## 1. Turing machines


The intuitive notions of *computation*
and *algorithm* are central to mathematics. Roughly speaking,
an algorithm is an explicit, step-by-step procedure for answering some
question or solving some problem. An algorithm provides *routine
mechanical instructions* dictating how to proceed at each
step. Obeying the instructions requires no special ingenuity or
creativity. For example, the familiar grade-school algorithms describe
how to compute addition, multiplication, and division. Until the early
twentieth century, mathematicians relied upon informal notions of
computation and algorithm without attempting anything like a formal
analysis. Developments in the foundations of mathematics eventually
impelled logicians to pursue a more systematic treatment. Alan
Turing's landmark paper "On Computable Numbers, With
an Application to the Entscheidungsproblem" (Turing 1936)
offered the analysis that has proved most influential.


A *Turing machine* is an abstract model of an idealized
computing device with unlimited time and storage space at its
disposal. The device manipulates *symbols*, much as a human
computing agent manipulates pencil marks on paper during arithmetical
computation. Turing says very little about the nature of symbols. He
assumes that primitive symbols are drawn from a finite alphabet. He
also assumes that symbols can be inscribed or erased at "memory
locations". Turing's model works as follows:


* There are infinitely many memory locations, arrayed in a linear
structure. Metaphorically, these memory locations are
"cells" on an infinitely long "paper
tape". More literally, the memory locations might be physically
realized in various media (e.g., silicon chips).
* There is a central processor, which can access one memory location
at a time. Metaphorically, the central processor is a
"scanner" that moves along the paper tape one
"cell" at a time.
* The central processor can enter into finitely many *machine
states*.
* The central processor can perform four elementary operations:
write a symbol at a memory location; erase a symbol from a memory
location; access the next memory location in the linear array
("move to the right on the tape"); access the previous
memory location in the linear array ("move to the left on the
tape").
* Which elementary operation the central processor performs depends
entirely upon two facts: which symbol is currently inscribed at the
present memory location; and the scanner's own current machine
state.
* A *machine table* dictates which elementary operation the
central processor performs, given its current machine state and the
symbol it is currently accessing. The machine table also dictates how
the central processor's machine state changes given those same
factors. Thus, the machine table enshrines a finite set of routine
mechanical instructions governing computation.


Turing translates this informal description into a rigorous
mathematical model. For more details, see the entry
on Turing machines.


Turing motivates his approach by reflecting on idealized human
computing agents. Citing finitary limits on our perceptual and
cognitive apparatus, he argues that any symbolic algorithm executed by
a human can be replicated by a suitable Turing machine. He concludes
that the Turing machine formalism, despite its extreme simplicity, is
powerful enough to capture all humanly executable mechanical
procedures over symbolic configurations. Subsequent discussants have
almost universally agreed.


Turing computation is often described as *digital* rather
than *analog*. What this means is not always so clear, but the
basic idea is usually that computation operates over discrete
configurations. By comparison, many historically important algorithms
operate over continuously variable configurations. For example,
Euclidean geometry assigns a large role to *ruler-and-compass
constructions*, which manipulate geometric shapes. For any shape,
one can find another that differs to an arbitrarily small
extent. Symbolic configurations manipulated by a Turing machine do not
differ to arbitrarily small extent. Turing machines operate over
discrete strings of elements (digits) drawn from a finite
alphabet. One recurring controversy concerns whether the digital
paradigm is well-suited to model mental activity or whether an analog
paradigm would instead be more fitting (MacLennan 2012; Piccinini and
Bahar 2013).


Besides introducing Turing machines, Turing (1936) proved
several seminal mathematical results involving them. In particular, he
proved the existence of a *universal Turing machine* (UTM).
Roughly speaking, a UTM is a Turing machine that can mimic any other
Turing machine. One provides the UTM with a symbolic input that codes
the machine table for Turing machine *M*. The UTM
replicates *M*'s behavior, executing instructions
enshrined by *M*'s machine table. In that sense, the UTM
is a *programmable general purpose computer*. To a first
approximation, all personal computers are also general purpose: they
can mimic any Turing machine, when suitably programmed. The main
caveat is that physical computers have finite memory, whereas a Turing
machine has unlimited memory. More accurately, then, a personal
computer can mimic any Turing machine *until it exhausts its
limited memory supply*.


Turing's discussion helped lay the foundations
for *computer science*, which seeks to design, build, and
understand computing systems. As we know, computer scientists can now
build extremely sophisticated computing machines. All these machines
implement something resembling Turing computation, although the
details differ from Turing's simplified model.


## 2. Artificial intelligence


Rapid progress in computer science prompted many, including Turing,
to contemplate whether we could build a computer capable of
thought. *Artificial Intelligence* (AI) aims to construct
"thinking machinery". More precisely, it aims to construct
computing machines that execute core mental tasks such as reasoning,
decision-making, problem solving, and so on. During the 1950s and
1960s, this goal came to seem increasingly realistic (Haugeland
1985).


Early AI research emphasized *logic*. Researchers sought to
"mechanize" deductive reasoning. A famous example was
the *Logic Theorist* computer program (Newell and Simon 1956),
which proved 38 of the first 52 theorems from *Principia
Mathematica* (Whitehead and Russell 1925). In one case, it
discovered a simpler proof than *Principia*'s.


Early success of this kind stimulated enormous interest inside and
outside the academy. Many researchers predicted that intelligent
machines were only a few years away. Obviously, these predictions have
not been fulfilled. Intelligent robots do not yet walk among us. Even
relatively low-level mental processes such as perception vastly exceed
the capacities of current computer programs. When confident
predictions of thinking machines proved too optimistic, many observers
lost interest or concluded that AI was a fool's
errand. Nevertheless, the decades have witnessed gradual progress. One
striking success was IBM's Deep Blue, which defeated chess
champion Gary Kasparov in 1997. Another major success was the
driverless car Stanley (Thrun, Montemerlo, Dahlkamp, et al. 2006),
which completed a 132-mile course in the Mojave Desert, winning the
2005 Defense Advanced Research Projects Agency (DARPA) Grand
Challenge. A less flashy success story is the vast improvement in
speech recognition algorithms.


One problem that dogged early work in AI is *uncertainty*.
Nearly all reasoning and decision-making operates under conditions of
uncertainty. For example, you may need to decide whether to go on a
picnic while being uncertain whether it will rain. *Bayesian
decision theory* is the standard mathematical model of
inference and decision-making under uncertainty. Uncertainty is codified
through *probability*. Precise rules dictate how to update
probabilities in light of new evidence and how to select actions in
light of probabilities and utilities. (See the
entries Bayes's theorem
and normative theories of rational choice: expected utility
 for details.) In the 1980s and
1990s, technological and conceptual developments enabled efficient
computer programs that implement or approximate Bayesian inference in
realistic scenarios. An explosion of Bayesian AI ensued (Thrun,
Burgard, and Fox 2006), including the aforementioned advances in
speech recognition and driverless vehicles. Tractable algorithms that
handle uncertainty are a major achievement of contemporary AI (Murphy 2012), and
possibly a harbinger of more impressive future progress.


Some philosophers insist that computers, no matter how
sophisticated they become, will at best *mimic* rather
than *replicate* thought. A computer simulation of the weather
does not really rain. A computer simulation of flight does not really
fly. Even if a computing system could simulate mental activity, why
suspect that it would constitute the genuine article?


Turing (1950) anticipated these worries and tried to defuse
them. He proposed a scenario, now called *the Turing Test*,
where one evaluates whether an unseen interlocutor is a computer or a
human. A computer *passes the Turing test* if one cannot
determine that it is a computer. Turing proposed that we abandon the
question "Could a computer think?" as hopelessly vague,
replacing it with the question "Could a computer pass the Turing
test?". Turing's discussion has received considerable
attention, proving especially influential within AI. Ned Block (1981)
offers an influential critique. He argues that certain possible
machines pass the Turing test even though these machines do not come
close to genuine thought or intelligence. See the
entry the Turing test for discussion of
Block's objection and other issues surrounding the Turing
Test.


For more on AI, see the entry
 logic and artificial intelligence.
 For much more detail, see Russell and
Norvig (2010).


## 3. The classical computational theory of mind


Warren McCulloch and Walter Pitts (1943) first suggested that
something resembling the Turing machine might provide a good model for
the mind. In the 1960s, Turing computation became central to the
emerging interdisciplinary initiative *cognitive science*,
which studies the mind by drawing upon psychology, computer science
(especially AI), linguistics, philosophy, economics (especially game
theory and behavioral economics), anthropology, and neuroscience. The
label *classical computational theory of mind* (which we will
abbreviate as CCTM) is now fairly standard. According to CCTM, the
mind is a computational system similar in important respects to a
Turing machine, and core mental processes (e.g., reasoning,
decision-making, and problem solving) are computations similar in
important respects to computations executed by a Turing machine. These
formulations are imprecise. CCTM is best seen as a family of views,
rather than a single well-defined
view.[1]


It is common to describe CCTM as embodying "the computer
metaphor". This description is doubly misleading.


First, CCTM is better formulated by describing the mind as a
"computing system" or a "computational system"
rather than a "computer". As David Chalmers (2011) notes,
describing a system as a "computer" strongly suggests that
the system is *programmable*. As Chalmers also notes, one need
not claim that the mind is programmable simply because one regards it
as a Turing-style computational system. (Most Turing machines are not
programmable.) Thus, the phrase "computer metaphor"
strongly suggests theoretical commitments that are inessential to
CCTM. The point here is not just terminological. Critics of CCTM
often object that the mind is not a programmable general purpose
computer (Churchland, Koch, and Sejnowski 1990). Since classical
computationalists need not claim (and usually do not claim) that the
mind is a programmable general purpose computer, the objection is
misdirected.


Second, CCTM is not intended metaphorically. CCTM does not simply
hold that the mind is *like* a computing system. CCTM holds
that the mind *literally is* a computing system. Of course, the
most familiar artificial computing systems are made from silicon chips
or similar materials, whereas the human body is made from flesh and
blood. But CCTM holds that this difference disguises a more
fundamental similarity, which we can capture through a Turing-style
computational model. In offering such a model, we prescind from
physical details. We attain an abstract computational description that
could be physically implemented in diverse ways (e.g., through silicon
chips, or neurons, or pulleys and levers). CCTM holds that a suitable
abstract computational model offers a literally true description of
core mental processes.


It is common to summarize CCTM through the slogan "the mind
is a Turing machine". This slogan is also somewhat misleading,
because no one regards Turing's precise formalism as a plausible
model of mental activity. The formalism seems too restrictive in
several ways:


* Turing machines execute pure symbolic computation. The inputs and
outputs are symbols inscribed in memory locations. In contrast, the
mind receives *sensory input* (e.g., retinal stimulations) and
produces *motor output* (e.g., muscle activations). A complete
theory must describe how mental computation interfaces with sensory
inputs and motor outputs.
* A Turing machine has infinite discrete memory capacity. Ordinary
biological systems have finite memory capacity. A plausible
psychological model must replace the infinite memory store with a
large but finite memory store
* Modern computers have *random access memory*: addressable
memory locations that the central processor can directly
access. Turing machine memory is not addressable. The central
processor can access a location only by sequentially accessing
intermediate locations. Computation without addressable memory is
hopelessly inefficient. For that reason, C.R. Gallistel and Adam King
(2009) argue that addressable memory gives a better model of the mind
than non-addressable memory.
* A Turing machine has a central processor that
operates *serially*, executing one instruction at a time. Other
computational formalisms relax this assumption, allowing multiple
processing units that operate in *parallel*. Classical
computationalists can allow parallel computations (Fodor and Pylyshyn
1988; Gallistel and King 2009: 174). See Gandy (1980) and Sieg (2009)
for general mathematical treatments that encompass both serial and
parallel computation.
* Turing computation is *deterministic*: total computational
state determines subsequent computational state. One might instead
allow *stochastic* computations. In a stochastic model, current
state does not dictate a unique next state. Rather, there is a certain
probability that the machine will transition from one state to
another.


