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Saturday, December 21, 2013

Geocentrism and the Three Sillies

The Three Sillies illustrated by Arthur Rackham, as found on Project Gutenberg.

Every once in a while, I come across the claim by some well-intentioned Catholic that Sacred Tradition demands that we adopt a geocentric model.  I'm not really sure what drives this kind of misconception; it may be an honest mistake about what the point of Church Teaching is, or it may be the same kind of thrill with being one of the few "in the know" that has led me to compare cryptozoology with pagan mystery religions.  Periodically these claims are disposed of by Catholic apologists, and it is not my intention to deal with the whole issue here.

Instead, I would like to deal with a more subtle argument that is occasionally tossed out by geocentrists.  Many of them have enough education to know that Einstein's General Theory of Relativity allows us to correctly describe the mechanics of the universe using essentially any coordinate system, including coordinate systems in which the Earth is non-rotating, stationary, and at the center.  Nature is satisfied with such a coordinate system.  If that coordinate system is as good as any other, surely preference should be given to the one used by the Church Fathers?

Of course, this misses the fact that Relativity specifically denies that the idea of a center is physically meaningful, and it is only physically meaningful distinctions that are in dispute.  No one will deny that the Earth is the center of the baseball universe, for example.  But is it true that just because it is possible to make correct physical descriptions in any coordinate system, no coordinate system can be said to be "better" than another?

The answers to a surprising number of basic questions can be found in folk tales. The one of relevance here is an English folk tale called, "The Three Sillies".  Please read the whole story, but the passage that most pertains to this case is as follows.


And the inn was so full that he had to share a room with another traveler. Now his room-fellow proved quite a pleasant fellow, and they forgathered, and each slept well in his bed.
But next morning, when they were dressing, what does the stranger do but carefully hang his breeches on the knobs of the tallboy!
"What are you doing?" asks young squire.
"I'm putting on my breeches," says the stranger; and with that he goes to the other end of the room, takes a little run, and tried to jump into the breeches.
But he didn't succeed, so he took another run and another try, and another and another and another, until he got quite hot and flustered, as the old woman had got over her cow that wouldn't go up the ladder. And all the time young squire was laughing fit to split, for never in his life did he see anything so comical.
Then the stranger stopped a while and mopped his face with his handkerchief, for he was all in a sweat. "It's very well laughing," says he, "but breeches are the most awkwardest things to get into that ever were. It takes me the best part of an hour every morning before I get them on. How do you manage yours?"
Then young squire showed him, as well as he could for laughing, how to put on his breeches, and the stranger was ever so grateful and said he never should have thought of that way.
"So that," quoth young squire to himself, "is a second bigger silly."
Although the stranger managed to eventually get his pants on each morning, the reader (or listener) is supposed to understand immediately that this is the wrong way to put on pants. 

What the geocentrists would have us do is just as silly.  Most mechanics does not really require Relativity; usually Newton's Laws are sufficient, and they can be taught to students with no more math than algebra and trigonometry.  

An example would be the trajectory of a satellite in a circular polar orbit.  For this, we can choose a coordinate system in which the Earth is stationary, but we still need to allow the Earth to rotate under the satellite.  Everything is easy to explain; the satellite is in uniform circular motion, and its centripetal force is supplied by gravity as specified by Newton's Law of Universal Gravitation.  

If we maintain that the Earth does not rotate, though, the eastward rotation of the Earth will have to be explained as a westward force on the satellite, a force that varies with latitude and is hard to justify in simple terms.  The real justification comes from a relativistic phenomenon called frame-dragging, in which the motion of a massive body "drags" the inertial reference frame with it.  In sane coordinate systems, this is hard to measure except near a massive, exotic object like a black hole or neutron star.  A very sensitive experiment, Gravity Probe B, was required to see this effect from the rotation of the Earth.  To calculate the relativistic forces from first principles requires coupled partial differential equations of tensors -- higher math than most people will ever see, let alone master.

The final answers for what will be observed are, of course, identical, just as the final result of putting on their pants was identical for the young man and the stranger.  Just as trying to jump into the pants is the wrong way to put on pants, though, the geocentric approach is the wrong way to do physics.  To top it off, the people with the attachment to the geocentric model always prove to have never even attempted to solve physics problems in what they insist is the uniquely correct way.

Gravity Probe B Confirms the Existence of Gravitomagnetism

Friday, December 20, 2013

What Do Neon Slugs Tell Us About Bigfoot?

Nothing, really.  How could any reasonable person expect them to?

Some fans of cryptozoology, though, think otherwise.

Mount Kaputar in Australia is apparently a kind of snail paradise, serving as home to a number of rare species of snails and slugs with little predation from vertebrates.  New Guinea may have its birds of paradise, but Mount Kaputar has the slugs of paradise -- large, hot pink slugs.  (Sexual selection drives the colors, patterns, and displays of the birds of paradise, but the reason for the hot pink color is not yet known.)  Until recently, these were thought to be just a variant of the red triangle slug, but careful morphological and genetic studies now indicate that these slugs should be grouped under a new species name.

Although this has nothing obvious to do with cryptozoology, it was reported on cryptozoology web sites (to which I will not here link), usually with the "therefore ..." unstated -- until, predictably, one of the regulars filled in what the take-home message was supposed to be:
You can have something like that crawling around, suddenly Bigfoot ain’t so farfetched, eh?
A statement like that isn't just pseudoscientific; it's pseudorational.  It's pseudoscientific because it seems intended to be the same kind of statement as when biologists say that, based on what we have discovered so far, probably around 90% of all species remain undescribed, whereas in fact, the writer's confidence that Bigfoot is a real, corporeal animal has no relationship whatsoever to science.

First a caveat regarding the word "species".  Sadly, determining what constitutes a species is not at all as straightforward as it seemed in my elementary school textbooks.  Those books said that two animals are of different species if they cannot produce offspring with each other or if, as with a horse and donkey producing a mule, the offspring is always infertile.  That is not really something that paleontologists can usually confirm or deny, though, so they have tended to create new species names whenever the differences seemed large enough to justify it -- in the process creating far too many species names, since sometimes two pieces of animals (for example, heads and legs) would be found separately and assigned different species, or juveniles would be mistaken to be different species than the adults, or the differences between males and females would be mistaken for the differences between different species.  Even when genetic information is available, though, the meaning of "species" has changed, as is clear when we are told that most of us are hybrids of two or three "species" of humans -- Homo sapiens, Neanderthals, and for many, the Denisovans.  (Even this is messed up, since according to taxonomic convention, Carl Linnaeus is the "type specimen" of Homo sapiens, meaning the species is defined by him.  Because he was a European, though, he must have had some Neanderthal ancestry, if the genetics studies are correct.)  In many cases, species boundaries are fuzzy, indistinct, and ultimately dependent on the varying criteria used to set them.
What can we infer from the discovery that the bright pink slugs of Mount Kaputar are a different species?  Extrapolations can be constructed on the basis of a "sample space".  A sample space has to be a collection of genuine possibilities, of which the observed cases can be said to be a typical sample.  For example, the trans-Neptunian object Sedna is near the limits of modern telescopes even when it is closest to the sun (at about 76 times the earth-sun distance), but it follows a very eccentric orbit.  Kepler's 2nd Law insures that Sedna is most often near its farthest point from the sun -- 937 times as far away as earth is.  Our sample space would consist of all possible numbers (consistent with current observations, that is) of Sedna-like objects at random locations on their orbits.  If Sedna is unique, we have been remarkably lucky to catch it in the brief period it is close enough to the sun for us to see.  It is much more likely that there are dozens, maybe even hundreds, of Sedna-like objects orbiting the sun.

