Tortoise: Hello, Achilles! I haven’t seen you since one of our last encounters discussing Godel, Escher, and Bach.
Achilles: That break has actually been quite enjoyable.
Tortoise: Really! Do I take that to mean that you do not enjoy my company?
Achilles: No, not at all; you are, in fact, among my closest friends. I just mean that all that incessant talking about recursion, logic, and the mathematics of paradox—well, frankly, it made my head hurt after awhile. I’ve been quite happy just relaxing and not thinking of anything even vaguely esoteric, thank you very much!
Tortoise: Well, that is understandable. I’m glad to hear that you do not tire of my presence, per se, because I have something on my mind which I’ve been longing to discuss.
Achilles: As long as it does not involve mysterious meta-mathematical ideas, I’m certainly open to the idea of a chat!
Tortoise: I don’t think it’s very meta-mathematical—a curious word, by the way; is the word “meta-mathematical” itself, meta-mathematical?
Achilles: That is exactly the sort of thing, Tortoise, that I’d prefer not to discuss. What was it that was really on your mind?
Tortoise: Well, I’m having a problem with my new year’s resolutions.
Achilles: That’s a relief! That seems like a very mundane thing to talk about, not meta-mathematical (whatever that means) in any way. What is the problem that you seem to be having?
Tortoise: It involves what seems to be a conflict between my new year’s resolution from last year, and the resolution I intend to make this year—perhaps “conflict” is not the right word, but to tell the truth, I can’t exactly put my finger on it.
Achilles: A conflict? What does your last year’s resolution have to do with it? In my experience, most people seem to make the same resolutions year after year, with the idea that this time they will keep the resolution.
Tortoise: I’ll explain. Last January, I made a new year’s resolution—we’ll call it resolution 2005. In fact, I still have the piece of paper on which I wrote it down.
Achilles: You certainly are organized! Very well, let’s hear what it was.
Tortoise: This was my resolution: I, Tortoise, resolve to break the resolutions I will make next year.
Achilles: That is a very curious thing to resolve.
Tortoise: What a thing to say! It is my life, after all, and, I think I can resolve any thing I like, no matter how strange it may seem to someone else.
Achilles: I suppose that is your prerogative. Still, I wonder if there was any reason for it?
Tortoise: Truth be told, I had observed that most people, myself included, tend not to keep their new year’s resolutions. So I decided to make it official—I would resolve not to keep whatever resolutions I would make the following year. Since it seemed almost certain that this would happen, I felt assured that this, resolution 2005, was a new year’s resolution that I could finally keep.
Achilles: That’s some interesting reasoning, to be sure. So, you have vowed to break the resolutions you will make this year—is this your conflict?
Tortoise: Nothing so simple as that. There is a resolution which I would like to make for this year—we’ll call it resolution 2006. I have written it down on a separate piece of paper; would you like to hear it?
Achilles: There seems little point, since you will need to break it in order to keep resolution 2005, but let’s hear it.
Tortoise: Alright, here is resolution 2006: I, Tortoise, resolve to keep the resolution I made last year.
Achilles: That seems simple enough. What is the problem with that?
Tortoise: That’s just it; I can’t quite determine it. Something about these two resolutions is bothering me; it seems as though I can neither keep nor break either of them!
Achilles: There is something strange about the pair. I’m quite sure that normal people do not make new year’s resolutions like these at all.
Tortoise: There you go again, impugning my resolutions. I don’t see what’s so wrong with them.
Achilles: All right, let’s solve the matter. I suppose instead of using the phrases “next year’s resolution” and “last year’s resolution” you could substitute resolution 2005 and resolution 2006, since that is really what we’re talking about.
Tortoise: What a wonderful idea! Very well then… I just happen to have a blank index card here. (writes down on the index card with a pencil.) There! My amended resolutions are:
Resolution 2005: Break Resolution 2006.
Resolution 2006: Keep Resolution 2005.
Achilles: I’m not sure that this helped at all. I’m more confused than ever.
Tortoise: It is quite perplexing.
Achilles: Something seems very familiar about this problem. Nevertheless, here’s what I see: in order to keep resolution 2005, you must break resolution 2006 … So, you must simply choose not to keep resolution 2006.
Tortoise: If only it were so simple! If I were actually able to break resolution 2006, then I would be keeping resolution 2005… which is exactly what resolution 2006 purposes. Therefore I would have actually have kept resolution 2006 (since its purpose has been accomplished), rather than break it as I had intended.
Achilles: …How do you suppose? I don’t think I follow you.
Tortoise: Fortunately, I have another index card. I’ll make a little chart on it, perhaps that will help. On this card I will write Resolution 2005 and Resolution 2006, each on their own line, and I will check off whether I have kept them or not with a “Y” or and “N”.
