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Rethinking the languageof thoughtSusan Schneider1∗ and Matthew Katz2

In this piece, we overview the language of thought (LOT) program, a currentlyinfluential theory of the computational nature of thought. We focus on LOT’sstance on concepts, computation in the central system, and mental symbols. Weemphasize certain longstanding problems arising for the LOT approach, suggestingresolutions to these problems. Many of the solutions involve departures from thestandard LOT program, i.e., the LOT program as developed by Jerry Fodor. Weclose by identifying avenues for future work. © 2011 John Wiley & Sons, Ltd.

How to cite this article:WIREs Cogn Sci 2011. doi: 10.1002/wcs.1155

INTRODUCTION

The computational paradigm in cognitive scienceaims to provide a complete account of our mental

lives, from the mechanisms underlying our memoryand attention to the computations of the singularneuron. However, what does the computationalparadigm amount to, philosophically speaking? Inthis piece, we discuss one influential approach toanswering that question: the language of thought(LOT) approach. LOT is one of two leading positionson the computational nature of thought, the otherbeing connectionism. According to LOT, humans andeven non-human animals think in a lingua mentis,an inner mental language that is not equivalentto any natural language. This mental language iscomputational in the sense that thinking is regardedas the algorithmic manipulation of mental symbols.In this piece, we outline the main features of theLOT position. We then consider several problemsthat have plagued the LOT approach for years: theconcern that LOT is outmoded by connectionism; anobjection by LOT’s own philosophical architect, JerryFodor, that urges that it is likely that cognition isnon-computational; and the failure of LOT to specifythe nature of mental representations that it invokesin its theory (such representations are called ‘mentalsymbols’). We respond to these problems, and in so

∗Correspondence to: sls@sas.upenn.edu1Department of Philosophy, University of Pennsylvania, Philadel-phia, PA, USA2Department of Philosophy and Religion, Central MichiganUniversity, Mt. Pleasant, MI, USA

doing, we sculpt a version of LOT that departs incertain ways from Fodor’s influential approach. Interalia, we outline an account of symbols that gives riseto a new theory of the nature of concepts (pragmaticatomism). We close with a discussion of an issuethat we believe requires further investigation, that ofmathematical cognition.

THE BASIC ELEMENTS OF THE LOTHYPOTHESIS

The idea that there is a LOT was mainly developedby Jerry Fodor, who defended the hypothesis in hisinfluential book, The Language of Thought (1975).1

The view that cognition is a species of symbolprocessing was in the air around the time thatFodor wrote the book. For instance, Allen Newelland Herbert Simon suggested that psychologicalstates could be understood in terms of an internalarchitecture that was like that of a digital computer.2

Human psychological processes were said to consistin a system of discrete inner states (symbols) thatare manipulated by a central processing unit (seealso Ref 3). The LOT or ‘symbol processing’position became the paradigm view in informationprocessing psychology and computer science until the1980s, when the competing connectionist view gainedcurrency. At that time the symbol processing viewcame to be known as ‘Classical Computationalism’ orjust ‘Classicism’.4,5

LOT consists in several core claims. (1) A firstclaim is that cognitive processes are the causalsequencing of tokenings of symbols in the brain.

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(2a) LOT also claims that mental representations havea combinatorial syntax. A representational systemhas a combinatorial syntax just in case it employsa finite store of atomic representations that maybe combined to form compound representations,which may in turn be combined to form furthercompound representations. (2b) Relatedly, LOT holdsthat symbols have a compositional semantics—themeaning of compound representations is a functionof the meaning of the atomic symbols, together withthe grammar.a (2c) LOT further claims that thinking,as a species of symbol manipulation, often preservessemantic properties of the thoughts involved.1,6 Animportant example of one such property is beingtrue. Consider the mental processing of an instanceof modus tollens. The internal processing is purelysyntactic, but it is nevertheless truth preserving. Giventrue premises, the application of the rule will result infurther truths.

(3) Finally, LOT asserts that mental operationson internal representations are causally sensitiveto the syntactic structure of the symbols. Thatis, computational operations work on any symbolor string of symbols satisfying a certain structuraldescription, transforming the symbol/string intoanother symbol/string that satisfies another structuraldescription. For instance, the system may employ anoperation in which it transforms any representationthat it recognizes as having the form (Q&R) intoa representation with the form (Q). In such anoperation, the meaning of the symbols plays no role atall. In addition, the physical structures onto which thesymbol structures are mapped are the very propertiesthat cause the system to behave in the way it does.6

