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## Don’t do algebra with numbers!

As a physics professor, I have certain pet peeves.  For example, I cringe when someone says that “gravity” is 9.8 m/s2 when they mean the acceleration due to gravity.  I’m annoyed if someone says that an object “weighs” 7 kg.  And I stifle a laugh if someone says that a roller coaster is exciting because it goes so “fast”—humans can only detect acceleration, not speed, which is why we don’t notice that we’re traveling something like 67,000 mph right now in our orbit around the sun.

“I feel the need for acceleration!”

But my biggest pet peeve may be students doing algebra with numbers.

Fellow physics professors will know exactly what I’m talking about, but for the uninitiated, here’s an example:

If you drop an object from a height of 20 m, how long will it take to hit the ground?

A student knows that a kinematics equation is needed, hits upon the correct one, Δyvi Δt + (1/2) a Δt2, and then correctly identifies Δy = –20 m, a = –9.8 m/s2, and vi = 0.  So far, so good.  They’ve studied their physics, right?  What happens next is sheer madness:

Sigh.

Over and over again I tell students, “don’t plug numbers in until the end.”  But students love plugging in numbers.  They feel they’re actually getting closer to the answer if they’re manipulating numbers.  On some level, they still feel uncomfortable with letters—as if manipulating letters isn’t really “math”.

How does this problem look in my answer key?  Like this:

You can now plug in values if you like…and get Δt = √[2(-20)/-9.8] = 2.02 s.

Which of these approaches is more beautiful, more powerful?  The approach you pick indicates whether you “get” algebra or not.  If you do algebra with numbers, the answer you get is very narrow and very specific, even if you do it correctly.  That hypothetical student could have gotten 2 seconds as an answer, and I would have given them full credit.  But their answer would have been ugly.

The second approach is beautiful, because it is completely general and applicable to multiple situations.  I try to tell students “Look!  You found the time to fall a certain distance.  You now know the answer no matter what the height is, and even no matter what planet you’re on, since g doesn’t have to be 9.8 m/s2.”  This is usually followed by a blank open-mouthed stare, much like Kristen Stewart in a Twilight movie.

There is a more practical reason to avoid doing algebra with numbers.  It’s simply that when you do algebra with numbers, other people cannot follow your work as easily.  And then, if you make a mistake, it’s harder for someone else to spot.  Quick: what algebra error did the student make above?  It takes a while to find the mistake.

My ultimate point is that students need experience seeing the power of algebra.  It’s all well and good that algebra classes stress real-world applications—else, why teach algebra in the first place?  But real-world doesn’t only mean with numbersE=mc2 is certainly a real-world application of algebra, and it’s a lot more elegant than saying that 378,000,000,000,000 Joules is released when a teaspoon of sugar with mass 4.2 grams  is converted to pure energy, given that the speed of light is 300,000,000 m/s.  The hard part, for us physics professors, is to help this spoonful of algebra go down.

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## The top 10 classical pieces you aren’t familiar with

Shostakovich makes the list twice.

Let’s face it, classical music isn’t all that popular compared to more “modern” genres.  I guess people just think that Rihanna has more to say about the human condition that J. S. Bach.  But part of the problem is that the same hackneyed classical pieces get played again and again, to the point of banality.  If I ever hear Vivaldi’s Four Seasons, Pachelbel’s canon, Boccherini’s minuet, Albinoni’s adagio (written by Remo Giazotto in the 20th century!), Mozart’s Eine kleine Nachtmusik, Orff’s “O Fortuna” from Carmina Burana, or Beethoven’s Für Elise on the radio, I turn the dial immediately.  Honestly, Rihanna’s Diamonds seems more fresh.

But it’s not the fault of classical music in general.  There’s a lot of crazy, beautiful, weird, awe-inspiring, emotional, frightening, uplifting classical music out there—you just have to know where to look.

Anyway, here’s my list of the top 10 classical pieces that are probably off your radar but shouldn’t be.  These pieces aren’t even that obscure (maybe someday I’ll post about the Busoni piano concerto, George Antheil’s Ballet mécanique, or Buxtehude or Meyerbeer or Piston or de la Guerre) but for whatever reason these pieces just aren’t played that often.  That’s a shame; they should be.  I hope you like them.

1.      Funeral of Amenhotep III (from the opera Akhnaten) (Philip Glass)

http://www.youtube.com/watch?v=otS9N7TgmhA

Amenhotep III’s son was originally crowned Amenhotep IV, but soon re-named himself Akhenaten and tried to ditch Egyptian polytheism and replace it with a sort of cult-of-the-sun.  It didn’t stick, and everyone forgot about him.  Except for Philip Glass, apparently.

