Entry 393: How Time-Travel Movies Can Resolve Paradoxes

There are real things going on in the world right now -- things that are not so great, like hurricanes and floods and racist attorneys general.  If you want to read about such things -- and you should; don't bury your head in the sand -- feel free to leave my blog and go to a legitimate news site right now and read about them (and donate some money if you can spare it).  I won't be offended.  But if you need a distraction, about a completely moot topic, keep reading.

Time-travel is a common plot element in science fiction.  It's one rife with paradoxes.  A few very notable examples can be found in the Terminator franchise.  In the original movie, Kyle Reese goes back in time to stop The Terminator, sent from the future by the evil machines, from killing Sarah Connor, so that she can give birth to John Connor, who will lead a successful human resistance against the machines.  The paradox arises when we learn that Kyle Reese is also John Connor's father -- but then how did John Connor get there to lead the resistance in the first place?!

In the second movie, The Terminator is now a good guy (Arnold was too marketable a star at that point to make him the enemy), who thwarts an attempt by a different time-traveling terminator to kill a now teenage John Connor.  Also in this movie, The Terminator destroys a chip that paved the way for the technological advances leading to the takeover by the machines (which was left by behind by the original bad Terminator).  But in so doing, wouldn't he have eliminated the very technology that created him, and thus wouldn't he disappear instantly after having done that?  (In the movie, he does not disappear, but instead dramatically destroys himself immediately after destroying the chip -- or maybe he has Sarah destroy him, because he's programmed not to destroy himself; either way, he gets melted in some sort of industrial lava).

These paradoxes are things I've wrestled with before, and I came up with a way to think about time and reality that will resolve them.  As always, these are probably not original ideas.  I'm sure if I Googled it I could find a dozen websites laying out these ideas better than I could.  But I'm not going to Google it.  I thought of it on my own, so I'm going to write it up on my own.  Also, I'm certainly not claiming this is how our physical universe actually works.  I'm just saying this is how movies could resolve their time-travel paradoxes -- and, who knows, maybe there are some that do this that I've never seen.

The gist of the idea is to think of time as comprised of discrete moments and the universe being comprised of discrete particles.  At any given moment t, we are in some state of reality based on the location (and other physical properties) of all the particles in the universe.  At the next moment t+1, we move to a new state in which at least one particle has changed it's location.  However, in addition to "reality", the state we are actually in, we also have billions and billions of "potential realities" that could have happened if a particle did something different than it actually did.  So we have one potential reality for possible every movement (within the laws of physics) of every particle in the universe.

For example, in the diagram, our reality is, say, blue, but if at least one particle had done something different, then we might have been on the red track or the green track or one of the orange tracks or one of the googols of other tracks not pictured.  Those other tracks don't actually exist, but they could have existed, if a particle (or particles) had done something different.

What determines which state is the next part reality?  How the particles know where to go next?  How we actually travel through time?  Good question.  Maybe it's random chance (God does play dice); maybe it's a divine hand; or, most intriguing to me, maybe we determine it -- maybe that's what free will is.  Our collective self-governance determines the next state of reality.  For the sake of resolving movie paradoxes, however, it doesn't really matter.

Thinking of time and the universe in this way, can instantly eliminate almost all time-travel paradoxes.  For example, in T2, once the terminators go back in time, their presence changes the state of the universe at that moment, thus putting us in a new reality (because some particles are in a different place).  So, say, the red track is now reality, and the blue track -- the one in which the machines take over -- is now a potential reality.  The goal now for the good guys is to make sure the machines also don't take over in the red reality.  (That is, in actual reality, which would be very close to the old (blue) reality since only a little bit has change.)  The Terminator succeeds by destroying the bad terminator (John lives), and the chip, making the human-enslaving technology now nonexistent in the new (red) reality.  And since he destroyed it in the red reality, not in the blue reality, in which he was created (which is now only a potential reality), he's not preventing his own existence.   He was made in the blue reality and "jumped" backwards to the red reality, where he lives, until he destroys himself a few minutes later.  Paradox resolved!

