School of Computer Science
Department of Philosophy
University of Birmingham
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A partial index of discussion notes is in
An example is the answer sketched by Douglas Hofstadter in
Hofstadter wrote, in the "Reflections" section following Thomas Nagel's "What is it like to be a bat?" (Chapter 24):
But wait-you can get your heart to stay on the proper side if, instead, you flip yourself head over heels, as if swinging over a waist-high horizontal bar in front of you. Now your heart is on the same side as the mirror-person's heart-but your feet and head are in the wrong places, and your stomach, although at approximately the right height, is upsidedown. So it seems a mirror can be perceived as reversing up and down, provided you're willing to map yourself onto a creature whose feet are above its head. It all depends on the ways that you are willing to slip yourself onto another entity. You have a choice of twirling around a horizontal or a vertical bar, and getting the heart right but not the head and feet, or getting the head and feet right but not the heart. It's simply that, because of the external vertical symmetry of the human body, the vertical self-twirling yields a more plausible-seeming you-to-image mapping. But mirrors intrinsically don't care which way you interpret what they do. And in fact, all they really reverse is back and front!
The well known psychologist and vision scientist Richard Gregory (who discovered and analysed many fascinating visual illusions, and founded the Bristol Exploratory, among other things), wrote a short review of The Mind's I in New Scientist December 1981, available online here.
In column 4 he criticises Hofstadter for not noticing that the left/right rotation of text in a mirror occurs because the text has been rotated about its vertical axis to face the mirror "giving left right reversal" whereas "twiddling" text around its horizontal axis gives up-down and not right-left reversal, adding "This has nothing whatever to do with mental mapping as Hofstadter claims it has on page 404". It is not clear whether he interpreted Hofstadter's analysis correctly.
The same happens if you attach a mirror to a ceiling, so that it is horizontal and facing down. There is a restaurant in Vienna (or was when A.S. last visited over 20 years ago) where the walls and some of the ceilings are covered in mirrors, including the ceiling of the men's toilet above the urinals, producing a somewhat disconcerting anti-gravity effect for anyone looking up!
So, to get the vertical flip, instead of rotating yourself, or imagining rotating yourself, just rotate the mirror through 90 degrees until it is horizontal, facing down from above, or facing up from below the viewer.
Note that if a mirror is not parallel to or perpendicular to a person's long axis, then the reflection of the person will not be parallel to the person.
However, in all cases Hofstadter's point about asymmetric body parts not being congruent with their mirror images is correct. E.g. in the reflection, your left hand is congruent with your right hand outside the reflection (and vice versa). That is independent of where the mirror is and where you are. This is related to the suggestion explained in the discussion headed "What a mirror really does" below.
Notice that a vertical mirror will reverse the word "Ambulance" from left to right even if the mirror is not facing the word, as sketched below (with apologies for poor perspective drawing -- perhaps fixed later):
However a horizontal mirror will not reverse the text left to right, but top to bottom, as depicted here:
Apologies for inaccurate geometrical relationships --
the reflections in the vertical mirror should be made to
tilt away to right, or the mirror should be made to appear
perpendicular to the wall, with the view-point changed.
And just to prove that it works with a real vertical mirror to one side:
Or with two mirrors: one on the floor, and one on the side:
Each of the four pictures could be an original picture of a 3-D configuration. Without the labels provided you could not tell which of the four is the original, since the objects shown could be stuck to the surfaces, and therefore information about gravity cannot be used to distinguish the pictures that have been reflected up/down.
NOTE: all the reflections are reversible. E.g. the top right scene if reflected horizontally becomes the top left scene. However, two scenes depicted in the same row or the same column are, like a left and right hand, incapable of being superimposed (even after rotations). They differ by one reflection. The scenes that differ by two reflections, e.g. top left and bottom right, or top right and bottom left, can be superimposed, like two left hands, or two right hands.
The video is misleading in one important respect. What a mirror does (as opposed to how we describe what it does) is entirely independent of how the object reflected in the mirror has been presented. The actual transformation produced by a mirror depends only on the orientation of the mirror, though how we think about it may depend on how we describe ourselves, e.g. as having a top and a bottom, a left side and a right side, a front and a back.
One of the commentators referred to this image, which almost makes this point:
If the mirror is tilted at 45 degrees to a person's vertical axis it will reflect a vertical person as a horizontal one, neither parallel to nor perpendicular to the mirror.
Physics Girl's demonstration is excellent at debunking one straightforward source of confusion. Mirrors do not reverse side-to-side any more than they reverse up-and-down. But it doesn't go any way towards addressing a deeper puzzle about mirrors. This puzzle is independent of how the thing to be reflected in the mirror is presented (right way up, upside down, etc.) It concerns instead the actual transformation produced by a mirror. And solving it is not just a matter of debunking it; the solution turns out to require drawing both on facts about the nature of space and on facts about the environment that human beings evolved in.
