Can we provide a visual model of the hierarchy of concepts in addition to the descriptive, as was done by Ayn Rand ?
Would it be possible to model the hierarchy of concepts as a mathematical graph or network (set of vertices and edges, or nodes and links) ? Suppose every concept is a node, and links are directed according to dependency between a concept that is based on more earlier concept (directed from later concept to earlier concept). Does this graph illustrate units ? Can the graph be modified to be show more information my coloring nodes or adding weights to the links ? Can we read off more information from the graph ? Could we read off distinguishing characteristics from it ? Can we read off additional information from a "dual" of this graph, in which every node becomes an link, and every link becomes a node ?
Would it be possible to create a visual presentation of just one definition of some concept ? To indicate all the referent concepts and link them, while labeling their relationship on the links ?
Would a graph of all definitions be basically the same thing as a graph a of all concepts ?
It might be possible to construct a "tree graph" of concepts by sifting through a dictionary and identifying the defining genus of every concept in the dictionary. The genus would identify the link from a concept to its definitional "parent." Do the same for the "parent," and so on. One would find that many concepts would have the same genus. One might also find that some concepts belong to more than one genus, although only one genus would be considered the defining genus of the concept.
First-level abstractions, of course, would not have any further concepts as units. The units of first-level abstractions would be directly sensory-perceptual concretes.
Having done that, however, the result probably would be nearly as big and complex as the original dictionary. What, then, would such a visual model add to man's knowledge and understanding, beyond what he already accomplishes with dictionaries and the natural operation of his inherently human capacity for cognitive integration? (If the question is actually asking about objectivity and how to achieve it, it could be asked explicitly as a question by itself, assuming that the questioner is making a conscientious effort to understand and not merely demanding that others try to convince him while he himself remains cognitively passive or actively disintegrating.)
Some observers might be interested in making machines more visibly "intelligent" in interacting with man and/or aiding man's efforts at more highly automated observation and cognition. Some kind of computerized "tree graph" representing a dictionary might prove useful in an effort to enhance machine "intelligence." It wouldn't necessarily approximate human intelligence very closely, however, since man's knowledge is always contextual (hence, often varying from person to person) and continually expanding or restructuring as man learns.
A comment asks about circularity ("cycles") in a hierarchy of concepts. Certainly an ordinary dictionary will yield many examples of circular definitions, which is one of the problems (deficiencies) with ordinary dictionaries. The kind of conceptual hierarchy identified by Objectivism can never be circular, by the nature of what an objective, reality-based hierarchy is. Hierarchy arises in connection with abstraction from abstractions, i.e., integrating "lower level" abstractions into higher level ones (or subdividing wider concepts into narrower categories, starting with an already formed wider concept first, to arrive at subdivisions of it). "Lower level" means closer to perceptual concretes. Higher level means farther away from perceptual concretes in terms of the number of levels of abstraction needed to reach the higher level abstractions. If the whole hierarchy is grounded in perceptual concretes, as it needs to be to be considered objective, then it has a definite "stopping point" in the process of tracing the integrations back to their roots one level at a time. There is always "bedrock" in a valid conceptual hierarchy, consisting of perceptual concretes. For additional overview, refer to the topic of "Hierarchy of Knowledge" in The Ayn Rand Lexicon. For greater detail, refer to ITOE Chapter 3, "Abstraction from Abstractions," and OPAR Chapter 4, "Objectivity," section titled, "Knowledge as Hierarchical."
Modern mathematics deals with abstract ideas such as "sets" and "classes," which are not necessarily grounded in anything concrete. That is a key difference between modern math and objective human cognition.
Update: Integration and Subdivision
One key point about abstraction from abstractions is that it proceeds in two directions: not just integrating narrower concepts into wider ones, but also subdividing wider concepts into narrower categories. To depict the resulting hierarchy of logical interdependence of concepts in some kind of graphical form, one would need to account for both directions of abstraction from abstractions. On thinking about this a little more fully and reviewing the main Objectivist references (especially ITOE2 Chap. 3 and 5 and related material in the Appendix), I don't think that simply looking at the genus and differentia of concepts would necessarily be enough. The genus of a concept could be either a wider category that was formed by integrating a number of narrower ones, or it could be a wider category that had to come first to allow the narrower subdivisions to be identified. We would need to know more than just the genus-versus-instance relation in order to identify the level of abstraction of the concepts. Level of abstraction, in turn, is the essence of what Objectivism denotes by the "hierarchy of knowledge." (See "Hierarchy of Knowledge" in the Objectivist Lexicon.) The hierarchy involved here is epistemological, i.e, pertaining to the logical order in which concepts have to be formed or understood.
ITOE Chap. 3 (partially excerpted in the Lexicon) discusses "table" versus "desk," with "desk" identified as a more abstract subdivision of "table." That puts "desk" at a higher level in the hierarchy of abstraction than "table," even though "desk" is a type of table (a subdivision of table).
"Workspace" doesn't quite seem to fit here (and isn't mentioned in ITOE or anywhere else in the main, electronically searchable literature of Objectivism). "Workspace" to me means simply a "space" in which to "work," which doesn't have to be a type of table. If the questioner is hypothesizing that we integrate tables with other types of "working spaces" ("spaces for working") to form the compound concept "workspace," then "workspace" is a higher-level abstraction than "desk" or "table" and integrates both, along with many other "spaces" that are not tables.
In trying to classify "workspace" as an instance of "desk," the questioner actually seems to be referring to the process of clearing a space in which to work on a table top. But that does not make the whole table an instance of "workspace," since "workspace" really applies only to part of the table in that interpretation. Also, if "workspace" denotes an area that is somewhat larger than a table, in which a table resides for working on, then again the concept "workspace" is referring in that perspective to more than just the table. "Workspace" is actually denoting a relationship between the table and the space in which someone works, whether the space is on top of the table or surrounding it.
If I've misunderstood Objectivism in any of the foregoing, I'm open to others' informed clarifications. In any case, note that the "hierarchy of knowledge," as Objectivism uses that expression, can never be circular. Concepts always have a definite order of logical dependence, starting from perceptual concretes and proceeding onward from there. And trying to represent that dependence with an analogy to a living tree has definite limitations, such as whether the perceptual concretes should be represented by the tree's roots or the tree's leaves. I personally like the formulation, "Look for the tree among the leaves," as an expression of the importance of integration and identifying essentials. But injunctions to pay attention to the roots of one's "trees" has great value for living, as well, whenever one finds oneself or others talking about trees "floating in the air."