(Although, one thing that bothers me about this is that it seems like it can create energy out of nothing.  Every time a time-traveler comes back to a certain moment, they are new energy in reality that wasn't there before.  I think you can resolve this, by saying that time travel works by swapping energy from the two states.  So if somebody goes from state S021 to state S0 in the diagram above, an equal amount of energy must be swapped from S0 to S021, somehow.  That's what time travel is: an exchange of energy between moment-states.)

Resolving how Kyle Reese can be John Connor's father is a bit more difficult, but it still can be done.  Here's how.  Say, Kyle was born in the blue reality, and there is no John Connor.  For some reason, he goes back in time, makes sweet love to Sarah Connor, and she births John Connor, and all of this happens in the red reality.  Then, for other unknown reasons, Kyle goes forward into the future of the red reality (theoretically possible for real!).  He ends up in the middle of the robot apocalypse, but his own son, who is now his age, successfully leads the human rebellion.  So the machines send The Terminator back in time to kill Sarah, and so Kyle goes back in time to stop him and knock-up Sarah again, and all this happens in, say, the green reality -- The Terminator is this story.  There we go!  Air-tight!  That is, until I think of a logical flaw in bed, unable to sleep, at 2 am tomorrow morning.

Until next time...

Entry 392: Week One Is In The Books

Lil' S1's first week at his new school went pretty well.  He adapted almost seamlessly, but  the same cannot be said for his mother.  She can get so stressed over things like this.  I know why.  She says why.  It's because she has "working mom guilt," and she feels like if everything doesn't go perfectly (and it never does) it's somehow her fault (even though it's not).  I understand this.  Women are unfairly held to a much higher standard when it comes to parenting than men are.  If a woman sends her kid to school in dirty clothes, she gets the side eye; if a man does it, he gets applauded for taking his kid to school.  It doesn't help that the daycare we use reinforces this double standard.  Both S and I are there once a day, but whenever there's a problem the workers always talk to her about it -- even if it's about me!  I usually do drop-off in the morning, and once I read the weather report wrong (I got Columbia, SC, where S's parents live confused with Columbia, MD, where I work), so I dressed Lil' S2 in too little clothing, and I didn't bring a jacket.  On this particular day, I also was doing pickup, so I went there and got him, and they didn't say anything to me, and then I took him back the next day (appropriately dressed this time), and they still didn't say anything to me, and then when S picked him up, they told her that yesterday I didn't dress him properly.  They had two chances to tell me directly, but instead they put in on S.

That's the type of shit she has to deal with, so I try to be sympathetic, but it's not always easy, because a) sympathy is not my strong suit; b) sometimes she takes it out on me, like if I don't do everything exactly the way she wants it done, then she gets mad at me.  As an example, Lil' S1 apparently has "homework" that he has to do during aftercare and then turn in the next day.  So, one morning last week, I inadvertently take it out of his backpack before school, and S sees it when she gets home and gets annoyed and tells me to make sure it gets handed in the next day.  Fine.  I make sure it's in his backpack the next morning.  But then that evening, S notices that it's still in there.  It never got handed in.  So she gets really annoyed with me because I didn't physically hand the paper to his teacher.  So I get annoyed back because I think she's way overreacting.  It's just a stupid worksheet for a kindergartner -- how important could it be?  Also, why is it incumbent on me to hand in his homework?  How is it helping the student if the parent is the one responsible for it?  Commonsensically, one would assume that if this homework is really important to his teacher then she would ask him to hand it in, no?

Anyway, the next day, I told his teacher explicitly that his homework was in his backpack, and she was like, "Oh... yeah... okay... well, I'll have to get him a folder... uh, I'll take care of it... we'll get it sorted."  It was obviously as important to her as I figured.  There also was a mix-up with school lunch one day last week, but I won't go into it, other than to say Lil' S1 did not go hungry.