CCTM claims that mental activity is "Turing-style
computation", allowing these and other departures from
Turing's own formalism.


### 3.1 Machine functionalism


 Hilary Putnam (1967) introduced CCTM into philosophy. He
contrasted his position with *logical behaviorism*
and *type-identity theory*. Each position purports to
reveal the nature of mental states, including propositional attitudes
(e.g., beliefs), sensations (e.g., pains), and emotions (e.g.,
fear). According to logical behaviorism, mental states are behavioral
dispositions. According to type-identity theory, mental states are
brain states. Putnam advances an opposing *functionalist*
view, on which mental states are functional states. According to
functionalism, a system has a mind when the system has a 
suitable *functional organization*. Mental states are states
that play appropriate roles in the system's functional
organization. Each mental state is individuated by its interactions
with sensory input, motor output, and other mental states.


Functionalism offers notable advantages over logical behaviorism
and type-identity theory:


* Behaviorists want to associate each mental state with a
characteristic pattern of behavior--a hopeless task, because
individual mental states do not usually have characteristic behavioral
effects. Behavior almost always results from distinct mental states
operating together (e.g., a belief and a desire). Functionalism
avoids this difficulty by individuating mental states through
characteristic relations not only to sensory input and behavior but
also to one another.
* Type-identity theorists want to associate each mental state with
a characteristic physical or neurophysiological state. Putnam casts
this project into doubt by arguing that mental states are *multiply
realizable*: the same mental state can be realized by diverse
physical systems, including not only terrestrial creatures but also
hypothetical creatures (e.g., a silicon-based Martian). Functionalism
is tailor-made to accommodate multiple realizability. According to
functionalism, what matters for mentality is a pattern of
organization, which could be physically realized in many different
ways. See the entry 
multiple realizability
 for further discussion of this argument.


Putnam defends a brand of functionalism now called *machine
functionalism*. He emphasizes *probabilistic automata*,
which are similar to Turing machines except that transitions between
computational states are stochastic. He proposes that mental activity
implements a probabilistic automaton and that particular mental states
are machine states of the automaton's central processor. The
machine table specifies an appropriate functional organization, and it
also specifies the role that individual mental states play within that
functional organization. In this way, Putnam combines functionalism
with CCTM.


Machine functionalism faces several problems. One problem,
highlighted by Ned Block and Jerry Fodor (1972), concerns
the *productivity of thought*. A normal human can entertain a
potential infinity of propositions. Machine functionalism identifies
mental states with machine states of a probabilistic automaton. Since
there are only finitely many machine states, there are not enough
machine states to pair one-one with possible mental states of a normal
human. Of course, an actual human will only ever entertain finitely
many propositions. However, Block and Fodor contend that this
limitation reflects limits on lifespan and memory, rather than (say)
some psychological law that restricts the class of humanly
entertainable propositions. A probabilistic automaton is endowed with
unlimited time and memory capacity yet even still has only finitely
many machine states. Apparently, then, machine functionalism
mislocates the finitary limits upon human cognition.


Another problem for machine functionalism, also highlighted by
Block and Fodor (1972), concerns the *systematicity of
thought.* An ability to entertain one proposition is correlated
with an ability to think other propositions. For example, someone who
can entertain the thought *that John loves Mary* can also
entertain the thought *that Mary loves John*. Thus, there seem
to be systematic relations between mental states. A good theory should
reflect those systematic relations. Yet machine functionalism
identifies mental states with unstructured machines states, which lack
the requisite systematic relations to another. For that reason,
machine functionalism does not explain systematicity. In response to
this objection, machine functionalists might deny that they are
obligated to explain systematicity. Nevertheless, the objection
suggests that machine functionalism neglects essential features of
human mentality. A better theory would explain those features in a
principled way.


While the productivity and systematicity objections to machine
functionalism are perhaps not decisive, they provide strong impetus to
pursue an improved version of CCTM. See Block (1978) for additional
problems facing machine functionalism and functionalism more
generally.


### 3.2 The representational theory of mind


Fodor (1975, 1981, 1987, 1990, 1994, 2008) advocates a version of
CCTM that accommodates systematicity and productivity much more
satisfactorily. He shifts attention to the *symbols*
manipulated during Turing-style computation.


An old view, stretching back at least to William of
Ockham's *Summa Logicae*, holds that thinking occurs in
a *language of thought* (sometimes
called *Mentalese*). Fodor revives this view. He postulates a
system of mental representations, including both primitive
representations and complex representations formed from primitive
representations. For example, the primitive Mentalese words JOHN,
MARY, and LOVES can combine to form the Mentalese sentence JOHN LOVES
MARY. Mentalese is *compositional*: the meaning of a complex
Mentalese expression is a function of the meanings of its parts and
the way those parts are combined. Propositional attitudes are
relations to Mentalese symbols. Fodor calls this view *the
representational theory of mind* (*RTM*). Combining RTM
with CCTM, he argues that mental activity involves Turing-style
computation over the language of thought. Mental computation stores
Mentalese symbols in memory locations, manipulating those symbols in
accord with mechanical rules.


A prime virtue of RTM is how readily it accommodates productivity
and systematicity:


*Productivity*: RTM postulates a finite set of primitive
Mentalese expressions, combinable into a potential infinity of complex
Mentalese expressions. A thinker with access to primitive Mentalese
vocabulary and Mentalese compounding devices has the potential to
entertain an infinity of Mentalese expressions. She therefore has the
potential to instantiate infinitely many propositional attitudes
(neglecting limits on time and memory).


*Systematicity*: According to RTM, there are systematic
relations between which propositional attitudes a thinker can
entertain. For example, suppose I can think that John loves Mary.
According to RTM, my doing so involves my standing in some
relation *R* to a Mentalese sentence JOHN LOVES MARY, composed
of Mentalese words JOHN, LOVES, and MARY combined in the right way. If
I have this capacity, then I also have the capacity to stand in
relation *R* to the distinct Mentalese sentence MARY LOVES
JOHN, thereby thinking that Mary loves John. So the capacity to think
that John loves Mary is systematically related to the capacity to
think that Mary loves John.


By treating propositional attitudes as relations to complex mental
symbols, RTM explains both productivity and systematicity.


CCTM+RTM differs from machine functionalism in several other
respects. First, machine functionalism is a theory of mental
states *in general*, while RTM is only a theory of
propositional attitudes. Second, proponents of CCTM+RTM need not say
that propositional attitudes are individuated functionally. As Fodor
(2000: 105, fn. 4) notes, we must
distinguish *computationalism* (mental processes are
computational) from *functionalism* (mental states are
functional states). Machine functionalism endorses both doctrines.
CCTM+RTM endorses only the first. Unfortunately, many philosophers
still mistakenly assume that computationalism entails a functionalist
approach to propositional attitudes (see Piccinini 2004 for
discussion).


Philosophical discussion of RTM tends to focus mainly
on *high-level human thought*, especially belief and
desire. However, CCTM+RTM is applicable to a much wider range of
mental states and processes. Many cognitive scientists apply it to
non-human animals. For example, Gallistel and King (2009) apply it to
certain invertebrate phenomena (e.g., honeybee navigation). Even
confining attention to humans, one can apply CCTM+RTM
to *subpersonal processing*. Fodor (1983) argues that
perception involves a subpersonal "module" that converts
retinal input into Mentalese symbols and then performs computations
over those symbols. Thus, talk about a language of *thought* is
potentially misleading, since it suggests a non-existent restriction
to higher-level mental activity.


Also potentially misleading is the description of Mentalese as
a *language*, which suggests that all Mentalese symbols
resemble expressions in a natural language. Many philosophers,
including Fodor, sometimes seem to endorse that position. However,
there are possible non-propositional formats for Mentalese
symbols. Proponents of CCTM+RTM can adopt a pluralistic line, allowing
mental computation to operate over items akin to images, maps,
diagrams, or other non-propositional representations (Johnson-Laird
2004: 187; McDermott 2001: 69; Pinker 2005: 7; Sloman 1978:
144-176). The pluralistic line seems especially plausible as
applied to subpersonal processes (such as perception) and non-human
animals. Michael Rescorla (2009a,b) surveys research on *cognitive
maps* (Tolman 1948; O'Keefe and Nadel 1978; Gallistel 1990),
suggesting that some animals may navigate by computing over mental
representations more similar to maps than sentences. Elisabeth Camp
(2009), citing research
on baboon social interaction (Cheney and Seyfarth
2007), argues that
baboons may encode social dominance relations through non-sentential
tree-structured representations.


CCTM+RTM is schematic. To fill in the schema, one must provide
detailed computational models of specific mental processes. A complete
model will:


* describe the Mentalese symbols manipulated by the process;
* isolate elementary operations that manipulate the symbols
(e.g., *inscribing a symbol in a memory location*); and
* delineate mechanical rules governing application of elementary
operations.


By providing a detailed computational model, we decompose a complex
mental process into a series of elementary operations governed by
precise, routine instructions.


CCTM+RTM remains neutral in the traditional debate between
physicalism and substance dualism. A Turing-style model proceeds at a
very abstract level, not saying whether mental computations are
implemented by physical stuff or Cartesian soul-stuff (Block 1983:
522). In practice, all
proponents of CCTM+RTM embrace a broadly physicalist outlook. They
hold that mental computations are implemented not by soul-stuff but
rather by the brain. On this view, Mentalese symbols are realized by
neural states, and computational operations over Mentalese symbols are
realized by neural processes. Ultimately, physicalist proponents of
CCTM+RTM must produce empirically well-confirmed theories that explain
how exactly neural activity implements Turing-style computation. As
Gallistel and King (2009) emphasize, we do not currently have such
theories--though see Zylberberg, Dehaene, Roelfsema, and Sigman
(2011) for some speculations.


Fodor (1975) advances CCTM+RTM as a foundation for cognitive
science. He discusses mental phenomena such as decision-making,
perception, and linguistic processing. In each case, he maintains, our
best scientific theories postulate Turing-style computation over
mental representations. In fact, he argues that our *only*
viable theories have this form. He concludes that CCTM+RTM is
"the only game in town". Many cognitive scientists argue
along similar lines. C.R. Gallistel and Adam King (2009), Philip
Johnson-Laird (1988), Allen Newell and Herbert Simon (1976), and Zenon
Pylyshyn (1984) all recommend Turing-style computation over mental
symbols as the best foundation for scientific theorizing about the
mind.


## 4. Neural networks


 In the 1980s, connectionism emerged as a prominent rival to
classical computationalism. Connectionists draw inspiration from
neurophysiology rather than logic and computer science. They employ
computational models, *neural networks*, that differ
significantly from Turing-style models. A *neural network* is a
collection of interconnected nodes. Nodes fall into three
categories: *input* nodes, *output* nodes,
and *hidden* nodes (which mediate between input and output
nodes). Nodes have activation values, given by real numbers. One node
can bear a *weighted connection* to another node, also given by
a real number. Activations of input nodes are determined exogenously:
these are the inputs to computation. *Total input activation*
of a hidden or output node is a weighted sum of the activations of
nodes feeding into it. Activation of a hidden or output node is a
function of its total input activation; the particular function varies
with the network. During neural network computation, waves of
activation propagate from input nodes to output nodes, as determined
by weighted connections between nodes.


