The concept of intelligibility is often used by philosophers but not often analyzed. The OED gives this simple definition of “intelligible”: “able to be understood”, but it follows that up with a definition proper to philosophy: “able to be understood only by the intellect”. The intellect is the faculty that makes things intelligible; without it nothing would be. Clearly, intelligibility is a relational concept: something (we are not told what) is intelligible only if it is understood (possibly potentially) by someone, or by someone’s intellect. Logically, the concept is like being perceivable or knowable: a non-mental entity is said to be perceivable or knowable or intelligible in relation to a mind equipped with certain cognitive powers. We might paraphrase the dictionary definition by saying that something is intelligible if and only if it is graspable by the intellect (not by the senses or the faculty of knowing): it is intellectual apprehension. But what is that exactly? One tradition has it that intellectual apprehension belongs to the domain of the a priori—mathematics, logic, the forms, maybe philosophy itself. But it would be wrong to limit the concept to these areas, since the empirical world can be intelligible too, i.e. understood by the intellect. This is a special kind of cognition, distinct from perception and knowledge—superior, deeper, more penetrating. In it the world is made peculiarly transparent to the mind, not in a blur or superficially or inadequately. It is, we might say, an elevated type of insight, not to be identified with simply knowing brute facts.
What would count as paradigm instances of the intelligible? Mathematics is the standard example, both pure and applied. Numbers and geometric figures are inherently intelligible, but so is their application to the world: when we describe the world mathematically we make it intelligible, because we make it graspable by the intellect. Mathematical laws are the central examples. This doesn’t mean that everything in physics is intelligible, but the common assumption is that physics is our best hope of rendering the world intelligible–and mathematics is central to physics. However, there is another area of intelligibility that should be mentioned—belief-desire psychology. We make actions intelligible by relating them to beliefs and desires: actions are intelligible in the light of the beliefs and desires that lead to them. We render the action rational by describing it in this way, and hence we understand it; there is nothing brute or opaque about it. Just as with mathematics, there is a whiff of the a priori: the idea of rationality as a normative domain is invoked–hence the practical syllogism. In both cases an ideal structure is brought to bear on concrete reality, thereby rendering it intelligible. This is not something we detect with our senses; we apply it, by means of the intellect, to the things we perceive. We render the world intelligible rather than see it to be so.
I am inclined to suppose that these are the only cases of intelligibility in the natural world. Mere perceptual knowledge is never by itself intelligible knowledge; nor is causal knowledge, since it merely tells us what causes what, not what abstract principles underlie the causal connections. Physics might tell us that gravity causes motion, but without a mathematical law this does not render nature intelligible. Animals can have perceptual and causal knowledge, but they don’t have intelligible knowledge of the kind delivered by mathematical physics. Cartesian mechanism was supposed to make the physical world intelligible and it was a quantitative account of matter and motion. The case of biology is interesting: it is not generally thought of as having an a priori component, but it is supposed to provide understanding of the biological world. Does Darwin’s theory make evolution intelligible? It has a mathematical side because it uses the notion of frequency—advantageous traits lead to greater frequency in the population than disadvantageous ones. And we can quantify many aspects of animal behavior and genetic propagation. But there is also the teleological notion of function, which brings biology close to psychology: the heart beats as it does because that is its function (compare: a person beats a drum because he desires to). It is as if bodily organs desire to do what it is their function to do. Moreover, the standard way of understanding natural selection is analogous to intentional selection by agents—nature selects certain organisms to survive as selective breeders do. So biology falls under teleological conceptions and hence inherits the intelligibility that belongs to such conceptions (it is the same for theories that attribute evolution to God’s design). Thus biology is not a clear counterexample to the thesis that mathematics and teleology are the sole types of intelligibility. Merely knowing that clouds cause rain does not render the cloud-rain nexus intelligible: we must either treat it mathematically or conceive it teleologically to do that. In fact, the mathematical method works in this case because we can describe clouds as aggregates of water droplets subject to mathematically describable forces—rain thereby becomes intelligible.
Are these two modes of intelligibility unrelated? They certainly seem so at first glance: mathematics is one thing, psychology another. But perhaps there are some significant commonalities or areas of overlap. Psychology has its quantitative aspects, both scientific and common sense; in particular, we have the ideas of strength of desire and degree of belief. These are formalized in decision theory, a mathematical theory; so ordinary psychological understanding is capable of mathematical formulation in addition to being teleological. There is also a good deal of mathematics in the psychology of perception and elsewhere. We thus have a kind of double intelligibility in psychology, though the mathematical component is not as salient as it is in physics. In the case of mathematics itself, we can ask whether it has any teleological dimension, any built-in purpose. Formalists might say so, relying on the notion that mathematics reduces to symbolism and symbolism has a purpose; an instrumentalist view of mathematics would then be indicated. The same can be said for intuitionism: mathematics is a mental construction and that construction has a purpose—it is a kind of mental artifact that we employ in certain ways. Mathematics has a purpose and our application of it to the empirical world is the fulfillment of that purpose. Platonism, however, seems to banish purpose from mathematics, viewing it as a non-human objective realm of reality that pre-dates human existence. But that is not so clear on reflection: for there is an uncanny fit between mathematics as an abstract inquiry and the nature of empirical reality. For example, numbers are remarkably useful for counting objects, and geometry seems tailor made for describing objects in space. Is this just a happy accident? One could swear that mathematics was designed so as to be applied in these ways. Yet, according to Platonism, mathematical reality follows its own internal rules and was not constructed by human minds in any way. Its usefulness is therefore entirely contingent, extrinsic to its inner nature. Thus Platonism pulls away from the idea that mathematics has a purpose that is realized in its applications.
