Happily, Edward Skidelsky throws some light on this problem for me, as he devotes Chapter 6 of his Cassirer book to the differing encounters of Cassirer and Wittgenstein with the logical positivists who dominated the Vienna Circle.
The author begins the chapter by foregrounding the cordial relations between Cassirer and the positivists, and then defining their differences:
...although Cassirer and the logical positivists both strove to advance the cause of enlightenment, they envisaged this task quite differently... [The positivists'] ambition was to establish a rule separating sentences of science from sentences of metaphysics or pseudoscience.
In contrast, Cassirer's "aim was not to sequester science but rather to reveal its continuity with the other forms of culture."
After further exploration of the positivists' philosophical backgrounds, Skidelsky proceeds to analyze the influence of Wittgenstein's 1922 TRACTATUS on the Vienna Circle, and finds it "decisive."
[Wittgenstein's] doctrine that logic is nothing but a set of tautologies, combined with the logicist claim that all mathematics is reducible to logic, yielded the highly satisfactory conclusion that mathematics too is nothing but a set of tautologies. The main obstacle to radical empiricism was thereby removed.
However, Skidelsky stresses that while Wittgenstein was concerned only with the exercise of pure logic, the positivists chose to extend his meaning into the domain of epistemology. In other words, if Wittgenstein wrote:
To understand a proposition means to know what is the case if it is true
The positivists extended this to mean:
To understand a proposition means to know how to establish that it is true
Thus the positivists took the question back into the "truth-finding" direction which also governed Russell and Frege's transformations of symbolic logic. However, Skidelsky finds that Wittgenstein did not perfect his "sentences of science" for the same purpose as the positivists.
Wittgenstein's purpose in tightening the bounds of sense was not to destroy what lay on the other side but, on the contrary, to guard it against colonization by science... Thus by a curious irony, a technique arising out of a highly individual, mystical vision of the world ended up in the service of an anti-individualist, antimystical political agenda.
I cannot speak to the veracity of Skidelsky's findings on Wittgenstein's motives. I will note that my principal response to the TRACTATUS was that I too assumed that the author shared the purpose of the positivists: to devalue "sentences of metaphysics or pseudoscience."
Skidelsky then devotes most of the remainder of Chapter 6 to Cassirer's rejection of the positivist creed. He expatiates on why Cassirer's believe in a Goethean concept of "unifying reason" was out of step with the times, which insisted on either pure empiricism (Comte, Mach) and pure "irrationalism" (Nietzsche, Bergson). He concludes that Wittgenstein understood the standards of the "logicists," as Cassirer never completely did.
It is this that makes [Wittgenstein's] critique of scientism so much more powerful than Cassirer's. He has, so to speak, passed through and out the other side of the logicist mill, whereas Cassirer has not even entered it. Wittgenstein's humanism is radical; Cassirer's remains, in the last resort, that of a distinguished representative of a particular cultural tradition.
I cannot deny that Wittgenstein, even today, is viewed with more approval than Cassirer. Yet I must ask: how many persons interested in philosophy are even aware of Wittgenstein's "mystical vision," or his critique of scientism, and how many have made the same assumptions that the Vienna Circle did, translating pure logic into empiricist epistemology? Cassirer may not be understood by the average readers of philosophy today; he may well be regarded as "old hat." But do these readers understand that Wittgenstein opposed empiricist scientism?
I tend to doubt it, and I'm tempted to make a survey of philosophy blogs to determine how many people today write of "Wittgenstein, anti-empiricist." Wittgenstein's focus upon a logic denuded of and distanced from all sensuous content is at base allied to the language used by science: what Philip Wheelwright insightfully terms "steno-language:"
…meanings that can be shared in exactly the same way by a very large number of persons—in general, by all persons using the same language or the same group of inter-translatable languages. Examples are so obvious that they may be mentioned without explanation. Common words like child, parent, dog, tree, sky, etc., are steno-symbols, and their accepted meanings are steno-meanings, because what each of the words indicates is a set of definable experiences (whether actual or only possible) which are, in certain recognizable respects, the same for all who use the word correctly. (Metaphor and Reality, p. 33.)
While I wouldn't agree with McLuhan that the medium is always the message, in this case McLuhan's rule fully applies. Words without sensuous content are steno-symbols, whether they are used for the purpose of science, logic, or philosophy. One may disagree with Cassirer, or creatively misread him. But his constant insistence on the "expressive" function of humankind-- an expressivity that lines up well with Wheelwright's "plurisignative" language-- makes it well-nigh impossible to ignore Cassirer's critique of empiricism.
On a closing note, at the end of Skidelsky's Chapter 6 he makes this observation:
The logical positivists... were heirs to Wittgenstein's foundationalism. Their goal was an a priori theory of meaning in general, a firm standard against which to measure the particular theories embodied in the special sciences. To Cassirer, of course, it looked as if they had simply sublimated the structure of one particular special science-- physics-- into the form of meaning as such.
Interestingly, in my readings of biological theorist Stuart A. Kauffman-- beginning here-- Kauffman finds his branch of science equally hamstrung by modern monocausalists who insist that biology must reduce down to physics. It increasingly begins to seem like even true science has little use for the reductive qualities of scientism.