The Limits of A Priori Thinking (Column 376)

With God’s help

Disclaimer: This post was translated from Hebrew using AI (ChatGPT 5 Thinking), so there may be inaccuracies or nuances lost. If something seems unclear, please refer to the Hebrew original or contact us for clarification.

Some time ago Doron raised a question in the Q&A about the expansion of space. As is known, contemporary physics teaches us that the universe is expanding, meaning that the objects within it are receding from one another over time. Doron’s question was: into what is the universe expanding? What is there around it such that the expansion “comes at its expense”? In particular, if we understand that space itself was created in the Big Bang and from then on has been expanding (growing), then clearly there is nothing around it into which it could expand. The question is whether it is possible to define the expansion of space when we have no surrounding place into which—and at whose expense—it is carried out. In the course of the discussion the question arose whether understanding necessarily involves perception or visual imagination, and I wanted to touch on that here.

My answer and the discussion about it

There I answered Doron’s question that there is no necessity to posit the existence of a place around space into which it expands. Space itself grows from within itself and dilates (one can call this intrinsic growth—internal/structural). When looking from the inside it becomes larger and larger, but there is no viewing it from the outside. Note: it’s not that we cannot observe this process from the outside, but rather that there simply is no “outside” from which one could observe space—and of course one cannot observe that very “outside” itself.

Doron argued against me that the concept of expansion logically includes within it the existence of a substrate into which the expansion takes place. Therefore, in his view, one cannot speak of expansion as I described it. In his remarks he mentioned a thought experiment we can conduct, in which we are to imagine that indeed there is no such external substrate, but then—he claimed—we will necessarily return, by logical constraints, to the conclusion that no such expansion exists. I tried to propose to him the example of inflating a balloon (used by many to describe this phenomenon), whose surface area is constantly increasing, yet not directly at the expense of some external space. At least on the two-dimensional plane (the balloon’s surface itself), one can see intrinsic growth. Indeed, on the three-dimensional plane, the space within which that surface is situated/embedded, this is of course grasped differently. In the terminology of a great halakhic decisor (“the last posek,” shlita, there in the thread), this is swelling rather than expansion.

This remark raised the question whether the inability to imagine something is a logical constraint. That is, does what cannot be imagined necessarily not exist? Think, for example, of an image made of light with wavelengths beyond the visible spectrum. We cannot imagine it, since any sight of ours is composed of light in the visible wavelengths (otherwise it has no “appearance.” Think: what color would that image have?). Does that mean it doesn’t exist? Certainly not. The same goes for sounds at frequencies outside the audible range (as is known, there are such frequencies that dogs do respond to). One can make a similar claim regarding God. Some have wanted to say that since we have no way to imagine and describe Him—then the talk is necessarily empty (the spaghetti monster, an imaginary friend). Here too I claim this is incorrect, since to claim that something exists there is no necessity that we be able to imagine it. I brought there another example, from the emergence of language. There too the lexicon (and likewise the rules of grammar) grows over time, but that does not happen at the expense of something else. One could quibble and say that the realm of concepts that cannot be described is an external entity, and a term that does not describe a concept that “exists,” in some sense, cannot really arise. But with respect to grammar the situation is certainly different.

In short, my claim was that the world of things that exist is divided into three categories: those we experience in reality (sensory cognition); those not directly experienced by us but which we can imagine visually or by other sensory imagination; and those we cannot imagine (beyond cognition and imagination).

Example: A bird on the road

A good friend of mine, Rabbi Avi Bleidtstein, once gave me an illuminating example that can broaden our discussion a bit. Think of a bird standing on the road on which I’m driving. There is a phenomenon whereby the car gets very close to the bird and it does not move. Only when I’m about half a meter away does it “remember” to fly off and escape. Human beings, of course, would flee at a much greater distance from the car. The question is why it takes the bird so long to fly away. Does it have a slower reaction time compared to humans? Not likely. Does it see less well? That too is probably not true. So why does this happen anyway?

It turns out the answer is very simple. The bird both moves much faster than we do and escapes upward rather than sideways, and therefore it’s an easier and quicker process. That is, we should measure the distance from the car at which the bird will flee in terms of reaction time. In those terms, suppose a person standing on the road steps aside one second before impact. The bird’s distance is likewise a distance of one second from impact. Except that for it, because of its speed, a distance of half a meter gives it sufficient time to escape, which is not the case for humans.

