This chapter never made it into the final version of my children's book The Outer Limits (vol 2 of The Philosophy Files, now part of The Complete Philosophy Files). For Whittermarsh Jr.
4. The Mirror Puzzle
Sometimes it is the things that are most
familiar to us that turn out to be the most deeply puzzling. Take mirrors, for
example. How many times do you see yourself reflected in a mirror each day?
1.ILLUSTRATE:
BOY LOOKING INTO A MIRROR
At least ten or twenty times, I
should think. Most of us never stop to think about what we see. But, as you are
about to discover, mirrors are very strange and puzzling things.
An adventure in the mirror
Aisha and Kobir are visiting
Kobir’s auntie. Auntie Anaximander lives in an enormous, fusty old house deep
in the moors.
2. THE BIG SPOOKY HOUSE IN THE
MOORS. LIGHTNING.
It’s a wild and stormy night and
the phone and power lines are down. Auntie Anaximander has gone off in her car
to report the powercut leaving Kobir and Aisha all alone in the dark house.
3: ILLUSTRATE: NERVOUS KOBIR AND
AISHA AT DOOR OF CREEPY COUNTRY HOUSE. STORY WEATHER. THEY ARE WAVING OFF
AUNTIE A WHO IS DRIVING A MORRIS MINOR. NB KOBIR IS OLD PHILOSOPHY FILES
CHARACTER WITH GOATIE BEARD HAS A RING ON HIS RIGHT HAND AND WATCH ON HIS LEFT
AND A HIS-LEFT-SIDE HAIR PARTING.
They light a candle and decide to
explore. After a while, Aisha and Kobir come across a huge echoing hall.
4. ILLUSTRATE: KOBIR AND AISHA
ENTERING SPOOKY ROOM AS DESCRIBED, HUNG WITH OIL PAINGINGS .HOLDING CANDLE.
There’s no furniture, just lots of
dusty oil paintings of Auntie Anaximander’s dead relatives, and, at the very
end of the room, a huge mirror stretching from floor to ceiling.
Aisha and
Kobir walk up to the mirror and look at their reflections.
5 ILLUSTRATE: KOBIR AND AISHA,
HOLDING CANDLE, BEFORE THE HUGE BAROQUE MIRROR, WITH CANDLE. STARING AT
THEMSELVES.
As they have nothing better to do,
they decide to sit down on the carpet in front of the mirror and play cards.
Outside, the wind howls and rumbles over the chimney tops. But inside it is
deathly quiet. The candle casts flickering shadows up the walls. Aisha opens
her notebook to keep score and starts to shuffle the cards. While she is shuffling,
Kobir stares distractedly into the vast mirror. And the more he stares, the
more perplexed he starts to look.
KOBIR: That’s really, really weird.
AISHA: What is?
KOBIR: Take a look at our reflection.
Kobir and Aisha both stare into the
mirror.
KOBIR: Notice anything odd?
AISHA: Odd?
KOBIR: You can see a mirror
version of yourself can’t you? And a mirror version of me.
AISHA: Of course.
KOBIR: But there’s something very,
very peculiar about our mirror selves. They are reversed.
AISHA: Reversed?
Kobir
stands up in front of the mirror.
KOBIR: Yes. Suppose this wasn’t a mirror but a big sheet of glass. And
suppose I was actually over there, where the mirror-version of myself appears
to be standing. Then my right hand, the one with the ring on, would be where my
left hand actually appears in the mirror. And my left hand, the one with the
watch on, would be where my right hand appears.
6.
ILLUSTRATE: KOBIR STANDING NEXT TO SEATED AISHA BEFORE MIRROR. WATCH ON LEFT
HAND AND RING ON RIGHT. HAIR PARTED TO HIS LEFT. HE IS LOOKING IN THE BIG, FULL
LENGTH MIRROR.
Kobir walks
closer to the mirror, so that he’s staring right into the face of his
mirror-self.
Kobir: And look, my left eyebrow – the one I’m raising – appears where
my right eyebrow would be.
7. ILLUSTRATE:
KOBIR’S FACE AND ITS REFLECTION, WITH EYEBROW RAISED AS DESCRIBED. HAIR PARTED
TO HIS LEFT.
Isn’t that odd?
Kobir
is right about the reversal. When we look at Kobir in the mirror we see this:
8.
ILLUSTRATE: KOBIR STANDING LOOKING AT US, RING ON LEFT HAND AND WATCH ON RIGHT.
