Kuhn, Wittgenstein, Gintis: Questioning the Paradigm
Abstract: I briefly review Thomas Kuhn’s notion of “paradigm” and relate it to some aspects of Wittgenstein’s remarks on mathematics and Herbert Gintis’s work in contemporary game theory.
I. Thomas Kuhn and the Paradigm
In short, what I take Kuhn to be trying to achieve in The Structure of Scientific Revolutions is a retelling of the traditional story of scientific history that views the development of science as a cumulative, “progress”-oriented endeavor aimed at discovering and extending boundaries of the unknown. Assumed in this retelling—and we will unpack this more with Dummett and Wittgenstein—is the breakdown of the “Platonist” belief in the ultimate objective foundations of mathematical inference and the development of a constructivist treatment of scientific “invention” in its place. The constructivist argument, in turn, takes seriously the relevance of cultural conventions in the “invention” of method. Kuhn structures this retelling through the identification and analysis of the basal structure of scientific knowledge: the paradigm.
My first question might be what Kuhn means by this elusive term. Early on (p. 10), Kuhn defines the paradigm in relation to “normal science,” arguing that paradigms are systems of rules/practices that 1. commit follower-practitioners not only to certain methods of inference, experience and deduction, but to certain ways of viewing the world (more on this below); and 2. retain a certain open-endedness, granting practitioners a qualified body of data that enables systematic testing against said rules. Normal science occurs between these two poles: experiments are designed to test the predicative apparatus promoted through the particular scientific tradition, while at the same time acknowledging the lingering questions the apparatus invites. To this end, built into every paradigm is a substantial basis from which experiments can measure predictions/expectations against actual events, as well as an intellectual horizon that encourages further research and focuses the attention of researchers on particular sets of problems. No scientist tests the unknown; the paradigm opens a new arena of experimentation as it simultaneously closes certain problems off entirely—this is the process of general “puzzle-solving” the operations of normal science are devoted to exploring.
Throughout Kuhn’s work, however, his definition of a paradigm does tend to fluctuate. By the postscript, Kuhn acknowledges his ambiguous use and ultimately opts to define it closer to what I think someone like Randal Collins is getting at in his Sociology of Philosophies, e.g. as a “disciplinary matrix” that operates through channels of instruction which reproduce techniques qua the repeated use of “exemplars.” What is less ambiguous, however, is Kuhn’s treatment of scientific revolutions. For Kuhn, the proceedings of normal science necessarily encounter, either through accident or (beneficial) error, “anomalies”; for the most part, scientists within traditions tend toward a certain type of legitimated conservatism—anomalies are, more often than not, chalked up to insignificant displacements of expectation or mere user error. Still, the persistence of anomalies build up over time, and this persistence signals, under the right conditions, the moment Kuhn defines as a “crisis.” Where previously the paradigm had “made the anomalous the expected” (p. 53), the accumulation of new anomalies challenge the resilience of the paradigm; in this period, new theories germinate in an attempt to reconcile traditional expectations with new measurements of data. The revolutionary phase is inaugurated as one of these theories institutionalizes itself is a society of practitioners—this process, however, is fairly “messy,” and may take generations (Collins makes this point) to define its own parameters and stabilize.
These shifts are not mere transformations of application, nor are they the extension of traditional knowledge into untapped territory. They are, rather, metamorphoses in worldview, alterations in the interrelations among rules that permit a kind-of narrowed “scope” to a practitioners range of focus. New regions of activity—and their corresponding modes of seeing the world—are activated as others fall into obscurity; invoking Wittgenstein, Kuhn argues: “What were ducks in the scientist’s world before the revolution are rabbits afterwards. The man who first saw the exterior of the box from above later sees its interior from below. Transformations like these, though usually more gradual and almost always irreversible, are common concomitants of scientific training.” (p. 111) Arguing that the paradigm in some sense stands as the backcloth to perception itself, alterations in the structural preconditions of a paradigm translate into a literal re-visioning of the world itself. In moments of scientific revolution, we alter not only the application of the rules, but the criterion by which those applications fundamentally relate to technique.
In the business of identifying similarities, my major questions in relation to Kuhn involve the limitations of applying this notion of a paradigm to other fields of research. How might we formally relate the paradigm to Bourdieu’s habitus, for example? Apart from Kuhn’s own surprise at the popularity of the term, how might we “use” his notion of a paradigm to explain cultural trends, philosophical changes over time, and so forth? This is clearly what Collins is interested in exploring, but I am trying to ask something different: whether we can appropriate this paradigm model to other fields. That is, is there something about the paradigm that is unique to what Wittgenstein calls “mathematical statements”? In some sense, appropriating the language of the paradigm into other humanities-based disciplines runs the risk of explaining everything, and thereby explaining nothing; how are we to take Kuhn-as-practitioner of a paradigmatic method? In what sense has he offered a unifying theory, and it what sense does it design its own boundaries?
II. Wittgenstein and Proofs: Lectures on the Foundations of Mathematics
Imagine you pull up to a 4-way intersection replete with stop signs. You wait your turn as the person to your right goes; when it is your turn, the person to your left cuts in front of you. Your reaction is immediate, nearly visceral: if we take the intersection as a good example of a conventional “game,” then the essence of your felt injustice might be defined as something like the sense that, in playing this “intersection game,” the person who cut you off didn’t make the appropriate move at the appropriate time. We might say that they didn’t play the game “well,” that you wouldn’t know what they take to be the rules of the game at all if this is how they act.