What would be a corresponding sample space for these slugs?  As with the case of Sedna, it would have to be built of a population of possibilities that are related to the discovery about the slugs, so how would the discovery be described?
  • The existence of the pink slugs had been known already; in that sense they were not a new discovery at all.  
  • They are obviously somewhat different in appearance than red triangle slugs, but not so different in coloration or size to be obviously a different species.   
  • It could not have been "ethno-known" that they are a different species from red triangle slugs, since "species", with all its caveats and technicalities, does not entirely correspond to any concept outside modern biology.
  • The slugs are restricted to a remote, relatively inaccessible, and geographically limited range.
  • This was an evolutionary, not a revolutionary, change for science.  That's not an insult -- almost all science is that way -- but it serves to put the discovery into context.
Reasonable inferences, then, would include the following.
  • Genetic studies will show that other isolated populations of invertebrates with no visible differences are in fact different species.  (This happens all the time.)
  • Genetic studies will show that other isolated populations of land animals with no visible differences are in fact different species.  (Again, this is common.  It happened just recently with crocodiles, even though their populations do not appear to be as completely isolated.)
Every time the sample space is expanded, though, the inferences become less useful.  We're approaching the season of predictions for 2014, so we can take an example from the kinds of predictions psychics make. 

  • An actor who played a central character in MASH will die in 2014.  This has a very small "sample space", making it an interesting prediction.  It might well come true, but probably has no more than a 10% chance of coming true at random.
  • An actor who played a character on at least one episode of MASH will die in 2014.  This includes many more actors, so it is easier for this to come true. However, the implications of this for Alan Alda aren't quite so scary.
  • An actor who played a character on TV during the 1970's or 1980's will die.  It would be something like a miracle if this did not happen.  It's a prediction so safe as to be utterly useless.


Now let's look at the characteristics of a discovery of Bigfoot.
  • The mere existence of Bigfoot is not recognized by science at all.
  • Anything that could be called a "Bigfoot" would be obviously different from any animal known to have ever existed.  It would be larger and heavier than known human relatives like Paranthropus, and with different feet.  No other ape is known to be fully bipedal, including Gigantopithecus.
  • It is claimed that Bigfoot is ethno-known, on the basis of both modern alleged sightings and American Indian folklore.
  • Alleged Bigfoot sightings come from every U.S. state other than Hawaii and several of the Canadian provinces.  Depending on whether cryptids like the Almas, Yeti, and Yowie are considered the same thing as Bigfoot, this range might be extended to cover much of the world.
  • The verified discovery of a real Bigfoot would be a revolutionary discovery for (at least) primatology, evolutionary biology, and anthropology.  It would easily be worth a Nobel Prize.
In order to expand the sample space to include the discovery of Bigfoot, the sample space would have to include the addition of a new species of any land animal, important or unimportant, known or unknown, obviously different of visually indistinguishable, with a range of any size whatsoever.  With a sample space that big, only thing that can be said is that new species of animals remain unnamed, a fact as indisputable as that someone who has appeared on TV will die in the next 12 months.  The implications of this for the existence of Bigfoot, though, are even less than the implications of "an actor will die" for Alan Alda.

Wednesday, November 13, 2013

Soldiers and Umbrellas

To expand a bit on yesterday's post:  A soldier makes you free in the same way an umbrella makes you dry -- an umbrella that someone else is holding.  An umbrella can protect you from some of the external threats to your dryness, but not all of them -- it can't stop you from getting wet from the side or below.  Also, if you are already wet, it cannot really dry you off.  Likewise, a soldier can give you partial protection from external threats to your freedom, though some threats might be seen as sneaking in "from the side".  For example, censorship, rationing, the draft, and the like are real limitations to freedom, and these are often imposed in the face of external threats to freedom.  What's more, if you do not have the habits, culture, and internal disposition of freedom, a soldier is not really able to provide you with them.  

But perhaps most importantly, what an umbrella actually does depends on the skill and intentions of the person holding it.  An umbrella can be used to keep someone dry, but it can also be used to make someone even wetter than he was before (by channeling water onto him or dumping the accumulated water on him).  In a similar way, what a soldier actually does depends on the skill and intentions of the government employing him. 

Tuesday, November 12, 2013

Right-Wing Version of "You Didn't Build That"


Remember this?  When it came out, "conservatives" (by pretty much any definition) and Republicans reacted with outrage, as well they might.  

Yesterday, however, was Veterans Day, one of several days a year which generate warm hearts and fuzzy thinking from "conservatives".  On such a day it is not hard to find items like this gem:
It is the soldier, not the reporter, Who has given us freedom of the press.  It is the soldier, not the poet, Who has given us freedom of speech.  It is the soldier, not the organizer, Who has given us the freedom to demonstrate.  It is the soldier, Who salutes the flag, Who serves beneath the flag,  And whose coffin is draped by the flag, Who allows the protester to burn the flag.

This is exactly the same thing Obama was saying.

Saturday, November 2, 2013

The Aliens Are Among Us!

I just learned something that will take some time to process.  I had used the "Nyan Cat" before as an example of how alien Japanese culture is.  Although I still maintain that Japanese culture really is that alien, it turns out that the Nyan Cat was not created by someone from Japan, but by Christopher Torres, who is from Dallas, TX.  The song comes from Japan, but the song is not really the weird part of the video.

This is the kind of discovery that occasionally makes me feel as though I have gone to bed in one universe and awakened in a slightly different parallel universe.

Saturday, September 28, 2013

Inevitability -- The Keyword for Our Age

Nevertheless, I think that with us the keyword is "inevitability," or, as I should be inclined to call it, "impenitence." We are subconsciously dominated in all departments by the notion that there is no turning back, and it is rooted in materialism and the denial of free-will. Take any handful of modern facts and compare them with the corresponding facts a few hundred years ago. Compare the modern Party System with the political factions of the seventeenth century. The difference is that in the older time the party leaders not only really cut off each other's heads, but (what is much more alarming) really repealed each other's laws. With us it has become traditional for one party to inherit and leave untouched the acts of the other when made, however bitterly they were attacked in the making. -- G.K. Chesterton
I have no real feeling for the underlying political attitudes at the time when Chesterton wrote this, but "inevitability" certainly has oppressed political thinking in America for the past several decades.  When I was a child, for example, everyone took it as inevitable that either the Cold War would end in a nuclear holocaust, or at best we would have a bipolar standoff that would last for centuries.  When that turned out to be wrong and America was (according to many) the "last superpower", it became inevitable that American-style democracy would be embraced everywhere; this was the thesis of Francis Fukuyama's 1992 book The End of History and the Last Man.  During the late 1990's, it became inevitable that the stock market would only go up -- the dot-coms had brought about a "new paradigm" of economics.  