HAIR PARTED TO HIS RIGHT.
But
if Kobir was actually standing where his image appears to be standing, we would
see this:
9.
IDENTICAL IMAGE BUT WATCH ON LEFT HAND AND RING ON RIGHT. HAIR PARTED TO HIS
LEFT.
See?
Kobir’s left and right sides have been reversed.
Why do mirrors reverse left
to right, but not top to bottom?
Aisha
just wants to play cards. She’s irritated by Kobir’s question.
AISHA: Why is it odd?
KOBIR: Well, mirrors reverse us left
to right. So why don’t they also reverse us top to bottom.
AISHA: Top to bottom?
KOBIR: Yes, in the mirror, what’s
top and bottom is the same. Take a look: my head is still at the top and my
feet are still at the bottom. But my left and right sides are swapped round.
AISHA: True.
KOBIR: So my question is: Why
do mirrors reverse one way, but not the other? It’s very, very strange.
Kobir is
correct. In a mirror image, what’s at the top and what’s at the bottom remains
unchanged. But left and right are switched. That is why the word “ambulance” is
painted in reverse on the front of an ambulance.
10.
ILLUSTRATE: AMBULANCE WITH WORD “AMBULANCE” REVERSED ACROSS THE FRONT.
The word is
seen by other motorists in their rear view mirrors. Because mirrors reverse
left to right, and because the word “ambulance” is itself reversed left to
right, a rear view mirror has the effect of turning the word the correct way
round so it can easily be read.
11.
ILLUSTRATE: SAME AMBULANCE SEEN IN REAR VIEW MIRROR: NOW THE WORD IS RIGHT WAY
ROUND (DRIVER ON OTHER SIDE, OF COURSE)
An ancient and infernal puzzle
Why do mirrors reverse left and right, but
not top and bottom? Some of the world’s greatest minds – including that of the
Ancient Greek philosopher Plato – have struggled with and been defeated by this
infernal mystery.
12.ANCIENT
GREEK IN BEARD AND ROBE LOOKING IN MIRROR AND THINKING BUBBLE: “WHY DO
MIRRORS REVERSE LEFT/RIGHT BUT NOT TOP/BOTTOM?”
Can Kobir
and Aisha do any better? Can you?
Perhaps. But before we try to come
up with an explanation that might
work, let’s take a look at a few explanations that definitely don’t work.
“Doesn’t it depend on which we up we are?”
Aisha can
now see the puzzle.
AISHA: Hmm. I’m not sure why mirrors do what they do. Now you mention
it, I suppose that is rather
peculiar.
KOBIR: It’s weird!
Suddenly,
Aisha thinks she has the solution.
AISHA: Doesn’t it
all depend on which way up we are? If
we lay down, then it would be top and
bottom that are reversed, not left and right.
KOBIR: I don’t
think so. Let’s try.
Kobir moves the candle safely to
one side. Then they both lie on their sides and stare at themselves.
13.ILLUSTRATE; T AND A LOOKING INTO
MIRROR AS THEY LAY ON THEIR SIDES. KOBIR IS WIGGLING FINGERS OF RIGHT HAND (THE
ONE WITH RING).
KOBIR: You see? If
I was over there, my right hand – the one with the ring – would be where my
left hand appears. My left and right sides are still reversed.
AISHA: True. And
our tops and bottoms are still the right way round.
KOBIR: So the
mirror flips things around left to right but not top to bottom no matter what
way up we may happen to be in front
of it.
AISHA: Hmm. I
guess you’re right.
KOBIR: I am
right. The reversal has nothing to do with which way up we are. Actually it doesn’t have anything to do with which way up
the thing we are looking at is, or
which way up the mirror is, either.
Mirrors always reverse left to right
but not top to bottom.
You can confirm this for yourself.
Here’s the word “ambulance” reversed from left to right:
14.ILLUSTRATE:
“AMBULANCE” REVERSED L/R
Try putting this page up in front of a
mirror. Because the mirror also reverses left and right, the word gets switched
round the right way again, doesn’t it?
15.ILLUSTRATE:
MIRROR IN FRONT OF REVERSED WORD “AMBULANCE”, IN MIRROR THE WORD IS ROUND THE
RIGHT WAY.
Now let’s try turning you, the observer to see if that has any
effect on how the word is reversed. Hold the book upright, but turn your head
sideways like this.
16.ILLUSTRATE. KID BEFORE MIRROR:
HE IS HOLDING BOOK UPRIGHT WITH HEAD TILTED OVER SIDEWAYS.