2.      String quartet no. 8, movement 2 (Dmitri Shostakovich)

http://www.youtube.com/watch?v=PjvTTfbpWjY

Shostakovich dedicated this piece “to the victims of fascism and war”.  The 2nd movement is particularly chilling.

3.      Sinfonia Concertante for violin and viola (Wolfgang Mozart)

http://www.youtube.com/watch?v=uL3xZ8Qlhro

I know this isn’t that obscure, but I think this is the most profound thing Mozart wrote and it doesn’t get much attention.  When the viola comes in at about 2:27, you realize it was already there for about 6 seconds without you realizing it.  Sublime.

4.      9th Symphony, 2nd movement (Anton Bruckner)

http://www.youtube.com/watch?v=_1pw34abVEk

Savage, rhythmic chords in one section, eerie icy elfin menace in the other.  Listen to this in the dark if possible.

5.      Mt. St. Helens Symphony, movement 3 (volcano) (Alan Hovhaness)

http://www.youtube.com/watch?v=HqKghAUknOY

A musical depiction of the eruption of Mt. St. Helens, belying the idea that Hovhaness only wrote “soothing” stuff.

6.      Concerto Grosso 3 (Alfred Schnittke)

http://www.youtube.com/watch?v=ZvzdS4z6FKc

Insanity.  What can one say?  Madness.  If anyone ever makes a Cthulhu movie, this should be the soundtrack.

7.      B minor scherzo (Frédéric Chopin)

http://www.youtube.com/watch?v=nJhacebV24k

Chaotic.  Good times.  Oh, and there’s a lullaby in the middle!

8.      Piano concerto No. 2, movement 2 (Dmitri  Shostakovich)

http://www.youtube.com/watch?v=JlMHjo7Jwhk

Might be the most soothing music that I am aware of.

9.      Piano Sonata No. 2, movement 4 (Frédéric Chopin)

http://www.youtube.com/watch?v=JJEQqcw-830

Someone (Rubinstein?) described this as wind over your grave.  And ya gotta love the ending.

10.   Symphony No. 1, movement 4 (William Walton)

http://www.youtube.com/watch?v=RRYHIgu-dpY

After three dark and brooding movements, there is finally triumph.  Why isn’t Walton better known?

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## Let’s ignore Grover Norquist

Should we pledge to enact sanctions against ancient Carthage?

It’s time to start marginalizing Grover Norquist.

Haven’t heard of him?  That’s because he hasn’t really done anything noteworthy.  Sure, he got an M.B.A. from Harvard, and he did write speeches for the U.S. Chamber of Commerce for one year in the 1980’s, but other than that he’s done nothing except be a lobbyist.  He’s never had an elected position.  His reputation is based on lobbying.

Have I said he’s just a lobbyist?

Now, to the guy’s credit, he’s good at his job, and he wields power through his personal instrument Americans for Tax Reform.  That’s the lobbying group he founded.  Its only purpose is to advocate for Norquist’s world view.  Part of that world view is to lower tax rates in America, and I won’t comment on whether that’s a good idea or not…that’s a problem for economists to sort out.  But part of that world view is getting politicians (by scare tactics and intimidation) to commit to a “no tax raises” pledge.

I can’t think of anything sillier than a politician making such a pledge.  What is this, the days of Hamilton and Burr at Weehawken?

First of all, tactically, it’s always better to have options than to not have options.  If you pledge to never raise taxes, ever, then you’re a fool, plain and simple.  You’re locking yourself into a position that might make no sense at some point in the future.  When taking such a pledge, you’re saying, basically, the following: “I don’t think raising taxes is a good idea.  In fact, I feel strongly that it’s a bad idea.  But I am also convinced that I will never change my mind; I will never let new data change my mind; even if the circumstances change, it is logically inconceivable that I will ever change my mind; and even if I want to change my mind I won’t be able to because I am locked into a pledge.”  By taking a pledge, you are thumbing your nose at a future self (and potential wiser self) and forcing them down a path they might not agree with.

[Of course, there’s another reason to take such a pledge: you may not agree with it, but you take the pledge anyway in order to get elected.  Anyone who falls into that category is beneath contempt.]

What if scientists took pledges?  Newtonian physics was on very firm footing in 1904.  What if every physicist signed a pledge saying that Newtonian physics was 100% correct and was never to be doubted ever again?  What, then, would have happened with patent clerk Einstein in 1905?

Suppose everyone in Congress took the Norquist pledge.  And then suppose that aliens visited Earth, and offered to give us an unlimited source of clean energy.  The catch is, we have to raise taxes on upper incomes by, say, 1%, in order to pay for distribution costs.  I guess we’d have to say, “Sorry, we all took a ‘pledge’ so we can’t do it.  Fealty to Grover Norquist and his 18th century ‘pledge’ takes precedence over the country, over science, over common sense, and over anything else you can think of.  Have fun with your infinite energy, rest of the world.”