Thus interpreted, the mirror puzzle is not puzzling. Indeed, a new puzzle presents itself: why did people find the mirror puzzle puzzling? The answer to the new puzzle is not far to find. There are other possible interpretations of the mirror puzzle which lack any such straightforward answer. Here is one of them:
So why is it that mirrors reverse left-handedness and right-handedness but not up and down? To answer that question, we need to think a bit more about what transformation a mirror really performs.
Perhaps the most general way to think about what the mirror does is this:
The puzzle is resolved; mirrors flip left-handedness and right-handedness because they turn things inside-out. But this resolution immediately invites a new puzzle:
Three-dimensional space (or at least regions of it that are accessible for us) constitute what mathematicians call an orientable surface. On an orientable surface, a consistent notion of clockwise rotation is definable in a continuous way over all points. This property of a three-dimensional space is provably equivalent to it being impossible to move a left glove around along any path (without turning it inside out) to bring it into perfect congruence with a right-handed glove. So, in our three-dimensional space the chirality of an object is preserved no matter how much it is moved around. In fact, chirality is preserved no matter how much an object is moved around or stretched or squeezed in any direction.
Some objects -- those which have at least one axis of rotational symmetry -- can be brought into congruence with their chiral opposites. A sphere of uniform colour, for example, looks just the same in a mirror. And a cylinder could also be brought into congruence with its mirror image. These objects are achiral. Most objects, though, do not have any axes of rotational symmetry. Even cylindrical or spherical objects, if their surface is marked with a design with no non-trivial rotational asymmetries (like an 'L') can be distinguished from their chiral opposites. They are chiral.
Now that we understand these notions, we can apply them to restate clearly and answer our remaining puzzle:
The answer to this question is not about mirrors, or about the mathematics of chirality, or about anything so abstract. It's about the environment that humans evolved in, and the physiology and form of life that we adopted in response to this environment.
Hands are the most familiar pairs of chiral opposites; but of course feet, ears, and even eyes also qualify.
This gives us an easy way to distinguish between front and back, and top and bottom, which we lack with respect to side-to-side. Your front has the eyes on; the top has the hair on; the bottom has the feet on. If as an evolving human you want to tell a friend from which side of them the tiger is approaching, you can say 'face-side' or 'hair-side', or the equivalent in your developing language. But you can't say 'hand-side' -- your friend has two hands, and will not know to which one of them you mean to refer. So arbitrary labels -- left and right -- will have been needed early on for practical purposes.
The side-to-side axis is where an arbitrary label is useful. If I want to tell someone in a 3d maze to go forwards, I just say 'forwards'. To make them go down a ladder, I just say 'down'. But if I want them to turn one way rather than another, I need to have some (arbitrary) labelling system. So here we need terms like 'left' and 'right'.
If humans had not had the (approximate) symmetry we do have, and instead always had a big blue mark on the hand that we now call the left hand, we could, instead of 'left side' and 'right side' have talked about the 'blue side' and the 'non-blue side', or something like that.
Supplementing the directional axes up/down, side/side, forward/backwards, an agreed distinction between left and right allows us to fully characterize the shapes of things including their chirality. Since one arbitrary choice is enough to achieve full expressive adequacy with respect to orientations, there is no need for any other contrastive pairs to be associated with any other axis. This resolves the last part of our puzzle.
How contingent is our understanding of the difference between left and right, and its relation to chirality? The following parable is an attempt to answer this question.
For us, orientation in the horizontal plane is not easily detectable, unless we have a compass or can see the sun as a reference point, or some well known landmark that looks different from different directions. But we can easily detect which direction is up and which is down, using gravity (at least at normal locations on earth).
Our imagined creatures, lacking this ability, nevertheless have a clear sense of a distinction between the forwards and backwards directions (we assume for simplicity that their direction of perception coincides with their habitual direction of motion), but there would for them be no privileged horizontal plane. Accordingly, where we face a two-dimensional problem of orienting ourselves in the horizontal plane that we ordinarily occupy, they would face a three-dimensional problem of orienting themselves in space.
Our creatures would still need to describe the difference between a left glove and a right glove. They would do it not via an arbitrary choice of orientation in the side-by-side axis, since they have no such axis but via a choice of direction of roll: counter-clockwise as opposed to clockwise rotation in their field of view.
A glove is a right glove if it fits the right hand, namely the hand for which the perceived direction of rotation from thumb through fingers is clockwise when held with palm facing away from the viewpoint.
When you were small, and learning how to tell the time on an analogue clock face, you learned the distinction between these two directions. (Perhaps you learned it earlier still.) In doing so, you probably remembered it via the hand moving from left to right as it goes over the top, then from right to left as it passes below. But the creatures in our less restricted environment have no privileged sense of 'above' or 'below' that distinguishes either term from 'to the side'; so they would have to remember it without reducing it to a combination of a stipulated left-right distinction and a known up-down distinction.
We leave you with the following thought (with apologies to Wittgenstein):
Whether and how two physicists located far apart in the universe can agree on the difference between right and left, or between clockwise and counter-clockwise rotation, without being able to see each other, or to see and identify a common structure, such as a galaxy or a constellation, is a well known problem. E.g. see:
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