Is a person free in imagination and bound in intellect?

The misunderstanding regarding the bird stems from the fact that we judge the bird in our own terms, and therefore its behavior seems strange to us. But if we step into its “shoes” (or head), things become entirely comprehensible. Of course we have no principled impediment to imagining the bird’s situation; it’s only a matter of inattention to its different parameters. The inattention stems from our operating within a set of data and a worldview different from its own, and we tend to analyze the world in our terms and within our own perceptual and cognitive framework. Yet in this example, once we grasp this, it is quite easy to imagine the bird’s “considerations” from its point of view. I will use this example to broaden the canvas. The limits of our thinking prevent us from understanding phenomena, but there are situations in which the limits of our imagination and cognition (such as the limits of light or sound frequencies) can sometimes even prevent us entirely from imagining something—and that does not mean it does not exist.

R. Yisrael Salanter, in his Iggeret HaMussar, writes that “a person is free in his imagination and bound in his intellect.” His intent is that reason is limited by constraints of thinking and logic, whereas imagination is free and unshackled. But now you can see that even in imagination a person is not entirely free. There are not insignificant constraints on our imagination. On second thought, in a certain sense the intellect is actually freer than imagination. Imagination is bound to our visual perceptions. Thinking is subject to logic, but anything within logic can be thought—even if it cannot be imagined. It is true that what is beyond logic apparently cannot exist at all. But what cannot be imagined (visually or by another sense), so long as it is logically and conceptually defined, can certainly exist.

Imagination and conceptual thinking

Therefore it is clear that in many cases our ability to free ourselves from the shackles of imagination is through conceptual thinking. Thus, for example, one can think of a reality with a number of dimensions different from three, even though we cannot imagine it. We do this with the tools of conceptual-mathematical thinking. We define the concept of a dimension, and then we can extend and generalize it as we wish, beyond the three familiar to us from our immediate experience. The fact that we cannot imagine it does not mean it is undefined, and therefore there is no reason to deny the possibility of the existence of such a reality.

The same holds regarding the swelling of space, as opposed to expansion. It is indeed difficult to imagine a situation in which space expands without there being a surrounding substrate to receive that expansion, but conceptually the matter is well-defined. Therefore this is only a limitation of imagination, not a logical-conceptual limitation. Hence there is no impediment to thinking that such a thing is indeed possible. It seems to me that with effort one might even be able to imagine something like this (simply focus the imaginative gaze into the sphere and not outward, or into the balloon’s surface without imagining the volume around it).

The book Flatland (flatland), by Edwin Abbott, was written at the end of the nineteenth century. It takes the conceptual analysis of a two-dimensional world and tries to enliven it into a story in which things happen and beings live and act—and which can also be imagined in some sense. Think, for example, that in a two-dimensional world the fence that cannot be crossed is a line. If you draw a line all the way across, a two-dimensional being cannot cross it in any way to the other side (it cannot rise above it via a third dimension, as might seem to us obvious). By understanding, as a three-dimensional being, what happens in a two-dimensional world, we can abstract and generalize, and try to understand how a four-dimensional world would look, or how we and our three-dimensional world would appear from the point of view of beings who live in a four-dimensional world.

I’ll note that in column 174 I showed illustrations that help us imagine a mathematical creature called a “Klein bottle” (an extension of the “Möbius strip” to four dimensions—a bottle with no inside and outside) via its projections onto our three-dimensional world. Here too this is an application of conceptual analysis and thinking that tries to create a reality with a visual dimension that we can imagine. Another example is the field of the topology of knots. Topology teaches us that in a four-dimensional world knots do not exist (roughly put: you cannot tie a string in a way that cannot be undone by a simple pull; see the section “Generalizations” there). This is already a less trivial result, and if you think about it you will see that it cannot really be imagined. Our attempt to imagine it will not present us with a complete visual picture of the situation (because one cannot imagine four dimensions), but at most a vague and general impression (a three-dimensional knot that gets undone by a pull, without getting into details, for there truly is no way to untie every knot in three dimensions). And yet, one can reach this conclusion through conceptual thinking. Does that mean a four-dimensional world in which there are no knots is impossible? There is no reason to assume so. I am not claiming here that it actually exists, only that there is no impediment to its existing. We return to the conclusion that the ability to imagine is not a condition for the existence of something, and even conceptual analysis is not such a condition (though the absence of logical contradiction—certainly is).