I think S is feeling especially guilty right now because she has to go to Africa for about a week for work on Wednesday.  This trip cropped up quickly, so we didn't have time to bring in reinforcements (her parents) like we usually do when she goes away.  It's just going to be me -- which is fine, I actually don't mind at all.  One week isn't that long.  It won't be that much more work for me.  But she feels guilty about it, so she volunteered to get the kids out of the house for a few hours everyday this long weekend and let me have some free time.  I certainly won't say no to that!  That's actually how I prefer to work, in general.  I like to work really intensely for a while and then take long a break.  That was one of my favorite parts about school.  I wish I could work on that schedule now, like put in 60 hour weeks for a month and then take two weeks off.  It's tough to do with kids, though, and it's even tougher to do when you work for a company that won't let you do it.

In other news, I turned 40 recently.  It's cool.  I don't really get excited by birthdays or round numbers, though.  I'm generally happy with my life, which is the most important thing.  I don't mind getting older.  I don't like the random aches and pains, but that's about it.  I do kinda wish I had thrown a big 40th birthday party for myself while I was in the Sea-Tac region, just so that I could get a bunch of friends and family in one place, a rarity these days, but I also hate planning, so it didn't happen.  Maybe I'll do it next year.  It would be kinda funny to celebrate 41 as if it's some great milestone.  It's just as good as 40, if you think about it.

Alright, that's all I got today.  Until next time...

Entry 391: Old School vs. New School

Big news this week at the G & G household: Lil' S1 was accepted to a DC public charter school.  Applicants are selected by lottery, so it's mostly luck, but also S was extremely persistent in checking his status (i.e., bothering the school on a daily basis), and I think that might have helped (it certainly didn't hurt).  She's really happy about this.  I'm more "meh."  Actually "meh" is the wrong exclamation, because it connotes a lack of interested.  It's not that I don't care; I'm ambivalent.  I have strong feelings on both sides.  I haven't totally reconciled them, but I'm going to try to lay them out in this entry.

The first, most important thing is that I wanted this to happen because S wanted it to happen.  She's obsessed with getting our kids into the "best" school possible.  Rankings and tiers and reputation and things of that nature mean a lot to her, and I don't think she would ever be content sending our kids to the "normal" neighborhood school, when these "better" charter schools exist as an option.  She's got a bit of Tiger Mom in her that way.

I'm totally fine with her viewpoint on this -- I think it will probably serve our kids well in the long run -- but as you can probably guess from my use of quotation marks above, I feel differently about things.  I'm a socialist when it comes to schooling.  I'm not about my kids being as advantaged as possible, but rather about them being part of a system that works for everybody and one in which everybody works for it.  Is that too idealistic?  Maybe.  But it's how I feel.

I also question whether or not "better" schools are actually better.  I'm of the mindset that education is much more about what you, as a parent and a student, put into it, than it is about where you go to school.  But it could be I'm too biased by my own personal experience on this.  I never went to an elite school -- it was public high school and state universities for me -- but I feel I attained the same level of academic success and amassed the same skills I would have if I went to a posh private high school and an Ivy League university.  (I don't have the same connections as I would have in that case, but that's another story.)  My feeling is that the best schools don't make the best students, but rather the best students make the best schools.  Harvard is great because all the smartest kids choose to go to Harvard.  If suddenly they chose to go to Middle Tennessee State, then Middle Tennessee State would be great (and Harvard less so), even if nothing else about the two universities changed.  That's my broad opinion on the matter.  It could be total wrong, but it's worth noting that in his book Everybody Lies, Seth Stephen-Davidowitz highlights a big-data study that lends some support to this theory.

So, I'm not particularly worried about my sons lagging behind academically no matter where they go to school.  I want them to go somewhere that's safe and clean and comfortable (and of course it has to have some baseline of academic standards), but also I want them to go somewhere they (and we, as parents) can contribute to the local community.  And therein lies my ambivalence with sending him to a charter school.

I haven't totally wrapped my head around charter schools yet.  On the one hand, they really can be a savoir for low-income families who are "trapped" in districts with failing schools.  On the other hand, they can further disadvantage those students who don't have parents willing work to the system -- that is to say, exactly those students who need it most.