Here an ingenious theist may spot his chance: God designed mathematics to be both an objective abstract structure and imbued with purpose! His relation to mathematics is like our relation to our machines: both are objective constituents of reality but both are also purposive. Mathematics is objective-cum-functional. So even Platonism may be understood (at a stretch) to incorporate a teleological dimension, though obscurely so; in which case, it shares something with psychology. The two are not then completely separate conceptually, though it would obviously be wrong to reduce one to the other. If so, we have a more unified or integrated theory of intelligibility than we might have hoped for: our two paradigm cases turn out to have more in common than appears at first sight. We might even speak of the “teleological-mathematical” as the cornerstone of intelligibility: this joint conceptual structure is the key to making nature intelligible to ourselves—transparent to the intellect. Where it applies we have intelligibility–otherwise we don’t.
And what about unintelligibility? What are the paradigm cases of that? Nonsense is surely at the top of the list—garbled speech, ungrammatical sentences, and rampant non sequitur. Here we can make no sense of what we hear: the words don’t add up to a semantically coherent whole. The purpose of words is to join with other words according to rules to produce meaningful sentences, but in nonsense speech this purpose breaks down. The combinatorial power of grammar, itself a type of computational structure, fails to apply to nonsensical products. There is an absence of both fulfilled purpose and mathematical order: it is like saying, “Zero plus addition over prime number equals infinity”—mathematical nonsense. In nonsense abstract form and purpose fail to apply. And the same is true of actions in general: a person’s actions are said to be unintelligible when we can discern no purpose in them, when even the abstract structure belief and desire fails to apply. We also find unintelligibility in science—quantum theory being the prime example. The idea of God playing dice with the universe is an expression of teleological chaos at the root of things—what could God’s purpose be in playing cosmic dice? We feel we cannot make sense of things if no agency could ever act as reality is thought to demand. In the case of the mysteries of mind we also use the notion of unintelligibility—for example, the nexus of consciousness and the brain is said to be unintelligible. Here again we have no coherent mathematics to apply and it is difficult to see how the brain can fulfill the purpose of producing consciousness. If we could see how an agent could build a brain so as to mathematically guarantee that consciousness would be the result, then we would regard the psychophysical nexus as intelligible; but we lack any such understanding, so we declare the connection unintelligible, at least for now. Thus the scaffolding of intelligibility applies in some areas but not in all, more or less dramatically. We can bring it to bear in some cases but not in every case. It is the idea (and ideal) of making intellectual sense—conformity to the paradigms of mathematics and commonsense psychology being the model.
We try to extend the paradigms into various corners of the world; sometimes we succeed, sometimes not. If we lacked these conceptual structures, nothing would be intelligible to us—we would at best have perceptual and causal knowledge (as we may presume is the state of animal cognition). The special type of comprehension that we call intellectual understanding is constituted by these two types of thinking—the mathematical and the teleological. They afford us a kind of transparency and order not available otherwise. And notice that they are not perceptually based: we don’t see the world as mathematically or teleologically ordered; we bring these notions tothe given, rather than deriving them from it. They are not licensed by strict empiricism. There is something projective at work here—imposed, self-generated. We make the world intelligible; we don’t find it to be so—except in the sense that we discover that things turn out that way. We apply our intellect to empirical reality and thereby render it intelligible; we don’t have impressions of intelligibility as we have impressions of color and shape. Intelligibility is not a sense datum.
Some strains of thought have it that the world is actually not intelligible at all, not as an objective trait of reality. Instead we force it into an appearance of intelligibility by imposing our own minds on it. Thus nothing is inherently teleological or mathematical: there is no purpose in psychology and the physical world is not a mathematical structure. Ideally, we should banish both ways of thinking from psychology and physics: no goals and no numbers. One need not agree with this point of view to appreciate its motivation: goals and numbers are not part of the given but a conceptual apparatus that we bring to bear in order to organize the facts. They are how weunderstand things not how things are in themselves (phenomenal not noumenal). Thus we cannot really make sense of the world, only our apprehension of it; in itself the world is without sense, not subject to intellectual comprehension at all. It is all, as the saying goes, just one damn thing after another, without rhyme or reason. To say that the world is intelligible can only mean that we can apply the apparatus of mathematics and teleology to it in order to organize our knowledge, but it is quite indifferent to these invocations. This is certainly an intelligible position to take on intelligibility, to be set beside the more realist position that goals and numbers are part of the fabric of objective reality. I won’t attempt to decide the issue, though I incline to the realist view.
 The same is true of exercises of theoretical reasoning, i.e. acquiring beliefs by rational processes: here we use logic, a normative discipline, as a means of rendering belief formation intelligible. It is like applying mathematics to empirical reality.
 It is sometimes supposed that the touchstone of intelligibility is conformity to commonsense categories and principles, as with the idea that causation works only by physical contact. But conformity to common sense is neither necessary nor sufficient for intelligibility: much of physics is not part of common sense but quite intelligible (the same is true of pure mathematics), and some commonsense categories are not intelligible (at least presently)—such as consciousness and free action, arguably. Moreover, common sense has little to do with specifically intellectual knowledge as opposed to practical knowledge.
 We can compare mathematics with morality in this respect: moral realism can be combined with a functional view of morality. Its truths are not dependent on the human mind, but they fit human life remarkably well, as if designed to do so. Morality is not irrelevant to human life, even if its basis is extra-human. Indeed, we need to be able to combine both moral realism and moral relevance in order to give a satisfactory account of morality, as we also do in the case of mathematics.
 I have not discussed the extremely controversial question of whether philosophy itself renders the world intelligible. The question turns on the nature of philosophy and the form of its findings. Let me just remark that concepts have a purpose and that the notion of a calculus of concepts is not to be rejected out of hand. On the other hand, philosophy is the domain of mystery par excellence.