See? It doesn’t matter which way up
you are. The mirror still flips the
letters back the right way. It still reverses them left to right to make them
readable again.
Now let’s
try rotating the book 90 degrees, like this.
17.ILLUSTRATE: SAME KID BEFORE
MIRROR, NOW WITH HEAD UPRIGHT BUT HOLDING THE BOOK SIDEWAYS ON.
Does that have any effect on how
the word appears? No. The word “ambulance” is still legible. So it doesn’t matter which way up the word is
either.
Maybe you’re thinking that the solution
has something to do with which way up the
mirror is. But it doesn’t. Turn the mirror upside down, and the image will
remain exactly the same.
So it
doesn’t matter which way up we are,
which we up the thing we are looking at
is, or even which way up the mirror
is, a mirror always reverses left to
right but not top to bottom.
18.ILLUSTRATE: KOBIR AND AISHA
LOOKING AT THEMSELVES IN MIRROR 9STILL IN THE SPOOKY ROOM): BIG THOUGH BUBBLE
WITH QUESTION MARK IN IT COMING OUT OF THEIR HEADS. THEY LOOK PERPLEXED.
The more you grapple with this
mystery, the deeper and more profound it seems to become. In fact, the more you
think about mirrors, the more they seem to take on a rather spooky, almost magical quality. Just why do mirrors do what they do?
I think I know the solution, or at least a
part of it.
Aisha
and Kobir heard a distant bang followed by echoing footsteps. After a minute or
so, Auntie Anaximander’s damp features appear round the doorway.
25.ILLUSTRATE:. ANAXIMANDER IN
RAINCOAT, LOOKING ROUND DOOR OF ROOM.
ANAXIMANDER: Ah.
There you are! Power should be back on soon. What have you been up to?
Auntie Anaximander takes off her
raincaot and shakes out the raindrops, spraying them across the room. A few
droplets land on the pages of Aisha’s open notebook.
26. ILLUSTRATE: RAINDROPS ACROSS
THIS PAGE.
Aisha and Kobir explain that they
had been discussing why mirrors reverse left and right, but not top and bottom.
Auntie Anaximander, who loves puzzles such as this, sits down on the carpet beside
them. She thinks hard for a moment or two.
ANAXIMANDER: You
know, in a sense, a mirror doesn’t reverse anything at all.
AISHA: In what
sense?
ANAXIMANDER: Well
suppose we hold a clock up in front of a mirror, like this.
Anaximander draws this picture on a page
of Aisha’s notebook.
27.ILLUSTRATE: CLOCK AND ITS
REFLECTION IN MIRROR
ANAXIMANDER: Now
let’s draw arrows linking each number on the clock face with the same number
reflected in the mirror.
Anaximander adds arrows
like this:
28.ILLUSTRATE: AS ABOVE BUT WITH
ARROW DRAWN FROM EACH NUMBER ON CLOCK FACE TO IT’S IMAGE IN THE MIRROR (I WILL
PROVIDE ROUGH)
ANAXIMANDER:
These arrows show that the way the mirror reflects is entirely symmetrical in every direction. The arrows don’t cross
over top to bottom. But neither do they
cross over left to right.
KOBIR: You know,
I think you’re right. In a sense, nothing
gets reversed!
Auntie Anaximander is right. The way in which the mirror
reflects what’s in front of it is perfectly symmetrical. It is not as if a
mirror reflects rays of light differently depending on whether they are coming
from your left and right sides rather than your top and bottom. The light is
reflected in the same way no matter where it happens to land on the mirror.
So the puzzle has absolutely nothing to
do with how light behaves.
KOBIR: But hang
on. We said, didn’t we, that the left and right are swapped round? And that top
and bottom are unchanged?
ANAXIMANDER: Yes.
That is how we would normally
describe what we see.
AISHA: Normally?
ANAXIMANDER: If
we were to replace this mirror with a sheet of glass, and you were to walk
round the glass and stand on the other side, looking towards me, in the exact
spot you currently see your mirror-self standing, then what would I, standing
over here, see?
KOBIR: You would
see my right hand to your left, and my left hand to your right.
29.ILLUSTRATE: WE ARE LOOKING AT
KOBIR BEHIND BIG SHEET OF GLASS. AUNTIE ANAXIMANDER AND AISHA ARE THIS SIDE,
WATCHING.