My point has nothing to do with the merits (or lack thereof) of the pledge.  I have a problem with the idea of such a pledge itself.  A pledge is indicative of an anti-science mentality; a tendency towards dogmatism; a lack of mental flexibility—and those are not traits I want to see in our country’s leaders.  Leaders need to keep everything on the table.  You have to decide based on current data what the best course for the country is.  You cannot let a decision made 20 years ago affect your thinking today.  I’m sure that 2200 years ago I might have been in favor of sanctions against Carthage; I may have even signed a pledge to that effect.  Today, though, that pledge wouldn’t mean very much…

Let’s all agree to never mention Grover Norquist again.  He’s irrelevant.  He’s a lobbyist, and his only purpose is to push his own agenda.  His tax foundation doesn’t do scientific research, doesn’t create jobs, doesn’t build things, doesn’t design things, doesn’t contribute to science, or culture, or human knowledge, or service, or humanity.  Norquist himself is not a super villain.  He’s just a random dude with a loud megaphone.  Luckily, we have the ability to ignore him if we like.  Maybe then he’ll just go away.

Then again, probably not.  After all, he is a lobbyist.

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## Einstein’s “Imagination is more important than knowledge”

Poor Einstein.  Is there anyone else who is misquoted more often?  Is there anyone else to whom more nonsense is attributed?

I have no desire to rehash things that Einstein said about “God”.  Einstein was by all accounts an atheist, an agnostic, or a pantheist—depending upon your definitions—and various religious apologists have been trying to co-opt the man for years by misquoting him.  Others have already discussed this at length.

My goal today is to tackle that old chestnut, “Imagination is more important than knowledge,” as seen on T-shirts, bumper stickers, and even on the packaging of the Albert Einstein action figure.  Did Einstein really say this, and if so, what did he mean?

Here’s the quote in context:

“At times I feel certain I am right while not knowing the reason.  When the [solar] eclipse of 1919 confirmed my intuition, I was not in the least surprised.  In fact I would have been astonished had it turned out otherwise.  Imagination is more important than knowledge.  For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution. It is, strictly speaking, a real factor in scientific research.”  [From A. Einstein, Cosmic Religion: With Other Opinions and Aphorisms, p. 97 (1931).]

So Einstein did say this.  However, I maintain that the full quote in context has a different feel to it than the quote in isolation.

When I see “Imagination is more important than knowledge” on a bumper sticker, I think this: “Flights of fancy and imagination are more important than learning stuff.  So why should I study?  Einstein didn’t study.  He just sat around and daydreamed and came up with the most remarkable breakthroughs about the workings of our universe.  Imagination is more important than learning all the proofs and figures ranged in columns before me.  So I am going to follow good ol’ Einstein and daydream about being Batman.”

The New Age meaning of the quote is this: “I’d rather daydream than study.”  It’s Walt Whitman’s “learn’d astronomer” nonsense all over again.

In context, it’s clear that Einstein was talking about doing science.  Imagination is more important in making scientific breakthroughs than knowledge, but that doesn’t mean that knowledge is not important.  Einstein worked very, very hard to learn an awful lot of physics.  By all accounts, it took him almost 10 years to flesh out general relativity, during which time he had to acquire a lot of mathematical knowledge about Riemannian geometry and tensor analysis.  The “intuition” that Einstein developed during this time frame is what allowed him to be so confident of the results of Eddington’s expedition.  What Einstein calls “intuition” is just knowledge that has become so ingrained that you are no longer cognizant of it.

Einstein may have been more famous than most of his contemporaries, and it was probably due to his superior imagination.  But take Einstein’s imagination today and give it to a twenty-five year old high school dropout, and he’d be lost in obscurity, stocking shelves at Wal-Mart.  Imagination is more important than knowledge.  But only slightly more.

[Note: my book Why Is There Anything? is now available for download on the Kindle!]

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## Marco Rubio and the age of the Earth

Recently, Senator Marco Rubio told an interviewer from GQ that he wasn’t qualified to say how old the Earth was:

GQ: “How old do you think the Earth is?”
Marco Rubio: “I’m not a scientist, man. I can tell you what recorded history says, I can tell you what the Bible says, but I think that’s a dispute amongst theologians and I think it has nothing to do with the gross domestic product or economic growth of the United States. I think the age of the universe has zero to do with how our economy is going to grow. I’m not a scientist. I don’t think I’m qualified to answer a question like that. At the end of the day, I think there are multiple theories out there on how the universe was created and I think this is a country where people should have the opportunity to teach them all. I think parents should be able to teach their kids what their faith says, what science says. Whether the Earth was created in 7 days, or 7 actual eras, I’m not sure we’ll ever be able to answer that. It’s one of the great mysteries.”

