In the Wikipedia article for Computational Theory of the Mind (NOT to be confused with Computational Representational Understanding of Mind which includes Connectionism), it is mentioned that alternatives to this philosophical view are Connectionism and Dynamicism (though there is some debate as to how different Dynamicism is). Why is this? Since they still use mathematical models run by computers, which still take in information and still process it, they seem to still fall under the Computational Theory of the Mind. What distinction am I missing here?
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The Computational Theory of Mind is not that the mind does some form of computation in the wide sense of computation. Rather, look at the examples for the CToM given in the Wikipedia article; people like Fodor, Pinker, Marr. Their view is very much the opposite to the Connectionist position of West Coast scientists like Rumelhart, Elman and McClelland. Both assume the brain does some form of computation, but they differ greatly about what kind of computation it is doing. Fundamentally, by the CToM, cognition is symbol manipulation and the mind Von Neumann-like (with a central-processor accessing memory), and to Connectionism, it isn't.
Historically, much of the foundations of CToM as it is currently used was first and best articulated by David Marr (although the basic thought is of course much older). Marr assumed cognition can be described on 3 distinct levels;
This framework is basically what much of modern CToM operates within.
Fodor or Pinker assume minds operate like a specific form of computer: somewhat akin to a Von Neuman machine. Specifically, minds do symbol manipulation. Symbols are distinct, abstract cognitive entities which are assigned some meaning. Typically, they represent something. So to a CToM person, the mind works by taking a symbol, such as a low-level representation of a small part of the outside world, and performing an operation on it. In vision, it might take low-level visual items and add them together to construct a 3D representation of the outside world. In language, it might take lexical units and operate on them using linguistic rules, yielding a linguistic/semantic representation.
Importantly, in all of these, the thing which is operated on (the symbol) and the thing which does the operation (some mental module) are distinct. Symbols are manipulated via mental modules showing rule-like, algorithmic behaviour. In at least some theories, each manipulation within a module follows in sequence after another.
Connectionism, in contrast, does not assume any of these, and it was in fact originally proposed as a direct answer and alternative to the CToM. In connectionism, the 3 levels approach is often not seen as foundational, or even useful; to study how neurons work is to study how cognition works. In Connectionism, there is no central processing unit and no distinct symbol. There is no difference between "data" (symbols, representation) and the processor. Processing and symbols are implicitly distributed amongst neurons; processing doesn't happen in modules and in sequence, but in parallel. The central unit is not the module, but the network. Neuronal systems don't implement specific rule-like operations like a "normal" computer program, but deal with the world by finding patterns in the input and linking it to dynamically created response patterns. In some frameworks, representationality is disputed.
The same spiel also goes for the Dynamical Systems take on cognition, because when compared to CToM, the two - Dynamicism and Connectionism - are united in what distinguishes them from the CToM. When comparing Dynamicism to the CToM at length, van Gelder immediately and explicitly aligns connectionism and dynamicism when comparing them to
Dynamicism, like connectionism, is non-symbolic, not sold on the Von Neumann model, and emphasizes adaptive processes (such as learning) over static, modular faculties. Moreover, Dynamicism emphasizes interactions with the outside world as well as embodied, non-symbolic phenomena.
As a post-script, I want to note that Marr is perhaps a bit straw manned when he is (by people on the CToM side, usually!) painted as the Arch Symbolist. Marr developed his foundational work together with Poggio, who has worked with neural networks and stochastic models a lot, and is sometimes presented as a Dynamicist(!).
Also: to add to the confusion, CToM research usually features much less interest in explicit computational models than connectionist work. Connectionist research almost always features computer implementations of models, and dynamicism can be heavy on the formulas. CToM research often prefers qualitative descriptions.