Moving from Levels & Reduction to Dimensions & Constraints

Moving from Levels & Reduction to Dimensions & Constraints David Danks (ddanks@cmu.edu) Department of Philosophy, 135 Baker Hall, Carnegie Mellon University Pittsburgh, PA 15213 USA Abstract Arguments, claims, and discussions about the “level of description” of a theory are ubiquitous in cognitive science. Such talk is typically expressed more precisely in terms of the granularity of the theory, or in terms of Marr’s (1982) three levels (computational, algorithmic, and implementation). I argue that these ways of understanding levels of description are insufficient to capture the range of different types of theoretical commitments that one can have in cognitive science. When we understand these commitments as points in a multi-dimensional space, we find that we must also reconsider our understanding of intertheoretic relations. In particular, we should understand cognitive theories as constraining one another, rather than reducing to one another. Keywords: Level of description; Marr; Philosophy of cognitive science; Reduction; Intertheoretic constraint Limitations of Levels It is customary within science to talk about our theories as falling at different “levels of description”: biology is at a higher level of description than chemistry, which is itself at a higher level than physics. Moreover, talk of levels is not restricted to the relationships between these large-scale domains of science; a sub-symbolic model of causal cognition can be said to be at a lower level of description than some symbolic model of the same cognition or behavior. “Levels talk” is particularly widespread in the cognitive sciences (as noted by many authors, such as Bechtel, 1994; Bickle, 1998; Marr, 1982). The proliferation of talk about levels is quite unsurprising, given the many different methodologies used to develop theories of human behavior and cognition. At the same time, exactly what is meant by a “level” is often left somewhat vague. Levels of description are sometimes identified with the ontological granularity of a theory, where its level is determined (largely) by its objects. This characterization misses important distinctions, however, such as the difference between a rational analysis that says how one should act, and a process model that describes the cognitive mechanisms generating behavior. One of the most precise characterization of levels in cognitive science—and certainly the most influential such characterization—was given by Marr (1982), and captured this key distinction. Marr’s three levels characterize information-processing devices in general, and processes in the human mind more specifically. The computational level identifies the input and output of the process, as well as constraints on the types of computation done on the input to get the output. The algorithmic level (also called the representation level) specifies an implementation of the computational theory, as well as the representation of the input and output of the process. Finally, the implementational level describes the physical realization of the representation and the algorithm. Roughly speaking, the computational level specifies what problem is being (appropriately) solved; the algorithmic level explains how it is solved; and the implementational level gives the details of the physical substrate that does the solving. As a concrete (non-cognitive) example, we can understand a word-processing program as (i) a process for entering, editing, and rendering text documents (the computational level); (ii) a bunch of lines of code that produce the appropriate behavior (the algorithmic level); or (iii) changes of 1’s and 0’s in the internal memory registers of the computer (the implementational level). As a more cognitive example, consider the problem of learning causal structure from observational data (e.g., Cheng, 1997; Griffiths & Tenenbaum, 2005). A computational-level model of this problem would characterize the relevant inputs (case-by-case observations or a summary of a sequence of such cases), the output that should result given such input (a representation that can be used for causal inference, decision-making, explanation, etc.), and any relevant cognitive constraints (though in practice, computational-level models rarely incorporate such constraints). An algorithmic-level model would characterize the internal representations and cognitive processes by which we humans happen to solve this challenge. And an implementation-level model would show how the relevant computations are performed in particular brain regions (e.g., frontal cortex as suggested by Fletcher, et al., 2001 or Satpute, et al., 2005). Marr’s three levels were a significant advance in part because they are based on the recognition that the mathematical or computational specification of a cognitive theory significantly underdetermines the commitments that are implied by it. A Bayesian model of causal learning could, for example, be at the computational or algorithmic level, depending on the intended interpretation of the terms in the model. Moreover, these differences in interpretation (and so commitments) can matter: whether some experiment or behavioral measure is a test of a model depends in part on the commitments of that model. Marr’s levels were also intended to help show that there can be distinct models of the same phenomenon that are not competitors. That is, models M 1 and M 2 can be incompatible (whether mathematically or ontologically) and yet both be correct as long as they are at different levels. For example, Bayesian and associationist models of causal learning are mathematically incompatible—they posit different representations and different learning processes—but can