We all know how those predictions fared.  Other "inevitable" outcomes have also failed to materialize.  A century ago, Prohibition was going to put a permanent end to alcohol abuse in the USA.  It has been "inevitable" on several occasions that Islam would conquer all of Christendom; we hear that it is again "inevitable".   Yet Chesterton was right in another place to describe a Christian hero as, "It is he that saith not 'Kismet'; it is he that knows not Fate."

Today, of course, it is inevitable that "gay marriage" will become a permanent fixture of American life.  The military has been required by their current commander-in-chief to embrace and celebrate homosexuality.  Many ecclesial communities now simulate marriage ceremonies for homosexual couples.  The Republican Party is backing away from support for "traditional" marriage, and a former editor of First Things says the Church should give up on marriage.

Well, maybe it will become a permanent fixture -- not because "gay marriage" will endure forever, but because America is highly unlikely to endure to the end of time.  Regardless, the future comes from two main sources:  the choices we make  each day, and realities that exist without regard for our choices.  The nature of the human being and the nature of marriage belong to the second category, which is why "gay marriage" is not truly marriage, and any legal or popular acceptance is only the acceptance of a fiction.   Whether we choose to accept it, either on a legal or a popular level, is of course a choice, but it is not a choice that will make it impossible for future generations to make a different choice, probably a choice that more accurately represents reality.

So what is behind all this conviction of inevitability, particularly among politicians?  It's easy to come up with several possibilities, and each probably makes some contribution.

  • If the future is inevitable, the politician cannot be held responsible for it.
  • Politicians are only faking their commitment to the moral well-being of America.  (The moral well-being of a country is a temporal good, and the government does have a responsibility to promote it.)  Frankly, they are probably faking their commitment to the other temporal goods as well.
  • The politicians have no moral courage.
Another possibility is suggested by a story by Hans Christian Andersen: 
"But he has nothing at all on!" at last cried out all the people. The Emperor was vexed, for he knew that the people were right; but he thought the procession must go on now! And the lords of the bedchamber took greater pains than ever, to appear holding up a train, although, in reality, there was no train to hold.
Page 234 of Andersen's fairy tales (Robinson)

Andersen was right about this.  How often have we stuck with a policy that was manifestly stupid and obviously not working, yet a president decided we must "stay the course" because we would lose face if we admitted that he was wrong?  The whole world sees that he has made a mistake, but in his pride he hopes that if he stubbornly refuses to acknowledge it, no one will notice.  Foolish pride removes his freedom to reverse course and makes it "inevitable" that he will persist in stupidity.

Friday, September 27, 2013

The Internet Is the Wall of a Truck Stop Restroom

"The American Catholic" has a recent blog post by Donald R. McClarey in which he complains that Popular Science has stopped accepting comments on their web site.  "This is too funny.  Science began as a search for truth," writes McClarey, who apparently thinks that many advances in science depended on comments by the general public in response to magazine articles.  McClarey simply cannot accept that the public comments are ill-informed and frequently insulting.  It must be a political cover-up! 

I was aware of and acknowledged the irony when I wrote in response, but I thought it really should be pointed out that, at least in principle, Popular Science had it right.  

Look at the comments underneath a typical Youtube video -- say, a history documentary.  You'll find incendiary and bigoted remarks representing the crazy right, the crazy left, and the just plain crazy crazy.  Or look at the comments left on the web pages for major newspapers.  When Aaron Hernandez was arrested for murder, it was reported in a sober and straightforward way by sources like ESPN, Sports Illustrated, and Yahoo! Sports.  The comments below the stories, though, were full of gleeful anticipation of Hernandez being raped in prison.

We've all seen that kind of writing before -- the walls of a filthy restroom in either a bad part of town or on the interstate.  This is really not the place to go for a serious pursuit of any truth, whether scientific, philosophical, or theological.

I also pointed out that the Holy See does not make it possible for visitors to their web pages to leave comments on, for example, papal encyclicals -- yet the Holy See is very much pursuing truth.  Why does the Holy See cut itself off from such a vast source of "wisdom"?  Because there is too much cumulative experience there to make that mistake.

So how did "The American Catholic", great champion of free speech and the wisdom of blog comments, respond?  They deleted my comment.

Sunday, September 15, 2013

It Is Better for Me to Fall into the Hands of the LORD

And David said to Gad: I am on every side in a great strait: but it is better for me to fall into the hands of the Lord, for his mercies are many, than into the hands of men. -- 1 Chronicles 21:13

Amen!  And may we not even fall into what they call mercy.

Saturday, September 14, 2013

"Standing on My Head"

Chesterton said he liked to stand on his head because the unusual perspective allowed him to see things as they are (or might be), not as he merely expected them to be.  I had an analogous experience while driving yesterday.  I saw part of a bridge over the Ohio River, but I did not immediately recognize it as part of a bridge.  What I saw was something like this
but what I thought I saw was more like this
Rollercoaster limit heide park germany


This naturally got me thinking -- "Wouldn't it be cool to use the same trusses that support a bridge to also support a roller coaster?  I wonder if this has ever been done?"

Well, I found no evidence that it has been done, which is hardly a surprise.  It would be outrageously cool and could be the signature piece of the town that had it, but that is really the only thing it has going for it.

  • Bridges are not infrequently hit by barges.  This could have disastrous consequences for a roller coaster on the bridge.
  • If the coaster cars became stuck at the top, a rescue could really only be attempted from the surface of the bridge, and it would require closing the bridge.
  • Drivers would be distracted, and perhaps dangerously startled, by the roller coaster.
  • Objects thrown or accidentally dropped from the roller coaster would present a hazard to drivers.
In spite of all the good sense arguments against it, I'd still like to see someone build one.

Sunday, September 8, 2013

What Good Does It Do to Pray for Peace?

What good does it do to pray for peace, as the Pope has requested?  At first glance, it might appear to do no good at all.  After all, at least one party to a war must (to some degree) will the war, and one of the clearest but strangest facts of theology is that God will not remove our free will.  He allows us to choose evil; He allows us to hate others.

Friedrich Overbeck - Praying Monk

But think again about times recently when you may have been grouchy or unreasonable.  We do ultimately choose how to behave, but there are many influences on our moods.  Maybe you did not sleep well the night before; maybe you had an upset stomach; maybe you did not know when the mechanic would finish with your car or how much it would end up costing.  Such exterior circumstances have nothing to do with free will, but they can make it easier or harder to exercise your will in a good way.

I assume that the good effect of prayer is likely to be (in many cases) some change in these external circumstances.  A president who is feeling too tired to deal with complaints that he has not taken action may get better rest; a combatant who feels invincible might experience a sudden reminder of his own mortality; a skillful compromise which no one had foreseen might be suggested.  As the saying goes, peace would be given a chance, though that chance may always still be refused.

The other benefit, of course, is the benefit of all prayer:  it reminds the one one praying that God is ultimately in charge of this as of all things.