In a *feedforward network*, weighted connections flow only
in one direction. *Recurrent networks* have feedback loops, in
which connections emanating from hidden units circle back to hidden
units. Recurrent networks are less mathematically tractable than
feedforward networks. However, they figure crucially in psychological
modeling of various phenomena, such as phenomena that involve some
kind of memory (Elman 1990).


Weights in a neural network are typically mutable, evolving in
accord with a *learning algorithm*. The literature offers
various learning algorithms, but the basic idea is usually to adjust
weights so that *actual outputs* gradually move closer to
the *target outputs* one would expect for the relevant
inputs. The *backpropagation algorithm* is a widely used
algorithm of this kind (Rumelhart, Hinton, and Williams 1986).


Connectionism traces back to McCulloch and Pitts (1943), who
studied networks of interconnected *logic gates* (e.g.,
AND-gates and OR-gates). One can view a network of logic gates as a
neural network, with activations confined to two values (0 and 1) and
activation functions given by the usual truth-functions. McCulloch and
Pitts advanced logic gates as idealized models of individual
neurons. Their discussion exerted a profound influence on computer
science (von Neumann 1945). Modern digital computers are simply
networks of logic gates. Within cognitive science, however,
researchers usually focus upon networks whose elements are more
"neuron-like" than logic gates. In particular, modern-day
connectionists typically emphasize analog neural networks whose nodes
take continuous rather than discrete activation values. Some authors
even use the phrase "neural network" so that it
exclusively denotes such networks.


Neural networks received relatively scant attention from cognitive
scientists during the 1960s and 1970s, when Turing-style models
dominated. The 1980s witnessed a huge resurgence of interest in neural
networks, especially analog neural networks, with the
two-volume *Parallel Distributed Processing* (Rumelhart, McClelland, and the PDP research group, 1986; McClelland, Rumelhart, and the PDP research group, 1987) serving as a
manifesto. Researchers constructed connectionist models of diverse
phenomena: object recognition, speech perception, sentence
comprehension, cognitive development, and so on. Impressed by
connectionism, many researchers concluded that CCTM+RTM was no longer
"the only game in town".



In the 2010s, a class of computational models known as *deep neural
networks* became quite popular (Krizhevsky, Sutskever, and Hinton
2012; LeCun, Bengio, and Hinton 2015). These models are neural
networks with multiple layers of hidden nodes (sometimes hundreds of
such layers). Deep neural networks--trained on large data sets
through one or another learning algorithm (usually
backpropagation)--have achieved great success in many areas of
AI, including object recognition and strategic game-playing. Deep
neural networks are now widely deployed in commercial applications,
and they are the focus of extensive ongoing investigation within both
academia and industry. Researchers have also begun using them to model
the mind (e.g. Marblestone, Wayne, and Kording 2016; Kriegeskorte
2015).



For a detailed overview of neural networks, see Haykin (2008). For a
user-friendly introduction, with an emphasis on psychological
applications, see Marcus (2001). For a philosophically oriented
introduction to deep neural networks, see Buckner (2019).


### 4.1 Relation between neural networks and classical computation


Neural networks have a very different "feel" than
classical (i.e., Turing-style) models. Yet classical computation and
neural network computation are not mutually exclusive:


* *One can implement a neural network in a classical
model*. Indeed, every neural network ever physically constructed
has been implemented on a digital computer.
* *One can implement a classical model in a neural
network*. Modern digital computers implement Turing-style
computation in networks of logic gates. Alternatively, one can
implement Turing-style computation using an analog recurrent neural
network whose nodes take continuous activation values (Graves, Wayne,
and Danihelka 2014, Other Internet Resources; Siegelmann and Sontag
1991; Siegelmann and Sontag 1995).


Although some researchers suggest a fundamental opposition between
classical computation and neural network computation, it seems more
accurate to identify two modeling traditions that overlap in certain
cases but not others (cf. Boden 1991; Piccinini 2008b). In this
connection, it is also worth noting that classical computationalism
and connectionist computationalism have their common origin in the
work of McCulloch and Pitts.


Philosophers often say that classical computation involves
"rule-governed symbol manipulation" while neural network
computation is non-symbolic. The intuitive picture is that
"information" in neural networks is globally distributed
across the weights and activations, rather than concentrated in
localized symbols. However, the notion of "symbol" itself
requires explication, so it is often unclear what theorists mean by
describing computation as symbolic versus non-symbolic. As mentioned
in SS1, the Turing formalism places very few
conditions on "symbols". Regarding primitive symbols,
Turing assumes just that there are finitely many of them and that they
can be inscribed in read/write memory locations. Neural networks can
also manipulate symbols satisfying these two conditions: as just
noted, one can implement a Turing-style model in a neural network.


Many discussions of the symbolic/non-symbolic dichotomy employ a
more robust notion of "symbol". On the more robust
approach, a symbol is the sort of thing that represents a subject
matter. Thus, something is a symbol only if it has semantic or
representational properties. If we employ this more robust notion of
symbol, then the symbolic/non-symbolic distinction cross-cuts the
distinction between Turing-style computation and neural network
computation. A Turing machine need not employ symbols in the more
robust sense. As far as the Turing formalism goes, symbols manipulated
during Turing computation need not have representational properties
(Chalmers 2011). Conversely, a neural network can manipulate symbols
with representational properties. Indeed, an analog neural network can
manipulate symbols that have a combinatorial syntax and semantics
(Horgan and Tienson 1996; Marcus 2001).


Following Steven Pinker and Alan Prince (1988), we may distinguish
between *eliminative connectionism* and *implementationist
connectionism*.


Eliminative connectionists advance connectionism as a rival to
classical computationalism. They argue that the Turing formalism is
irrelevant to psychological explanation. Often, though not always,
they seek to revive the *associationist* tradition in
psychology, a tradition that CCTM had forcefully challenged. Often,
though not always, they attack the mentalist, nativist linguistics
pioneered by Noam Chomsky (1965). Often, though not always, they
manifest overt hostility to the very notion of mental
representation. But the defining feature of eliminative connectionism
is that it uses neural networks as *replacements* for
Turing-style models. Eliminative connectionists view the mind as a
computing system of a radically different kind than the Turing
machine. A few authors explicitly espouse eliminative connectionism
(Churchland 1989; Rumelhart and McClelland
1986; Horgan and Tienson
1996), and many others incline towards it.


Implementationist connectionism is a more ecumenical position. It
allows a potentially valuable role for both Turing-style
models *and* neural networks, operating harmoniously at
different levels of description (Marcus 2001; Smolensky 1988). A
Turing-style model is higher-level, whereas a neural network model is
lower-level. The neural network illuminates how the brain implements
the Turing-style model, just as a description in terms of logic gates
illuminates how a personal computer executes a program in a high-level
programming language.


### 4.2 Arguments for connectionism


Connectionism excites many researchers because of the analogy
between neural networks and the brain. Nodes resemble neurons, while
connections between nodes resemble synapses. Connectionist modeling
therefore seems more "biologically plausible" than
classical modeling. A connectionist model of a psychological
phenomenon apparently captures (in an idealized way) how
interconnected neurons might generate the phenomenon.



When evaluating the argument from biological plausibility, one should
recognize that neural networks vary widely in how closely they match
actual brain activity. Many networks that figure prominently in
connectionist writings are not so biologically plausible (Bechtel and
Abrahamsen 2002: 341-343; Bermudez 2010: 237-239;
Clark 2014: 87-89; Harnish 2002: 359-362). A few
examples:


* Real neurons are much more heterogeneous
than the interchangeable nodes that figure in typical connectionist
networks.
* Real neurons emit discrete spikes (action potentials) as
outputs. But the nodes that figure in many prominent neural networks,
including the best known deep neural networks, instead have continuous
outputs.
* The backpropagation algorithm requires that weights between nodes
can vary between excitatory and inhibitory, yet actual synapses cannot
so vary (Crick and Asanuma 1986). Moreover, the algorithm assumes
target outputs supplied exogenously by modelers *who know the
desired answer*. In that sense, learning
is *supervised*. Very little learning in actual biological
systems involves anything resembling supervised training.



On the other hand, some neural networks are more biologically
realistic (Buckner and Garson 2019; Illing, Gerstner, and Brea
2019). For instance, there are neural networks that replace
backpropagation with more realistic learning algorithms, such as a
reinforcement learning algorithm (Pozzi, Bohte, and Roelfsema
2019, Other Internet Resources) or an unsupervised learning algorithm
(Krotov and Hopfield 2019). There are also neural networks whose nodes
output discrete spikes roughly akin to those emitted by real neurons
in the brain (Maass 1996; Buesing, Bill, Nessler, and Maass 2011).


Even when a neural network is not biologically plausible, it may
still be *more* biologically plausible than classical
models. Neural networks certainly seem closer than Turing-style
models, in both details and spirit, to neurophysiological
description. Many cognitive scientists worry that CCTM reflects a
misguided attempt at imposing the architecture of digital computers
onto the brain. Some doubt that the brain implements anything
resembling digital computation, i.e., computation over discrete
configurations of digits (Piccinini and Bahar 2013). Others doubt that
brains display clean Turing-style separation between central processor
and read/write memory (Dayan 2009). Neural networks fare better on
both scores: they do not require computation over discrete
configurations of digits, and they do not postulate a clean separation
between central processor and read/write memory.


Classical computationalists typically reply that it is premature to
draw firm conclusions based upon biological plausibility, given how
little we understand about the relation between neural, computational,
and cognitive levels of description (Gallistel and King 2009; Marcus
2001). Using measurement techniques such as cell recordings and
functional magnetic resonance imaging (fMRI), and drawing upon
disciplines as diverse as physics, biology, AI, information theory,
statistics, graph theory, and dynamical systems theory,
neuroscientists have accumulated substantial knowledge about the brain
at varying levels of granularity (Zednik 2019). We now know quite a
lot about individual neurons, about how neurons interact within neural
populations, about the localization of mental activity in cortical
regions (e.g. the visual cortex), and about interactions among
cortical regions. Yet we still have a tremendous amount to learn about
how neural tissue accomplishes the tasks that it surely accomplishes:
perception, reasoning, decision-making, language acquisition, and so
on. Given our present state of relative ignorance, it would be rash to
insist that the brain does not implement anything resembling Turing
computation.


Connectionists offer numerous further arguments that we should
employ connectionist models instead of, or in addition to, classical
models. See the entry connectionism
for an overview. For purposes of this entry, we mention two additional
arguments.


The first argument emphasizes *learning* (Bechtel and
Abrahamsen 2002: 51). A vast range of cognitive phenomena involve
learning from experience. Many connectionist models are explicitly
designed to model learning, through backpropagation or some other
algorithm that modifies the weights between nodes. By contrast,
connectionists often complain that there are no good classical models
of learning. Classical computationalists can respond by citing
perceived defects of connectionist learning algorithms (e.g., the
heavy reliance of backpropagation upon supervised training). Classical
computationalists can also cite Bayesian decision theory, which models
learning as probabilistic updating. More specifically, classical
computationalists can cite the achievements of *Bayesian cognitive
science*, which uses Bayesian decision theory to construct
mathematical models of mental activity (Ma 2019). Over the past few
decades, Bayesian cognitive science has accrued many explanatory
successes. This impressive track record suggests that some mental
processes are Bayesian or approximately Bayesian (Rescorla
2020). Moreover, the advances mentioned
in SS2 show how classical computing systems
can execute or at least approximately execute Bayesian updating in
various realistic scenarios. These developments provide hope that
classical computation can model many important cases of learning.


