In 1926 Otto Neurath, writing of the Bauhaus movement and its aspiration to transform social and individual life, argues that the key to the success of the Bauhaus movement, the promise it holds out of entering “the promised land,” will be assured if only we “will seize upon the formation of the new form of life as a technical achievement.” (cited in Origins of Logical Empiricism, p. 33). A promising future, therefore, is to be ushered in by way of technological advancements and improvements, much as Dick Fosbury, for example, gave to high jumpers a more promising future as a consequence of the technological improvement he made with respect to the belly roll. The Fosbury flop was a technological improvement of such a magnitude that it wasn’t long after he debuted his new style at the 1968 Mexico City Olympics that it was adopted by all other high jumpers (and is still the dominant style today, despite some minor technical improvements in the syntax of the Fosbury style [more on this below]).
This story is incomplete, however, for it fails to address the equally important role played by qualitative transformations, a point stressed both by Moritz Schlick and Gilles Deleuze.
In an early, 1910 essay ("The Boundaries of Scientific and Philosophical Concept-Formation"), Schlick distinguishes scientific from philosophical concept-formation by attributing to the former the task of establishing quantitative concepts and to the latter the task of creating qualitative concepts (thanks to Eric's essay for the reference). My hesitation with this distinction is that it appears to imply that there is very little room for philosophical activity within science. Now while this may be quite in line with Schlick's (and Carnap's) efforts to provide a technical, formal path for philosophy that can do very well without metaphysics - as Neurath puts it, the modern, technical form of life "is becoming ever more rigorously mechanised and leaves less room for metaphysical ideas." However, I think one can maintain, as Schlick would later recognize as well, that there is room for interplay between philosophy and science and we can make such room precisely by maintaining the quality/quantity distinction, though perhaps not in the sense intended by Sclick in 1910.
To see how this can be done, we can turn to Deleuze's short essay, "Les intercesseurs" / "Mediators" (Pourparlers, 165-184; Negotiations, 121-134), where the quality/quantity distinction is equally important for understanding Deleuze's claim that despite the unique creative productions of philosophy, science, and art - philosophy creates concepts, science creates functions, and art creates sensory aggregates - it is nonetheless possible "for a concept, an aggregate and a function to interact."
An important example for Deleuze that is used to illustrate the possible interactions of philosophy, science and art, and an example that will recur again, and significantly so, in A Thousand Plateaus, is that of a Riemannian space. A Riemannian space is well defined in relation to functions but they also involve, as Deleuze puts it, “setting up little neighboring portions that can be joined up in an infinite number of ways; it made possible, among other things, the Theory of Relativity.” These “neighboring portions” that can be joined up are what Deleuze will in other places discuss as the power of “and,” a power that is integral to his understanding of transcendental empiricism (as I argue here). As Deleuze goes on to argue, such neighboring portions are present in cinema (he thinks Bresson’s films are exemplary in this regard) and in philosophy (as in the case of Quine [see below]). The power of “and” is thus the power of "neighboring portions" (or mediators) that can be “joined up in an infinite number of ways” and don’t predetermine how they actually come to be joined up. Riemann’s developments in differential geometry did not predetermine Einstein’s Theory of Relativity.
Returning to Schlick’s qualitative/quantitative distinction, we can further clarify or supplement his distinction by way of this concept of “neighboring portions.” As Schliesser pointed out (in the essay linked above), a field of inquiry is “on its way to being a science” when it becomes capable of deploying quantitiative concepts as part of a “stable, theory-mediated measurement practice.” As I understand this, a qualitiative transformation occurs when one has created a “neighboring portion” that can be joined up in infinite ways – this is the qualitiative component of creation, whether what is created be concepts, functions, or sensory aggregates. Such neighboring portions, however, are inseparable from current practices, or from current concepts, functions, and sensory aggregates, which are actualized sets of normative spaces (in the Sellarsian sense of "space of reasons") which predetermine in a stable manner a set of ways in which things are to be joined up. The belly flop style of high jumping prior to Fosbury would be an example of an established normative space within which there were minor technical advances and quantitiative improvements (higher jumps).
The Fosbury revolution (as the video above correctly describes it) did not, however, simply represent a quantitative improvement upon the belly roll. It did indeed represent a quantitative, technical improvement, but more importantly we have here a qualitative transformation, a new syntax that is based on the earlier style (e.g., the run up remains much the same) but breaks with the estabished style in a fundamental way. What Fosbury did, in short, was to draw upon the "neighboring portions" of space inseparable from the belly roll style such that it became a qualitatively transformed style, a new syntax, but one that held out great promise for technological and quantitative improvement (hence the rapid, unanimous adoption of the style). The history of sports, Deleuze argues, is filled with such inventors, inventors whose inventions, alas, become the basis for technical mastery and quantitative improvement that more often than not bests the efforts of the inventors themselves. Deleuze then offers a summary of the history of sports that, I would argue, could be equally well extended to the history of philosophy, science, and art:
"it doesn’t matter: the history of sports runs through these inventors [qualitative transformations], who amount in each case to the unexpected, a new syntax, tranformations, and without whom the purely technological advances [quantitative improvements] would have remained quantitative, irrelevant and pointless.”
In the case of the history of philosophy we can take the case of Quine. Prior to Quine the logical empiricists practiced philosophy in what was then the dominant style among analytic philosophers. With his “Two Dogmas of Empiricism” essay, however, Quine brought forth a qualitative transformation of philosophic practice with his concept of synonymy. This concept presented "neighboring portions" of philosophic space that connected with the logical empiricists but qualitatively transformed it (the power of "and") and did so with much promise for technological improvement as one explores the space of possible connections opened up by Quine's concept. Much of the history of analytic philosophy in the latter half of the twentieth century can then be seen as a series of quantitative, technical improvements upon the qualitative concept Quine created.