Because here's what happens, or at least here's what happened to us at our school, which I assume is what happens a lot of places.  Lil' S1 started going to our neighborhood school two years ago for PK3.  It's a decent school, overall.  The teachers are mostly good, and I like the principal, but it does have some shortcomings.  It's old and needs to be renovated; the thermostats don't work correctly, so students complain about it being hot or cold in their classrooms; the kindergarten classrooms are glorified storage areas, which don't even have windows; and also sometimes it can seem disorganized when dropping off or picking up students.  It supposedly is in line for a major overhaul, but DC keeps delaying the funding.  Now it's set for 2020 or something like that.

The school is a mix of students I will call "advantaged" and "disadvantaged," for lack of betters terms.  The parents of some advantaged students see these shortcomings and want to upgrade (which is totally understandable), so they apply to charter schools or pay for private schools or move to the Maryland or Virginia suburbs.  The parents of the disadvantaged students can't or won't do this -- they might be poor or overworked or absent or have substance abuse issues or they just didn't win the charter school lottery (and even applying for charter schools has a cost associated with it in terms of time and incidental expenses).  So now the ratio of advantaged to disadvantaged kids is much lower than it was before, and this of course makes the school worse (and cruelly the school has less money to fix things, because funding is dependent on enrollment numbers).  Probably the best thing for a struggling student is to be immersed in a school of mostly non-struggling students.  And that's how schools should work in my opinion.  The privileged students should buoy the underprivileged.

But that's not how it works.  Instead all the other parents of advantaged students look around and see everybody else leaving and decided it's in their best interests to leave as well.  Everybody could just stay and work together to make their school better, but they don't, and it makes the remaining students worse off.  (A friend of mine framed it in terms of the prisoner's dilemma.)  And by the way, this doesn't necessarily break down racially the way you might expect.  It's true that the vast majority of the disadvantaged children are black and brown, but so are many of the advantaged children.  It's not "white flight"; it's "engaged parent" flight.  And that's the worst part, because it is precisely those students who don't have engaged parents who need good schools the most.

So here were are.  We are sending our kid to charter school, and I'm feeling very conflicted about it.

Until next time...

Entry 390: To U.P. and Beyond! (But Not That Far Beyond)

The G & G clan spent the past few weeks in University Place, Washington visiting my family.  First things first, U.P. really needs to change its name.  I know many people are hesitant to change things because change is scary and things from the past always seem better than they were.  But University Place is the stupidest name ever for a city containing exactly zero universities.  College Station, Texas and State College, Pennsylvania make sense because they are home to two of the largest colleges in the U.S.; University Place, Washington makes no sense.  It got its named because the University of Puget Sound once purchased land in the city (then an unincorporated region of Pierce County) and was going to build a campus there, but never did.  (The private high school Charles Wright Academy was built on the land instead, but Private High School Place is an even worse name.)  So somehow during the period of time after the land was purchased by UPS and before it was sold back to the county, people started calling it University Place, and it stuck.  My question: How long was the construction pending?  Months?  Years?  Whatever the case, we are now apparently stuck with an absurd name for an otherwise lovely city.

I've heard rumblings that somebody somewhere tried/is trying/is going to try to change the name to Chambers Bay, but I don't know much about it.  Other than Chambers Bay, which would be a fine name, here are a few other suggestions:

• Whiter Tacoma
• Roundabout Station
• Younger Fircrest
• Gary Larsonland
• Dead Crab City

The last one is a reference to Titlow Beach, where we spent an afternoon, and Lil' S1 used the occasion to wade into the water and "collect" dead crabs.  He got a massive handful and wanted to bring them home with him.  It was pretty gross.

[A very hazy view from Chambers Bay.  This is before the smoke from the Canadian wild fires had completely dissipated.]