Sunday, August 25, 2013

What I Mean by "Chestertonian"

I few weeks ago I noted that the word "conservative" now comes loaded with too many implications I cannot accept.  Since I am not willing to embrace a police state if only people will concede me the coveted title of "conservative", it is necessary to find some other label.  For now, none seem to fit quite so well as "Chestertonian", which automatically implies a great deal to anyone familiar with the writings of G.K. Chesterton, whose cause for canonization as a saint has finally been taken up.

G. K. Chesterton at work

Now it might be objected that what I really mean is "Distributist", and indeed Distributism covers a good deal of what I mean.  Distributism favors private property, but not monopolies and not usury.  Capitalism is an economic philosophy that sees the purpose of money as making even more money; Distributism is an economic philosophy that sees the purpose of money is the good of people.  Nor is Distributism by any means Socialism.  However, Distributism is really only an economic idea, and I mean something more general.

Well, since Distributism is based on Catholic social doctrine, and I want a more general term, maybe I should just call myself "Catholic"?  No, for three reasons.

  1. My intention is to describe a political / cultural position.  This will have nothing really to do with Catholic sacramental theology, for example.  Protestants, Jews, etc. could probably agree with most or all of what I mean.
  2. On the other hand, I do mean something particular to Western civilization.  Although most Catholics share this cultural background, the word catholic means universal, and there are Churches in communion with Rome that exist in very different cultures.  I still think that Indian, Chinese, Japanese, and other cultures could be somehow "baptized" as was done with the cultures of pagan Greece and Rome.
  3. Finally, it is not helpful to describe my politics as "Catholic" when Biden and Pelosi, among others, also claim that name.
Here, then, is what I mean by "Chestertonian":

  1. It starts with an anthropology and ethics consistent with the Catholic understanding of what a human person is.
  2. It furthermore includes the cultural legacy of what used to be called Christendom, which for the past thousand years or so has been from Russia to Greece to Spain to Ireland, and in the past has certainly included Syria, Egypt, and North Africa.  That cultural heritage is largely derived from ancient Greece and Rome, but contributions from Celtic, Slavic, and Germanic cultures have also left their mark.  All of these cultures have been harmonized to Christianity, so that they are complementary to each other rather than simply being in conflict.
  3. Finally, though, I give preference to the specific practices and attitudes of my own country and region.  These include a preference for individualism and for the frontier; a mountain man is more archetypically American than is a banker or a baker.  Implicit in this is a conviction that anyone can prove his worth and that barriers preventing people from improving their lots in life must be minimized; likewise, there is an implicit distaste for those whose status is achieved only through an accident of birth.  American culture also takes an attitude towards government that is decidedly more vigilant than deferential.  Finally, it also includes a hopefulness that we really can leave the world a better place than we found it.  That hopefulness has often been expressed in rash and unwise actions, but it gives rise to a willingness to take up a noble cause in spite of difficulties.


Tuesday, July 30, 2013

How to Set Up Physics Problems

Physics is a difficult subject.  It can be intimidating, particularly when you have little experience with the subject, as is the case with anyone taking an introductory physics class.  Physics problems will look much less scary, though, once you learn to break them up into small, manageable pieces.  The following is the advice I give to students in my intro classes.

More than 50 years ago, G. Polya described the main parts of solving math problems in his book How to Solve It.  These same steps help with the solutions of physics problem.

1. First, make sure you really understand the problem.  You should be able to paraphrase the problem without adding any new assumptions or removing any of the constraints of the original problem.  Identify all the quantities you are given or asked for and name them with variables.  You should also make a sketch to help you understand (and explain) how things relate to each other geometrically.

2. Next, devise a plan.  This means identifying the equations and principles you will need to solve the problem.  In most cases, you will have to use more than one equation, with the output of one equation becoming the input of another.  Make sure you see these relationships.


3. Once you have reduced the physics problem to a math problem, use your math skills to solve it. Your math prerequisites should be enough for you to do this, but the lectures and textbook will contain important refreshers and explanations for techniques you may not have quite mastered. Even though you are probably not used to doing so, it is important to keep all your units at each step.  This will help you in two main ways.

 a) It makes you think about the numbers you are using.  In introductory physics, we use the acceleration 9.8 m/s2 very frequently, but the velocity 9.8 m/s will only appear by chance.  If you see the velocity 9.8 m/s in your calculations, you are probably making a mistake, so double-check.  You won't notice this if you only write 9.8. 
 b) It performs the same check for numerical problems as dimensional analysis does for algebraic problems.

 4. Having carried out the math, the problem is solved, but it is still important to look back at your results and perform any checks you can.  

 a) Sometimes you may see that the answer is obviously wrong.  For example, a test several years ago asked how long it would take a diver off a 10-meter platform to hit the water, and a student answered 300 seconds.  That's five minutes to fall about thirty feet!  An answer like that is obviously wrong.
 b) Check to see whether your answer satisfies the constraints and conditions of the problem. For example, the quadratic equation you must solve for the diver problem above has two solutions:  1.43 seconds before the diver steps off the platform or 1.43 seconds after the diver steps off the platform.  However, the diver did not shoot up through the surface of the water to the level of the platform; before the dive, she was just standing on the platform, not in free fall.  Only the second answer meets the conditions of the problem.
 c) If your problem has a numerical answer, you could make an order-of-magnitude estimate to see if the answer you give is in the right ballpark.  These are most important if you are dealing with a problem involving very large numbers (like are found in problems relating to the planets) or very small numbers (as are found in problems involving atoms or electrons).
 d) If your problem requires an algebraic answer, you could perform dimensional analysis. Dimensional analysis can't tell you that your answer is right, but it can sometimes tell you that your answer is wrong.

The problem-solving sheets will help guide you through these steps.  In practice, the greatest difficulty most students have is with either step 1 or step 2, so let's look at those in more detail.


Understand the Problem


Read through the problem once to get the general idea, but then go back and look for keywords.


Example 1:  Spider-Man falls 10.0 m off a building ledge before a strand of his web starts to slow him. The strand of web brings him to a stop in 2.0 m.  What is the acceleration (assumed constant) as the web catches Spidey?

In this example, there are 3 key positions: the top of the building, where Spider-Man starts falling from rest; the point 10.0 m below that where the web starts to stretch; and the point 2.0 m below that (12.0 m) below the top of the building where he again comes to rest.

Since Spidey is just falling off the building, his initial vertical velocity will be zero when he is at the top of the building.  For the first 10.0 m he is freely falling, so his (vertical) acceleration is a constant -9.8 m/s2 over that distance.  For the last 2.0 m he again has a constant acceleration (because both gravity, which continues to exert a force on him, and the tension in the web are  both constant – not a realistic assumption for the web, as a matter of fact), but here the acceleration is not given.  It is what we are looking for.

Notice how useful a sketch would be in keeping this straight in your mind!  Such a sketch does need to show (not necessarily to scale) the two distances through which Spidey falls, but it does not need to contain elaborate artistic details.  A stick figure for Spider-Man is just right!

If sketches are useful for one-dimensional problems, they are indispensable for two-dimensional problems.


Example 2:  A spring-loaded toy cannon shoots a small metal ball off the top of a table with an initial speed of 4.75 m/s.  The ball is fired at an angle of 60o from the horizon at an initial  height of 1.10 m. How far does the ball move horizontally by the time it hits the floor?