This is so absurd on so many levels, I need to parse the quotation out bit by bit—partly to stretch out this blog post, of course, but partly to delve more deeply into the mind that is Marco Rubio.

“I’m not a scientist, man.”

Obviously.  But GQ wasn’t asking you to demonstrate the Earth’s age using science.  They were asking you how old the Earth is, which admittedly is code for: “Do you believe in the most basic science?”  And apparently, you do not.

“I can tell you what recorded history says,”

What does recorded history have to do with the age of the Earth?

“I can tell you what the Bible says,”

But unfortunately the Bible doesn’t actually say how old the Earth is.

“but I think that’s a dispute amongst theologians…”

Wait…what?!  Theologians?  We’re talking about the age of the Earth.  If geologists can’t answer this question, no one can.  Why?  Because geology is the study of the Earth.

“…and I think it has nothing to do with the gross domestic product or economic growth of the United States.”

So?

“I think the age of the universe has zero to do with how our economy is going to grow.”

True, but the interviewer is really asking if you have any knowledge about science and the scientific method.  If you don’t believe the arguments for a 4 billion-year-old Earth, if you distrust that much the collective human body of knowledge, then you may not listen to economists when they tell you that printing a quadrillion dollar bills and passing them out to everyone would be a bad idea.

“I’m not a scientist. I don’t think I’m qualified to answer a question like that.”

Do you have to be a historian to answer the question, when did the Crimean War start?  Do you have to be a biologist to answer the question, what does RNA stand for?  Do you have to be a mathematician to say that pi is irrational?  Are you really claiming, Mr. Rubio, that you have to be an expert to answer the most basic, settled questions that humans have successfully answered?  Are you really saying that you’re not qualified to look up an answer on Wikipedia?

“At the end of the day, I think there are multiple theories out there on how the universe was created…”

He’s right, check out these 90 different theories from religions around the world…

“and I think this is a country where people should have the opportunity to teach them all.”

Including, no doubt, the story of the water beetle Dâyuni  that made the Earth from mud, and the story of how Buga set fire to the water…oh, and don’t forget how Mbombo vomited the moon and the sun!  Make sure elementary school curricula cover them all!

“I think parents should be able to teach their kids what their faith says, what science says.”

I’m not sure what his point is here.  Is there a movement to ban what parents can and cannot teach their children?

“Whether the Earth was created in 7 days, or 7 actual eras, I’m not sure we’ll ever be able to answer that. It’s one of the great mysteries.”

Yes, a mystery, like how the tides work or how magnets work.  I guess the real mystery is why anyone takes Marco Rubio seriously.

(Photo credit: NASA)

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## Stay away from quantum mechanics, Deepak Chopra

Schrodinger’s cat

I am hesitant, sometimes, to expound upon the many-worlds interpretation of quantum mechanics, for fear of something I call the Deepak Chopra effect.  (I won’t give you a hyperlink to the guy, because I don’t want to increase any traffic to any website he’s associated with.)  The Deepak Chopra effect is this:

If you talk about some weird aspect of quantum mechanics, they will come.

Who are “they”?

They are the Deepak Chopras of the world: people who make money by peddling vague new age philosophies.

Suppose you’ve made up some sort of new religion.  You want followers, people to buy your books and watch your DVD’s and attend your seminars and drink your Flavor Ade and buy your T-shirts.  (Yes, Deepak Chopra sells T-shirts.)  What better way to attract attention, to give your puerile ideas a veneer of respectability than to cloak them in the mystique of quantum mechanics?  Quantum mechanics is weird—everyone knows that—but almost no one really knows the details.  PhD physicists don’t grow on trees, after all.  Therefore, if you appeal to quantum mechanics to cover up the stench of your ideology, you will most likely get away with it.

I’m tempted to write some computer code that invents Chopra-esque prose.  The output would look like this:

Your [mind] and [eternal light] have been exquisitely formed by [the cosmic warmth] to help you fulfill [your potential matrix] and your [soul capability].  This is because [wave-particle duality] and [the principle of decoherence] prove that your [neo-human consciousness] transcends the [body-mind paradox] to inhabit the [weak nuclear force] under the auspices of [string theory].

Easy, right?  Yet Deepak Chopra is the one worth \$80 million dollars.  Sigh.

So back to the many-worlds interpretation.  What could someone like Chopra ever do with such an idea?  How could he co-opt the multiverse to scratch out a few more ducats?  I shudder to think on it.