The second argument emphasizes *speed of
computation*. Neurons are much slower than silicon-based
components of digital computers. For this reason, neurons could not
execute serial computation quickly enough to match rapid human
performance in perception, linguistic comprehension, decision-making,
etc. Connectionists maintain that the only viable solution is to
replace serial computation with a "massively parallel"
computational architecture--precisely what neural networks
provide (Feldman and Ballard 1982; Rumelhart 1989). However, this
argument is only effective against classical computationalists who
insist upon serial processing. As noted
in SS3, some Turing-style models
involve parallel processing. Many classical computationalists are
happy to allow "massively parallel" mental computation,
and the argument gains no traction against these researchers. That
being said, the argument highlights an important question that any
computationalist--whether classical, connectionist, or
otherwise--must address: How does a brain built from relatively
slow neurons execute sophisticated computations so quickly? Neither
classical nor connectionist computationalists have answered this
question satisfactorily (Gallistel and King 2009: 174 and 265).


### 4.3 Systematicity and productivity


Fodor and Pylyshyn (1988) offer a widely discussed critique of
eliminativist connectionism. They argue that systematicity and
productivity fail in connectionist models, except when the
connectionist model implements a classical model. Hence, connectionism
does not furnish a viable alternative to CCTM. At best, it supplies a
low-level description that helps bridge the gap between Turing-style
computation and neuroscientific description.


This argument has elicited numerous replies and
counter-replies. Some argue that neural networks can exhibit
systematicity without implementing anything like classical
computational architecture (Horgan and Tienson 1996; Chalmers 1990;
Smolensky 1991; van Gelder 1990). Some argue that Fodor and Pylyshyn
vastly exaggerate systematicity (Johnson 2004) or productivity
(Rumelhart and McClelland 1986), especially for non-human animals
(Dennett 1991). These issues, and many others raised by Fodor and
Pylyshyn's argument, have been thoroughly investigated in the
literature. For further discussion, see Bechtel and Abrahamsen (2002:
156-199), Bermudez (2005: 244-278), Chalmers
(1993), Clark (2014: 84-86), and the encyclopedia entries on
 the language of thought hypothesis
 and on 
 connectionism.


Gallistel and King (2009) advance a related but distinct
productivity argument. They emphasize *productivity of mental
computation*, as opposed to *productivity of mental
states*. Through detailed empirical case studies, they argue that
many non-human animals can extract, store, and retrieve detailed
records of the surrounding environment. For example, the Western scrub
jay records where it cached food, what kind of food it cached in each
location, when it cached the food, and whether it has depleted a given
cache (Clayton, Emery, and Dickinson 2006). The jay can access these
records and exploit them in diverse computations: computing whether a
food item stored in some cache is likely to have decayed; computing a
route from one location to another; and so on. The number of possible
computations a jay can execute is, for all practical purposes,
infinite.


CCTM explains the productivity of mental computation by positing a
central processor that stores and retrieves symbols in addressable
read/write memory. When needed, the central processor can retrieve
arbitrary, unpredicted combinations of symbols from memory. In
contrast, Gallistel and King argue, connectionism has difficulty
accommodating the productivity of mental computation. Although
Gallistel and King do not carefully distinguish between eliminativist
and implementationist connectionism, we may summarize their argument
as follows:


* Eliminativist connectionism cannot explain how organisms combine
stored memories (e.g., cache locations) for computational purposes
(e.g., computing a route from one cache to another). There are a
virtual infinity of possible combinations that might be useful, with
no predicting in advance which pieces of information must be combined
in future computations. The only computationally tractable solution is
symbol storage in readily accessible read/write memory
locations--a solution that eliminativist connectionists
reject.
* Implementationist connectionists can postulate symbol storage in
read/write memory, *as implemented by a neural
network*. However, the mechanisms that connectionists usually
propose for implementing memory are not plausible. Existing proposals
are mainly variants upon a single idea: a recurrent neural network
that allows reverberating activity to travel around a loop (Elman
1990). There are many reasons why the reverberatory loop model is
hopeless as a theory of long-term memory. For example, noise in the
nervous system ensures that signals would rapidly degrade in a few
minutes. Implementationist connectionists have thus far offered no
plausible model of read/write 
memory.[2]


Gallistel and King conclude that CCTM is much better suited than
either eliminativist or implementationist connectionism to explain a
vast range of cognitive phenomena.


Critics attack this new productivity argument from various angles,
focusing mainly on the empirical case studies adduced by Gallistel and
King. Peter Dayan (2009), John Donahoe (2010), and Christopher Mole
(2014) argue that biologically plausible neural network models can
accommodate at least some of the case studies. Dayan and Donahoe argue
that empirically adequate neural network models can dispense with
anything resembling read/write memory. Mole argues that, in certain
cases, empirically adequate neural network models
can *implement* the read/write memory mechanisms posited by
Gallistel and King. Debate on these fundamental issues seems poised to
continue well into the future.


### 4.4 Computational neuroscience


 *Computational neuroscience* describes the nervous system
through computational models. Although this research program is
grounded in mathematical modeling of individual neurons, the
distinctive focus of computational neuroscience is *systems* of
interconnected neurons. Computational neuroscience usually models
these systems as neural networks. In that sense, it is a variant,
off-shoot, or descendant of connectionism. However, most computational
neuroscientists do not self-identify as connectionists. There are
several differences between connectionism and computational
neuroscience:


* Neural networks employed by computational neuroscientists are
much more biologically realistic than those employed by
connectionists. The computational neuroscience literature is filled
with talk about firing rates, action potentials, tuning curves,
etc. These notions play at best a limited role in connectionist
research, such as most of the research canvassed in (Rogers and
McClelland 2014).
* Computational neuroscience is driven in large measure by
knowledge about the brain, and it assigns huge importance to
neurophysiological data (e.g., cell recordings). Connectionists place
much less emphasis upon such data. Their research is primarily driven
by behavioral data (although more recent connectionist writings cite
neurophysiological data with somewhat greater frequency).
* Computational neuroscientists usually regard individual nodes in
neural networks as idealized descriptions of actual
neurons. Connectionists usually instead regard nodes
as *neuron-like processing units* (Rogers and McClelland 2014)
while remaining neutral about how exactly these units map onto actual
neurophysiological entities.


One might say that computational neuroscience is concerned mainly
with *neural computation* (computation by systems of neurons),
whereas connectionism is concerned mainly with abstract computational
models *inspired* by neural computation. But the boundaries
between connectionism and computational neuroscience are admittedly
somewhat porous. For an overview of computational neuroscience, see
Trappenberg (2010) or Miller (2018).


Serious philosophical engagement with neuroscience dates back at
least to Patricia Churchland's *Neurophilosophy*
(1986). As computational neuroscience matured, Churchland became one
of its main philosophical champions (Churchland, Koch, and Sejnowski
1990; Churchland and Sejnowski 1992). She was joined by Paul
Churchland (1995, 2007) and others (Eliasmith 2013; Eliasmith and
Anderson 2003; Piccinini and Bahar 2013; Piccinini and Shagrir
2014). All these authors hold that theorizing about mental computation
should begin with the brain, not with Turing machines or other
inappropriate tools drawn from logic and computer science. They also
hold that neural network modeling should strive for greater biological
realism than connectionist models typically attain. Chris Eliasmith
(2013) develops this neurocomputational viewpoint through
the *Neural Engineering Framework*, which supplements
computational neuroscience with tools drawn from control theory
(Brogan 1990). He aims to "reverse engineer" the brain,
building large-scale, biologically plausible neural network models of
cognitive phenomena.


Computational neuroscience differs in a crucial respect from CCTM
and connectionism: it abandons multiply realizability. Computational
neuroscientists cite specific neurophysiological properties and
processes, so their models do not apply equally well to (say) a
sufficiently different silicon-based creature. Thus, computational
neuroscience sacrifices a key feature that originally attracted
philosophers to CTM. Computational neuroscientists will respond that
this sacrifice is worth the resultant insight into neurophysiological
underpinnings. But many computationalists worry that, by focusing too
much on neural underpinnings, we risk losing sight of the cognitive
forest for the neuronal trees. Neurophysiological details are
important, but don't we also need an additional abstract level
of computational description that prescinds from such details?
Gallistel and King (2009) argue that a myopic fixation upon what we
currently know about the brain has led computational neuroscience to
shortchange core cognitive phenomena such as navigation, spatial and
temporal learning, and so on. Similarly, Edelman (2014) complains that
the Neural Engineering Framework substitutes a blizzard of
neurophysiological details for satisfying psychological
explanations.



Partly in response to such worries, some researchers propose an
integrated *cognitive computational neuroscience* that connects
psychological theories with neural implementation mechanisms
(Naselaris et al. 2018; Kriegeskorte and Douglas 2018). The basic idea
is to use neural network models to illuminate how mental processes are
instantiated in the brain, thereby grounding multiply realizable
cognitive description in the neurophysiological. A good example is
recent work on neural implementation of Bayesian inference (e.g.,
Pouget et al. 2013; Orhan and Ma 2017; Aitchison and Lengyel
2016). Researchers articulate (multiply realizable) Bayesian models of
various mental processes; they construct biologically plausible neural
networks that execute or approximately execute the posited Bayesian
computations; and they evaluate how well these neural network models
fit with neurophysiological data.


Despite the differences between connectionism and computational
neuroscience, these two movements raise many similar issues. In
particular, the dialectic from SS4.4
regarding systematicity and productivity arises in similar form.


## 5. Computation and representation


Philosophers and cognitive scientists use the term
"representation" in diverse ways. Within philosophy, the
most dominant usage ties representation to intentionality, i.e., the
"aboutness" of mental states. Contemporary philosophers
usually elucidate intentionality by invoking *representational
content*. A representational mental state has a content that
represents the world as being a certain way, so we can ask whether the
world is indeed that way. Thus, representationally contentful mental
states are *semantically evaluable* with respect to properties
such as truth, accuracy, fulfillment, and so on. To illustrate:


* Beliefs are the sorts of things that can be true or false. My
belief *that Emmanuel Macron is French* is true if Emmanuel
Macron is French, false if he is not.
* Perceptual states are the sorts of things that can be accurate or
inaccurate. My perceptual experience *as of a red sphere* is
accurate only if a red sphere is before me.
* Desires are the sorts of things that can fulfilled or
thwarted. My desire *to eat chocolate* is fulfilled if I eat
chocolate, thwarted if I do not eat chocolate.


Beliefs have truth-conditions (conditions under which they are
true), perceptual states have accuracy-conditions (conditions under
which they are accurate), and desires have fulfillment-conditions
(conditions under which they are fulfilled).


In ordinary life, we frequently predict and explain behavior by
invoking beliefs, desires, and other representationally contentful
mental states. We identify these states through their representational
properties. When we say "Frank believes that Emmanuel Macron is
French", we specify the condition under which Frank's
belief is true (namely, that Emmanuel Macron is French). When we say
"Frank wants to eat chocolate", we specify the condition
under which Frank's desire is fulfilled (namely, that Frank eats
chocolate). So folk psychology assigns a central role
to *intentional descriptions*, i.e., descriptions that identify
mental states through their representational properties. Whether
scientific psychology should likewise employ intentional descriptions
is a contested issue within contemporary philosophy of mind.


