With certain significant exceptions, every concept can be defined and communicated in terms of other concepts. The exceptions are concepts referring to sensations, and metaphysical axioms.
Sensations are the primary material of consciousness and, therefore, cannot be communicated by means of the material which is derived from them. The existential causes of sensations can be described and defined in conceptual terms (e.g., the wavelengths of light and the structure of the human eye, which produce the sensations of color), but one cannot communicate what color is like, to a person who is born blind. To define the meaning of the concept “blue,” for instance, one must point to some blue objects to signify, in effect: “I mean this.” Such an identification of a concept is known as an “ostensive definition.”
Ostensive definitions are usually regarded as applicable only to conceptualized sensations. But they are applicable to axioms as well. Since axiomatic concepts are identifications of irreducible primaries, the only way to define one is by means of an ostensive definition—e.g., to define “existence,” one would have to sweep one’s arm around and say: “I mean this.”
Now while Rand admits that most concepts are defined in terms of other concepts, she fails to note any infinite regress problems that might arise as a consequence. However, an Objectivist could argue that all concepts are ultimately defined by ostensive definitions, thus preventing the regress from stretching into infinity. Peikoff, in his Objectivism: The Philosophy of Ayn Rand, adopts just such a position:
Materialists sometimes regard the concept of "consciousness" as unscientific on the grounds that it cannot be defined. This overlooks the fact that there cannot be an infinite regress of definitions. All definitions reduce ultimately to certain primary concepts, which can be specified only ostensively; axiomatic concepts necessarily belong to this category. The concept of "matter," by contrast, is not an axiomatic concept and does require a definition, which it does not yet have; it requires an analytical definition that will integrate the facts of energy, particle theory, and more. To provide such a definition is not, however, the task of philosophy, which makes no specialized study of matter, but of physics. As far as philosophic usage is concerned, "matter" denotes merely the objects of extrospection or, more precisely, that of which all such objects are made. In this usage, the concept of "matter," like that of "consciousness," can be specified only ostensively. [34, italics added]Note how Peikoff's acknowledgement that definitions must be reducible to "primary" concepts that can be "specified" ostensively is brought up in the context in a discussion, not of definitions, but of materialism. It's merely an aside, casually tossed to the reader like a bone to a dog. He doesn't even attempt to establish this aside, but takes it for granted, as if it were obviously true and therefore beyond all doubt and discussion. But it is not obviously true; on the contrary, it is, as Hume would say, a palpable absurdity.
Many concepts (or, rather, words) cannot be defined ostensively; at least not in any meaningful sense. Nor is it plausible that they can be defined in terms of words that can be defined ostensively. That is because the terms don't actually refer to things that can be pointed at, but to psychic states or to inferred objects or to complex processes taking place over a period of time. Can the concepts of utility, or neutrino, or simultaneity, or recession be understood, defined, and/or "specified" through a regression to the primary concepts upon which they are (allegedly) ultimately based? Of course not. How do we know this? Because no one actually explains what they mean by defining all their terms (including all the terms used in their definitions) until they reach "primary" concepts that can be "specified" ostensively. Knowledge doesn't work that way. A person either understands a concept or he doesn't. If he doesn't understand the concept, pointing at something will only help if the concept directly refers to the thing pointed at. If the thing refers to something that cannot be pointed at, pointing is no longer an option, even if you began hunting, through the regress of definitions, for terms that can be "specified" ostensively.
Partly this is due to the sheer impracticality of such an enterprise. The understanding of a complex concept is itself very complex. It cannot be reduced to "primaries" defined ostensively, because there is often an immense amount of inference and sheer conjecture going on behind the scenes. Indeed, there are likely unconsious cogitative processes going on that Objectivism not only knows nothing of, but (by implication) denies altogether. Again, knowledge does not work in the simplistic fashion imagined by Aristotle, Rand, and Peikoff. There are no simple hierarchies that can be reduced to a perceptual base that can be defined (or "specified") ostensively. If I don't know what an elephant is, it can be "specified" for me by pointing at an actual elephant. But that won't ever work for utility, or neutrino, or simultaneity, or recession, among a multitude of other concepts that decorate and inform our cerebrations. If I know what those things are but don't know the word used to specify them, a definition might help. But in that case, the definition is being used to assist me with word usage rather than understanding. If I don't know what these words refer to, if I have no understanding of them, no definition will ever make up for this lack, since a definition merely provides the same meaning (i.e., the same understanding) in different words.
There is another issue here as well. Popper argued that definitions lead to an infinite regress. But in practice that is not the case. Since there are only a finite number of words, the attempt to define all words in terms of other words leads, not to infinity, but merely to words previously defined. In other words, it leads to a circle -- which is to say, question begging. Concepts must be understood apart from definitions if they are to be understood at all. Definitions, in any case, don't define concepts; as I have repeatedly insisted, definitions define only word usage. They match words with meanings. To understand a concept is to understand its meaning. To understand a concept's definition is merely to know the name of the concept.