People have been using the strange ideas of physics for a long time now, with predictable results.  Take this garbage:  “What the #\$*! Do We Know!?”  I wish I had been blogging in 2004 when this farce came out; my review of the movie would have been six words: “Nothing about physics, that’s for sure.”  Yet part of the blame rests with physicists themselves: they bandy about strange ideas amongst themselves, with nary a thought about how the public at large will perceive said ideas.

Consider Schrödinger’s cat, for example, the popular notion of which is as follows: a cat can be alive and dead at the same time!  Weird!  And yet when Schrödinger first proposed this “paradox” his intent was to attack the Copenhagen interpretation of quantum mechanics, by pointing out an obvious absurdity.  I know of no physicist on the planet who seriously believes a cat can be alive and dead at the same time.  And yet Chopra and others like him point to such quantum weirdness and use it to excuse all manner of hooey.

But what about many worlds?  Isn’t it just as crazy, just as loony, as anything Chopra peddles to the masses?

Well, no.  It’s weird, sure.  But it is based in peer-reviewed science, and is an active topic of investigation to this day.  (I doubt anyone’s in a lab somewhere, trying to verify Chopra’s claims.)  Many worlds is an interesting mathematical structure to explain our universe, but it doesn’t really affect anyone’s life.  It’s not even relevant to how anyone should live their life.  It should certainly never be used to prop up a shaky religion.

My advice to you, Dr. Chopra, is to quit using physics to bolster your claims.  After all, I don’t use your only field of expertise (endocrinology) to support my idea that Matt Damon is really a cyborg, do I?  Then again, maybe I could start a religion—

[Note: my book Why Is There Anything? is now available for download on the Kindle!]

(Photo credit: http://en.wikipedia.org/wiki/File:Schrodingers_cat.svg)

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## Get off that train! A different way to teach special relativity

Einstein circa 1905

There are a lot of people who, to this day, deny the truth of Einstein’s special relativity (SR).  I’m not even referring to the OPERA superluminal neutrino debacle—an anomaly that was eventually found to be caused by a misconnected fiber optic cable.  No, I’m referring to laymen who deny SR because it goes against common sense.

A Google search will find such people readily.  Most of the time, their arguments aren’t even worth refutation, since it’s obvious in most cases that they haven’t mastered even the simplest algebra, much less sophomore-level physics.  (I am planning to use this gem in my Modern Physics class in the spring as a homework problem: for 20 pts., find the elementary flaw in this person’s logic.)  However, as a working physicist, I sometimes find myself dismissing such people too readily: it is easy, and self-gratifying, to call such people cranks.

A person who doubts SR is not necessarily a crank.  After all, relativity is very counter-intuitive, and our brains have been exquisitely fine-tuned by natural selection to perceive the world as inherently classical.  In fact I will go so far as to say that if you accept SR whole-cloth, without any mathematical or scientific background, then you’re basically showing a blind faith in science in the same way that Iotians have a blind faith in “The Book”.  I would rather beginning physics students showed some skepticism; it makes their final “conversion” that much more intellectually pleasing.

I think the main problem with perceptions of SR is the way it is normally presented.  My thesis is this: most physicists are teaching it wrong.  And as a consequence, many people who have studied SR come away with a misguided notion of what SR is all about.

The old way to teach SR begins with Einstein’s two postulates.  The first is that the laws of physics should be the same, in any inertial reference frame.  The second is that the speed of light is the same for all inertial observers.  There is then an obligatory picture of a train and lightning bolts, and talk about how simultaneity isn’t preserved in SR.  This leads (usually after a lengthy derivation) to time dilation and length contraction.  And then, out of the blue, there might be talk of the twin paradox and the obligatory pole vaulter in the barn.

Shudder.  Such a pedagogically confusing approach!  No wonder very few first-time SR students “get it” at all.

The original train picture from Einstein’s 1916 book

This approach has a long history.  In Relativity: The Special and General Theory (1916), by Einstein himself (!) the discussion begins with the two postulates, and there is then a diagram of a train and a discussion of simultaneity (see above).  Seriously?  I’m not blaming Einstein, mind you; I’m blaming the textbook authors today who can’t let go of that stupid train.  It’s been almost 96 years.  Get over it.  Hop off that train, please.  There are more intuitive approaches that are easier for the layman to grasp.

Here’s the approach I use in my classes.  This is not the only approach, of course, nor do I claim it is the best approach.  However in my experience (admittedly, just one data point) this approach is a better way to get students to gradually accept SR.  The trick is to present information one plausible chunk at a time, and then only gradually derive all the weird stuff.  Thus, without realizing it, the students have been convinced of the truth of SR despite themselves.  If you start with simultaneity and time dilation and length contraction then half of the students will get turned off immediately (because their common-sense alarms will be blaring full-force).