In addition to learning my oldest son has a penchant for deceased crustaceans, I also learned that he is good at swimming and bad at rollerskating.  We used the pool of some family friends and Lil' S1 was swimming by himself for relatively long stretches.  I really want him to get to the point where if he fell into the deep end of a pool, he could swim to the wall unassisted, without incident.  I think he could do it now, but I'm not completely sure.

Rollerskating is a completely different story.  I thought he might be good at it, because he has good balance, when it comes to climbing and jumping and stuff like that, but he was a disaster on skates.  (My nephew G had a birthday party at the skating rink the night before we flew back to DC.)  It was like that scene in Bambi, where he's learning to walk -- legs flailing in every direction at once -- only there was no success at the end.  After a few minutes, he started crying and demanding his skates be removed, and that was the end of that.  It's fine.  I do wish he would have tried a bit longer though.  I don't really care if my son is good at rollerskating or not, but I do want him to have some perseverance.  He kept saying, "I don't want to rollerskate.  I'm not good at it," which makes me a bit nervous, as I don't want him to get in the habit of just giving up on things he isn't instantly good at it.  But he's not even five years old yet, so... yeah.

The past few times I've been back to U.P., the weather has been immaculate and this trip was no exception.  I don't know if it's global warming or luck or some combination thereof, but almost everyday it was clear skies and temps in the 70s and 80s.  That makes it SO much better.  The worst part about living in the Pacific Northwest (west of the Cascades) is the constant rain and overcast skies.  (I heard it was a particularly bad fall and winter.  I think my dad said it rained everyday in October.)  Absent that, there is no place I would rather be.

[My dad watering the shrubs with Lil' S1]

But it wasn't all sunshine and moonbeams.  My uncle B died a few days before we left.  This doesn't evoke a particularly strong response in me, as I barely knew him.  My dad's side of the family has always been a bit, let's say, odd, especially so my uncle B.  His is a sad story.  He came back from the Vietnam War with obvious mental and emotional problems, and it seems as if his life never really got on track because of it.  He always seemed to be "searching" for something, and he tried to fill it with things like astrology and Kabbalah and whatever other mystical hocus-pocus was popular at the time.   He was a hard guy to like, as he was constantly holding grudges against other family members -- my dad included -- over trivial matters.  To me, he was always a harmless weird old guy, whom we would see sometimes when we came to visit and sometimes not, depending on his current mental state.  I'm certainly not happy that he died, but I'm not sad either.  I didn't know him.  I feel about the same way you probably do reading this right now.  Maybe it's sad, at a macro level, that I didn't have a stronger relationship with him, but since there was never a relationship there to begin with, it doesn't feel sad that one doesn't exist.  It's like getting sad because an imaginary friend is no longer speaking to you.

I actually was more sad when I found out that the my friend JW's uncle M had also died recently.  He committed suicide a few months ago.  Unlike with my own uncle, I actually hung out with M from time to time.  I used to go to their family lake house quite a bit, and it was like a second home for M.  He was a good guy.  He definitely didn't seem like somebody who would take his own life, but a lot of suicidal people keep that side hidden from even their closest confidants, let alone their nephew's friend whom they see once a quadrennial.  M wasn't married and didn't have any kids.  And apparently he was in chronic pain because of a bad back injury.  I'm certainly not in favor of suicide, especially at a relatively young age (I think M was in his 50s), but if you are in constant pain, and you don't see any relief on the horizon, and you don't have any kids to raise or a spouse to support, then okay, I guess.  I mean, not okay.  It's still sad.  It's still something I would try to talk somebody out of.  But it's more okay than it could be, I suppose.

Alright, happier news: I saw my old friend and college roommate TB for the first time in a few years.  Since I last saw him, he got divorced, took time off from his job, had something approaching a breakdown, got help, and turned things back around.  He seems to be doing genuinely well now.  And it was great to see him.  He's one of those guys that just makes you feel good to be around.  I really wish we lived in the same area.  I'm bad at corresponding, and he's worse than me.  So it's rare when we get together.  (The only reason our meeting happened this time is because I saw he was in the area on Facebook.)  But when we do, it's like old times again.  That's how you know who your true friends are.  When you get together after years apart, how long does it take for the initial air of formality to dissipate?  If the answer is instantaneously, then that's a true friend.

