case your elementary school reading list was a trifle incomplete.]

The Nine-Point Circle

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Just for fun, my Geometry class and I decided that we would do a team Blog about this interesting result from Euclidean Geometry: Feuerbach's Nine Point Circle.  We're going to work on this in a leisurely fashion over a long period, so please check in from time to time to see how far we have gone.  We're planning to put in lots of pictures, to make the geometric reasoning clearer, but since it is being written "by committee", it will probably show the usual signs of Committeeishness.

There are a few terms that some readers might not recognize.
+ Vertex: this means the corners.  In the triangle ABC, the vertex A is just the point A, and so on.  The plural is vertices.
+ The foot of a perpendicular.  A perpendicular is from a point to a line.  The foot of a perpendicular is the point where it hits the line to which we're drawing the perpendicular.
+ An altitude is a perpendicular from a vertex to the opposite side.


Introduction
The Nine-Point Circle Theorem is an interesting result in Euclidean Geometry, having to do with a circle that that passes through six important points on any triangle.  Every triangle has several important points associated with it, and usually these points have little to do with each other.  But it just so happens that someone discovered that six of them all lie on a circle.  Furthermore, it turns out that there are three more relatively unimportant points that also lie on this circle.

To explain the points and their significance, we show them in a sequence of diagrams below.

First, we show the midpoints of each of the sides.  We indicate these in RED.

Next, we show the feet of the altitudes from each vertex to the opposite side. We show these in GREEN.

Incidentally, this sketch illustrates that the altitudes meet at one point, which is called the orthocenter, shown as O below.

Finally, we show that the points that lie midway between the orthocenter and the vertices also lie on the circle; we show these in PINK.

And now, the moment you’ve all been waiting for: the actual circle:

We shall actually prove that these points lie on a circle (though it obviously does, according to the picture).

The proof of the existence of the 9-point circle is based on two previous theorems.

The first of these is the Mid-Point Theorem, which says that if XYZ is a triangle, and P is the midpoint of XY, and Q is the midpoint of XZ, then

(i) PQ = 1/2 YZ, and
(ii) PQ is parallel to YZ.

The proof of this is not difficult.

Let XYZ be a triangle, and let P and Q be the midpoints, as described above.  Consider the diagram at right.

To prove this theorem, we need a construction.  Extend PQ to point R, in such a way that PQ and QR are congruent (i.e., equal in length).  Join ZR.  Now triangles PQX and RQZ are congruent by "Side-Angle-Side".

Angles XPQ and ZRQ are congruent by Corresponding Parts.  Using the Alternate Angle Theorem, we know that lines XPY and RZ are parallel.

Consider the second diagram.  As you can see, XP, RZ, and PY are all congruent.  There is a result that says that if PY and RZ are both parallel and congruent, then PR and YZ are also both parallel and congruent.  It also means that the length of PQ is half of the length of YZ!

The second result we need is the interesting fact that if UV is the diameter of a circle, and if it is a side of a triangle whose third vertex, W, is on one of the semicircles, then angle W will be a right angle.  We give a diagram; the result follows from a little angle-chasing (notice that there are two isosceles triangles in the figure).