1. Talk about classical (Galilean) relativity.  That is, discuss how the laws of physics should be the same in any (inertial) reference frame you choose.
2. Talk about coordinate transforms: how you can take the spatial coordinates of an object (x,y,z) and find what the coordinates (x’,y’,z’) would be in a different coordinate system.
3. Talk about how some coordinate transforms are “good” and some are “bad”.  For example, a translation in space such as x’ = x – L preserves distance, but a rescaling transform like x’ = ax does not.
4. Mention how the good ol’ Pythagorean theorem s2 = x2 + y2 + z2 gives you an invariant quantity (s2) that is preserved under “good” transforms.
5. Mention that the (experimental) behavior of light throws a monkey wrench into this analysis.  For whatever reason, all observers measure the same speed c for light, and this actually makes things a little harder.  (Don’t do any math at this point!)
6. Here you should start talking about time as being a 4th dimension.  The earlier you introduce the idea of an event P as a point P=(x,y,z,ct) in space-time, the better.
7. State Einstein’s postulate about the speed of light.
8. Show that the light postulate implies that s2 = x2 + y2 + z2 is no longer an invariant quantity, when talking about transforms as applied to space-time.
9. If s2 = x2 + y2 + z2 is no longer invariant, can we modify the formula in any way so as to make s2 invariant, while still preserving the light postulate?  The answer is yes; and so you should derive the 4D version of the distance formula, s2 = x2 + y2 + z2c2t2.

To me, this is the core idea of SR.  Everything else follows from the invariant interval s2.  One should no longer think of our existence as being 3D; time represents another “direction”.  And it turns out that the time you perceive depends upon your vantage point (time is “relative”), just like position.

For example, suppose you are looking at a row of trees.  From one location, the trees are lined up in front of you (they all share the same x-coordinate).  From another vantage point, they are separated by 1 m each (x=0, 1m, 2m, 3m, etc.)  No one, not even Galileo, would find this controversial.

But now imagine that you think of time as just another “direction”.  Why is it so hard to believe that your time coordinate could have one value in one reference frame, and another value in a different frame?  Why is it so hard to believe that events that are simultaneous in one frame are not simultaneous in another?

Time dilation and length contraction follow from this in a straightforward way.  And they are much easier to visualize if you buy into the paradigm (I’ll say it again) that time is another “direction”, and therefore relative just like position.

[Caveat: I do know that time is special, in the sense that there’s a minus sign in the s2 = x2 + y2 + z2c2t2 formula.  That minus sign is crucial.  But discussing its importance should be deferred to a later (pun-intended) time.]

If you’re interested, here’s the rest of my SR program:

10.  Discuss space-like, time-like, and light-like intervals, and the ideas of proper length and proper time.
11.   Show how the Galilean boost (which is a “good” transform in classical relativity) must be modified into the Lorentz boost in order to preserve s2.
12.   Show how the Lorentz boost implies length contraction and time dilation.
13.   Discuss relative velocity in SR.
14.   Discuss so-called paradoxes like the twin paradox and the pole vaulter paradox.

The discussion can then go into advanced topics: momentum, energy, E=mc2, forces, etc.  However, with the foundation I have described, I believe these topics are much easier to present.

I’m sure there are some great professors out there who have had great success with the “traditional” program of SR instruction.  I’m sure Feynman could teach circles around me, even with the train and lightning bolts.  But I prefer this different approach, as I have presented it, and I hope others will realize that there’s more than one way to explain special relativity.

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## Don’t let the f-word bother you…

Please don’t be offended!

A Socratic dialogue

[Inside PASCAL’s home.  SOCRATES is watching television, while PASCAL is kneeling in the corner, praying.]

SOCRATES: They just bleeped a word on my favorite show!

PASCAL: Well, there are certain words you can’t say on TV.

SOCRATES: Like “fuck”, apparently.

PASCAL: Dear Lord!

SOCRATES: What, you don’t like hearing that word?

PASCAL: Heavens, no!

SOCRATES: And I bet you think that we shouldn’t be able to say “fuck” on television?

PASCAL: Certainly not!

SOCRATES: But why not?  The word simply means “sex” or “intercourse” and you can say those words on the air.

PASCAL: The word f—, uh, the word you mentioned, is obscene.  It is vulgar.

SOCRATES: What exactly does that mean?  Would the word, if heard, harm the listeners in some way?

PASCAL: Of course.  People’s sensibilities would be harmed.  Decent people would hate to hear such rubbish.

SOCRATES: But why?  Does the sound of a voiceless labiodental fricative preceding a mid-back, unrounded vowel and followed then by a voiceless velar stop somehow produce a dangerous resonance pattern that could damage the eardrums of unsuspecting people?