The hint for how we go about this problem comes from the wording of that last sentence.  The solution has two main parts.  (1)  “How far does the ball move horizontally by [a certain] time?”  “How far” is the final result we are looking for, but to get it we need first to find out how much time passes until we stop measuring.  That part comes from the second question:  (2) “At what time does the ball hit the floor?”

As for the sketch(es), in this case you need to show the parabolic trajectory of the ball as it leaves the cannon.  The initial height needs to be shown, as well as the horizontal distance you are seeking and the initial velocity vector.  You should also show (probably in an expanded view) a right triangle showing the initial velocity vector together with its horizontal and vertical components.

Devise a Plan

 1. Write out all the equations you think might be useful in the solution.

 2. Circle every variable that you already know.

 3. Draw a box around the final variable you are looking for. 


 4. If you find one equation in which all the variables are either circled or boxed, you can use that equation to solve the problem.  

 5. If not, see if you have a pair of equations in which 
 a) in one equation, everything is circled but one variable
 b) in the other equation, everything is circled or boxed but one variable, and 
 c) the unmarked variable is the same in both a) and b).  

 6. Draw a parallelogram around the unmarked variable from step 5.  This is your intermediate result.


 7. Find the intermediate result from the equation that has everything circled except for the variable in the parallelogram. 

 8. Use value of the intermediate result in the other equation to find your final result. 

That sounds terribly complicated, but the examples already given (the Spider-Man problem and the projectile problem) should help clarify what I mean. 

Example 1: First some definitions.  Let's call the top of the building point A, the point 10.0 m below that where the web first starts to stretch B, and the point 2.0 m below that where Spidey stops again point C.  Some quantities are defined at these points and will use a subscript A, B, or C; others are defined over the interval AB or the interval BC, so they will use those subscripts.  Both intervals involve constant acceleration, so we can write out our equations for constant acceleration.

1. 
ΔyAB vA tAB + ½ aAB tAB2
2. vBvA aAB tAB
3. vB2vA2 + 2 aAB ΔyAB
4. ΔyBCvB tBC + ½ aBC tBC2
5. vCvBaBC tBC
6. vC2vB2 + 2 aBC ΔyBC

The only things we know are 


ΔyAB
 = -10.0 m,
vA = 0 m/s,
aAB= -9.8 m/s2,
ΔyBC = -2.0 m, and
vC = 0 m/s,

so draw circles around those variables.  We are looking for aBC, so draw a box around that. None of the equations have all the variables in circles or boxes, so we follow Step 5 and look for a pair of equations that share one unmarked variable.  Equations 3 and 6 satisfy the requirements, with 
vB as the shared unmarked variable.  Draw a parallelogram around it; it is an intermediate result we will need.  Equation 3 now has only circles and the parallelogram; it can be solved for  vB, then Equation 6 can be solved for aBC.

Example 2:  To simplify things, I'll assume you have already calculated the x- and y-components of the initial velocity.  In the x-direction you have constant velocity, and in the y-direction you have constant acceleration, so the equations you might use are

1. Δyviy t + ½ ay t2
2. vfyviyay t
3. vfy2viy2 + 2 ay Δy
4. Δx = vix

We know

vix = (4.75 m/s) cos 60o = 2.375  m/s,
viy = (4.75 m/s) sin 60o = 4.1136  m/s,
Δy = -1.1 m, and
ay = -9.8 m/s2,

so draw circles around those variables.  We are looking for Δx, so draw a box around that.  Once again, when we come to Step 4, there are no equations that have only circled and boxed variables.  However, Equations 1 and 4 are a pair satisfying the requirements of Step 5, with t as the intermediate result, so draw a parallelogram around t.  At this point you should be able to see that you can solve Equation 1 for 
t and then substitute that intermediate value into Equation 4 to find Δx.

Of course, more complicated problems may require you to chain together three equations connected by two intermediate results, four equations connected by three intermediate results, or whatever.  For example, the problem in Example 1 could have given Spider-Man's mass and asked for the net force on him while the web is slowing him, which would add 

7. FBC = m aBC,


making 
aBC an intermediate result connecting Equations 6 and 7, just like vB connects Equations 3 and 6.  In principle you can go through the same steps, starting with circles and boxes, but if you need multiple intermediate results you're probably better off using different-colored pens to keep the different parallelograms separate.  In practice, you will find that, with experience, you can intuitively identify the equations you will need.

Suggestion:  Use the problem-solving sheets to fully work out Examples 1 and 2. 

Saturday, July 27, 2013

Why I Am Not a Libertarian

It's not just because many libertarians are, in theory if not in practice, libertines, but rather because we have different views on the appropriate role of government.  To lift a line from the Catholic Encyclopedia, "The goal of the State is the temporal happiness of man...."  However, virtue is a part of temporal happiness, not just eternal happiness (which is the goal of the Church, not the State).  As a result, the State has an obligation to promote virtue, something with which a libertarian would not agree.  That is not to say that the State should be coercive in the promotion of virtue; both justice and prudence place strong limits on what the State should do.

One non-coercive tool for promoting virtue is the "bully pulpit".  Like any tool of government, it can be (and often is) abused, but it is appropriate for there to be some public statements about the character we the people aspire to have.

Anyhow, that is one reason why I am not a libertarian.  Yet at the same time, I would want someone with libertarian habits to always be on hand to ask (to paraphrase the WW2 slogan), "Is this law really necessary?"  Even the most well-intended law may be (or become) too burdensome, intrusive, or punitive.



Friday, July 26, 2013

Why I Can No Longer Call Myself "Conservative"

I cannot call myself "conservative" because that term has come to be loaded with too many connotations that simply do not apply to me.  I realize that there are many people in the same situation who still use the term "conservative", but eventually a break must be made.  A 2-second word followed by a 5-minute clarification just does not make sense.  

Here are a few of the connotations to which I was referring.

1.  Most people think "conservative" means "Republican".  Well, I am not a Republican.  I sometimes vote for Republican candidates, yes, but the only time I voted for a major-party contender in the presidential general election was 1988.  The GOP sometimes calls people with my beliefs part of their "base", but they actually see us a means of getting into political office.  Once in office, their priorities turn out to be rather different from those of their "base".  This has been a problem for many years, but it appears to be getting worse. 

1988 GOP Convention


2.  It seems to be expected that a "conservative" favors draconian immigration laws.  This is not the place for a full discussion of immigration, but there does not seem to be any "side" of that discussion with which I can fully agree.  I agree that illegal immigration is, well, illegal, and under most circumstances is also morally problematic.  The same is true of speeding, though; respect for the law is of course important, but so is a sense of perspective regarding the gravity of the offense.  Illegal immigration has been going on so long and in such numbers that any serious effort to deport all illegal aliens would require a police state, and the prospect of becoming a police state is much more frightening than the problem of illegal immigration.

