Critiques of the real vs. nominal definition distinction

Real definition points to Socratic dialogues: Socrates' search about definitions is not simply about words meaning, but about the essence or real nature of a thing, sought through the method of questioning (the Socratic method) to find universal truths for virtues like justice or piety.

According to Plato's Socrates, the understanding of "real definitions" was crucial for true knowledge. See e.g. Euthyphro, 6d: "you did not give me sufficient information before, when I asked what holiness was, but you told me that this was holy which you are now doing, prosecuting your father for murder. [...] this is not what I asked you, to tell me one or two of the many holy acts, but to tell the essential aspect, by which all holy acts are holy.

True knowledge is not knowing some relevant properties of facts, but grasping "the essential aspect". See Realism.

If you are a Nominalist, there is no essence, and thus definitions must be nominal.

See Real and nominal definitions:

Whether the search for an answer to the Socratic question “What is virtue?” is a search for real definition or one for nominal definition depends upon one’s conception of this particular philosophical activity. When we pursue the Socratic question, are we trying to gain a clearer view of our uses of the word ‘virtue’, or are we trying to give an account of an ideal that is to some extent independent of these uses? Under the former conception, we are aiming at a nominal definition; under the latter, at a real definition.

Dictionary definitions, explaining the meaning of terms through context: their use in speech, are "nominal".

The Logic of definitions can be summarized as a way to abbreviate an already known concept/description (the definiens: the right-hand side of the definition) with a new term (the definiendum: the left-hand side).

When we define one as the successor of zero we have already introduced, in this case by way of axioms, the concept successor and the term zero (the number that is not a successor).

In order to avoid circularity, that we find in a dictionary, we have to start with some basic terms that are undefined and whose elucidation is made through context: the axioms governing them.

Further attempts to define also basic term must either relies on some more fundamental theory: e.g. the set-theoretic foundations of arithmetic, or fails: the (presumably interpolated) Euclid's definitions of point and line.

Regarding Wittgenstein, see e.g. Tractatus (1921):

3.26 A name cannot be dissected any further by means of a definition: it is a primitive sign. 3.261 Every sign that has a definition signifies via the signs that serve to define it; and the definitions point the way.[...] Names cannot be anatomized by means of definitions. [...] 3.263 The meanings of primitive signs can be explained by means of elucidations. Elucidations are propositions that contain the primitive signs.

This view is quite consistent with that expressed by the "second Wittgenastein" into the Philosophical Investigations (1953):

§6 An important part of the training will consist in the teacher's pointing to the objects, directing the child's attention to them, and at the same time uttering a word [...] (I do not want to call this "ostensive definition", because the child cannot as yet ask what the name is. I will call it "ostensive teaching of words".[...])

And §29 [...] how he [the learner] 'takes' the definition is seen in the use that he makes of the word defined.

I have had my Chambers dictionary for almost fifty years. One of its idiosyncrasies is to provide one-word definitions that simply refer the reader to yet another word they don't know and need to look-up elsewhere in its 1,500 pages. They are classic nominal definitions- their purpose is to tell you about what the word means and how it is used. Real definitions have a different target- they are trying to tell you something about things, rather than trying to tell you something about the meaning of words.

Clearly there is a degree of overlap between the two terms- at some point you can only explain the meaning of a word by referring to the properties of the things it points at. An example might be the dictionary definition of 'router' as a woodworking tool with a rotating cutter head. That tells you what the word means by telling you something about the real properties of routers; but the purpose of the definition is to explain the word 'router' not to explain what routers are.

From there we might say that nominal definitions are the typical stuff of dictionaries, while real definitions might be found in an encyclopaedia.

Another good example might be the word 'photon'. You can provide a nominal definition as a quantum of electromagnetic radiation, perhaps, but books could be written about a real definition.