John Rawls was a philosopher who became famous for a thought experiment, in which the only way to reach a fair arrangement is if people don't know which side they'll end up on. This thought experiment is played out in today's Doctor Who special, and it's at the center of the Doctor's own dilemma.
Spoilers ahead...
Rawls believed that if we could have a "veil of ignorance," we'd all opt to create a perfectly fair society. In other words, if you somehow had your memory wiped and didn't know whether you were a rich person or a poor person, you'd opt to create a society where you'd be okay in either eventuality.
In "The Day of the Doctor," the assembled Doctors impose a veil of ignorance on the humans and the Zygons, to force them to negotiate from a position of total neutrality. The Zygons, handily enough, are shape-shifters who want to take over the Earth. The humans have a nuclear warhead which they're going to use to destroy the Zygons, along with all the super-weapons they've gotten their suckers on.
Luckily, though, the vault where the Zygons and humans are collected has memory-wiping devices in the ceiling. And the Zygons have already turned themselves into perfect duplicates of the humans in the room. So the Doctors arrange it so that nobody in the room knows whether they're a human or a Zygon, and thus everybody will have to work together.
This, in turn, is a means of avoiding a terrible choice — sacrifice millions to save billions, thanks to that nuclear warhead — which echoes the choice the Doctor made four hundred years ago, to destroy his home planet and his own people to save the universe from the ravages of the Time War.
The Doctor is a shapeshifter
The choice of Zygons, the red sucker-covered beasties that only appeared in 1975's "Terror of the Zygons," as the episode's villains, seems like an odd one. (Although bear in mind, Steven Moffat doesn't really seem to like to include proper villains in his "event" episodes. No villain appears in "The Big Bang," and the Silence are more a momentary distraction, in "The Wedding of River Song.")
But in fact, the Zygons are a curiously appropriate choice, for an episode celebrating the birthday of a shape-shifter. For that, of course, is what the Doctor is. He's worn many faces, but he's also been many people, over the course of his centuries roaming the universe. And just as the Zygons present us with the dilemma of figuring out who's "real" and who's a "copy," the anniversary special is all about figuring out who's a real Doctor and what it means to be the Doctor.
Even as the Doctors are imposing a veil of ignorance on the humans and Zygons in the Black Vault, their past self (John Hurt) is dealing with a veil of ignorance of his own. He can't know his own future, or what sort of man he'll become if he goes through with his plan to commit mass murder to stop the Time War.
We see John Hurt's War Doctor in the middle of the last day of the Time War, as Gallifrey is falling to the Daleks with a lot of pew-pew-pew. This looks quite different than what we heard described in previous episodes, more of a rout for the Time Lords in fact — there's no glimpse of the Time Lords getting power mad, not to mention the Horde of Travesties or the Could-Have-Been-King or whatnot.
Hurt seems more intent on using a gun to carve the words "NO MORE" into a Gallifreyan wall than stopping the Daleks, although he does use his TARDIS as a battering ram against a bunch of them. Then he breaks into the Time Lord weapons vault and steals the Moment, a weapon so sophisticated it developed a conscience. After the War Doctor lands in the middle of nowhere and drags the Moment miles away from the TARDIS, its interface appears to him in the guise of... Rose Tyler. Or more accurately, Bad Wolf Rose.
She decides to show him his future selves, Christmas Carol-style, so he can see what sort of man he'll become if he commits this atrocity. And actually? It's not that bad. Both the Tenth and the Eleventh Doctors feel terrible about the choice they made, in the past, and they can barely stand to look at the War Doctor. But the good news is, after doing the unthinkable, the War Doctor learns that he can go back to being just "the Doctor" again.
The Moment tells the War Doctor he'll one day count all the children who were on Gallifrey at the time when he destroyed the planet, and when he asks his future selves about it, the Eleventh Doctor has no idea how many there were. And the Tenth Doctor gives an actual number. Because after four hundred years, the Eleventh Doctor has actually moved on, while the Tenth Doctor is still mired in grief and guilt. The Tenth Doctor is outraged to see how much his future self has forgotten about the horror they inflicted. They're "the man who regrets and the man who forgets," as the Moment puts it.
And in a bit of a masterstroke, Moffat turns his own cute "Timey Wimey" catch-phrase into a sign that the post-Time War Doctors have become too infantilized. (And likewise, when the Moment wants to show the War Doctor his future, she clearly expects something dreadful and admonishing, but instead she gets... a fez.) The future Doctors can't face being grown up, because when they think about being an adult, they picture John Hurt's weathered, monstrous not-Doctor.
In the end, the War Doctor sees that his future selves' regret over his actions will lead them to become more heroic than ever, and save tons of planets. So he's reinforced in his resolve to wipe everything out and end the war, the only way he can.
And meanwhile, the Tenth and Eleventh Doctors realize, at last, that they can't dismiss the War Doctor as a false self, a kind of Zygon Doctor — because even though they posture about regretting their choice, they would do it again. The Doctor is the one who does what must be done, when there's no other way.
