In philosophy, the computational theory of mind (CTM) refers to a family of views that hold that the human mind is an information processing system and that cognition and consciousness together are a form of computation. Warren McCulloch and Walter Pitts (1943) were the first to suggest that neural activity is computational. They argued that neural computations explain cognition. The theory was proposed in its modern form by Hilary Putnam in 1967, and developed by his PhD student, philosopher and cognitive scientist Jerry Fodor in the 1960s, 1970s and 1980s. Despite being vigorously disputed in analytic philosophy in the 1990s due to work by Putnam himself, John Searle, and others, the view is common in modern cognitive psychology and is presumed by many theorists of evolutionary psychology. In the 2000s and 2010s the view has resurfaced in analytic philosophy (Scheutz 2003, Edelman 2008).
The computational theory of mind holds that the mind is a computational system that is realized (i.e. physically implemented) by neural activity in the brain. The theory can be elaborated in many ways and varies largely based on how the term computation is understood. Computation is commonly understood in terms of Turing machines which manipulate symbols according to a rule, in combination with the internal state of the machine. The critical aspect of such a computational model is that we can abstract away from particular physical details of the machine that is implementing the computation. This is to say that computation can be implemented by silicon chips or neural networks, so long as there is a series of outputs based on manipulations of inputs and internal states, performed according to a rule. CTM, therefore holds that the mind is not simply analogous to a computer program, but that it is literally a computational system.
Computational theories of mind are often said to require mental representation because 'input' into a computation comes in the form of symbols or representations of other objects. A computer cannot compute an actual object, but must interpret and represent the object in some form and then compute the representation. The computational theory of mind is related to the representational theory of mind in that they both require that mental states are representations. However, the representational theory of mind shifts the focus to the symbols being manipulated. This approach better accounts for systematicity and productivity. In Fodor's original views, the computational theory of mind is also related to the language of thought. The language of thought theory allows the mind to process more complex representations with the help of semantics. (See below in semantics of mental states).
Recent work has suggested that we make a distinction between the mind and cognition. Building from the tradition of McCulloch and Pitts, the Computational Theory of Cognition (CTC) states that neural computations explain cognition. The Computational Theory of Mind asserts that not only cognition, but also phenomenal consciousness or qualia, are computational. That is to say, CTM entails CTC. While phenomenal consciousness could fulfill some other functional role, computational theory of cognition leaves open the possibility that some aspects of the mind could be non-computational. CTC therefore provides an important explanatory framework for understanding neural networks, while avoiding counter-arguments that center around phenomenal consciousness.
Computational theory of mind is not the same as the computer metaphor, comparing the mind to a modern-day digital computer. Computational theory just uses some of the same principles as those found in digital computing. While the computer metaphor draws an analogy between the mind as software and the brain as hardware, CTM is the claim that the mind is a computational system.
'Computational system' is not meant to mean a modern-day electronic computer. Rather, a computational system is a symbol manipulator that follows step by step functions to compute input and form output. Alan Turing describes this type of computer in his concept of a Turing Machine.
One of the earliest proponents of the computational theory of mind was Thomas Hobbes, who said, "by reasoning, I understand computation. And to compute is to collect the sum of many things added together at the same time, or to know the remainder when one thing has been taken from another. To reason therefore is the same as to add or to subtract." Since Hobbes lived before the contemporary identification of computing with instantiating effective procedures, he cannot be interpreted as explicitly endorsing the computational theory of mind, in the contemporary sense.
At the heart of the Computational Theory of Mind is the idea that thoughts are a form of computation, and a computation is by definition a systematic set of laws for the relations among representations. This means that a mental state represents something if and only if there is some causal correlation between the mental state and that particular thing. An example would be seeing dark clouds and thinking “clouds mean rain”, where there is a correlation between the thought of the clouds and rain, as the clouds causing rain. This is sometimes known as Natural Meaning. Conversely, there is another side to the causality of thoughts and that is the non-natural representation of thoughts. An example would be seeing a red traffic light and thinking “red means stop”, there is nothing about the color red that indicates it represents stopping, and thus is just a convention that has been invented, similar to languages and their abilities to form representations.
The computational theory of mind states that the mind functions as a symbolic operator, and that mental representations are symbolic representations; just as the semantics of language are the features of words and sentences that relate to their meaning, the semantics of mental states are those meanings of representations, the definitions of the ‘words’ of the language of thought. If these basic mental states can have a particular meaning just as words in a language do, then this means that more complex mental states (thoughts) can be created, even if they have never been encountered before. Just as new sentences that are read can be understood even if they have never been encountered before, as long as the basic components are understood, and it is syntactically correct. For example: “I have eaten plum pudding every day of this fortnight.” While it's doubtful many have seen this particular configuration of words, nonetheless most readers should be able to glean an understanding of this sentence because it is syntactically correct and the constituent parts are understood.
A range of arguments have been proposed against physicalist conceptions used in Computational Theories of Mind.
An early, though indirect, criticism of the Computational Theory of Mind comes from philosopher John Searle. In his thought experiment known as the Chinese room, Searle attempts to refute the claims that artificially intelligent systems can be said to have intentionality and understanding and that these systems, because they can be said to be minds themselves, are sufficient for the study of the human mind. Searle asks us to imagine that there is a man in a room with no way of communicating with anyone or anything outside of the room except for a piece of paper with symbols written on it that is passed under the door. With the paper, the man is to use a series of provided rule books to return paper containing different symbols. Unknown to the man in the room, these symbols are of a Chinese language, and this process generates a conversation that a Chinese speaker outside of the room can actually understand. Searle contends that the man in the room does not understand the Chinese conversation. This is essentially what the computational theory of mind presents us—a model in which the mind simply decodes symbols and outputs more symbols. Searle argues that this is not real understanding or intentionality. This was originally written as a repudiation of the idea that computers work like minds.
Searle has further raised questions about what exactly constitutes a computation:
the wall behind my back is right now implementing the WordStar program, because there is some pattern of molecule movements that is isomorphic with the formal structure of WordStar. But if the wall is implementing WordStar, if it is a big enough wall it is implementing any program, including any program implemented in the brain.
Objections like Searle’s might be called insufficiency objections. They claim that computational theories of mind fail because computation is insufficient to account for some capacity of the mind. Arguments from qualia, such as Frank Jackson’s Knowledge argument, can be understood as objections to computational theories of mind in this way—though they take aim at physicalist conceptions of the mind in general, and not computational theories specifically.
There are also objections which are directly tailored for computational theories of mind.
Putnam himself (see in particular Representation and Reality and the first part of Renewing Philosophy) became a prominent critic of computationalism for a variety of reasons, including ones related to Searle's Chinese room arguments, questions of world-word reference relations, and thoughts about the mind-body relationship. Regarding functionalism in particular, Putnam has claimed along lines similar to, but more general than Searle's arguments, that the question of whether the human mind can implement computational states is not relevant to the question of the nature of mind, because "every ordinary open system realizes every abstract finite automaton." Computationalists have responded by aiming to develop criteria describing what exactly counts as an implementation.  
Roger Penrose has proposed the idea that the human mind does not use a knowably sound calculation procedure to understand and discover mathematical intricacies. This would mean that a normal Turing complete computer would not be able to ascertain certain mathematical truths that human minds can.