*Intentional realism* is realism regarding
representation. At a minimum, this position holds that
representational properties are genuine aspects of mentality. Usually,
it is also taken to hold that scientific psychology should freely
employ intentional descriptions when appropriate. Intentional realism
is a popular position, advocated by Tyler Burge (2010a), Jerry Fodor
(1987), Christopher Peacocke (1992, 1994), and many others. One
prominent argument for intentional realism cites *cognitive science
practice*. The argument maintains that intentional description
figures centrally in many core areas of cognitive science, such as
perceptual psychology and linguistics. For example,
perceptual psychology describes how perceptual activity transforms
sensory inputs (e.g., retinal stimulations) into representations of
the distal environment (e.g., perceptual representations of distal
shapes, sizes, and colors). The science identifies perceptual states
by citing representational properties (e.g., representational
relations to specific distal shapes, sizes, colors). Assuming a
broadly scientific realist perspective, the explanatory achievements
of perceptual psychology support a realist posture towards
intentionality.


*Eliminativism* is a strong form of anti-realism about
intentionality. Eliminativists dismiss intentional description as
vague, context-sensitive, interest-relative, explanatorily
superficial, or otherwise problematic. They recommend that scientific
psychology jettison representational content. An early example is W.V.
Quine's *Word and Object* (1960), which seeks to replace
intentional psychology with behaviorist stimulus-response psychology.
Paul Churchland (1981), another prominent eliminativist, wants to
replace intentional psychology with neuroscience.


Between intentional realism and eliminativism lie various
intermediate positions. Daniel Dennett (1971, 1987) acknowledges that
intentional discourse is predictively useful, but he questions whether
mental states *really* have representational properties.
According to Dennett, theorists who employ intentional descriptions
are not *literally* asserting that mental states have
representational properties. They are merely adopting the
"intentional stance". Donald Davidson (1980) espouses a
neighboring *interpretivist* position. He emphasizes the
central role that intentional ascription plays within ordinary
interpretive practice, i.e., our practice of interpreting one
another's mental states and speech acts. At the same time, he
questions whether intentional psychology will find a place within
mature scientific theorizing. Davidson and Dennett both profess
realism about intentional mental states. Nevertheless, both
philosophers are customarily read as intentional anti-realists. (In
particular, Dennett is frequently read as a kind
of *instrumentalist* about intentionality.) One source of this
customary reading involves *indeterminacy of
interpretation*. Suppose that behavioral evidence allows two
conflicting interpretations of a thinker's mental states.
Following Quine, Davidson and Dennett both say there is then "no
fact of the matter" regarding which interpretation is correct.
This diagnosis indicates a less than fully realist attitude towards
intentionality.


Debates over intentionality figure prominently in philosophical
discussion of CTM. Let us survey some highlights.


### 5.1 Computation as formal


Classical computationalists typically assume what one might
call *the formal-syntactic conception of computation*
(FSC). The intuitive idea is that computation manipulates symbols in
virtue of their formal syntactic properties rather than their semantic
properties.


FSC stems from innovations in mathematical logic during the late
19th and early 20th centuries, especially
seminal contributions by George Boole and Gottlob Frege. In
his *Begriffsschrift* (1879/1967), Frege effected a
thoroughgoing *formalization* of deductive reasoning. To
formalize, we specify a *formal language* whose component
linguistic expressions are individuated non-semantically (e.g., by
their geometric shapes). We may have some intended interpretation in
mind, but elements of the formal language are purely syntactic
entities that we can discuss without invoking semantic properties such
as reference or truth-conditions. In particular, we can
specify *inference rules* in formal syntactic terms. If we
choose our inference rules wisely, then they will cohere with our
intended interpretation: they will carry true premises to true
conclusions. Through formalization, Frege invested logic with
unprecedented rigor. He thereby laid the groundwork for numerous
subsequent mathematical and philosophical developments.


Formalization plays a significant foundational role within computer
science. We can program a Turing-style computer that manipulates
linguistic expressions drawn from a formal language. If we program the
computer wisely, then its syntactic machinations will cohere with our
intended semantic interpretation. For example, we can program the
computer so that it carries true premises only to true conclusions, or
so that it updates probabilities as dictated by Bayesian decision
theory.


FSC holds that *all* computation manipulates formal
syntactic items, without regard to any semantic properties those items
may have. Precise formulations of FSC vary. Computation is said to be
"sensitive" to syntax but not semantics, or to have
"access" only to syntactic properties, or to operate
"in virtue" of syntactic rather than semantic properties,
or to be impacted by semantic properties only as
"mediated" by syntactic properties. It is not always so
clear what these formulations mean or whether they are equivalent to
one another. But the intuitive picture is that syntactic properties
have causal/explanatory primacy over semantic properties in driving
computation forward.


Fodor's article "Methodological Solipsism Considered as
a Research Strategy in Cognitive Psychology" (1980) offers an
early statement. Fodor combines FSC with CCTM+RTM. He analogizes
Mentalese to formal languages studied by logicians: it contains simple
and complex items individuated non-semantically, just as typical
formal languages contain simple and complex expressions individuated
by their shapes. Mentalese symbols have a semantic interpretation, but
this interpretation does not (directly) impact mental computation. A
symbol's formal properties, rather than its semantic properties,
determine how computation manipulates the symbol. In that sense, the
mind is a "syntactic engine". Virtually all classical
computationalists follow Fodor in endorsing FSC.


Connectionists often deny that neural networks manipulate
syntactically structured items. For that reason, many connectionists
would hesitate to accept FSC. Nevertheless, most connectionists
endorse a *generalized formality thesis*: computation is
insensitive to semantic properties. The generalized formality thesis
raises many of the same philosophical issues raised by FSC. We focus
here on FSC, which has received the most philosophical discussion.


Fodor combines CCTM+RTM+FSC with intentional realism. He holds that
CCTM+RTM+FSC vindicates folk psychology by helping us convert common
sense intentional discourse into rigorous science. He motivates his
position with a famous abductive argument for CCTM+RTM+FSC (1987:
18-20). Strikingly, mental activity tracks semantic properties
in a coherent way. For example, deductive inference carries premises
to conclusions that are true if the premises are true. How can we
explain this crucial aspect of mental activity? Formalization shows
that syntactic manipulations can track semantic properties, and
computer science shows how to build physical machines that execute
desired syntactic manipulations. If we treat the mind as a
syntax-driven machine, then we can explain why mental activity tracks
semantic properties in a coherent way. Moreover, our explanation does
not posit causal mechanisms radically different from those posited
within the physical sciences. We thereby answer the pivotal
question: *How is rationality mechanically possible*?


Stephen Stich (1983) and Hartry Field (2001) combine CCTM+FSC with
eliminativism. They recommend that cognitive science model the mind in
formal syntactic terms, eschewing intentionality altogether. They
grant that mental states have representational properties, but they
ask what explanatory value scientific psychology gains by invoking
those properties. Why supplement formal syntactic description with
intentional description? If the mind is a syntax-driven machine, then
doesn't representational content drop out as explanatorily
irrelevant?


At one point in his career, Putnam (1983: 139-154) combined
CCTM+FSC with a Davidson-tinged *interpretivism*. Cognitive
science should proceed along the lines suggested by Stich and Field,
delineating purely formal syntactic computational models. Formal
syntactic modeling co-exists with ordinary interpretive practice, in
which we ascribe intentional contents to one another's mental
states and speech acts. Interpretive practice is governed by holistic
and heuristic constraints, which stymie attempts at converting
intentional discourse into rigorous science. For Putnam, as for Field
and Stich, the scientific action occurs at the formal syntactic level
rather than the intentional level.


CTM+FSC comes under attack from various directions. One criticism
targets *the causal relevance of representational content*
(Block 1990; Figdor 2009; Kazez 1995). Intuitively speaking, the
contents of mental states are causally relevant to mental activity and
behavior. For example, my desire to drink water rather than orange
juice causes me to walk to the sink rather than the refrigerator. The
content of my desire (*that I drink water*) seems to play an
important causal role in shaping my behavior. According to Fodor
(1990: 137-159), CCTM+RTM+FSC accommodates such
intuitions. Formal syntactic activity *implements* intentional
mental activity, thereby ensuring that intentional mental states
causally interact in accord with their contents. However, it is not so
clear that this analysis secures the causal relevance of content. FSC
says that computation is "sensitive" to syntax but not
semantics. Depending on how one glosses the key term
"sensitive", it can look like representational content is
causally irrelevant, with formal syntax doing all the causal
work. Here is an analogy to illustrate the worry. When a car drives
along a road, there are stable patterns involving the car's
shadow. Nevertheless, shadow position at one time does not influence
shadow position at a later time. Similarly, CCTM+RTM+FSC may explain
how mental activity instantiates stable patterns described in
intentional terms, but this is not enough to ensure the causal
relevance of content. If the mind is a syntax-driven machine, then
causal efficacy seems to reside at the syntactic rather the semantic
level. Semantics is just "along for the ride". Apparently,
then, CTM+FSC encourages the conclusion that representational
properties are causally inert. The conclusion may not trouble
eliminativists, but intentional realists usually want to avoid it.


 A second criticism dismisses the formal-syntactic picture as
speculation ungrounded in scientific practice. Tyler Burge (2010a,b,
2013: 479-480) contends that formal syntactic description of
mental activity plays no significant role within large areas of
cognitive science, including the study of theoretical reasoning,
practical reasoning, and perception. In each case, Burge argues, the
science employs intentional description *rather than* formal
syntactic description. For example, perceptual psychology individuates
perceptual states not through formal syntactic properties but through
representational relations to distal shapes, sizes, colors, and so
on. To understand this criticism, we must distinguish *formal
syntactic description* and *neurophysiological
description*. Everyone agrees that a complete scientific
psychology will assign prime importance to neurophysiological
description. However, neurophysiological description is distinct from
formal syntactic description, because formal syntactic description is
supposed to be multiply realizable in the neurophysiological. The
issue here is whether scientific psychology should
supplement *intentional descriptions*
and *neurophysiological descriptions* with *multiply
realizable, non-intentional formal syntactic* descriptions.


### 5.2 Externalism about mental content


 Putnam's landmark article "The Meaning of
'Meaning'" (1975: 215-271) introduced
the *Twin Earth thought experiment*, which postulates a world
just like our own except that H2O is replaced by a
qualitatively similar substance XYZ with different chemical
composition. Putnam argues that XYZ is not water and that speakers on
Twin Earth use the word "water" to refer to XYZ rather
than to water. Burge (1982) extends this conclusion
from *linguistic reference* to *mental content*. He
argues that Twin Earthlings instantiate mental states with different
contents. For example, if Oscar on Earth thinks *that water is
thirst-quenching*, then his duplicate on Twin Earth thinks a
thought with a different content, which we might gloss as *that
twater is thirst-quenching*. Burge concludes that mental content
does not supervene upon internal neurophysiology. Mental content is
individuated partly by factors outside the thinker's skin,
including causal relations to the environment. This position
is *externalism about mental content*.


Formal syntactic properties of mental states are widely taken to
supervene upon internal neurophysiology. For example, Oscar and Twin
Oscar instantiate the same formal syntactic manipulations. Assuming
content externalism, it follows that there is a huge gulf between
ordinary intentional description and formal syntactic description.


Content externalism raises serious questions about the explanatory
utility of representational content for scientific psychology:


*Argument from Causation* (Fodor 1987, 1991): How can mental
content exert any causal influence except as manifested within
internal neurophysiology? There is no "psychological action at a
distance". Differences in the physical environment impact
behavior only by inducing differences in local brain states. So the
only causally relevant factors are those that supervene upon internal
neurophysiology. Externally individuated content is *causally
irrelevant*.


*Argument from Explanation* (Stich 1983): Rigorous
scientific explanation should not take into account factors outside
the subject's skin. Folk psychology may taxonomize mental states
through relations to the external environment, but scientific
psychology should taxonomize mental states entirely through factors
that supervene upon internal neurophysiology. It should treat Oscar
and Twin Oscar as psychological
duplicates.[3]


Some authors pursue the two arguments in conjunction with one
another. Both arguments reach the same conclusion: externally
individuated mental content finds no legitimate place within causal
explanations provided by scientific psychology. Stich (1983) argues
along these lines to motivate his formal-syntactic eliminativism.