3.  It seems to be expected that a "conservative" will always favor management and oppose unions.  When the huge salaries and bonuses of executives are challenged, the stereotypical conservative will say that (a) their contracts are negotiated, so who are we to question the market?  and (b) contracts are sacred and must be honored.  Where union salaries and benefits are concerned, though, the market suddenly becomes much less infallible and contracts much less sacred.  Somewhere at the root of this is the idea that workers should be grateful for their wages as they would be grateful for a gift -- as though the wealthy and powerful were in fact entitled to the labor of men and women and would be justified in forcing them to work for nothing. 


4.  It seems to be expected that a "conservative" favors an executive branch of ever-increasing strength.  This is usually justified as needed to "get tough on crime" or to "fight terrorism".  The system of checks and balances simply does not work anymore, and several explicit parts of the Constitution are now routinely ignored.  The fact that so much of this was done under the GOP puts the lie to their claim to believe that the original intent of the authors of the Constitution should be the normative interpretation.  Instead, we move closer and closer to being a full-fledged police state.


5.  It seems to be expected that a "conservative" thinks that if anyone in the unfortunate incident was a bad guy, it was Trayvon Martin, and that George Zimmerman is at least manifestly innocent and possibly a hero.  This one has me baffled.  A man with a gun overtakes and confronts an unarmed man who is minding his own business.  The man with the gun is not a policeman or even a security guard; he is answers to no one.  The confrontation takes place on a public street.  The confrontation escalates, and the man with the gun kills the man with no gun.  Prima facie, the man who (a) prepared for a fight by bringing the gun, (b) initiated the confrontation, and (c) killed a man has done something wrong.


The only real explanation I can come up with for the "conservative" position is that Obama and Al Sharpton came out strongly against Zimmerman.  For many people, that fact alone means that Zimmerman must be a good guy.


On the other hand, if the prosecution is not able to make its case to the satisfaction of the jury, there should be no second trial in federal court.  We should not allow end runs around the protection against double jeopardy because, once again, it pushes us further in the direction of becoming a police state.  (Do you note a theme?)  So do, of course, attempts by the president of the US and the governor of Florida to influence the outcome of the trial.


..ooOoo..

Of course, it is not right to simply be negative.  If I can't call myself a conservative, I ought to say what I should call myself, and if I don't agree with much of what people associate with conservatives, I should state what principles I do hold.

Having given this some thought, I think I'll call myself a Chestertonian.  I'll have to explain what I mean by that in a separate post.

Thursday, July 25, 2013

George H. W. Bush Goes Bald to Support 2-Year Old with Leukemia

If you know me at all, you know how disappointed I am in how much power and pride have accumulated to the Executive Branch.  It is all the more refreshing, then, to see that a former president still has some humility and his priorities in the right order.

Tuesday, July 23, 2013

Yes, There Are Stupid Questions

We have all heard the proverb, "There is no such thing as a stupid question."  Teachers use this to encourage students to ask the questions they must ask if they are to learn.  As long as the question is an honest question, the proverb is probably true.



Not all questions are honest, though, and dishonest questions are indeed stupid questions that should not be answered. 


Answer not a fool according to his folly, lest thou be made like him. 
-- Proverbs 26:4

I was reminded of this some time ago in an exchange of comments on another blog which had posted something about the FDA making "Plan B" available without a prescription to girls of any age -- no matter how young.  I had stated, in response to some thread, that the pro-life consideration is not the only reason for opposing this action by the FDA.  Someone responded by asking, "If the embryo were not a person, would it still be wrong?"  I refused to play that game.



From the context it was clear we were not discussing horse shoe crab embryos; we were discussing embryonic human beings.  The question was not, "Are human embryos persons?" or "How do you know human embryos are persons?"  Those questions would be legitimate from someone who really wanted to know.  That was not what was being asked, though.  Instead, I was asked to pretend that one group of humans, namely those who are still in the embryonic stage, are not persons.

Not only is this offensive -- exactly as it would be offensive to deny the personhood of any other class of humans -- it renders any further discussion pointless.  If you were to grant me that 1=2, I could show that the US debt is zero, because I would be able show that any number is equal to any other number.  By choosing a particularly bad starting point, you would have destroyed any possibility of a meaningful arithmetic.  In the same way, treating human nature as subject to arbitrary redefinition renders ethics meaningless.

Sunday, July 14, 2013

The Biggest Cultural Divide Today

As a follow-up to a long series of comments on one of my earlier posts, I think it's worth going into what I consider the single biggest divide in today's society:  whether or not one agrees that there is some sort of Natural Law.  This divide does not always follow familiar markers.  For example, it might seem that at least all Christians would agree that there must be a law written on the heart (Romans 2:15), but it is not hard to find professed Christians who reject such an idea.  At the same time, it might seem that atheists would not believe this, and certainly many do not, but many deists, agnostics, and outright atheists have always believed in a Natural Law.  This brings up several distinct but related questions.


Authority of Law SCOTUS

1.  Are some actions really, objectively wrong?  The question is not, "Are some actions wrong according to the standards of my culture?" nor, "Are some actions wrong in the opinions of most people?" nor any such variation.  Are some actions really wrong, meaning the actor cannot "opt out" of the ethical condemnation?  Is any culture which says otherwise objectively in error?


Many people say No; actions may make us uncomfortable or be disapproved by a society or by the traditions of a culture, but there is no other meaning to "right" or "wrong".  These people would say all rules ultimately are merely agreements we make among each other -- we "play ethics" like we play baseball, so that just like we could change the rules of baseball, nothing prevents us from reformulating our ethical systems.

The answer given here is absolutely fundamental.  If ethics is not a serious subject -- if it is just a game the rules of which we are free to change -- then ethics cannot guide society; when society wants to do something different, it will simply change the rules of ethics.

It should be obvious that I am among those who believe that yes, an objective right and wrong does exist and apply to certain choices.

2.  Assuming we agree that some actions are really, objectively wrong, do they constitute a coherent whole?  If so, what are the basic principles around which these are organized?

At the very beginning of the Nicomachean Ethics, Aristotle says that the objective is happiness, then he notes immediately that people disagree about what "happiness" really means, a disagreement that certainly is still found today.  Is happiness merely a life of pleasure, even if that means being "fat, happy, and stupid" -- even if it means being like the Eloi in H. G. Wells' "The Time Machine"?  Or was Odin right to sacrifice one eye for wisdom?  Does happiness consist primarily in what we experience, or in what we become -- so that for serious wrongdoing "the real and final punishment is having to be the person you are"?

And whose happiness are we talking about?  Is it "every man for himself," or is it the happiness of the whole society?  If it is the happiness of society, does that extend to the very old and the very sick?  Does it extend to criminals?  Does it extend to infants?  The unborn?  What about animals?  Is it OK for a society to utterly crush an innocent person if that would increase the total happiness?  If "it is expedient for us, that one man should die for the people, and that the whole nation perish not," does that make it right?

Although Aristotle's thinking has great merit, I think a better starting point comes from the two greatest commandments.


I.  Thou shalt love the Lord thy God with thy whole heart and with thy whole soul and with thy whole mind.
II.  Thou shalt love thy neighbour as thyself.