Back to the expansion of space: the inability to imagine absence

When I visually imagine the expansion of space, there must be something outside it present before my eyes (even a vacuum). Otherwise, what is supposed to be in that place into which space expands before the expansion occurs? This is a limitation of visual imagination and of the temporal features of the process. It seems to me this is the reason for the feeling that there must be an external substrate in order to imagine expansion.

I think underlying this lies our inability to imagine, visually, absence. Note, this is not about imagining a vacuum—that is, empty space—but imagining the absence of space. That is something else entirely. An analogy: trying to imagine what a blind person “sees” in his awareness. We tend to think that the blind person sees before him a screen that is all black (as with us when our eyes are closed). But I think this is a mistake. Black is a color like any other, and seeing black is seeing something. The blind person does not see; it is not that he sees “nothing.” That is something else entirely. In column 340 I mentioned that people sometimes play with the question of a point’s length. Many will answer that its length is 0. But if we do not enter overly convoluted mathematical definitions, then that answer is incorrect. A point has no length. Mind you: it’s not that its length is 0, but rather that it has no length. The concept of length does not apply to it (I think this is the source of the illusion in the question whether 0 is a number or an absence).

In my article here I noted that there are two kinds of opposite or negation: oppositional negation and privative negation. Oppositional negation creates an object with opposite properties, for example: when we have a particle with positive charge, oppositional negation makes it a particle with negative charge. The relation between the two sides in oppositional negation is inversion, like the relation between 1 and (−1). By contrast (is this an annihilating or an opposing inversion?), privative inversion is like the relation between a particle charged positively and a neutral particle. A neutral particle is a particle devoid of charge. To reach such a state, we must annul the positive particle’s charge, not invert it. This is analogous to the relation between 1 and 0. But even the 0 here can have two meanings: absence of charge, or a charge whose magnitude is zero. Perhaps we can illustrate this via a particle that can be charged but whose charge is currently 0, versus something that cannot be charged at all—that is, for which the concept of charge is irrelevant. For example, what is the charge of the color red? A meaningless question, because a color does not have a charge of 0; the concept of charge is irrelevant to it.

When I close my eyes I see black. One might say I see an empty image (of colors)[1]. I do indeed see, but what stands before me is empty—there is no image. By contrast, the blind person does not see at all. He has no sight, and therefore even if he stands before a non-empty image he will not see it. And this is the secret of the difficulty regarding the expansion of space. We, as sighted humans, when we look in our imagination at the process of space expanding (swelling), necessarily see the expanding space as something, some object (not merely empty space), and consequently we also see around the expanding object empty space. But to see empty space expanding is not possible at all, even irrespective of the question of an outside. Note that to see empty space does not mean not to see at all. In this sense, my claim is that if we were to look at expanding space we would not see anything around it (and not: we would see “nothing” around it). But a sighted person cannot be in such a state. When he looks at expanding space there is necessarily something around it into which and at whose expense it expands. The conceptual analysis I have done here allows us to understand that despite the limits of our imagination, there is no impediment to such a thing existing and occurring in reality.

The limits of representation

I’ll preface with an apology for entering the relation between signifier and signified, a topic beloved by our postmodern cousins (who, as usual with them, say not a little nonsense and reach absurd conclusions about it as well).

In column 220 I discussed the question of what Torah is. I claimed there that the Torah is not the text of the Five Books. The Five Books are an expression, in our language, of the Torah—but the Torah is something more abstract, of which the text, and even the ideas the text expresses (such as the commandments), are only one representation. And note: I do not mean that the words formulate the ideas and the ideas are Torah. I mean that even the ideas are a formulation or representation of the Torah by means of a system of concepts and principles from our world. Thus, for example, the commandment to honor father and mother is not Torah. It is a representation of a more abstract idea, whose expression in our world is the commandment to honor parents.