Many philosophers respond to such worries by promoting *content
internalism*. Whereas content externalists favor *wide
content* (content that does not supervene upon internal
neurophysiology), content internalists favor *narrow content*
(content that does so supervene). Narrow content is what remains of
mental content when one factors out all external elements. At one
point in his career, Fodor (1981, 1987) pursued internalism as a
strategy for integrating intentional psychology with
CCTM+RTM+FSC. While conceding that wide content should not figure in
scientific psychology, he maintained that narrow content should play a
central explanatory role.


Radical internalists insist that *all* content is narrow. A
typical analysis holds that Oscar is thinking not about water but
about some more general category of substance that subsumes XYZ, so
that Oscar and Twin Oscar entertain mental states with the same
contents. Tim Crane (1991) and Gabriel Segal (2000) endorse such an
analysis. They hold that folk psychology always individuates
propositional attitudes narrowly. A less radical internalism
recommends that we recognize narrow content *in addition to*
wide content. Folk psychology may sometimes individuate propositional
attitudes widely, but we can also delineate a viable notion of narrow
content that advances important philosophical or scientific
goals. Internalists have proposed various candidate notions of narrow
content (Block 1986; Chalmers 2002; Cummins 1989; Fodor 1987; Lewis
1994; Loar 1988; Mendola 2008). See the
entry narrow mental content for an
overview of prominent candidates.


Externalists complain that existing theories of narrow content are
sketchy, implausible, useless for psychological explanation, or
otherwise objectionable (Burge 2007; Sawyer 2000; Stalnaker
1999). Externalists also question internalist arguments that
scientific psychology requires narrow content:


*Argument from Causation*: Externalists insist that wide
content can be causally relevant. The details vary among externalists,
and discussion often becomes intertwined with complex issues
surrounding causation, counterfactuals, and the metaphysics of mind.
See the entry mental causation for
an introductory overview, and see Burge (2007), Rescorla (2014a), and
Yablo (1997, 2003) for representative externalist discussion.


*Argument from Explanation*: Externalists claim that
psychological explanation can legitimately taxonomize mental states
through factors that outstrip internal neurophysiology (Peacocke
1993; Shea, 2018). Burge observes that non-psychological sciences often
individuate explanatory kinds *relationally*, i.e., through
relations to external factors. For example, whether an entity counts
as a heart depends (roughly) upon whether its biological function in
its normal environment is to pump blood. So physiology individuates
organ kinds relationally. Why can't psychology likewise
individuate mental states relationally? For a notable exchange on
these issues, see Burge (1986, 1989, 1995) and Fodor (1987, 1991).


Externalists doubt that we have any good reason to replace or
supplement wide content with narrow content. They dismiss the search
for narrow content as a wild goose chase.


Burge (2007, 2010a) defends externalism by analyzing current
cognitive science. He argues that many branches of scientific
psychology (especially perceptual psychology) individuate mental
content through causal relations to the external environment. He
concludes that scientific practice embodies an externalist
perspective. By contrast, he maintains, narrow content is a
philosophical fantasy ungrounded in current science.


Suppose we abandon the search for narrow content. What are the
prospects for combining CTM+FSC with externalist intentional
psychology? The most promising option emphasizes *levels of
explanation*. We can say that intentional psychology occupies one
level of explanation, while formal-syntactic computational psychology
occupies a different level. Fodor advocates this approach in his later
work (1994, 2008). He comes to reject narrow content as otiose. He
suggests that formal syntactic mechanisms implement externalist
psychological laws. Mental computation manipulates Mentalese
expressions in accord with their formal syntactic properties, and
these formal syntactic manipulations ensure that mental activity
instantiates appropriate law-like patterns defined over wide
contents.


In light of the internalism/externalism distinction, let us revisit
the eliminativist challenge raised in SS5.1:
what explanatory value does intentional description add to
formal-syntactic description? Internalists can respond that suitable
formal syntactic manipulations determine and maybe even constitute
narrow contents, so that internalist intentional description is
already implicit in suitable formal syntactic description (cf. Field
2001: 75). Perhaps this response vindicates intentional realism,
perhaps not. Crucially, though, no such response is available to
content externalists. Externalist intentional description is not
implicit in formal syntactic description, because one can hold formal
syntax fixed while varying wide content. Thus, content externalists
who espouse CTM+FSC must say what we gain by supplementing
formal-syntactic explanations with intentional explanations. Once we
accept that mental computation is sensitive to syntax but not
semantics, it is far from clear that any useful explanatory work
remains for wide content. Fodor addresses this challenge at various
points, offering his most systematic treatment in *The Elm and the
Expert* (1994). See Arjo (1996), Aydede (1998), Aydede and Robbins
(2001), Wakefield (2002); Perry (1998), and Wakefield (2002) for
criticism. See Rupert (2008) and Schneider (2005) for positions close
to Fodor's. Dretske (1993) and Shea (2018, pp. 197-226) pursue
alternative strategies for vindicating the explanatory relevance of
wide content.


### 5.3 Content-involving computation


 The perceived gulf between computational description and
intentional description animates many writings on CTM. A few
philosophers try to bridge the gulf using computational descriptions
that individuate computational states in representational terms. These
descriptions are *content-involving*, to use Christopher
Peacocke's (1994) terminology. On the content-involving
approach, there is no rigid demarcation between computational and
intentional description. In particular, certain scientifically
valuable descriptions of mental activity are both computational and
intentional. Call this position *content-involving
computationalism*.


Content-involving computationalists need not say that all
computational description is intentional. To illustrate, suppose we
describe a simple Turing machine that manipulates symbols individuated
by their geometric shapes. Then the resulting computational
description is not plausibly content-involving. Accordingly,
content-involving computationalists do not usually advance
content-involving computation as a general theory of computation. They
claim only that *some* important computational descriptions are
content-involving.


One can develop content-involving computationalism in an
internalist or externalist direction. *Internalist
content-involving computationalists* hold that some computational
descriptions identify mental states partly through
their *narrow* contents. Murat Aydede (2005) recommends a
position along these lines. *Externalist content-involving
computationalism* holds that certain computational descriptions
identify mental states partly through their *wide*
contents. Tyler Burge (2010a: 95-101), Christopher Peacocke
(1994, 1999), Michael Rescorla (2012), and Mark Sprevak (2010) espouse
this position. Oron Shagrir (2001, forthcoming) advocates a
content-involving computationalism that is neutral between internalism
and externalism.


Externalist content-involving computationalists typically cite
cognitive science practice as a motivating factor. For example,
perceptual psychology describes the perceptual system as computing an
estimate of some object's size from retinal stimulations and
from an estimate of the object's depth. Perceptual
"estimates" are identified representationally, as
representations of specific distal sizes and depths. Quite plausibly,
representational relations to specific distal sizes and depths do not
supervene on internal neurophysiology. Quite plausibly, then,
perceptual psychology type-identifies perceptual computations through
wide contents. So externalist content-involving computationalism seems
to harmonize well with current cognitive science.


A major challenge facing content-involving computationalism
concerns the interface with standard computationalism formalisms, such
as the Turing machine. How exactly do content-involving descriptions
relate to the computational models found in logic and computer
science? Philosophers usually assume that these models offer
non-intentional descriptions. If so, that would be a major and perhaps
decisive blow to content-involving computationalism.


Arguably, though, many familiar computational formalisms allow a
content-involving rather than formal syntactic construal. To
illustrate, consider the Turing machine. One *can* individuate
the "symbols" comprising the Turing machine alphabet
non-semantically, through factors akin to geometric shape. But does
Turing's formalism *require* a non-semantic individuative
scheme? Arguably, the formalism allows us
to individuate symbols partly through their contents. Of course, the
machine table for a Turing machine does not explicitly cite semantic
properties of symbols (e.g., denotations or truth-conditions).
Nevertheless, the machine table can encode mechanical rules that
describe how to manipulate symbols, where those symbols are
type-identified in content-involving terms. In this way, the machine
table dictates transitions among content-involving states without
explicitly mentioning semantic properties. Aydede (2005) suggests an
internalist version of this view, with symbols type-identified through
their narrow
 contents.[4]
 Rescorla (2017a) develops the view in
an externalist direction, with symbols type-identified through their
wide contents. He argues that some Turing-style models describe
computational operations over externalistically individuated Mentalese
symbols.[5]


In principle, one might embrace both externalist content-involving
computational description *and* formal syntactic description.
One might say that these two kinds of description occupy distinct
levels of explanation. Peacocke suggests such a view. Other
content-involving computationalists regard formal syntactic
descriptions of the mind more skeptically. For example, Burge
questions what explanatory value formal syntactic description
contributes to certain areas of scientific psychology (such as
perceptual psychology). From this viewpoint, the eliminativist
challenge posed in SS5.1 has matters
backwards. We should not assume that formal syntactic descriptions are
explanatorily valuable and then ask what value intentional
descriptions contribute. We should instead embrace the externalist
intentional descriptions offered by current cognitive science and then
ask what value formal syntactic description contributes.


Proponents of formal syntactic description respond by
citing *implementation mechanisms*. Externalist description of
mental activity presupposes that suitable causal-historical relations
between the mind and the external physical environment are in
place. But surely we want a "local" description that
ignores external causal-historical relations, a description that
reveals underlying causal mechanisms. Fodor (1987, 1994) argues in
this way to motivate the formal syntactic picture. For possible
externalist responses to the argument from implementation mechanisms,
see Burge (2010b), Rescorla (2017b), Shea (2013), and Sprevak
(2010). Debate over this argument, and more generally over the
relation between computation and representation, seems likely to
continue into the indefinite future.


## 6. Alternative conceptions of computation


 The literature offers several alternative conceptions, usually
advanced as foundations for CTM. In many cases, these conceptions
overlap with one another or with the conceptions considered above.


### 6.1 Information-processing


It is common for cognitive scientists to describe computation as
"information-processing". It is less common for proponents
to clarify what they mean by "information" or
"processing". Lacking clarification, the description is
little more than an empty slogan.


Claude Shannon introduced a scientifically important notion of
"information" in his 1948 article "A Mathematical
Theory of Communication". The intuitive idea is that information
measures *reduction in uncertainty*, where reduced uncertainty
manifests as an altered probability distribution over possible states.
Shannon codified this idea within a rigorous mathematical framework,
laying the foundation for *information theory* (Cover and
Thomas 2006). Shannon information is fundamental to modern
engineering. It finds fruitful application within cognitive science,
especially cognitive neuroscience. Does it support a convincing
analysis of computation as "information-processing"?
Consider an old-fashioned tape machine that records messages received
over a wireless radio. Using Shannon's framework, one can
measure how much information is carried by some recorded
message. There is a sense in which the tape machine
"processes" Shannon information whenever we replay a
recorded message. Still, the machine does not seem to implement a
non-trivial computational 
model.[6]
 Certainly, neither the Turing machine
formalism nor the neural network formalism offers much insight into
the machine's operations. Arguably, then, a system can process
Shannon information without executing computations in any interesting
sense.


Confronted with such examples, one might try to isolate a more
demanding notion of "processing", so that the tape machine
does not "process" Shannon information. Alternatively, one
might insist that the tape machine executes non-trivial computations.
Piccinini and Scarantino (2010) advance a highly general notion of
computation--which they dub *generic
computation*--with that consequence.