PASCAL: You mock me.

SOCRATES: Not at all.  I am simply at a loss to understand how any word, any single morpheme, could possibly harm anyone, apart from the word being delivered at a dangerous volume level.

PASCAL: You misunderstand.  It is not physical harm that such language can cause, but mental harm.  Distress, if you will.

SOCRATES: But how does such distress come about?  Do the concepts cause distress?

PASCAL: Why, yes.

SOCRATES: So the word “sex” offends you.

PASCAL: Well…

SOCRATES: Let me rephrase.  Would the word “sex” offend an ordinary person, a person that…perhaps…had not written the Pensées?

PASCAL: I would think not.  No.

SOCRATES: So the content of a word like “fuck” is not what offends, after all.

PASCAL: OK then, it is not the content of the word that offends, but the word itself.  The word is offensive.

SOCRATES: For what reason?  Is there some intrinsic property of the word that offends you…perhaps the way the word rolls off one’s tongue?  Or perhaps you hate the word because it scores 13 points in a game of Scrabble, and 13 is an unlucky number?

PASCAL: The word is offensive.  Everyone knows that.

SOCRATES:  But do you mean that literally?  Everyone?  Would a person who speaks no English be offended?

PASCAL: Of course not.

SOCRATES: But why not?

PASCAL: They wouldn’t know what the word means.

SOCRATES: But we agreed that being offended doesn’t come from the meaning of the word.

PASCAL: Oh, uh…well, they don’t know it’s a vulgar word.  They haven’t been raised in an English-speaking culture, so they don’t know that the word isn’t proper.

SOCRATES: So being offended by “fuck” must be learned.

PASCAL: Of course.  One must learn correct etiquette, the right manners.  Why should language be different?

SOCRATES: So you would agree, would you not, that being offended by a word like “fuck” is a completely arbitrary, learned behavior?

PASCAL: I would not say arbitrary.  After all, that particular word denigrates what is actually a solemn bond between man and wife.  That word cheapens the act itself.  That can’t be tolerated.

SOCRATES: It can’t be tolerated?  How come?  What would happen if you did show tolerance?

PASCAL:  The foundations of society rest on certain conventions, certain morals.  If we allow vulgarities, the whole framework is weakened significantly.

SOCRATES: And then what would happen?

PASCAL: Society would become unraveled; in its place would be—

SOCRATES: A better society?

PASCAL: —anarchy.

SOCRATES: Can you really believe that allowing certain words to become less vulgar would mean that all society would crumble?

PASCAL: But there is nothing we can do.  There are always words that will be vulgar; you cannot make them less so.

SOCRATES: Why not?  We’ve already shown that a word like “fuck” offends only because society arbitrarily says so.  Well, we are society, all of us, are we not?  What if we simply said that “fuck” is hereby no longer a “dirty” word.  It means, simply, “sexual intercourse”—if taken as a verb—or “I am very displeased” if taken as an interjection.  How could such an action harm society?  Wouldn’t it help, in the sense that no one will ever again feel mental distress when that particular word is pronounced within earshot?

PASCAL: Such a thing could never happen.  Proper people will always be aghast at the mentioning of obscenities.

SOCRATES: Ah, but there’s the rub: how will they know which words are obscenities and which are not?  What if society refuses to sort this out for them?

PASCAL:  You propose an end to all that’s decent!

SOCRATES: Not at all, my bridge-dangling friend.  I propose to remove an arbitrary system of “word regulation” from our society, because doing so will make our society that much better.  No one will ever be offended by “dirty” words again, because there will be no dirty words!  We would have a society in which censorship would be nonexistent, and in which freedom of expression would reach new, unexplored levels.

PASCAL: You are persuasive, but servants of the Devil often are.  But now I think I see what you are doing.  In grand Orwellian fashion, you propose to make odious words acceptable by simply indoctrinating people.  Brainwashing them.

SOCRATES:  No, that’s your jurisdiction.  Look at it this way: I am not trying to make murder more acceptable, simply by calling “murder” by another name such as “wexelflugen”.  You cannot change concepts by switching their labels around.  “A rose by any other name…”  But you can change the connotations a given word has, if you want, because words are just placeholders for concepts around us.  Why not take all the so-called vulgarities and throw them out the window?  Who says that our language has to have any vulgarities at all?  Why don’t we all just agree that no word will ever again bother us, ever offend us?  Don’t people have enough control of their own minds to make such a decision…a decision that would benefit everyone?

PASCAL: Well, be that as it may, some concepts would remain offensive.  There’s nothing you can do about that.