In those remarks I noted that every abstract thing must be represented to us in some way for us to be able to understand and handle it. Except that here I must add that this need for representation raises two kinds of difficulties: 1) We find it difficult to grasp things we have no way to represent. 2) The representation leads us to infer certain conclusions, and at times we fail to notice that they pertain to the representation and not to the thing itself. There are phenomena that arise from the representation, while we attribute them to the things themselves (because we do not know how to grasp things except through the representation). I will now illustrate these two phenomena.

Example of phenomenon 1: the board and domino puzzle[2]

Think of a square board of 100 by 100 squares. It can be covered by 5,000 rectangular domino tiles, each covering two squares on the board. Now cut off the two squares at the ends of one diagonal. Can the trimmed board be covered by 4,999 domino tiles? The area is of course the same, yet it is not obvious whether this can actually be done or not. Try to think about it, and I assume you will find this is not a simple riddle.

The answer is of course negative. To see this easily, we should color the board black and white, like a chessboard. Every domino, no matter how you place it, always covers one black square and one white square. But the trimmed board contains an unequal number of black and white squares (either we cut off two blacks or two whites, depending on which diagonal we did this along), and therefore it cannot be fully covered by such a number of domino tiles.

What’s nice here, beyond the charming riddle and the elegant solution, is that coloring the board helped us notice the solution easily. Following the advice of my friend Shmuel Keren, I presented the riddle on a board that is not 8×8 squares and is not colored, to distance the chess association and thus make it less likely that you would think to color the board (for then the solution is obvious and simple). On an uncolored board this riddle is very hard to solve, and on a colored board it is exceedingly simple. If we did not know such board games at all, it would not occur to us to color the board, and then our chances of solving the riddle would drop dramatically.

But note that color is not truly required essentially for the riddle nor for the solution. It is required only didactically, to illustrate the solution (and to get us to think of it in the first place). Without coloring the board we would not have noticed the property (which is mathematical and abstract in itself, and exists also on an uncolored board) that ties together all the “black” or all the “white” squares. In that situation it would not occur to us to think that each domino covers one square of each type. The coloring added here a parameter that represents that hidden and abstract property, and thereby helped us solve the riddle. This demonstrates phenomenon 1: sometimes one cannot, or at least finds it hard to, think of something in itself—or even to grasp that it exists—without having a (visual or other) representation for it.

Example of phenomenon 2: the mirror paradox

In column 220 I presented the mirror paradox, and I will briefly revisit it here (I explained there that one not versed in abstract, mathematical or physical thinking may struggle to understand why there is a paradox). When I stand before a mirror it swaps right and left. A person wearing a watch on his right hand will see before him himself with the watch on the left hand. But the mirror does not swap up and down. The mirror is symmetric, and there seems to be no difference in it between the horizontal and vertical axes. So how does this “miracle” happen? Why are right and left swapped while up and down are not?

I explained there that the reason is that the relation between right and left is not like the relation between up and down. The relation between right and left derives from our body’s structure. It is impossible to teach someone where the right side is if he is not endowed with that capacity beforehand. In reality there is no difference you can point to between right and left. By contrast, the difference between up and down is very easy to explain (by the relation or proximity to the earth). That is, the up-down axis has a root in reality, whereas the right-left relation does not. The mirror turns nothing into anything else, but for us right and left are reversed in the image reflected in the mirror, because these are our definitions. In other words, besides the mirror and the figure before it there is another factor in this game: the observer (= us). The one who introduces the asymmetry between right and left is the observer, not the mirror.

Once again we saw that the representations of the image in terms of up-down or right-left influence the way we perceive it. In this case they led us into a paradox that at first glance seems insoluble. Therefore this example expresses phenomenon 2—that is, there are our conclusions that pertain to the representation and not to the thing itself.

As a marginal note I will remark that in the end it turns out that even in reality there is a right-left axis—that is, it is not truly only a subjective matter of ours. There are physical properties determined by the right-hand rule or left-hand rule (defined via screw rotations). Thus, for example, there is a difference in the properties of a particle that spins on its axis to the right or to the left; that is, there is a physical outcome to this difference. This is an indication that we are dealing with a property that exists in the world itself and not only in our system of thought and representation. Our body structure enables us to grasp this relation, and perhaps with a different body structure (that is not symmetric between right and left) we would not know how to point it out and acknowledge its existence—and certainly not understand it. In this sense there is a similarity to the domino riddle, for here too the representation of the right-left distinction enables us to notice and understand it, and in that sense there is also an expression here of phenomenon 1.