 A second prominent notion of information derives from Paul
Grice's (1989) influential discussion of *natural
meaning*. Natural meaning involves reliable,
counterfactual-supporting correlations. For example, tree rings
correlate with the age of the tree, and pox correlate with
chickenpox. We colloquially describe tree rings as carrying
information about tree age, pox as carrying information about
chickenpox, and so on. Such descriptions suggest a conception that
ties information to reliable, counterfactual-supporting
correlations. Fred Dretske (1981) develops this conception into a
systematic theory, as do various subsequent philosophers. Does
Dretske-style information subserve a plausible analysis of computation
as "information-processing"? Consider an
old-fashioned *bimetallic strip thermostat*. Two metals are
joined together into a strip. Differential expansion of the metals
causes the strip to bend, thereby activating or deactivating a heating
unit. Strip state reliably correlates with current ambient
temperature, and the thermostat "processes" this
information-bearing state when activating or deactivating the
heater. Yet the thermostat does not seem to implement any non-trivial
computational model. One would not ordinarily regard the thermostat as
computing. Arguably, then, a system can process Dretske-style
information without executing computations in any interesting
sense. Of course, one might try to handle such examples through
maneuvers parallel to those from the previous paragraph.


A third prominent notion of information is *semantic
information*, i.e., representational
content.[7] Some
philosophers hold that a physical system computes only if the
system's states have representational properties (Dietrich 1989;
Fodor 1998: 10; Ladyman 2009; Shagrir 2006; Sprevak 2010). In that
sense, information-processing is *necessary* for
computation. As Fodor memorably puts it, "no computation without
representation" (1975: 34). However, this position is
debatable. Chalmers (2011) and Piccinini (2008a) contend that a Turing
machine might execute computations even though symbols manipulated by
the machine have no semantic interpretation. The machine's
computations are purely syntactic in nature, lacking anything like
semantic properties. On this view, representational content is not
necessary for a physical system to count as computational.


It remains unclear whether the slogan "computation is
information-processing" provides much insight. Nevertheless, the
slogan seems unlikely to disappear from the literature anytime soon.
For further discussion of possible connections between computation and
information, see Gallistel and King (2009: 1-26), Lizier,
Flecker, and Williams (2013), Milkowski (2013), Piccinini and
Scarantino (2010), and Sprevak (forthcoming).


### 6.2 Function evaluation


 In a widely cited passage, the perceptual psychologist David Marr
(1982) distinguishes three levels at which one can describe an
"information-processing device":



> 
>  *Computational theory*: "[t]he device is
> characterized as a mapping from one kind of information to another,
> the abstract properties of this mapping are defined precisely, and its
> appropriateness and adequacy for the task as hand are
> demonstrated" (p. 24).
> 
> 
>  *Representation and algorithm*: "the choice of
> representation for the input and output and the algorithm to be used
> to transform one into the other" (pp. 24-25).
> 
> 
>  *Hardware implementation*: "the details of how the
> algorithm and representation are realized physically" (p.
> 25).
> 
>  


Marr's three levels have attracted intense philosophical
scrutiny. For our purposes, the key point is that Marr's
"computational level" describes a mapping from inputs to
outputs, without describing intermediate steps. Marr illustrates his
approach by providing "computational level" theories of
various perceptual processes, such as edge detection.


Marr's discussion suggests a *functional conception of
computation*, on which computation is a matter of transforming
inputs into appropriate outputs. Frances Egan elaborates the
functional conception over a series of articles (1991, 1992, 1999,
2003, 2010, 2014, 2019). Like Marr, she treats computational
description as description of input-output relations. She also claims
that computational models characterize a purely *mathematical*
function: that is, a mapping from mathematical inputs to mathematical
outputs. She illustrates by considering a visual mechanism (called
"Visua") that computes an object's depth from
retinal disparity. She imagines a neurophysiological duplicate
("Twin Visua") embedded so differently in the physical
environment that it does not represent depth. Visua and Twin Visua
instantiate perceptual states with different representational
properties. Nevertheless, Egan says, vision science treats Visua and
Twin Visua as *computational duplicates*. Visua and Twin Visua
compute the same mathematical function, even though the computations
have different representational import in the two cases. Egan
concludes that computational modeling of the mind yields an
"abstract mathematical description" consistent with many
alternative possible representational descriptions. Intentional
attribution is just a heuristic gloss upon underlying computational
description.


Chalmers (2012) argues that the functional conception neglects
important features of computation. As he notes, computational models
usually describe more than just input-output relations. They describe
intermediate steps through which inputs are transformed into outputs.
These intermediate steps, which Marr consigns to the
"algorithmic" level, figure prominently in computational
models offered by logicians and computer scientists. Restricting the
term "computation" to input-output description does not
capture standard computational practice.


An additional worry faces functional theories, such as
Egan's, that exclusively emphasize *mathematical* inputs
and outputs. Critics complain that Egan mistakenly elevates
mathematical functions, at the expense of intentional explanations
routinely offered by cognitive science (Burge
2005; Rescorla
2015; Silverberg 2006; Sprevak 2010). To illustrate, suppose
perceptual psychology describes the perceptual system as estimating
that some object's depth is 5 meters. The perceptual
depth-estimate has a representational content: it is accurate only if
the object's depth is 5 meters. We cite the number 5 to identify
the depth-estimate. But our choice of this number depends upon our
arbitrary choice of measurement units. Critics contend that the
content of the depth-estimate, not the arbitrarily chosen number
through which we theorists specify that content, is what matters for
psychological explanation. Egan's theory places the number
rather than the content at explanatory center stage. According to
Egan, computational explanation should describe the visual system as
computing a *particular mathematical function* that
carries *particular mathematical inputs* into *particular
mathematical outputs*. Those particular mathematical inputs and
outputs depend upon our arbitrary choice of measurement units, so they
arguably lack the explanatory significance that Egan assigns to
them.


We should distinguish the functional approach, as pursued by Marr
and Egan, from the *functional programming paradigm* in
computer science. The functional programming paradigm models
evaluation of a complex function as successive evaluation of simpler
functions. To take a simple example, one might evaluate \(f(x,y) =
(x^{2}+y)\) by first evaluating the squaring function and then
evaluating the addition function. Functional programming differs from
the "computational level" descriptions emphasized by Marr,
because it specifies intermediate computational stages. The functional
programming paradigm stretches back to Alonzo Church's
(1936) *lambda calculus*, continuing with programming languages
such as PCF and LISP. It plays an important role in AI and theoretical
computer science. Some authors suggest that it offers special insight
into mental computation (Klein 2012; Piantadosi, Tenenbaum, and
Goodman 2012). However, many computational formalisms do not conform
to the functional paradigm: Turing machines; imperative programming
languages, such as C; logic programming languages, such as Prolog; and
so on. Even though the functional paradigm describes numerous
important computations (possibly including mental computations), it
does not plausibly capture computation *in general*.


### 6.3 Structuralism


Many philosophical discussions embody a *structuralist
conception of computation*: a computational model describes an
abstract causal structure, without taking into account particular
physical states that instantiate the structure. This conception traces
back at least to Putnam's original treatment (1967).
Chalmers (1995, 1996a, 2011, 2012) develops it in detail. He
introduces the *combinatorial-state automaton* (CSA) formalism,
which subsumes most familiar models of computation (including Turing
machines and neural networks). A CSA provides an abstract description
of a physical system's *causal topology*: the pattern of
causal interaction among the system's parts, independent of the
nature of those parts or the causal mechanisms through which they
interact. Computational description specifies a causal topology.


Chalmers deploys structuralism to delineate a very general version
of CTM. He assumes the functionalist view that psychological states
are individuated by their roles in a pattern of causal organization.
Psychological description specifies causal roles, abstracted away from
physical states that realize those roles. So psychological properties
are *organizationally invariant*, in that they supervene upon
causal topology. Since computational description characterizes a
causal topology, satisfying a suitable computational description
suffices for instantiating appropriate mental properties. It also
follows that psychological description is a species of computational
description, so that computational description should play a central
role within psychological explanation. Thus, structuralist computation
provides a solid foundation for cognitive science. Mentality is
grounded in causal patterns, which are precisely what computational
models articulate.


Structuralism comes packaged with an attractive account of
the *implementation relation* between abstract computational
models and physical systems. Under what conditions does a physical
system implement a computational model? Structuralists say that a
physical system implements a model just in case the model's
causal structure is "isomorphic" to the model's
formal structure. A computational model describes a physical system by
articulating a formal structure that mirrors some relevant causal
topology. Chalmers elaborates this intuitive idea, providing detailed
necessary and sufficient conditions for physical realization of CSAs.
Few if any alternative conceptions of computation can provide so
substantive an account of the implementation relation.


We may instructively compare structuralist computationalism with
some other theories discussed above:


*Machine functionalism*. Structuralist computationalism
embraces the core idea behind machine functionalism: mental states are
functional states describable through a suitable computational
formalism. Putnam advances CTM as an empirical hypothesis, and he
defends functionalism on that basis. In contrast, Chalmers follows
David Lewis (1972) by grounding functionalism in the conceptual
analysis of mentalistic discourse. Whereas Putnam defends
functionalism by defending computationalism, Chalmers defends
computationalism by assuming functionalism.


*Classical computationalism, connectionism, and computational
neuroscience*. Structuralist computationalism emphasizes
organizationally invariant descriptions, which are multiply
realizable. In that respect, it diverges from computational
neuroscience. Structuralism is compatible with both classical and
connectionist computationalism, but it differs in spirit from those
views. Classicists and connectionists present their rival positions
as bold, substantive hypotheses. Chalmers advances structuralist
computationalism as a relatively minimalist position unlikely to be
disconfirmed.


*Intentional realism and eliminativism*. Structuralist
computationalism is compatible with both positions. CSA description
does not explicitly mention semantic properties such as reference,
truth-conditions, representational content, and so on. Structuralist
computationalists need not assign representational content any
important role within scientific psychology. On the other hand,
structuralist computationalism does not preclude an important role for
representational content.


*The formal-syntactic conception of computation*. Wide
content depends on causal-historical relations to the external
environment, relations that outstrip causal topology. Thus, CSA
description leaves wide content underdetermined. Narrow content
presumably supervenes upon causal topology, but CSA description does
not explicitly mention narrow contents. Overall, then, structuralist
computationalism prioritizes a level of formal, non-semantic
computational description. In that respect, it resembles FSC. On the
other hand, structuralist computationalists need not say that
computation is "insensitive" to semantic properties, so
they need not endorse all aspects of FSC.


Although structuralist computationalism is distinct from CTM+FSC,
it raises some similar issues. For example, Rescorla (2012) denies
that causal topology plays the central explanatory role within
cognitive science that structuralist computationalism dictates. He
suggests that externalist intentional description rather than
organizationally invariant description enjoys explanatory
primacy. Coming from a different direction, computational
neuroscientists will recommend that we forego organizationally
invariant descriptions and instead employ more neurally specific
computational models. In response to such objections, Chalmers (2012)
argues that organizationally invariant computational description
yields explanatory benefits that neither intentional description nor
neurophysiological description replicate: it reveals the underlying
mechanisms of cognition (unlike intentional description); and it
abstracts away from neural implementation details that are irrelevant
for many explanatory purposes.