Achilles: You certainly are determined to make sense of these odd resolutions! That sounds simple enough, though.
Tortoise: Indeed I am; the ordeal is quite frustrating. Alright, here is what I have written:
Resolution 2005
Resolution 2006
Now, it seems to me that I cannot write a “Y” next to Resolution 2005 until I have broken Resolution 2006—is that correct?
Achilles: Absolutely; the resolution was to break resolution 2006, so you cannot check it off until you have broken it.
Tortoise: Very well. I will break Resolution 2006, and write an “N” next to it. Resolution 2006, you remember, was to keep Resolution 2005.
Achilles: Yes, yes, I remember.
Tortoise: So, to break Resolution 2006, I will have to not keep Resolution 2005 … so I will write an “N” next to Resolution 2005; and then, since I have .
Resolution 2005 N
Resolution 2006 N
There! What do you think of that?
Achilles: Well, I see a problem. Since the point of Resolution 2005 was, in fact, to break Resolution 2006—which you have done—then you have succeeded in keeping it, and should put a “Y” next to it.
Tortoise: Oh, of course. How silly of me. Here….
Resolution 2005 Y
Resolution 2006 N
Achilles: Hmm. Something isn’t quite right, now.
Tortoise: I feel the same way! Let’s see… well, my Resolution 2006 was to keep Resolution 2005—which I have done, so I should put a “Y” next to Resolution 2006 as well.
Achilles: …yes, I suppose…
Resolution 2005 Y
Resolution 2006 Y
Tortoise: That’s odd. The values beside the two resolutions have flipped since I first wrote them down.
Achilles: Yes, it is odd. And it is also incorrect; if you have kept Resolution 2006, then you have broken Resolution 2005, so you ought to put a “N” next to it.
Tortoise: (Examining the paper) Well, you’re absolutely right… here, let me make that change.
(Many hours later, after much erasing and rewriting of “Y”s and “N”s on the index card)
Tortoise: Oh, dear me. I find we have another contradiction. I will need to change this to a “N”…
Achilles: Stop, stop! I think I’ve begun to see a pattern here!
Tortoise: You’ve begun to see a pattern?
Achilles: Yes! These resolutions don’t seem to be able to be either true or false in relation to each other.
Tortoise: Neither of them are able to be true or false? I’m not sure how that’s possible, but continue.
Achilles: I propose that the universe will not allow you to make both of these resolutions. You can make one or the other of them, but not both.
Tortoise: Not allowed? That is sort of a totalitarian way to look at the concept of new year’s resolutions. How can a certain resolution not be allowed?
Achilles: It seems to me that either of them is alright, by itself, but if you make them both, in subsequent years, as you have done… well, you simply can’t.
Tortoise: Why not?
Achilles: (Pointing at the index card) Well, just look at what happens when you attempt to make them! You run into an endless loop of truths and falsehoods!
Tortoise: Well, that’s true, but is that sufficient to say that the two resolutions are not allowed to coexist? That seems a little extreme.
Achilles: It seems to me that there would have to be something wrong with our idea of logic if a conundrum like this is allowed to exist.
Tortoise: Hmm. It sounds like you are suggesting that the question Given these two New Year’s resolutions, can they be kept? is formally undecidable.
Achilles: That is not how I might have said it… but, let’s see. “Formally undecidable.” Yes, I think that is what I’m saying.
Tortoise: And you are saying that formally undecidable propositions ought not to be allowed?
Achilles: Absolutely not! It seems quite illogical to do so.
Tortoise: I, on the other hand, believe that any system of reasoning must necessarily contain the possibility of “formally undecidable propositions.”
Achilles: Well, that is where we must agree to disagree, I suppose. I urge you to abandon your current Resolution 2006 and replace it with something like “Do more exercise” or “Eat healthier”, both of which you could easily abandon without sparking an endless loop of conflict between your new years resolutions.
Tortoise: I’m not really certain I can abandon this problem simply by waving my hand and declaring that it isn’t allowed to exist, but very well; I will consider your suggestion. By the way, what is your new year’s resolution?
Achilles: Mine? Quite simply, I resolve to continue avoiding the sort of meta-mathematical conversations we used to have all the time, which made my head spin.
Tortoise: Well, that oughtn’t be too hard to keep.
Achilles: Not at all; well, shall we meet again at a random place and time in the new year?
Tortoise: That sounds like an excellent plan; I’ll be there.
NOTE: Achilles and the Tortoise are visiting courtesy of Greek philosopher Zeno. They have also been spotted among the writings of Lewis Carroll and Douglas Hofstadter.