I have tons of stories about TB from back in the day, but here's one of my favorites.  He was constantly struggling to make ends meet throughout college, and there were times when he literally had no money to his name.  Once, he came home with a $10 bill and said that he was getting paid tomorrow, but that this bill was all he had for a meal that night.  Back then, $10 in our smallish college city was more than enough to get a decent amount food, so we went to the grocery store, and here's what he bought: a giant bag of tortilla chips, a tub of french onion dip, two Rainier tallboys, and the movie Soap Dish.

I got to see some of my other friends while I was there as well.  My aforementioned friend JW just had a baby with his wife Y, so I got see their new little guy.  They've had really bad luck with pregnancies, so seeing them with a baby is really something special.  I also saw my friend JY and his family.  His only child is about to turn 14 (!), so he's a legit teenager.  It's such a different parenting world.  We went to a Mariners game, and he and his buddy could just go off on their own while the adults hung out among themselves.  Everybody says to enjoy your kids being young while you can, but, man, it would be nice to not have to constantly worry about entertaining them at every turn.

[We saw Edgar Martinez's number 11 retired by the Mariners.  If you look carefully, you can see the image of Edgar in his batting stance cut into the outfield grass.]

Seeing my family was great also, as always.  My brother and sister in-law also have a new little baby girl, Lil' A, and she's just as cute as you would imagine.  They recently moved from Seattle to U.P., so now my entire immediate family lives there (except us).  It makes it easier to see everybody when you come visit.  Lil' Q, my bro's oldest, is a few months younger than Lil' S1, and they get along pretty well.  Sure occasionally they have incidents -- like when Lil' S1 hit Lil' Q with a map at the zoo or when Lil' Q poked Lil' S1 in the eye during their sleep over ("I forgot I wasn't supposed to do that" was Lil' Q's defense) -- but for the most part they play together pretty well.

Alright, I wanted to write a bit more, but as usual I'm out of time.  S is away at the moment with the kids, and if I don't unpack my half of the suitcase before she gets home, it's not going to be pretty.  She has been asking me to do it for the last two days after all.

Until next time...

Entry 389: On Crossword Constructor Pay

I'm about to go on vacation, again -- a "real" vacation this time, not one of those long-weekend, fake vacations.  The whole G & G clan is going to the southern shores of glorious Puget Sound to visit family and friends for ten days or so.  I'm sure I will have much to write about when I get back.  Until then, please enjoy this entry at my other blog on crossword constructor pay.

Entry 388: I Will Post Again Sometime Soon

It's been a couple of weeks since my last post because we've had family in town and because we've been out of town.  I will post again sometime soon, but not right now.  In the meantime, please enjoy this picturesque vista of Ventosa Winery on Seneca Lake in Geneva, NY.

Until next time...

Entry 387: Deep Thoughts with DG

I had something stuck in my mental craw for a while.  I have a decent understanding of the basics of Einstein's theory of relativity, especially the "special" version.  One of the consequences of it is that time slows down and distances get shorter for objects moving at very high speeds.  At first this sounds totally insane (because it is), but once you accept that the speed of light is constant for all observers, always, then it has to follow.  The thought experiments that demonstrate this are not that difficult to follow if you're actually willing to sit down and think them through.

A further consequence is that time travel into the future is theoretically possible.  If we had a craft that could move close to the speed of light, somebody could get in, zoom out into space for, say, 16 years (their time), and then return to Earth.  Since time was moving "normally" here (not slowed down) more than 16 years would have elapsed for us -- let's say, 20 years, for the sake of example -- and thus our rocketperson would have effectively traveled 4 years into the future.  Totally trippy, but totally true.  (In fact, I believe this is how Ender time travels at the end of Ender's Game, one of the few science fiction/fantasy books I read and enjoyed -- even if the author is kind of a bigot).