The advantages of such a formulation are threefold.  First, ethics is immediately placed in a social context, which avoids the common pitfall of egocentricity.  It is not really clear that "intelligent self-interest" can consistently avoid a tendency towards selfishness.  Secondly, ethics is geared towards our neighbors -- specific people we actually meet -- not towards an airy abstraction like "mankind".  Thirdly, because the First Commandment is oriented towards an eternal, immutable, and omnipresent God, the principles (if not the people studying or implementing them) are not limited to fads or provincialism.  That is, I am not able to make the happiness of my own family, or nation, or race, or political party, or generation into the summum bonum without violating the First Commandment.

3.  Can we know what these basic principles really are?  If so, how?

Of course, the objection could be made that maybe universally applicable moral principles exist, but we cannot be sure what they are.  One may be agnostic about ethics.

One objection is that even if a core of fundamental principles tends to recur in culture after culture, for most of these principles -- perhaps all -- at least one culture can be found that does not hold that principle.  This is no doubt true, but it a statement of anthropology, not of ethics; it is about what people or cultures say, not about what actually is.

The traditional answer, which again I hold, is that we perceive whether actions are right or wrong with our consciences, just as we perceive positions, pressures, temperatures, etc. with our senses.  

Our senses can be dulled -- for example, after spending an hour or more in a dimly lit room, we may not realize how dark it is, or after spending time in a smelly dormitory, we may not notice the smell.  Our senses can also be deceived, especially in complicated situations where our attention is misdirected.  In spite of this, we do not simply give up on our senses altogether, but we use reason to develop a consistent view of the world that is either consistent with our senses or at least provides reasons why our senses are occasionally unreliable.

Our consciences can similarly be dulled (when we habitually ignore our consciences) or deceived (mostly in situations in which multiple principles are in play simultaneously).  The whole project of ethics is to develop a consistent moral worldview that is consistent with our consciences or at least provides reasons why our consciences are occasionally unreliable.

Saturday, July 13, 2013

Why I Find Materialism Unconvincing.

It is not uncommon to here people make statements like, "Consciousness and thought are emergent phenomena from the physical processes of the brain.  Consciousness and thought may not be present in the individual neurons, true, but neither is ferromagnetism present in individual atoms.  Bringing a lot of neurons together makes a qualitative difference, just as bringing a lot of iron atoms together does."

Gehirn, medial - beschriftet lat

I've never found this kind of statement to be at all convincing, and I think I should explain the foremost reason why. 

We start out with no ideas or experience of the world.  First come our sensory experiences:  we see, hear, smell, taste, and feel.  Rather quickly we form ideas about these experiences:  for instance, that the toy does not cease to exist just because we cannot see it any more.  

One of the first divisions we make is between things and people. (We undoubtedly think of plants as like things and animals as like people, at least at first.)  People are in some ways like things; they both persist in existence.  In other ways, though, they are different.  People have intentions; things do not.  People have knowledge; things do not.

When we are a little older, we give names to these.  What behaves like a thing is a material object, what has been called in philosophy for millenia a body.  People are said to have bodies, too, which accounts for why they are in so many ways like things; the differences are accounted for by saying that they have spirits, which have the properties not found in bodies (knowledge, intention, etc.) and lack those properties that are found in bodies (position, size, shape, color, etc.).

From this perspective, it is clear that if anything is to have wrong knowledge -- if anything is to be deceived -- it must be a spirit, not a body or a material object.  Material objects are what is left when we have abstracted away traits like knowledge, so they cannot even have mistaken knowledge.  The idea that "consciousness is an illusion of the body" is just nonsense.

This is, however, precisely the idea put forward by many philosophical materialists.  They believe the universe and everything in it to be bodies in the sense discussed above.  The awkward fact remains that each of us knows the universe only because we have knowledge, and we are aware of the universe only because we have awareness; as soon as we formulate the concepts of spirit and matter, it is first of all clear that we have spirits, and only secondarily clear that we also have bodies.  Without spirits we would not even know the meaning of matter.

What, then, are we to make of the fact that the condition of the brain has such an obvious influence on the ability to think?  First of all, I would say that this is probably analogous to the obvious influence that the condition of the eye has on the ability to see; yet no one would today say that vision is actually in the eye.  Secondly, the approach of Aristotelian philosophy to which we are heirs does not by any means deny the existence of a strong but mysterious connection between the spirit and the body; the soul is the form of the body.

How can anyone think this way in the information age?  Doesn't artificial intelligence prove that knowledge can emerge from carefully arranged material objects?  What about the Turing Test?

Material objects can obviously store and manipulate information.  A library of books is an example; an Egyptian temple wall covered with hieroglyphics is another.  Even without knowing how to read the symbols, we can quantify the information by calculating the Shannon entropy.  Books do not actually contain knowledge, though, because the symbols do not have meaning in themselves.  

The same is actually true about AI.  A machine running an AI program can respond in an appropriate way to external stimuli, but it does so without understanding the meaning of the stimuli or the purpose of the reactions; those are known only to the programmer and the user.  It is really no different than the speaker in a telephone handset; the speaker can transform electronic signals into the sounds of a human voice, but it neither has nor needs any understanding of what is being said.

As for the Turing Test, that really works better as a test for how gullible someone is.  For some people, an 8-ball would pass the Turing Test.

Finally, one might object that the whole problem arises from dividing the world into people and things in the first place.  Maybe things have thought and purpose, too, either in a way we might not understand or as a latent ability -- like a man who is blind because of a problem with his eyes, even though he has no problems with his visual cortex.  If you think that and that different objects remain distinct, you might be an animist; if you think all things together have a collective knowledge and purpose (both perhaps latent), you might be a pantheist.  Many people and many cultures have embraced some variation on this theme.  Such ideas may have no room for the separate existence of spirits, but they are still very different from materialism.

Wednesday, June 26, 2013

We have tried to heal Babylon, ...

... but she cannot be healed. -- Jeremiah 51:9

One of the nasty things about crossing an event horizon is nothing happens locally to indicate that a fateful threshold has been crossed and destruction can no longer be averted.  If you're busy gazing into your navel, you will remain clueless, though the signs can be seen clearly if you look to the distance.  For a two-meter tall human being, the difference in gravitational acceleration between his head and his feet as he crossed the Schwarzscshild radius of the supermassive black hole at the center of NGC 4889 (the largest known black hole) would only be about 5 x 10-11 m/s2 -- smaller than the same difference for a man standing on Earth by a factor of about 100,000.

Thursday, June 13, 2013

Not Advancing Because They Are Not Challenged

There was a story on NPR yesterday about how high school students today are reading books intended for much younger children rather than growing into meatier, more sophisticated books.  According to Eric Stickney of Renaissance Learning, "The complexity of texts students are being assigned to read has declined by about three grade levels over the past 100 years. A century ago, students were being assigned books with the complexity of around the ninth- or 10th-grade level. But in 2012, the average was around the sixth-grade level."  Students have always had difficulties with hard books, but rather than challenging them and -- heaven forbid! -- educating them, the response has all too often been to lower the expectations and find "fun" or "relevant" books at a lower level.

