The other day, as I was wandering down an old tenement block on New York's Lower East Side, I chanced on an odd little shop. It sold only one item: mirrors which do not reverse left and right-or True Mirrors, as the store calls them. There was one in the shop window. Looking at my reflection in it, I was appalled by how crooked my facial features seemed, how lopsided my smile was, how ridiculous my hair looked parted on the wrong side of my head. Then I realised that the image I was confronting was the real me, the one the world sees. The image of myself I am used to, the one I see when I look into an ordinary mirror, is actually that of an incongruous counterpart whose left and right are the reverse of mine.
There is nothing very strange about the fact that ordinary mirrors reverse left and right, is there? "Left" and "right" are labels for the two horizontal directions parallel to the mirror. The two vertical directions parallel to the mirror are "up" and "down." But the optics and geometry of reflection are precisely the same for all dimensions parallel to the mirror. So why does a mirror treat the horizontal and vertical axes differently? Why does it reverse left and right but not up and down?
This question might seem foolish at first. "When I wave my right hand, my mirror counterpart waves his left hand," you say. "When I wiggle my head, I should scarcely expect my counterpart to wiggle his feet." True enough, but you might plausibly expect your counterpart to appear upside down, with his feet directly opposite your head-just as his left hand is directly opposite your right hand.
Foolish or not, the issue has been vexing philosophers for at least half a century. As far as I can tell, it first arose in the early 1950s, as a sort of sidebar to discussions of Kant's theory of spatial relations. In his 1964 book The Ambidextrous Universe, the science populariser Martin Gardner stirred things up by arguing that the puzzle has a false premise. A mirror does not really reverse left and right at all, he claimed; instead, it reverses front and back along an axis perpendicular to the mirror. In his view, we merely "find it convenient" to call our image left/right reversed because we happen to be bilaterally symmetric. In 1970, the philosopher Jonathan Bennett published an article endorsing Gardner's supposed resolution of what by now had become the "mildly famous mirror problem."
His optimism was premature. In 1974, MIT philosopher Ned Block wrote a long, diagram-filled piece for the Journal of Philosophy arguing that "Why does a mirror reverse right/left but not up/down?" has at least four different interpretations. Block claimed that the four interpretations had been clumsily conflated by Gardner and Bennett; he also insisted that in two of the four, a mirror really does reverse left and right. Three years later, in an equally lengthy article in the Philosophical Review, an English philosopher named Don Locke declared that Block was only "half right." Mirrors actually reverse left and right in every relevant sense.
Reading these papers, and others published since, I begin to think that the mirror problem defies philosophical reflection. People can't agree on the most basic facts. For example: stand sideways to a mirror, shoulder to shoulder with your image. Your left/right axis is now perpendicular to the mirror's surface. Gardner and Bennett say that in this case and this case alone, a mirror really does reverse left and right. Block and Locke say that in this case and this case alone, left and right are the same direction for you and your mirror image. (Having just dashed to the bedroom mirror to try this, I think I'm in the Block-Locke camp. My right arm and my mirror counterpart's right arm both pointed east; on the other hand, he was wearing his watch on his right wrist, whereas I wear mine on my left.)
The clue to the mirror puzzle would seem to lie in some subtle disanalogy between left/right and up/down. Both of these pairs of directions are relative to the orientation of the body (unlike, say, east/west and skyward/earthward). But as any child will attest, left/right is much harder to master than up/down. The human body displays no gross asymmetries between its two sides. (There is, of course, the heart, but that is hidden.) So "left" and "right" have to be defined in terms of "front" and "head." Your left hand is the one which is to the west when you stand on the ground and face north. This would remain true even if a surgeon cut off your two hands and sewed them on to the opposite arms.
Left/right is thus logically parasitic on front/back, whereas up/down is not. And a mirror, everyone agrees, reverses front and back. That must be why it also reverses left and right-if indeed it does, which to this day remains unclear.
Fatigued by the debate? Then visit the little shop I discovered in New York and get yourself a True Mirror. But don't try shaving in front of the thing-your face will be a bloody mess.