The principle of causality

It seems very hard for us to imagine something occurring without a cause. When something happens we immediately search for what caused it. Yet here, if you think again, you will see there is no problem with imagination. Imagination can depict before our cognition an event that occurs just so, ex nihilo, without a cause. The search for a cause is an a priori principle of thought and not a constraint of our visual cognition or our imagination. The fact that we cannot imagine such an event is the result of a cognitive assumption and not of constraints on the mode of viewing and cognition or imagination themselves. If we exert ourselves, we have no problem imagining such things.

There are logical arguments on this matter as well, and therefore there is ostensibly a possibility that we are dealing here with a logical rather than visual-imaginative constraint. One such argument: if the thing happened just so, the question arises why it happened precisely here and now? Therefore it must have a cause that brought it about. But as expected (as with any valid logical argument), this begs the question, since the “why” is itself based on the principle of causality. Who says there must be a cause and that it cannot occur just like that without a cause? We assume the principle of causality in this argument, and therefore it is no wonder that its conclusion is a causal picture. It seems there is no logical constraint forcing us to grasp the world causally. This is apparently a psychological constraint (but, as noted, it compels thought, not imagination).

If so, it is not the result of a visual constraint nor of a logical one, but of a psychological one. This raises the question whether it might be a limitation of our conceptual thinking. That is, in reality the principle of causality is not true, but we are compelled to think so because of the structure of our thinking. In other words: if there is no logical necessity to say that every event must have a cause, and this is also not a constraint of visual cognition, then whence did we derive the principle of causality—that is, the assumption that everything must have a cause? Is it not merely an artificial product of our psychology? I will only remark here that in my view this is likely a discernment of intuition (which, I think, is part of cognition and not of our discursive thinking).

To sharpen the connection to our topic, I will bring here as an example Peter van Inwagen’s argument against the hypothesis of free will (libertarianism). Van Inwagen claims there are two possibilities: either an event has a cause or it does not. If it has a cause, then it happens deterministically; if not—then it happens randomly (without a cause). Either way, there is no possibility of an occurrence that is neither of these two, and therefore there is no state of free will. In his view, this is a state that exists only in our imagination—if even there. There is here a contradiction or a proof of non-existence on the logical-conceptual plane.

But he is mistaken. He assumes that an uncaused event is necessarily random, thereby of course begging the question. The libertarian’s claim is precisely this: that it is possible for an event to be without a cause and yet not random/arbitrary. For example, an event done for a purpose and out of planning—even if it has no cause. The person’s choice/decision is the cause, but the person’s choice has no cause—rather a purpose (I elaborated on this in my book Sciences of Freedom). As I explained there, if anything, I would remove from the picture the possibility of a truly random, uncaused event. We have no indication of that from experience or reason. Moreover, even events that have a causal relation do not arise from our experience (as David Hume explained). By contrast, events of free choice we experience (though we do not imagine them—certainly not visually) all the time. Therefore that is precisely the “mechanism” that best fits our experience.

Above I already noted that it is quite hard for us to imagine or think of a state in which some event happens without a cause (though this is a psychological constraint—not a logical or visual one). But we have no problem thinking of an event that happens by decision. In fact, this is the most basic experience of us all, including determinists—except that they feel a need to deny it because their reason does not acknowledge the existence of such a mechanism. Here, in fact, the ability to imagine (more precisely: to experience) is pushed aside in the face of their (faulty) conceptual analysis. In this sense, here is an example of a conceptual system (mistaken) that forces upon us mistaken cognitive conclusions, even against our immediate experience.

In follow-up columns I will touch on two topics related to this column: non-verbal thinking and Platonism. You have been warned!

[1] Actually, a true absence of colors is transparent, not black (and not white). And perhaps the difference between white and transparent better expresses the difference between the two kinds of non-existence.

[2] See Gomory’s theorem and also here.