I'm not sure where I read this view first though, as it certainly has been defended by several authors before (see Soames below). And I hadn't appreciated how much in tension it is with other widely held views in foundations of semantics, and of linguistics more generally, until I read Putnam 1985, "A Comparison of Something with Something Else" (which I didn't until sometime after 2000 or so: Putnam's article, as well as being quite funny, makes clear what his objections to Tarski are). In all of these cases, the Abstract View emerges as a response to three major criticisms of Tarski’s semantic conception of truth: these are
(i) the “modal objection” (which has a clear formulation in Putnam 1985);A recent discussion of some of these objections appears in a 2008 article, "Truth, Meaning and Translation" (in D. Patterson (ed.) 2008, New Essays on Tarski and Philosophy, OUP), by Panu Raatikainen, who points out that Putnam also attributes something like the Abstract View to Rudolf Carnap. (Whether Tarski himself held the view or something like it is an interesting matter of historical exegesis.)
(ii) the “use objection” (which also has a clear formulation in Putnam 1985);
(iii) the “non-explanatory list” objection (Field 1972, "Tarski's Theory of Truth").
The earliest formulation I know of the Abstract View appears in Scott Soames’ 1984 article, “What is a Theory of Truth?” (J. Phil). As Soames notes, this view was suggested to him by reading David Lewis 1975, "Languages and Language".
Here is Scott Soames:
We have, then, two major objections to Tarski. Field demands that semantic properties be dependent on speakers in a way in which Tarski's substitutes are not. A familiar sort of semantic theorist demands that meaning and truth conditions be contingent, but analytically connected, properties of a sentence in a manner in- compatible with Tarski. The only way to defend Tarski's philosophical interpretation of his work is to reject these demands.
Although this might initially seem to be a desperate strategy, it is not. Think of a standard first-order language $L$ as a triple $(S_L, D_L, F_L)$, where $S_L$ is a family of sets representing the various categories of well-formed expressions of $L$; $D_L$ is a domain of objects; and $F_L$ is a function that assigns objects in $D_L$ to the names of $L$, subsets of the domain to one-place predicates of $L$, and so on. Let J be a class of such languages. Truth can now be defined in nonsemantic terms for variable '$L$' in J in a straightforward Tarskian fashion.
The only significant change from before is that the notions of primitive denotation are no longer given language-specific list definitions, but rather are defined for variable '$L$' using the "interpretation" functions built into the languages. In particular,a name $n$ refers to an object $o$ in a language $L$ iff $F_L(n) = o$.The resulting truth predicate is just what is needed for metatheoretical studies of the nature, structure, and scope of a wide variety of theories.
What the truth definition does not do is tell us anything about the speakers of the languages to which it applies. On this conception, languages are abstract objects, which can be thought of as bearing their semantic properties essentially. There is no possibility that expressions of a language might have denoted something other than what they do denote; or that the sentences of a language might have had different truth conditions. Any variation in semantic properties (across worlds) is a variation in languages. Thus, semantic properties aren't contingent on anything, let alone speaker behavior.
What is contingent on speaker behavior is which language a person or population speaks and which expression a given utterance is an utterance of. Let $L_1$ and $L_2$ be two languages in J which are identical except for the interpretations of certain nonlogical vocabulary-perhaps the color words in $L_1$ are shape words in $L_2$. We can easily imagine a situation in which it is correct to characterize $L_1$, rather than $L_2$, as the language of a given population. To ask what such a characterization amounts to, and what would justify it, is to ask not a semantic question about the languages, but a pragmatic question about their relation to speakers. (Soames 1984, pp. 425-6. Emphasis above added)
Soames's footnotes are also interesting. He says that this way of looking at things was suggested to him by David Lewis 1975, "Languages and Language", and by remarks by Saul Kripke in a seminar on truth in 1982.
Footnote 24. This sort of construction is familiar from model theory. However, its use here is different from model-theoretic treatments. Here we are not defining truth in $L$ relative to a model, but rather truth in $L$ (simpliciter) for an enriched conception of a language. This way of looking at things was suggested to me from two sources: David Lewis's "Languages and Language," in K. Gunderson, ed., Minnesota Studies in the Philosophy of Science, VII (Minneapolis: U of Minnesota Press, 1975), pp. 3-35; and one of Saul Kripke's seminars on truth, Princeton, 1982.Given the Lewis-Soames-Kripke connection, one might be tempted to call this the Princeton View! Though it's safer to stick with the Abstract View.
Footnote 25. Note, $F_L$ is a purely mathematical object--a set of pairs, if you like. Thus, it does not incorporate any undefined semantic notions. This was one of the points noted by Kripke in the seminar mentioned in fn 24.
A dramatic consequence of this view (amongst others) is that nominalism about abstract entities becomes very hard to salvage if languages themselves are abstract entities. For the nominalist wishes to defend the view that there are no numbers, or sets, or functions, or structures, or sequences, or types, etc., while taking language as a given. However, if the Abstract View of language is anything like right, then this given is already abstract in the way that nominalists reject. This objection is not new (in effect, it appears in Quine's writings frequently after the joint 1947 paper, "Steps Toward a Constructive Nominalism", with Nelson Goodman, attempting to base a workable theory of syntax on concrete tokens; Quine seems to have given up this approach quite quickly) and is one that I've made to nominalists in talks and seminars for a long time (a version, discussing syntax and metalogic, appears in Ch. 1, Scs 1.7 and 1.8, of my 1998 PhD thesis, "The Mathematicization of Nature").
[Update, 10:38pm - some edits!]