ANAXIMANDER:
True. And of course, when we look at your mirror-image, your left hand appears
where your right hand would be if you were standing there, and your right hand
appears where your left would be. That’s
why we said that the mirror-version of you is reversed left to right. Correct?
KOBIR: Yes.
ANAXIMANDER: But
notice that we have just taken something
for granted: the axis about which we rotate you when we imagine you over
there.
AISHA: The what?
ANAXIMANDER: When
we turn something round, we rotate it about an axis, don’t we? A spinning top,
for example, rotates around a vertical
axis, doesn’t it?
30. ILLUSTRATE: SPINNING TOP WITH
VERTICAL DOTTED LINE LABELLED “VERTICAL AXIS”
And a car wheel
rotates around a horizontal axis.
31.ILLUSTRATE CAR WHEEL WITH DOTTED
LINE THROUGH MIDDLE LABELLED HORIZONTAL AXIS.
Now, when we
imagine you over there in the position your mirror-self seems to be in, we
mentally put you there by rotating you about a vertical axis, like this.
Anaximander draws this
diagram in the notebook:
32.ILLUSTRATE: SEMI CIRCLE DOTTED
ARROW SHOWING HOW KOBIR IS ROTATED BY AN INDICATED VERTICAL AXIS. SEE ROUGH.
KOBIR: That’s right. That’s how I would get over there, by walking round the
mirror like that.
ANAXIMANDER: Of
course. But what if we were to get you over there not by rotating you around a vertical axis, but a round a horizontal axis? What would happen then?
KOBIR: I’m not
sure. Let’s draw it to see.
Kobir now draws this picture:
33.ILLUSTRATE: AS ABOVE, BUT KOBIR
IS NOW ROTATED AROUND HOROIZONTAL AXIS, LEAVING HIM STANDING ON HIS HEAD BEHIND
THE MIRROR (I WILL PROVIDE ROUGHS)
ANAXIMANDER: Now
see what happens! You are standing on your head. And, compared to your mirror
image, your left and right sides aren’t
switched round.
34.SPLIT ILLUSTRATION: CAPTION 1
“MIRROR IMAGE” RING ON KOBIR’S LEFT, WATCH ON HIS RIGHT, PARTING TO HIS RIGHT.
CAPTION 2: “KOBIR STANDING ON HIS HEAD” KOBIR UPSIDE, BUT NOT REVERSED LEFT TO
RIGHT.
Your right hand
stays over to the right, just as it appears in the mirror. But top and bottom have been switched round!
AISHA: Wow!
You’re right.
ANAXIMANDER: So
you see, the reason we say mirrors reverse left and right but not top and
bottom is due to the fact that we take for granted a particular axis of rotation. But we could just as
easily choose a horizontal axis. Then it would be true to say that a mirror
reverses top to bottom but not left to right.
I think Auntie Anaximander is
correct. She really has hit on the explanation of why mirrors reverse left to
right but not top-to bottom. Yes, it’s true to say mirrors reverse left to
right, but only if we choose a vertical axis of rotation. Choose a horizontal
axis and they then reverse top to bottom.
Kobir furrows his brow.
KOBIR: But why do we take the vertical axis for
granted?
ANAXIMANDER:
Because people aren’t in the habit of flying through the air like pigeons and
settling on their heads. When people normally
rotate, it’s almost always about a vertical axis. So we just took for granted a vertical axis of
rotation in this case too.
Suddenly the electricity comes on. The
chandeliers above their heads flood the hall with light. Auntie Anaximander
jumps up and blows out the candle. She suggests they all go downstairs for some
hot chocolate. So they do.
35: ILLUSTRATE: AUNTIE, KOBIR AND AISHA
LEAVING THE NOW VERY BRIGHTLY-LIT HALL.
The door puzzle
I believe Auntie Anaximander is right:
this puzzle about why mirrors do what they do is created by our not noticing
what we have taken for granted. To
solve the puzzle, we need to take a step back and start questioning what we
taken for granted.
Actually, I have noticed a similar puzzle
about doors. Walk though a door that opens on your left and turn round to come
back through it, and the door now opens on your right. But pass through a door
that opens at the top (like a cat-flap) and turn to come back through it and
the door still opens at the top.
36.ILLUSTRATE: DOOR OPENS AT TOP
LIKE A CAT FLAP.
Why does passing through a door
reverse the way it opens from left to right, but not from top to bottom? What
explains the difference?