In the end, it's the future Doctors, not just the past Doctor, who lose their veil of ignorance and see their past choices for what they are.
You have to embrace the terrible choice before you can avoid it
Steven Moffat's big "event" episodes always feature the Doctor embracing the unthinkable, right before he finds a way around it. He embraces erasure in "The Big Bang" and death in "The Wedding of River Song," and then finds a loophole. But this time around, it seems especially significant — the Tenth and Eleventh Doctors have to embrace the terrible choice to burn everything, before they can avoid it.
They have to own that choice, and stop treating it as the work of some other person. So in the end, after they've sorted out the five Zygons that are intent on conquering the planet, the other two Doctors are led to visit their past self, thanks to the Moment, so all three of them can make the choice together.
And then, at the last possible moment, Clara proves that it's always the companions who save the Doctor from himself (as the Eleventh Doctor says, "this is what I'm like when I'm alone") and insists that the Doctor has to be better than this. (And we get a nice Terrance Dicks quote, that the Doctor is "never cruel or cowardly.") So of course, the Doctor comes up with another way, one that requires the help of all his past selves, plus his future self (Peter Capaldi cameo!).
The Doctor's big plan for defeating the Daleks and ending the Time War boils down to "duck and they'll catch each other in the crossfire." The Doctors gather all their TARDISes and use them to whisk Gallifrey into stasis, or into a pocket universe or something. The Daleks who are all shooting at Gallifrey will wind up shooting each other, and they'll all be destroyed. So it'll look as though Gallifrey and the Daleks were wiped out, but Gallifrey will be safe. (Maybe.)
Gallifrey can be preserved, like the people inside the Time Lord paintings, who are caught in a "slice of time."
Paintings don't ever stand still
This is yet another one of Steven Moffat's explorations of the motif of seeing and memory — the Zygons are "translated" inside the paintings in the Undervault of the National Gallery, and kept in suspended animation. The Zygons hide themselves there, so they can wait four hundred years until Earth is "worth conquering."
And the Doctors use a similar trick, to get inside the TARDIS-proofed Black Archive to stop Kate Lethbridge-Stewart from blowing up London. They go inside the painting of the Last Day of the Time War, which they've conveniently gotten smuggled inside the Black Archive already.
The paintings in this episode don't stand still, because people can escape from them — but also, because they change depending on who's looking at them, the way all paintings everywhere do. Works of art change their meaning over time, like shapeshifters. And the painting at the center of the episode has two titles, either "Gallifrey Falls" or "No More."
It takes the Curator of the National Gallery, who turns out to be a surprise Tom Baker cameo, to explain that the actual title of the painting is "Gallifrey Falls No More" — because the Doctors have saved Gallifrey, although they apparently also lost it. (Tom Baker milks this, and then milks the milk.)
[Update: a friend suggests that Gallifrey, itself, is actually preserved inside the painting "Gallifrey Falls No More" — although then, why would the Doctor have to go searching for Gallifrey? It's right there.]
But by this time, we're already teed up for the next terrible event — which is another awful war. The Eleventh Doctor claims that the reason why he's forgotten the number of children who've died on Gallifrey is because he's been dealing with brand new darkness, something the Tenth Doctor really shouldn't want to know about.
At the end of the episode, when the War Doctor and the Tenth Doctor are both about to forget the events of this adventure, the Eleventh Doctor finally confides that he's seen Trenzalore, his own final resting place. And that the Doctor is doomed to perish in a bloody war, involving millions of people. Of course, we've just seen how fate can be cheated, but the Eleventh Doctor insists that this will absolutely positively no question be different. Really. For real.
In any case, the Doctor's ever-shifting nature means that his heroism always has a bit of a veil of ignorance over it — he can never know who he'll be next, or whether he'll finally become a monster as a result of the choices he's made. [Edited to add: And that's the point of the veil of ignorance: You can't know who you'll be.]
That's the dilemma that John Hurt faces at the start of this episode, and its ending offers a more definitive answer: the Doctor is always a savior, no matter what guise he appears in. John Hurt's face gets to appear alongside all the other incarnations, at last, front and center.
But of course, you can't know how restoring Gallifrey from its saved state will turn out either — that could be another monstrous act, given how messed up the Time Lords were by the end.
Oh, and there were tons of random in-jokes, which are probably already being catalogued online. Coal Hill School. The clock reading 5:20, which was the original transmission time. "We're both reversing the polarity." And David Tennant's final words as the Doctor have apparently become a jokey catchphrase now.
In many ways, this was a standard Moffat "event" episode. There's some terrible thing, which is apparently inevitable, but which only appears to happen in the end, thanks to some sleight of hand. There's a wedding, after the Doctor has just insulted the bride. There's a bit of wordplay, which turns out to mean something different at the end than you thought. The meaning of the Doctor is questioned and then reinforced.
But John Hurt's emotional, raw performance as the wounded Doctor on the brink of committing a war crime, confronting his cuter, older selves, is such a standout that it makes the whole episode into something special.
Some GIFs via Doctor Who Tumblr.