Apropos my earlier rant on people who think that paradoxes are meaningless, I figured I ought to take a stab at giving some meaning to paradox. To this end I reformulated a paradox in my terms. I suppose I should called it the Mirror Paradox, though ‘Looking-Glass Paradox’ seems more lyrical and has an historical nod. My apologies to whoever actually came up with this first, though I am sure I haven’t heard it before…
In my room I have a full length mirror. If I look at the man in the mirror and point to him saying, “There I am!” then where am I? If I am the man on the other side of the mirror, then I am not sitting on this side of the mirror. However, the man on the other side of the mirror has just pointed at me and said that he is not on his side of the mirror, but on mine. So I am not on my side, nor is he on his side. But then neither of us are on our side or on our mirror self side.
Now with semantic paradoxes and the like, we don’t have an agreed upon framework for analyzing what is going on in a paradox. Many times it is a paradox that signals that some such theory is unsatisfactory. However, this paradox deals with locations of people, namely me and mirror me, and we do have a general consensus on judging objects’ locations: in physics we determine some object’s location with reference to some previously agreed upon location.
Let us ignore for the moment that mirrors do not actually open up into other dimensions that you could enter if only your reflection didn’t get in the way. All that is important is that we have an exact double of yourself that at the instant you declare that you are where he or she is, that person does the exact same thing.
Declaring your location relative to your reflection is no different than declaring your location relative to anything else. Your reflection simultaneously declaring its location relative you is likewise unproblematic on its own. However, since the two non-identical perspectives are associated with only one person, we have a disconnect between perspectives and the person who holds the perspectives.
This problem of perspective is most telling. In Russell’s Paradox, there is no problem, no obvious contradiction that is, until the question “Is this set self-membered?” has been asked and answered twice. The first time through is arbitrary, let us assume no: Russell’s set is not part of itself. Now we ask, “If it is not self-membered, then is this set not self-membered?” Now we answer yes and have arrived at a contradiction. There is no problem yet, we merely have to revise our assumption: let us assume that Russell’s set is included in itself. Of course, then we ask, “If it is self-memberd, then is this set not self-membered?” and our answer is no: contradiction. At this point the paradox exists, but not before. It required us to look at the one set from two perspectives, one in which it is self-membered and one in which it is not.
The comparison of assumptions and perspectives that is drawn here is a good one. Our perspective, in a different sense, is our background assumptions. When we have contradictory perspectives on a subject we have incompatible background assumptions. The Mirror Paradox pulls our background assumption of location out of the background. We all assume that one perspective is associated with one location, but when you declare that you are someplace else and your reflection does the same, then you end up with two perspectives.
We can’t tell before hand whether we can have more than one perspective or a set that is defined by non-self-membership. Therefore, since the problem occurs with the selection of assumptions or perspective, the meaning of paradoxes, semantic or otherwise, is that your fundamental background assumptions are problematic. Sure, each paradox will only pertain to that particular system that it exists in, but for that system it will signal the most important and deep underlying problems.
A side note: I thought of this while in bed last night and didn’t look at a mirror until this afternoon, even though I do have that full length mirror. Then I actually did point and say “There I am!” It was a bit of a strange experience because for some fraction of time I felt like me and my mirror self were in some sort of vortex with the rest of the world frozen outside. Almost needless to say I was a bit surprised if not shocked- I wasn’t expecting a reaction. When philosophy grabs you, even for an instant, it is spooky. I suggest you try this and see if you have the same reaction if only because I don’t think there are other paradoxes to actually participate in, save becoming a very methodological barber. How often do you get to participate in an experiment that isn’t prefaced by ‘thought’? Between the small mirror in my bathroom and my full length mirror, the full length elicited a better reaction, so use a full length one if you can.