The deeper Hegelian truth of a conceptual content can only be approached diachronically, via a historical recollective expressive genealogy. But in passing in the course of his world-historically groundbreaking interpretation, Brandom says Hegel rejects the very possibility of conveying a conceptual content by defining it, without saying what definition is or elaborating on what this denial means for the status of definition (Spirit of Trust, p.7). I find this to be ambiguous, and potentially a little misleading. At least within any given synchronic context and to some extent even more broadly, I believe definition in the sense of an Aristotelian “what it is” still has a positive role to play. It would not be reasonable to suppose that Brandom really means to ban the philosophical use of definitions; otherwise, we would have an extreme nominalism incompatible with his stated goals, which include what he calls conceptual realism. (See also Abstract and Concrete.)
The ambiguity in the passage has to do with how strong a sense we give to “conveying”. We should not expect a run-of-the-mill definitional representation to literally convey conceptual (inferential) content in its explicit form. But such a representation absolutely does address or concern conceptual content, and therefore can still “convey” that content in the weaker sense of referring to it or reliably picking it out. (We could also atypically construct definitions in terms of explicit material incompatibilities and consequences. These would presumably in a stronger sense convey the conceptual content isomorphic to them. We could even atypically construct definitions in terms of the current best expressive genealogy, so I don’t really see these as counterposed.)
I do not think Hegel would go so far as to deny the high pragmatic value of definition in synchronic contexts. This is part of the necessary moment(s) of determinacy (and Understanding) in the larger process of the development of Spirit. He just wants to make the larger point that diachronically, any realized ground-level definition is ultimately just a stopping point along the way. That does not mean we should not attempt to sum up the best understanding we have achieved at each moment. I think we are deontically obligated to do just that. Every ground-level definition is contextualized by its historical situation and therefore subject to change, but at every moment we should still strive to speak and act in accordance with the best definitions we can achieve. Representational clarity is imperfect and always dependent on other considerations in the background, but it is still a moment to be preserved.
We should distinguish the conceptual-content-related doing associated with developing a definition from the representation produced. Further, I find it difficult to separate a concern for definition from a methodological concern for problems of definition, as evinced by Plato and Aristotle for instance. From this perspective, definition has more to do with a line of questioning than a putative answer. The question of the “what” or conceptual content of things is actually far more substantial and interesting than those of mere fact or abstract existence. Even if it aims at a representation, definition as a practical task is all about inquiry into that whatness of things. The norm to which synchronic representation of whatness is responsible comes down to the best achievable view of the relevant difference and mediation, or material incompatibility and material consequence (as Brandom would put it) in the circumstances of that logical moment. This I think is actually independent of the diachronic moves of expressive genealogy.
Hegel’s “Substance that is also Subject” is explicitly presented as an extension of Aristotle’s (expressive meta) concept of ousia, and I think Aristotle anticipates even more than Hegel recognizes. (Expressive genealogy is distinctively Hegelian, but Substance certainly not, and Hegel himself notes in the History of Philosophy lectures that the concerns he groups under “Subject” were significantly addressed by Socrates, Plato, and Aristotle.)
If Brandom is right that Hegel intended to exclude such expressive metaconcepts from the general prognosis that all (ground-level) concepts eventually elicit their own negation, then it is at least logically possible that Aristotle’s metaconcept had already achieved the requisite stability to be incorporated by Hegel without negating the subordinate aspect of ousia that for Aristotle corresponds to a definition.
Without prejudice to claims about what Hegel added, I would argue that Hegel did in this way intend to incorporate all the multiple nuances of Aristotelian ousia, including the definitional one. With due respect for Brandom’s distinction between determination as Hegelian process and determinateness as Kantian/Fregean property (and the importance of the process as a superior point of view), I also think we need to forgivingly recollect all best attempts at determinateness. (See also Classification.)
I wonder what Brandom would say about the role of definitions in the articulation of mathematical conceptual content. The doing of mathematics seems to join the doing of history as problematic for simple subsumption under a genealogical approach as Brandom has described it. Mathematics needs definitions, and history needs to evaluate data without Whiggish filtering. (But Brandom does not exactly disallow either, and I can’t imagine that he would want to. The meaning of mathematical theorems can certainly be expressed in terms of material incompatibility and consequence, and the concepts used in non-Whiggish historiography could themselves be Whiggishly genealogically grounded.)
We should think about the functional inferential role of stipulative definitions, as well as the definitions of empirical concepts that I expect Brandom has foremost in mind. We could say that in both cases, the meaning sought by definition — as distinct from the definiens — is actually constituted through material incompatibility and material consequence. But a stipulative definition is a making rather than a taking. It in a sense starts a whole course of reasoning, whereas empirical concepts implicitly summarize results of reasoning.
Also, mathematical definition is mostly concerned with structures and structural properties. I believe a case could be made that in general, such structures and structural properties are expressive metaconcepts in much the same sense that logical concepts are.
I don’t think it’s historically right that expressive metaconcepts are a “discovery or invention” of German Idealism (p.5). Aristotle already had quite a few expressive metaconcepts, as at least partially exhibited in this blog. I believe Hegel himself recognized this.