Compare this immediate reaction to the reaction you have while reading “6 + 5 = 23.” I take this to be, in part, Wittgenstein’s point: that we treat mathematical statements in the same manner that we treat constellations of conventions, e.g. as bounded sets of techniques, rules and criteria of application. In so doing, Wittgenstein is implicitly rejecting the Platonism identified by Dummett and Chihara. Responding to the continued influence of Russell and Frege in mathematics, and figures such as Augustine in theories of language, Wittgenstein seems to be shifting the focus away from theories of math as a window into/reflection of objective phenomena and toward constructivist arguments that reduce the phenomenon of “proof” to the practice of technique. To this end, the operative question is not “what is a number?” and more “what are we doing when we use the term ‘number’?”
This turn to “ordinary language philosophy” seems itself to be Wittgenstein’s intervention into a tradition. While a reflection of his own paradigmatic influence, this shift is, I think, incredibly instructive. The rules of mathematics, for Wittgenstein, are related to technique in that they are technique; “You might say,” Wittgenstein begins, “that the relation between a proof and an experiment is that the proof is a picture of the experiment, and is as good as the experiment.” (p. 72) I take Wittgenstein to be offering a sort-of micro-analysis of what the paradigm in operation actually looks like; it is designed, Wittgenstein contends, to enable basic modes of comparison between disparate elements. It is in this fundamental condition of what is “analogous” that what we might call games of analogy emerge: we say to the student that their composition of the sequence “2 4 6 8 10 13” is incorrect on the grounds that the step between 10 and 13 is not analogous to the steps before hand, and further that this sequence is not analogous to the system we would have produced if equally tasked. A new proof, on this account, alters the criterion by which the rules are applied; it alters the use of “analogous” and the reactions we will have—Wittgenstein jumping after the Arabella as it falls into the pool, our reaction to being cheated at the intersection—in relation to the environment thereby conceived through those rules.
I totally disagree with Dummett’s treatment of Wittgenstein on this point. Dummett, as Chihara also identifies, seems to reduce Wittgenstein’s constructivism to the argument that really anyone could ultimately propose any convincing alteration at each step of the proof—that it permits far too much to occur. I don’t think that this is Wittgenstein’s point at all—I think he falls quite closer to Kuhn’s point as to why introducing quantum mechanics to ancient Greece would have not elected Athens into an age of unprecedented science and technology, but would have fallen utterly flat. The importance of Wittgenstein’s overall project, therefore, is to attune us to the question begged by identifying mathematics as a system of applied rules: what is the force of those rules? What constitutes their “compulsion” (see Chihara’s second essay)?
This is where I think Wittgenstein is at his most insightful and most ambiguous. I take him to answer something like: they spring from/interact with a specific domain of activity, e.g. the force of the utterance is grounded in a form of life and in the very way we use words. The proof, to be determined along axes of use/usefulness and not truth/falsity, seeds itself in a particular way of doing things, both enabling applications and exploration along those application-trajectories and discouraging and functionally disabling exploration of entirely different or potentially alternative routes. Mathematics is, in short, not a sequence of discoveries but a process of invention and reinvention, persistently conditioning would-be practitioners into a system of interconnected techniques bounded through the use of examples. Like Kuhn, these examples are enabled by the paradigm itself: the answer is always/already there, in some manner, latently implied by the very tools themselves. In this light, the proof for the calculation “3 x 4” is a proof of “3 x 4” insofar as it gets one to 12; it is a “picture” of how to use a certain method.
In criticizing Wittgenstein, I want to address Chihara’s point that Wittgenstein ultimately fails to refute the realist view. To this, I would add: doesn’t Platonism “sneak in the backdoor,” so to speak, in that we are left with the—pragmatic?—problem that math seems to work? E.g. why, when we use this invented system, does it work better than a religious doctrine? That is, I think that there is something fundamentally right in the things that Wittgenstein is saying; what does this condition of “rightness” suggest about the nature of the paradigm and its relation to the totality of experiential data? Can we say that Wittgenstein is creating a paradigm of his own?
Further, what does he mean by “mathematical statement” and how does it differ from experiential statements? By the factor of its “timelessness”? By their use? Is it reduced merely to the manner we use the criterion of “analogous”? Is the proof the same as a paradigm?
III. Herbert Gintis and Crisis
In some sense, I think Gintis’s The Bounds of Reason constitutes the manifestation of Kuhn’s crisis/revolutionary moment. One gets the sense that Gintis has identified a growing crisis in various fields of applied mathematics and, in accounting for their interrelated encounter of anomalies, is proposing a solution to the problem. His solution, it seems, is the elaboration of evolutionary game theory and an explication of its relation to the assumptions made in the biological, sociological, psychological and economic sciences. By splicing the assumptions of rational choice theory with computational gene-culture coevolution (and so forth), Gintis seems to suggest that these four major fields can be made structurally analogous.
It is here I find Gintis most confusing. It is unclear to me whether Gintis is proposing a new paradigm altogether, or whether he is merely identifying areas of overlap in which these various subfields can supplement or complement one another. In turn, one question might be: what is the difference between offering a theory of interrelated supplementation and offering a new paradigm? Is Gintis successful in his reworking? My feeling is: no, but for very telling reasons. What separates Gintis’ attempt from the creation of a new explanatory model, e.g. the kind-of revolution described by Kuhn? What moment in the history of these disciplines is Gintis inaugurating?