### 6.4 Mechanistic theories


The mechanistic nature of computation is a recurring theme in
logic, philosophy, and cognitive science. Gualtiero Piccinini (2007,
2012, 2015) and Marcin Milkowski (2013) develop this theme into a
mechanistic theory of computing systems. A *functional
mechanism* is a system of interconnected components, where each
component performs some function within the overall
system. *Mechanistic explanation* proceeds by decomposing the
system into parts, describing how the parts are organized into the
larger system, and isolating the function performed by each part. A
computing system is a functional mechanism of a particular kind. On
Piccinini's account, a computing system is a mechanism whose
components are functionally organized to process vehicles in accord
with rules. Echoing Putnam's discussion of multiple
realizability, Piccinini demands that the rules
be *medium-independent*, in that they abstract away from the
specific physical implementations of the vehicles. Computational
explanation decomposes the system into parts and describes how each
part helps the system process the relevant vehicles. If the system
processes discretely structured vehicles, then the computation is
digital. If the system processes continuous vehicles, then the
computation is analog. Milkowski's version of the mechanistic
approach is similar. He differs from Piccinini by pursuing an
"information-processing" gloss, so that computational
mechanisms operate over information-bearing states. Milkowski and
Piccinini deploy their respective mechanistic theories to defend
computationalism.


Mechanistic computationalists typically individuate computational
states non-semantically. They therefore encounter worries about the
explanatory role of representational content, similar to worries
encountered by FSC and structuralism. In this spirit, Shagrir (2014)
complains that mechanistic computationalism does not accommodate
cognitive science explanations that are simultaneously computational
and representational. The perceived force of this criticism will
depend upon one's sympathy for content-involving
computationalism.


### 6.5 Pluralism


We have surveyed various contrasting and sometimes overlapping
conceptions of computation: classical computation, connectionist
computation, neural computation, formal-syntactic computation,
content-involving computation, information-processing computation,
functional computation, structuralist computation, and mechanistic
computation. Each conception yields a different form of
computationalism. Each conception has its own strengths and
weaknesses. One might adopt a *pluralistic* stance that
recognizes distinct legitimate conceptions. Rather than elevate one
conception above the others, pluralists happily employ whichever
conception seems useful in a given explanatory context. Edelman (2008)
takes a pluralistic line, as does Chalmers (2012) in his most recent
discussion.


The pluralistic line raises some natural questions. Can we provide
a general analysis that encompasses all or most types of computation?
Do all computations share certain characteristic marks with one
another? Are they perhaps instead united by something like family
resemblance? Deeper understanding of computation requires us to
grapple with these questions.


## 7. Arguments against computationalism


CTM has attracted numerous objections. In many cases, the
objections apply only to specific versions of CTM (such as classical
computationalism or connectionist computationalism). Here are a few
prominent objections. See also the
entry the Chinese room argument for a
widely discussed objection to classical computationalism advanced by
John Searle (1980).


### 7.1 Triviality arguments


 A recurring worry is that CTM is *trivial*, because we can
describe almost any physical system as executing computations. Searle
(1990) claims that a wall implements *any* computer program,
since we can discern some pattern of molecular movements in the wall
that is isomorphic to the formal structure of the program. Putnam
(1988: 121-125) defends a less extreme but still very strong
triviality thesis along the same lines. Triviality arguments play a
large role in the philosophical literature. Anti-computationalists
deploy triviality arguments against computationalism, while
computationalists seek to avoid triviality.


Computationalists usually rebut triviality arguments by insisting
that the arguments overlook constraints upon computational
implementation, constraints that bar trivializing implementations. The
constraints may be counterfactual, causal, semantic, or otherwise,
depending on one's favored theory of computation. For example,
David Chalmers (1995, 1996a) and B. Jack Copeland (1996) hold that
Putnam's triviality argument ignores counterfactual conditionals
that a physical system must satisfy in order to implement a
computational model. Other philosophers say that a physical system
must have representational properties to implement a computational
model (Fodor 1998: 11-12; Ladyman 2009; Sprevak 2010) or at
least to implement a content-involving computational model (Rescorla
2013, 2014b). The details here vary considerably, and
computationalists debate amongst themselves exactly which types of
computation can avoid which triviality arguments. But most
computationalists agree that we can avoid any devastating triviality
worries through a sufficiently robust theory of the implementation
relation between computational models and physical systems.


*Pancomputationalism* holds that every physical system
implements a computational model. This thesis is plausible, since any
physical system arguably implements a sufficiently trivial
computational model (e.g., a one-state finite state automaton). As
Chalmers (2011) notes, pancomputationalism does not seem worrisome for
computationalism. What would be worrisome is the much stronger
triviality thesis that almost every physical system implements almost
every computational model.


For further discussion of triviality arguments and computational
implementation, see Sprevak (2019) and the
entry computation in physical systems.


### 7.2 Godel's incompleteness theorem


 According to some authors, Godel's incompleteness
theorems show that human mathematical capacities outstrip the
capacities of any Turing machine (Nagel and Newman 1958). J.R. Lucas
(1961) develops this position into a famous critique of CCTM. Roger Penrose
pursues the critique in *The Emperor's New Mind* (1989)
and subsequent writings. Various philosophers and logicians have
answered the critique, arguing that existing formulations suffer from
fallacies, question-begging assumptions, and even outright
mathematical errors (Bowie 1982; Chalmers 1996b; Feferman 1996; Lewis 1969, 1979; Putnam
1975: 365-366, 1994; Shapiro 2003). There is a wide consensus
that this criticism of CCTM lacks any force. It may turn out that
certain human mental capacities outstrip Turing-computability, but
Godel's incompleteness theorems provide no reason to
anticipate that outcome.


### 7.3 Limits of computational modeling


 Could a computer compose the *Eroica* symphony? Or discover
general relativity? Or even replicate a child's effortless
ability to perceive the environment, tie her shoelaces, and discern
the emotions of others? Intuitive, creative, or skillful human
activity may seem to resist formalization by a computer program
(Dreyfus 1972, 1992). More generally, one might worry that crucial
aspects of human cognition elude computational modeling, especially
classical computational modeling.


Ironically, Fodor promulgates a forceful version of this
critique. Even in his earliest statements of CCTM, Fodor (1975:
197-205) expresses considerable skepticism that CCTM can handle
all important cognitive phenomena. The pessimism becomes more
pronounced in his later writings (1983, 2000), which focus especially
on *abductive reasoning* as a mental phenomenon that
potentially eludes computational modeling. His core argument may be
summarized as follows:



(1)Turing-style
computation is sensitive only to "local" properties of a
mental representation, which are exhausted by the identity and
arrangement of the representation's constituents.
(2)Many mental
processes, paradigmatically abduction, are sensitive to
"nonlocal" properties such as relevance, simplicity, and
conservatism.
(3)Hence, we may have
to abandon Turing-style modeling of the relevant
processes.
(4)Unfortunately, we
have currently have no idea what alternative theory might serve as a
suitable replacement.

Some critics deny (1), arguing that suitable Turing-style
computations can be sensitive to "nonlocal" properties
(Schneider 2011; Wilson 2005). Some challenge (2), arguing that
typical abductive inferences are sensitive only to "local"
properties (Carruthers 2003; Ludwig and Schneider 2008; Sperber
2002). Some concede step (3) but dispute step (4), insisting that we
have promising non-Turing-style models of the relevant mental
processes (Pinker 2005). Partly spurred by such criticisms, Fodor
elaborates his argument in considerable detail. To defend (2), he
critiques theories that model abduction by deploying
"local" heuristic algorithms (2005: 41-46; 2008:
115-126) or by positing a profusion of domain-specific cognitive
modules (2005: 56-100). To defend (4), he critiques various
theories that handle abduction through non-Turing-style models (2000:
46-53; 2008), such as connectionist networks.


The scope and limits of computational modeling remain
controversial. We may expect this topic to remain an active focus of
inquiry, pursued jointly with AI.


### 7.4 Temporal arguments


 Mental activity unfolds in time. Moreover, the mind accomplishes
sophisticated tasks (e.g., perceptual estimation) very quickly. Many
critics worry that computationalism, especially classical
computationalism, does not adequately accommodate temporal aspects of
cognition. A Turing-style model makes no explicit mention of the time
scale over which computation occurs. One could physically implement
the same abstract Turing machine with a silicon-based device, or a
slower vacuum-tube device, or an even slower pulley-and-lever
device. Critics recommend that we reject CCTM in favor of some
alternative framework that more directly incorporates temporal
considerations. van Gelder and Port (1995) use this argument to
promote a non-computational *dynamical systems framework* for
modeling mental activity. Eliasmith (2003, 2013: 12-13) uses it
to support his Neural Engineering Framework.


Computationalists respond that we can *supplement* an
abstract computational model with temporal considerations (Piccinini
2010; Weiskopf 2004). For example, a Turing machine model presupposes
discrete "stages of computation", without describing how
the stages relate to physical time. But we can supplement our model by
describing how long each stage lasts, thereby converting our
non-temporal Turing machine model into a theory that yields detailed
temporal predictions. Many advocates of CTM employ supplementation
along these lines to study temporal properties of cognition (Newell
1990). Similar supplementation figures prominently in computer
science, whose practitioners are quite concerned to build machines
with appropriate temporal properties. Computationalists conclude that
a suitably supplemented version of CTM can adequately capture how
cognition unfolds in time.


 A second temporal objection highlights the contrast
between *discrete* and *continuous* temporal evolution
(van Gelder and Port 1995). Computation by a Turing machine unfolds in
discrete stages, while mental activity unfolds in a continuous time.
Thus, there is a fundamental mismatch between the temporal properties
of Turing-style computation and those of actual mental activity. We
need a psychological theory that describes continuous temporal
evolution.


Computationalists respond that this objection assumes what is to be
shown: that cognitive activity does not fall into explanatory
significant discrete stages (Weiskopf 2004). Assuming that physical
time is continuous, it follows that mental activity unfolds in
continuous time. It does *not* follow that cognitive models
must have continuous temporal structure. A personal computer operates
in continuous time, and its physical state evolves continuously. A
complete physical theory will reflect all those physical changes. But
our *computational* model does not reflect every physical
change to the computer. Our computational model has discrete temporal
structure. Why assume that a good cognitive-level model of the mind
must reflect every physical change to the brain? Even if there is a
continuum of evolving *physical* states, why assume a continuum
of evolving *cognitive* states? The mere fact of continuous
temporal evolution does not militate against computational models with
discrete temporal structure.


### 7.5 Embodied cognition


Embodied cognition is a research program that draws inspiration
from the continental philosopher Maurice Merleau-Ponty, the perceptual
psychologist J.J. Gibson, and other assorted influences. It is a
fairly heterogeneous movement, but the basic strategy is to emphasize
links between cognition, bodily action, and the surrounding
environment. See Varela, Thompson, and Rosch (1991) for an influential
early statement. In many cases, proponents deploy tools of dynamical
systems theory. Proponents typically present their approach as a
radical alternative to computationalism (Chemero 2009; Kelso 1995;
Thelen and Smith 1994). CTM, they complain, treats mental activity as
static symbol manipulation detached from the embedding environment. It
neglects myriad complex ways that the environment causally or
constitutively shapes mental activity. We should replace CTM with a
new picture that emphasizes continuous links between mind, body, and
environment. Agent-environment dynamics, not internal mental
computation, holds the key to understanding cognition. Often, a
broadly eliminativist attitude towards intentionality propels this
critique.


Computationalists respond that CTM allows due recognition of
cognition's embodiment. Computational models can take into
account how mind, body, and environment continuously interact. After
all, computational models can incorporate sensory inputs and motor
outputs. There is no obvious reason why an emphasis upon
agent-environment dynamics precludes a dual emphasis upon internal
mental computation (Clark 2014: 140-165; Rupert 2009).
Computationalists maintain that CTM can incorporate any legitimate
insights offered by the embodied cognition movement. They also insist
that CTM remains our best overall framework for explaining numerous
core psychological phenomena.