Don’t Think of a Square Circle!
Wittgensteinian Reflections on Imagining the Impossible
3.031 It used to be said that God could create anything except what would be contrary to the laws of logic. The truth is that we could not say what an 'illogical' world would look like.
—Wittgenstein, Tractatus Logico-Philosophicus, trans. D. F. Pears and B. F. McGuinness (London: Routledge & Kegan Paul)
It has been noted (by George Lakoff, among others) that if someone tells you “Don’t think of an elephant!”, it’s pretty hard not to do it. Though it would take empirical research to fill in the details, the answer as to why this is so does not seem hard to discern: To understand the order—to understand what one is not supposed to think of—one must understand the term ‘elephant’, and thus come to think of one. I’d wager that in addition to thinking of one an image of an elephant popped into your head as well. This is probably because the concept of an elephant is an empirical concept—no crisp, abstract definition of an elephant comes readily to mind, so a stereotypical image is needed to make it intelligible. By contrast, if someone were to tell you “Don’t think of the number 2!” it is less likely that an image would come up, unless you confuse the numeral ‘2’ with the number 2—or, at least, that the image would be unlikely to be constant for different people, or for the same person at different times.
What, though, if someone were to tell you “Don’t think of a square circle!” Is it so hard to comply in this case?
Well, maybe it is: Do I really know what it is I’m not supposed to think of? If not, I’m not really complying with the order, because I fail to understand it. I’m merely doing what it says. Nevertheless, what interests me here is not our concept of compliance, but rather that of conceiving or imagining the impossible.
If one can imagine or conceive of something, what then could one mean by saying that one can’t understand how it could be the case? Could it be, for instance, that I can conceive that a square circle exists, but not that the existence of a square circle is possible? It would seem not: It is not more impossible that it is possible that a square circle exists than that a square circle exists; so if I can conceive the latter, I should be able to conceive the former as well. If however, I cannot conceive of the existence of a square circle, I cannot conceive of its possibility either. That seems fine, but one could ask: If I cannot conceive of a square circle, nor of its possibility, how could I still conceive of its impossibility? That is, if I truly cannot conceive of it, how can I say that it is that, and not something else, or nothing at all, of which I am unable to conceive? If someone tells me that I cannot eat or drink on the subway, I know what it is that I am being forbidden to do. And if I am informed that no human being can run as fast as a cheetah, I know what is being declared to be impossible, and it is relatively easy to form an image of what the contrary would look like. But if someone tells me that it is impossible for a circle to be square, or identical to (i.e., the same thing as) a square, how am I to know what it is that is being said to not possibly be the case? For if you were to be told that snorogs are impossible, you would have every right to ask your alleged informant what a snorog is. If no answer were forthcoming, the most you could conclude is that you cannot know whether they are possible or not if you have no idea what they are supposed to be. The same should hold just as much for “square circles”: If I simply don’t know what the term means I should conclude, not that they cannot exist, but rather that I have no idea whether they can or not, if the term is meaningful at all. And if I do understand it, it seems—as was said above—that I must also be able to understand the claim that square circles are possible, as the possibility of something cannot be any more impossible than the thing itself. If I nevertheless say that they or their possible existence is inconceivable, I cannot mean that I do not understand the claim that they exist, or that their existence is possible. What, then, can I mean?
A natural answer would be something like: In virtue of understanding the term ‘square circle’, I “see” or “grasp” that they cannot exist. But what am I seeing here, and how do I see it?
The second question would involve us in the thorny details of epistemological debates surrounding a priori knowledge, which I will not enter into. By contrast, it might seem that an answer to the first question is trivial: I’m seeing that square circles cannot exist! But what we have said so far should make us suspicious of this answer. To be sure, it is a right answer to our question, if it’s true that I’m really seeing that, but it’s not the only possible answer, and certainly not the most helpful one. If someone were to tell you that they can see that snorogs cannot exist, and you were to ask them what exactly it is to see that, it would not be very informative to be told that it is simply to see that snorogs cannot exist! If you don’t know yourself what snorogs are, or what ‘cannot’ means in this context, such an answer won’t help you one bit. We have two problems: (1) If I am to “see that something cannot be the case” in virtue of understanding a term, ‘cannot’ should not mean: It is inconceivable that it is the case. What, then, does it mean? It must be more than mere nomic or physical impossibility, for one cannot tell something to be nomically or physically impossible merely in virtue of understanding a certain term. (2) How am I to tell whether I or someone else truly understands the term? For unless I have some means of doing so I am at a loss as to how to assess the claim of impossibility, and can make no progress.
I’ll leave the first question to one side (for now). As for the second, in all cases it should be possible to specify the meaning of a term somehow, to give some kind (not necessarily very exact and not necessarily very informative) of explanation or definition of what it is. In addition, for empirical terms it should be possible to imagine the corresponding entity or recognize it in experience.
Can one specify the meaning of ‘square circle’? It certainly seems so: One can define it as a circle that is also a square, or as a circle that is identical to (is the same thing as) a square. Alternatively, if the term ‘spherical cube’ is already understood, (as, e.g., a sphere that is also a cube, or a sphere that is identical to a cube) one could define a square circle as a 2-dimensional cross-section of a spherical cube that cuts through its center. Admittedly, these definitions are not very rigorous, but rigorous definitions cannot be given for most terms in ordinary use, and they are none the worse off for that.
One can note that as ‘square’ and ‘circle’ (or ‘sphere’ and ‘cube’) are empirical terms, ‘square circle’ should be one too. But whereas it has appeared easy to define what a square circle would be, it seems much more difficult to imagine what one would look like. Supposing one were shown a drawing of a disc with a large square hole in it, or a square solid with a large circular hole in it, one would naturally reply that that isn’t what one means . But what then does one mean? It is not as though I have some (vague) image of a square circle to which they fail to conform, but rather that no image suggests itself. Is it that it would look like something, only I don’t know what? Or is it that there is no such thing as “what it would look like”?
