Investigating the nature of what can be proved, what can be known, ect has been done before, by many others. But as to the nature of what can possibly be thought little has been said. And this is not because the nature of what is thinkable is never used in philosophy, on the contrary there are several arguments that I know of that turn on the nature of what is thinkable. The real problem is that defining what it means for something to be thinkable in general is probably impossible, the notion of thinkability is used in different contexts to mean different things. Thus I shall tackle the problem by cases, developing four definitions of what it could mean for something to be thinkable, and then showing what limits exist, if any, for that particular kind of thinkability.
One definition of the thinkable is that which we can say something about, or, more precisely, which we might conceive of as having a certain property. And this doesn’t commit us to having a specific conception of that thing, for example we might think of infinity only as that which is not finite, although we cannot conceptualize an infinite number of things. In this case it is easy to prove that everything is thinkable. Simply assume that there is some x that is not thinkable, but if that x isn’t thinkable we are saying something about it right here, that it isn’t thinkable. This contradicts our definition as to what it means to be thinkable or unthinkable, and hence there cannot be such an x, meaning that everything is thinkable in this way.
But this is an exceedingly loose definition of thinkable, one that we use rather rarely. Let me consider then the strictest possible definition of what it means for something to be thinkable: that it is fully understand. And, if you fully understand something and the laws by which it operates, then you can make perfect predictions about it. But there are limits on predictability. In rough terms no system can perfectly predict a system of equal or greater complexity, which means that no system, even the mind, can perfectly predict itself. But this tells us only that there are things that cannot be thought about for some minds, it doesn’t (yet) tell us whether there are things that universally cannot be thought. So let us consider a relation P between two systems that only holds if the first system can perfectly predict the second. Clearly this property is transitive (since if A can predict B and B can predict C than A can predict C simply by perfectly predicting B predicting C). And neither can it contain any loops, since it isn’t the case that a system can predict itself. Given this we can put the systems in a kind of order, in which there is either one system that cannot be predicted by any system or a set of systems that cannot be predicted by any system (even other systems in that set), a result that can be shown to follow mathematically. And thus, under our definition of what it means to be thinkable, one of these systems is something that, universally, cannot be thought.
This takes care of the extreme cases, and, as shown above, in one extreme case everything can be thought and in the other there are things that cannot be thought. But neither of these definitions of what is thinkable captures how we usually use the term. Usually our conception of what is thinkable depends on our notion of intentionality, meaning that we can think about anything with which we can have an intentional relation (intentionality, loosely speaking, being the ability of our thoughts to be about something). Unfortunately there are many definitions of intentionality, and I have neither the time nor the patience to cover each of them, so instead I will simply consider the question from the viewpoint of the two major theories about intentionality, which are that it is derived from causal relations and that it is derived from a match between the properties we conceive of an object as having and those it actually has.
The causal theory about intentionality holds that what we are thinking about is that which is the causal origin of our mental concept. This has the advantage of allowing the causal theory to describe intentional relations between us and things that we have misperceived. For example, if there is a squirrel outside, but I misperceive it as a cat, then when I am thinking about the cat I am really thinking about the squirrel. But this introduces a problem concerning fictional objects. For example, what if I am thinking about unicorns? Unicorns don’t actually exist, so I can’t be thinking about them, since no actual unicorn is the cause of my thoughts about unicorns. The causal theorist might say that I am really thinking about the animals that gave rise to the early legends, but even so this is to admit that I can’t be thinking about unicorns, only other real animals. And if the causal theorist responds to this by saying that the theory only covers particulars, and not kinds, then what about my thoughts concerning Jack, the unicorn who lives next door? Either way there are some things that we can’t think about under the causal theory (although we may live under the systematic delusion that we are thinking about them).
Finally let us consider the, “property match” theory of intentionality, which holds that we are thinking about an object when the properties we give it in thought pick that object out uniquely, or when there is some unique match that best fits the properties we give in thought to the object. This theory deals with the problem of non-existent objects by holding that there is some possible object that, if it existed, our thoughts would be about, and thus this possible object can be considered to be what we are thinking about. This is well and good, but consider two identical white balls that you are presented with. The balls are then put into a box and shaken around. Now, if the box isn’t opened, can you think about just one of the balls? Certainly you could say to yourself “I am thinking about just one of the balls”, but you wouldn’t be able to mentally assign properties to it that distinguish it from the other. But perhaps you believe that historical properties count as well, such as the property of being the ball that was previously to my right. Such properties are somewhat questionable, but even if they exist there are plenty of things they can’t help us distinguish between. Can we think about a particular electron on Jupiter, or a particular molecule of interstellar gas? According to the “property match” theory of intentionality we can’t, because when we try to think about these things our conception of them doesn’t distinguish one of them from the others, at best we can think of the group they belong to (“electrons on Jupiter” and “molecules of interstellar gas”). So, even under this theory of intentionality, there are things we can’t think about. It is, however, more of an open question as to whether there are things that are universally unable to be thought about. Certainly there are groups of items that don’t seem to allow us to meaningfully think of a particular member, and, moreover, which no actual thinker would be able to mentally distinguish (such as a particular atom of the gas cloud that formed shortly after the big bang). But in principle there could have been some observer there that could have made such distinctions, although all evidence points to the fact that there wasn’t. It seems likely that there might be some quantum effect that could fulfill this role, such as one member of a pair of entangled electrons, but I would be hesitant to commit to such an answer at the moment.
So, on all conceptions of what it means to be thinkable, except for the very first, there are some limits. Now these limits don’t normally intrude into our thought; it is not often that I find myself contemplating the exact nature of systems that no system can predict. Such considerations do come into play, however, when we form arguments based on the nature of what can, in principle, be thought, which occur in the philosophy of religion and in epistemology. Interestingly there were authors in the philosophy of religion who believed in god, but whose writing seem to imply that he can be thought of only in the first way (a simple* god, who has no properties, or about which nothing positive can be said**, who interacts with the world only as a first mover or as a condition for the world’s existence). But if that is the case it is hard to say what they believed in, because such a conception of god leaves very little content, certainly nothing that can distinguish god from various other entities, such as the universe itself, or “nothingness”. But I will leave the details as to how such accounts connect with what I have written here for a future date.
* A technical terms meaning having no parts and no accidents, a consequence of which is that something which is simple can have no properties
** The negative way, in which the only things you can say about god is that it is not finite, not material, ect.