SOCRATES: There isn’t?  Give me an example of something—not a word—that offends you.

PASCAL: OK.  Violence offends me.  Surely you will not try to tell me that we should all just agree that violence is acceptable, and that we should all live with violence happily.

SOCRATES:  You are right, I will not.  I abhor violence; the very idea of murder sickens me.

PASCAL:  A ha!  Then you agree with me.

SOCRATES: Well, yes and no.  “Violence” does not offend me, but violence does.

PASCAL:  Um.  I’m confused.  You just said—

SOCRATES: I said that I abhor violence.  But “violence” doesn’t bother me.

PASCAL: What is this, semantic trickery?

SOCRATES: I do not like violence; I dislike seeing graphic pictures of murder victims; I become nauseated at the thought of war; the sight of blood often makes me sick.

PASCAL: Then violence offends your soul.

SOCRATES: I suppose.  But talk of violence doesn’t bother me at all.

PASCAL: What do you mean?

SOCRATES: I experience certain physiological symptoms when exposed to violence, but if someone says “murder” or “rioting” or “rape” or “you may eat the flesh of kings” then I am not offended.  And what I mean by that is this: I do not like to see or hear violence as it is committed, nor do I enjoy thinking about violent acts, and I certainly am very angry that certain forms of violence still exist in this world.  However, the concept of violence, the abstraction, can’t “offend” me any more than the concept of volcanoes or tsunamis or great white sharks.  Maybe I don’t want to be near them, but how can they “offend” me?  And if you talk about them, why should that bother me?

PASCAL: Let me understand.  The word “murder” is not offensive.  And a discussion of murder is not offensive.  But if someone commits murder—

SOCRATES: Then yes, that would bother me.  Look: concepts like murder are too important as issues for people to ignore just because they’re afraid their sensibilities might be punctured.  And this applies to other concepts I dislike as well: bigotry, deception, subjugation—these words do not offend me, because I do not let them.  But the acts themselves very well might piss me off.

PASCAL: But if a word is used in a hateful way, such as a racial epithet…

SOCRATES: Again, don’t blame the word.  What I find offensive in such a case is the insensitivity (or downright malice) of the speaker, their intent to harm.  So saying “fuck you!” might be offensive not because that particular phrase was uttered, but because of the intent of the speaker.

PASCAL: But then it would be too hard to sort out what was offensive and what was not!  If you always had to check the context—

SOCRATES: Sorry, but you have to do that anyway.  “Pain” means one thing to us, and quite another thing to the French.

PASCAL: I am at a loss for words.

SOCRATES: Here is my manifesto.  Many people, when encountering an idea they find distasteful, turn up their noses and say they are offended.  I, however, cannot be offended by ideas—that is, by abstract words devoid of context.  Nothing in the abstract can offend me at all, and I am very proud of this fact.  Sure, there are things I don’t like to see (like rotting corpses) or hear (like fingernails on a blackboard) or smell (like a diaper full of shrimp) but these things don’t offend me.  Why should I hold a grudge against such phenomena?  I don’t like them, but that’s not a problem: I can shut my eyes or cover my ears or pinch my nose shut.  I choose not to participate in these sensory experiences—choose—and for quite understandable reasons.  My aversion to such things is instinctual, biological, untainted by arbitrary sociological constraints.  Now, in certain cases, I cannot choose to avoid something, in which case I am offended by the selfishness of the perpetrator.  If you smoke in an elevator with me, I will be offended because I have no choice but to breathe your noxious detritus.  If you commit murder or cheat on your taxes or beat your wife, I will be offended because you have committed a crime, and betrayed an implied covenant between you and the rest of society.  But I emphasize again that the words “murder” or “cheating” or “beating” can’t offend me, as they’re just words.  And so too, the word “fuck” cannot have any effect on me at all.

PASCAL: Well then.

SOCRATES: Being offended is a learned skill.  There is no evolutionary gain to having people grimace when they hear a collection of phonemes such as “clusterfuck”.  People are offended by such a word because they choose to be offended, but do not even realize that they have a choice.  The content of the word is irrelevant…else “group sex” would offend just as much.  But it does not.

PASCAL: Speak for yourself.

SOCRATES: You don’t look convinced.

PASCAL: Well, your discourse has offended me.

SOCRATES: Touché!

PASCAL: And I would wager that some part of you is offended, as well.  You just can’t admit it.

SOCRATES: You and your wagers.

PASCAL: Anyway, I’ve made some tea out of Queen Anne’s Lace.  Would you like some?

SOCRATES: Not really.  There’s something about that plant that offends me—

PASCAL: No matter.  I have to catch the carriage to Neuilly anyway.

SOCRATES:  And I must kill a rooster for Asclepius.  So long.

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