I couldn't help but see the parallel with what has happened to the Mass over the same period.  Latin was "too hard", so now we have the vernacular -- and specifically "translations" that were dumbed down in a way that frequently changed the meaning and lacked any kind of beauty.  When the translations were recently improved, there were many complaints by lazy priests that the laity could never understand big words like "ineffable" or "consubstantial", as though teaching the Faith were not the responsibility of priests.  And heaven preserve us from "relevant" liturgies.  If the argument has been that "relevant" liturgies in "simple language" would encourage participation and enhance understanding of the Faith, the results have been exactly the opposite. 

Wednesday, June 12, 2013

Human Evolution and Japanimation

I can't be the only one who thinks there are some disturbing similarities between this speculation about human evolution and the following video. 


Friday, June 7, 2013

A Physicist's Idea of Beauty

One of the criteria that is important in the search for new theories in physics is beauty.  Unfortunately, the aesthetic used to make this judgment is not exactly obvious and is rarely explained, even in physics courses; yet general agreement tends to be reached on the basis of shared experience with physics problems.  Mostly, though, "beauty" consists of symmetry.

The essence of symmetry is that you can do something that, in general, should make a change, but it doesn't.  Consider the small letter b.  If you hold it up to a vertical mirror, you get a different letter, d.  If you reflect b through a horizontal mirror, you get yet another letter, P.  Thus b does not have any reflection symmetries.  On the other hand, T is unchanged by a vertical mirror, so it is more symmetric.  X is even more symmetric, since it is unchanged by either a vertical or a horizontal mirror. 


A good example of symmetry in physics can be found in Newton's Laws.  Students learn early on that these laws work just as well in any inertial frame of reference -- a frame of reference that is not rotating and is related to any other inertial frame by only a constant velocity.  If you're a passenger in the front seat of a car driving at constant speed along a smooth, straight road, you can toss an apple to a friend in the back seat just exactly the way you would if the car were parked.  Both this moving car and the parked car are in inertial frames of reference -- at least when we ignore the rotation of the earth, which is too slow to make a noticeable difference on the apple's trajectory.

In many cases, symmetry is present as a pattern.  A good example are the three known "generations" of quarks and leptons.  One generation consists of the Up and Down quarks (which combine to make protons and neutrons), the Electron, and the Electron Neutrino.  Everyday matter consists mostly of these two quarks and the electron.  Then another generation was discovered, piece by piece:  the Charm and Strange quarks, the Muon, and the Muon Neutrino.  Aside from the neutrinos, whose mass is not well known, these act just like more massive versions of the first generation.  So when a Bottom quark, a Tauon, and a Tauon neutrino were discovered, physicists were convinced that there must be another quark to complete the set -- a Top quark.  They were right.

This kind of thing has happened so often that physicists would definitely fall for this cartoon gag.  The pattern must be completed!




But the history of physics is not an unbroken chain of successes.  Over the years the similarity between Newton's Law of Gravity and Coulomb's Law of electrical attraction and repulsion has inspired many physicists to try to combine them into a single law that would be in some sense more symmetric.  Einstein himself tried to do this, but it has always ended in failure.  Unifying gravity with electromagnetism -- and with the other fundamental forces -- is one of The Big Questions in physics today.

Then there is the magnetic monopole.  This is a hypothetical particle that would make magnetism more like electricity by being all "north pole" or all "south pole"; to keep the discussion simple, let's restrict our discussion to the "north pole" magnetic monopole.  From whichever direction you approached this particle, you would see that you were approaching the north magnetic pole.  This is not the case with familiar, real-world magnets; when you approach from one direction, you're getting closer to the north magnetic pole, but when you approach from the opposite direction, you're approaching the south magnetic pole. 


There are two basic reasons why many physicists think that magnetic monopoles probably do exist, even if they have resisted all efforts to find one so far.  (They may exist as possible particles, but be hard to produce in particle colliders and decay almost immediately when they are produced.)
1.  Magnetic monopoles would make the equations of electrodynamics much more symmetric.  Monopoles would act as sources for the magnetic field, similar to the way electric charges act as sources for electric fields and masses act as sources for gravitational fields.  Also, just like a current of electric charges generates a magnetic field, and current of magnetic monopoles would generate an electric field. 
2.  The existence of magnetic monopoles would explain why charges are quantized, coming only in units of the charge on an electron.

Interestingly enough, "magnetic monopoles" of a sort have been observed in synthetic materials.  This is less significant than it sounds.  The "magnetic monopoles" occur only as quasiparticles in materials, and they apparently are not monopoles of what is called the B-field (the magnetic field of interest, as opposed to the H-field, which includes the effects of the material's magnetization).  Also, materials can do strange things, bending the laws of physics beyond anything possible in a vacuum.  "Electron" quasiparticles can have effective masses different from the mass of an electron; these effective masses can even be negative.  Quasiparticles can have fractional electric charges, unlike real particles that can be observed in a vacuum.

Just a couple more points about the laws of classical electromagnetism. Maxwell's Equations were originally written as 12 separate equations involving the components of vectors.  At the time vectors were a comparatively new mathematical idea, and Maxwell was not used to them.  Oliver Heaviside rewrote them in vector form.  This was not only more compact, it was more "beautiful", in that the vector formulation freed the equations from dependence on a specific coordinate system.  Sometimes the geometry of the physical system is spherical (like a cloud of charge that depends only on distance from a point), sometimes it is cylindrical (like a long charged wire or a wire carrying current), and sometimes it is rectangular (like a rectangular waveguide).  It is very helpful to be able to use coordinate systems with the same symmetry as the thing you're trying to describe, so it seems "beautiful" to be able to write Maxwell's Equations in a way that works for any coordinate system. In practice, though, after you choose the appropriate coordinate system you still have to expand the vector equations into sets of scalar equations for the components of the vectors in order to solve the problem.  So you have a smaller set of equations to use, but they involve more complicated mathematical entities, and in order to actually use them you have to expand them back out again.  If this seems complicated, then you are beginning to understand how the scientific idea of "beauty" is a subtle, acquired taste that is hard to explain.

That's not the end.  In vector form, Maxwell's Equations require four equations, but they can be written in two equations if we use covariant (and contravariant) 4-vectors and rank-2 tensors. This formulation is more compact, but the complications are not really done away with, only hidden in more complicated mathematical entities.  The advantage to this formulation is that it can be seen at a glance that these equations will satisfy Einstein's Special Theory of Relativity; the equations are now explicitly independent of any specific inertial frame of reference.  Unfortunately -- you guessed it -- to use the equations you first have to choose an inertial frame of reference and unpack these into four equations, then choose a coordinate system and unpack them into (in principle) twelve equations.  Beauty comes at a price. 

You won't be surprised that these two equations can be combined into a single equation using even more hideously complicated mathematical entities by using something called Clifford Algebra.  We spent so little time on that in grad school that I no longer remember what the point of that formulation might be.  It is so far removed from being practical that almost no one seems to use it.

Beauty is an important guide in the development of theories, but beauty, as most adults have come sadly to know, can be deceptive.  Philosophers say that in the final analysis, beauty, truth, and goodness are one; this may be true, but it is very bad advice to pretend they are identical in any but an eschatological "final analysis".  The world is an astonishingly beautiful place, but we have no right to demand that it conform to our ideas of beauty.