Well, if the One True Logic is not paraconsistent (i.e., is not non-explosive), then contradictions entail everything. So if square circles are contradictory—and they seem to be, for squares are square and circles are not, while square circles are both squares and circles—then every counterfactual beginning “If there were square circles, they would look like…” is true. If Logic is not paraconsistent, then, nothing is easier than imaging what a square circle would look like: Take anything you please, such as a shoe, a ship, some sealing wax, a cabbage, or a king. If the One True Logic is paraconsistent, matters are less clear. But suppose some lover of paradoxes were to come to you, draw a square and a circle next to each other, and ask you whether a square circle would look like that.
“Like the square or the circle?”, you enquire.
“Like both,” he replies.
“Well,” you say, “a square circle is supposed to be a square that’s identical to a circle, but the figures you have drawn don’t look identical.”
“I grant that the figures I have drawn are not identical,” he replies, “but that’s the thing about representations: they need not share the properties of the things they represent. As for identity, I wasn’t aware that it looked like anything at all. If it did, one could tell by mere inspection whether properties are universals or tropes, by ostending different “instances” or “tokens” of the same type and checking to see—literally see—that the instances either do or do not have something numerically identical in common; and one could thus very easily settle that dispute. As things are this is not possible, so it seems safe to say we cannot perceptually experience relations of identity. So I think I can safely say that the circle and the square are depicted as being identical, even though the depicted square and the depicted circle don’t look identical.
“Identity may not look like anything,” you reply, “but difference certainly does, and the square depicted and the circle depicted look different.
“Naturally,” he replies, “for I have drawn a contradictory object, which is naturally both self identical and self-distinct. Since I have drawn a square which is identical to a circle, it differs even from itself, and it is this difference which you are picking up on.”
“Well,” you say, more hesitantly, “not only do they look different, they appear to be in different places. A square circle isn’t supposed to be a mereological sum of a square in one place and a circle in another, but one thing in one place which is both square and circular.”
“That,” he replies, “is merely a defect of the medium. I drew a square shape and a circular shape in different places, but they are intended to depict exactly the same location. And in any case, just as the depicted shape is both self-identical and self-distinct, the depicted location of the shape must also be both self-identical and self-distinct. The places of the shape(s) are thus both different and not different, and again it is the difference which you are picking up on.”
“But,” you sputter in exasperation, “I can see that the square and the circle are different from each other, and also that they are not different from themselves, but what I cannot see is that they are not different from each other!”
“But of course you can!” he replies, smiling, “for take the depicted square. You can surely see that it is not different from itself; that is, that the square is not different from the square. And since that square is identical to the depicted circle, it follows by Leibniz’s Law that the depicted circle looks not to be different from the depicted square!”
At this point I suspect you would give up, and let the lover of paradoxes go on his merry way. But who has won the hypothetical dispute? Why shouldn’t we say that his drawing of a square circle is a perfectly good one—that by looking at it we can now tell what one would look like? If we still insist, as I think most will, that his drawing isn’t a good one, I think the proper moral may well be a Wittgensteinian one: When we say that we cannot imagine a square circle, what we’re really doing (whether we realize it or not) is excluding any purported description of what one would look like from our language-game(s). We’re saying, in effect, “No matter what anything looks like, we refuse to call that ‘what a square circle looks like’, and no matter what anyone draws, we refuse to call it ‘a good drawing of a square circle.’ And this isn’t to say that there can’t be some strange looking drawings—one can simply look at a Penrose triangle or some of M.C. Escher’s works (or for a real life case, one can check out ‘the waterfall illusion’). One must get away from the idea that behind a grammatical rule there stands something that cannot be done—unless one only means that there is a norm that we shouldn’t speak in a certain way. Any image can be described in language in multiple ways, and what the case of the lover of paradoxes shows, if it shows anything, is that our current, actual language game forbids certain descriptions of certain phenomena. Perhaps, as in the case of the waterfall illusion, a language game which admits of inconsistent modes of description would be more appropriate (and note that I say more appropriate, for consistent modes of description are certainly also available). As with the description ‘the pitch of sweetness’, the (apparent) description ‘the look / appearance of a square circle’ has been denied a role in our language game. Just as ‘the pitch of sweetness’ is not something we fail to hear in any ordinary sense, ‘the look / appearance of a square circle’ is not something that we fail to see or imagine in any ordinary sense. Why such terms play no role in our language may be hard to say, but that they do not could be said to be shown by our reaction to the case of the lover of paradoxes: the simple fact is that we know how the investigation is to turn out before we begin to inquire. For it is not that the term ‘square circle’, like the term ‘snorog,’ simply calls up no image for us as a matter of fact—if someone were to introduce us to a community where things which were regularly called snorogs looked such-and-such a way, we would accept that easily—it is rather that we know in advance that we will refuse to apply the term ‘square circle’ to anything.
In this way we avoid the misleading idea of imagining the impossible as something that we are unable to do—that “our powers of imagination are unequal to the task” as Wittgenstein put it in the Investigations—as well as avoiding the equally misleading picture of logical or conceptual necessity whereby we in effect personify it as the bouncer at the door of Club Reality; of logical/conceptual necessity as a powerful force which keeps out the riffraff of impossibilia struggling to get in in order that they may exist. This, at any rate, is the best I think I can do to give content to Wittgenstein’s idea of logical/conceptual necessity as being basically linguistic.