So that's all well and good, but here's the part that started bothering me.  Speed is a relative concept.  In the scenario above, we think of our rocketperson as moving at a very high speed, and that's true, they are, relative to Earth.  We could just as easily think of the rocketperson as being still and the Earth moving at a very high speed with respect to them.  But if that's the case, then why doesn't time slow down on Earth?  How does science know that the rocketperson is supposed to go into the future relative to people on the Earth, rather than vice versa?  How does it know?!

This had been bothering me off and on for some time, but I started thinking about it a lot while listening to an episode of Startalk.  I was actually going to email the show to see if Neil deGrasse Tyson would talk about it, but before I did this, I thought "maybe Google has an answer for me.  Maybe somebody else has thought of this."  Somebody has.

In fact, this conundrum has a name, The Twin Paradox, and it is "arguably the most famous thought experiment in relativity theory."  Huh, how 'bout that?  (Traditionally the problem has been framed by considering two identical twins, one who stays on Earth and one who rockets off into space at a high speed before returning years later.  The traveler will be younger than the homebody upon returning.)  As I've said before, I have mixed emotions when I think of something interesting and then Google it to find out that people way smarter than me have already thought of the same thing many times over.  On the one hand, it sucks because it's not original; on the other it's cool because I came up with the same thing on my own as really smart people.  It's a weird sorta of simultaneous rejection and validation.

The article linked above does a really good job of resolving the paradox, but I'm still not completely satisfied.  I now understand (I think) how science knows the difference between the traveler and the homebody in this particular example.  The travel returns to the reference frame of the other, not vice versa.  One way to think of it (I think) is to add a third person who can observe the clocks of both twins from a stationary position in the universe.  The clock of the twin on Earth never changes with respect to this third person, the clock of the twin in the spacecraft does when he or she makes the return trip to Earth.  If the twin in space never turned around and the Earth somehow shot off into space and caught up with the traveler so that it's clock was moving at the same rate as the traveler, then (I think) the twin on Earth would be younger than the twin in the spacecraft.

But there are still two things that bother me about this problem: (1) What would happen if instead of traveling out into space away from Earth, the traveling twin just orbited Earth at a very high speed before returning -- would he or she be younger then?  (2) Why is it the traveling twin specifically who gets younger?  I understand there is a difference between the two, but why does this difference manifest itself one way and not the other way?

For (1), I really don't know, but my guess is "no".  For (2), I think I will just have to content myself with the answer "because that's how the math works".  And by the way, although that's not the most satisfying answer in the world, it's also not a terrible one.  There have been many, many times throughout my life in which I tried to conceptualize a bit of mathematics and couldn't really do it, so I just memorized the parts I needed and moved on, and then at random moment in the future, something would click, and it was like "Ah... So that's why that's like that!  That's where that formula comes from!"  As an example, I went through an entire year of high school calculus (and scored a 4 on the AP Exam, I might add) without really understanding conceptually what the derivative of a function is.  This is kinda amazing and kinda impressive in a weird way.  I knew how to calculate derivatives, but I didn't really get what they are or why they're important until I got to college and did something I had never done before -- read a calculus textbook.

This brings me to another topic, one I've thought about a lot before and probably even blogged about here before: I wonder if a "concepts driven" approach to math education is the correct one.  Back when I was a TA and taught community college, the emphasis was always on the concepts -- teach the concepts, the formulas have no meaning without the concepts, math isn't just rote memorization and manipulating symbols it's about learning concepts.  But when I think back over my own experience in learning math, it's almost the exact opposite, especially before I got to college.  I memorized almost all the formulas first and then the concepts came, usually at a much later date.

Admittedly, I'm not very well-versed in the research of math education.  It could be I'm atypical, and weighing my own experience too heavily as part of a larger group.  I'm open to being proven wrong about this, but for now I'm sticking up for memorization and rote learning in mathematics.

Okay, gotta go.  We're going to a friend's annual pool party.  Last year I jumped into the pool with my phone and wallet in my pocket.  The wallet dried out fine, the phone, not so much.

Until next time...