The
solution is much the same as Auntie Anaximander’s solution to the mirror
puzzle. When you pass through a left-opening door and turn around to come back
through it, you would normally rotate
about a vertical axis, like this:
37.ILLUSTRATE: AS FOR PREVIOUS
ILLUSTRATION BUT ONE: EXCEPT WITH AXIS OF ROTATION ADDED: SEE ROUGH.
But what if you were to rotate
about a horizontal axis, like this?
38.ILLUSTRATE: AS ABOVE, BUT PERSON
FLIPPED HORIZONTALLY, LEAVING THEM UPSIDE DOWN.
Then the door
that opened on the left would still open the left on the way back though it,
but a door that opened at the bottom would now open at the top. We say that
left and right are reversed but not top and bottom only because we take for
granted a particular axis of rotation.
In space,
where we are weightless, the axis of rotation about which we choose to rotate
when turning to come back through a door is less likely to be the vertical
axis. You could just as easily spin about a horizontal axis instead.
39.ILLUSTRATE: SPACE MAN LEAVING
SHIP THROUGH DOOR, THEN ENTERING THROUGH DOOR UPSIDE DOWN.
So, after
years in space, it might start to seem as natural to you to say that a door
that opens at the top opens at the bottom when you come back through it as it
does to say that a door that opens on the left opens on the right when you
return though it.
For
creatures that live in a weightless environment, where it’s as easy to rotate
about one axis as it is to rotate about the other, perhaps neither the mirror
puzzle nor the door puzzle would even be
puzzles!
The “ambulance” test
Perhaps you are still not convinced
by Auntie Anaximander’s solution. If so, here’s another little experiment you
can perform. We said that in a mirror the word “ambulance” is reversed left to
right and not top to bottom. So now write out, on a separate piece of paper
next to the word “ambulance”, the left-right reversed version of the word. It
should look like this:
40. ILLUSTRATE: SEE ROUGH.
You have kept what’s at the top at
the top and what’s at the bottom at the bottom, but the right hand end of the
word has become its left hand end.
But now
suppose that mirrors reversed, not left to right, but top to bottom. What would
that look like? What would we see then?
Try to draw
it. This time, write on a separate piece of paper underneath the word
“ambulance” how the word would look if what’s at the top becomes what’s at the
bottom, but the left and right ends remain unchanged. It should look like this:
41.ILLUSTRATE: SEE ROUGH
The word is flipped top to bottom
but not left right.
But now
look very closely at the two reversed
words.
42.ILLUSTRATE: THE TWO REVERSED
WORDS (SEE ROUGH), LABELLED “REVERSED TOP TO BOTTOM” AND “REVERSED LEFT TO
RIGHT”.
Notice anything peculiar?
The two reversed
versions of the word that you have written are actually exactly the same! Turn the second one round to check, like this:
43.ILLUSTRATE: SEE ROUGH (LABEL
LEFT RIGHT REVERSED AND TOP BOTTOM REVERSED)
The left-right reversed version of
the word just is the top-bottom
reversed version.
So why did
we call one left-right reversed and the other the top-bottom reversed? Because
we just assumed a different axis of rotation in each case. Whether we describe
an image as left/right reversed or top/bottom reversed all depends on what axis
of rotation we choose.
When science can’t help us
So I think we have solved the
puzzle (or at least we solved this version
of it).
Here’s an
interesting fact about the mirror puzzle: it can’t be solved by doing science. It can only be solved by thinking
philosophically. We didn’t have to any scientific research into how light and
mirrors behave. We didn’t have to investigate how our brains work.
44.ILLUSTRATE SCIENTISTS IN LAB
EXPERIMENTING WITH MIRRORS, LAZERS, BRAINS ETC.
Even if we
had done that sort of scientific research, it wouldn’t have solved the puzzle.
In order to solve the puzzle, we need to stop
doing science and start doing philosophy. It’s a puzzle that can be solved just by thinking.
45. ILLUSTRATE: ME SLUMPED IN
CHAIR, LABELLED “PHILOSOPHER AT WORK”
People
sometimes assume that all our questions can be answered by science. They would
just assume that the mirror puzzle must have a scientific solution. But it
turns out that the mirror puzzle is a puzzle that science can’t solve.
Sometimes,
only philosophy will do.
Comments
This isn't really a comment about this blogpost, but I didn't know where else to go with this.
I'm just reading your book "believing bullshit", which I genuinely enjoy. But here's the thing: You should fire your editor (or at least give him/her a good old spanking ^^).
I only speak English as a second language, but I still found about 4 mistakes in the first 170 pages. I do know that the occasional typo will slip through, but there should limits to this. Especially when you reference a book that has the word "intellectual" in its title, you should take care not to misspell, simply to avoid the irony. (I'm talking about "Intellectual Impostures" of course)
Other than that: Nice read, I'll finish it this evening.
Best wishes from Germany.
This also explains the [missed] point in the discussion that when they were on their sides their "left and right" directions, in an absolute sense, were what they were calling "up and down" before.
I am looking for a philosophy book for my 7 year old daughter and would be really grateful if you would recommend something. I haven't been able to find info about what age your childrens books are suitable for. My daughter is a great reader, so being able to read the text isn't an issue. Emotionally though she is still 7 :-)
Thanks and kind regards
Sara
You couldn't possibly be more wrong here. A few minutes on Wikipedia (look up mirror or mirror image) will inform you that the properties of a mirror are completely explained by the simple principle that "angle of incidence equals angle of reflection." Light rays bouncing off a mirror MUST obey this principle. No children are required to stand on their heads - no storms or Aunties are needed either. The concept is taught in any basic physics class and there's nothing more to it. Reflection of light, the fundamental physical principle at work here, has been understood for centuries. All of your statements about "axis of rotation" only obscure the correct understanding, since nothing is being rotated (the image in the mirror only seems to be rotated because of the approximate symmetry of the human body - a one-armed man would notice immediately that his mirror image is not a rotated version of himself). What is happening is reflection, not rotation. Your "fact" that science cannot explain how mirrors work is false and utterly ridiculous. I suggest before offering your bogus views about science you either learn some, or at least talk to someone who already has. Consider yourself lucky that this embarrassing chapter did not make it into the book.
A Physics Ph.D. from California
Ditto neil.
Yes mirrors reverse front back and yes the way light is reflected is clear but neither of these observations, as they stand, explain the asymmetry.
To suppose they do is to miss the point.
Nb richard dawkins rightly endorsed the explanation given in this chpt in our recent video discussion btw.
For your print example, suppose the paper the writing is on is transparent (an overhead projector transparency if you're as old as I am). Then the writing as seen in the mirror is exactly what you see through the paper. The mirror didn't reverse "AMBULANCE", you did when you held the paper facing away from you.
I suppose the misattribution of agency can be a philosophical problem, though [evolutionary] psychologists might disagree.
I reject the first question because mirrors don't do that (per my explanation). The correct physical model provides simpler explanations of what you see. For the second question, the reason people jump to the wrong physical model may be related to the [approximate] bilateral symmetry of humans and possibly to "hard wiring" of facial recognition (I am not a cognitive psychologist) or related to unexamined philosophical assumptions (I am not a philosopher [obviously]).
I guess I'm trying to understand why you believe this optical illusion has more philosophical significance than the "line length" or "rotating snake" illusions. The latter even affecting cats.
In order to see an object, a ray of light has to go from the object to your eye. This is a concept that is missing here. When that ray hits a mirror it bounces off according to "angle of incidence equals angle of reflection." Otherwise it travels in a straight line. That explains everything about plane mirrors. The branch of physics that deals with this is known as "optics."
An object's mirror image cannot be superimposed on the object by any rotation or movement. You can't describe mirror image symmetry as a combination of translations and rotations. Example: your left hand and your right hand.
I'm a great admirer of Richard Dawkins but this isn't his field. Ask an optical physicist.
I'm one but there are lots of others.
Paul
Knowing what happens otically does not explain the leftright topbottom asymmetry, which is explained as in the chpt.
I have to agree with Paul that the observed effects of a mirror are more simply and more accurately explained by physics and specifically optics. The way the question is posed and answered in your example relates more to human (mis?)perception than the physical nature of reflection.
You also present this as a philosophical problem and solution. However, the process you illustrate to formulate the solution appears scientific as it is based on observation, empirical testing, hypothesis generation and repetition of these elements. You propose this as an armchair exercise or a thought experiment, but it is dependent on actual observation and experimentation.
Picking up on the questions you addressed to Paul. In my opinion, philosophy is most effective in codifying and interpreting human thought, constrained as it is by language. The question and answer you propose don't address the physical nature of the universe, but rather our perception of it. Thus philosophy, in the questions it raises and the solutions it proposes, is essentially a study of human thought processes and human nature (our need to understand/define our place in the universe).
I would propose that science (though the two are inextricably linked) has and will have more success in determining the physical nature of the universe.