(1)–(3) combine to form a position called the‘Computational Theory of Mind’ (CTM). CTM holdsthat thinking is a computational process involvingthe manipulation of semantically interpretablestrings of symbols that are processed according toalgorithms.2,4,7–11Together, LOT and CTM aim toanswer the age old question, ‘how can rationalthought be grounded in the brain’? Their answeris that rational thought is a matter of the causalsequencing of symbol tokens that are realized inthe brain. And further, these symbols, which areultimately just patterns of matter and energy, haveboth representational and causal properties. And asnoted, the semantics mirrors the syntax. So thinkingis a process of symbol manipulation in which thesymbols have an appropriate syntax and semantics(roughly, natural interpretations in which the symbolssystematically map to states in the worldb).4,10

LOT is controversial, to be sure. For one thing,LOT is by no means the only approach to the

format of thought. As we shall now see, some regardconnectionism as rendering LOT obsolete.

RESPONDING TO THECONNECTIONIST CHALLENGE

Cognitive science is only in its infancy, butvarious cognitive scientists, such as Jeffrey Hawkins(computational neuroscience) and Paul and PatriciaChurchland (philosophy) claim that they already seethe outlines of a final theory, glimmers of a singularformat of thought exhibited throughout the entirebrain, from the simplest sensory circuit to the mostcomplex computation of the prefrontal cortex. Inessence, the mind is a species of connectionist network.For instance, Paul Churchland writes:

. . . we are now in a position to explain how ourvivid sensory experiences arise in the sensory cortexof our brains: how the smell of baking bread, thesound of an oboe, the taste of a peach, and the colorof a sunrise are all embodied in a vast chorus ofneural activity. . . And we can see how the maturedbrain deploys that framework almost instantaneously:to recognize similarities, to grasp analogies, and toanticipate both the immediate and the distant future(Ref 12, p. 3).

Connectionists claim that thinking is determinedby patterns of activation in a neural network. Considerthe following extremely simple connectionist network(Figure 1).

Each circle (or ‘unit’) can represent either a singleneuron or a group of neurons. There are three layers of

Outputpattern

Outputunits

Hiddenunits

Inputunits

Inputpattern

FIGURE 1 | A simple network.

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WIREs Cognitive Science Rethinking the language of thought

units: an input layer, a middle or ‘hidden’ layer, andan output layer. Computation flows upward, withthe smaller arrows specifying connections betweenunits. Each hidden or output unit carries a numericalactivation value that is computed given the values ofthe neighboring units in the network, according to afunction. The input units’ signals thereby propagatethroughout the network, determining the activationvalues of all the output units. Of course, actualmodels of perceptual and cognitive functions are farmore complex than the simple illustration given here,exhibiting multiple hidden layers and feedback loops.But this is the bare bones idea.

It is important to note some of the differencesbetween such networks and classical computers, fromwhich LOT was originally inspired. First, while classi-cal computers possess distinct memory and processingunits, connectionist networks do not, being comprisedentirely of simple nodes and the connections betweenthem. Second, while classical computers are serialmachines, processing representations in step-wisefashion, network processing occurs in parallel. Activa-tion values of the nodes in a network are continuouslyupdated until the network ‘settles down’ into a steadystate in which activation values are no longer beingupdated. Third, while processing in classical comput-ers is concentrated in the central processing unit, it iswidely distributed in a connectionist network. Indeed,semantic interpretation is distributed across nodes,and compound representations are not formed byconcatenation, as in classical machines. This makesit difficult, if not impossible, to explain features ofthought such as productivity and systematicity withina connectionist framework (more on this below).

The connectionist framework has had manysuccess stories, especially in domains involving therecognition of perceptual patterns, and indeed,we find connectionist work on pattern recognitionpromising. However, when it comes to consideringthe implications of connectionism for the formatof higher thought, even connectionists are wont toadmit that existing models are only highly simplifieddepictions of how a given perceptual or cognitivecapacity works. As new information emerges aboutthe workings of the brain, connectionists will of courseadd more sophistication to their models. And thusconnectionists suspect that higher thought will bedescribable given the resources of a computationalneuroscience that is based on, and represents a moresophisticated version of, connectionist theorizing. Putbluntly, it will be ‘more of the same’ but with addedbells and whistles.

But will it really be more of the same? Manyadvocates of LOT deny this possibility. Their main

objection arises from the very reason they believe thebrain computes in LOT: thought has the crucial andpervasive feature of being combinatorial. First, con-sider the thought the beer in Munich is better thanon Mars. You probably have never had this thoughtbefore, but you were able to understand it. The keyis that the thoughts are built out of familiar con-stituents, and combined according to rules. It is thecombinatorial nature of thought that allows us tounderstand and produce these sentences on the basisof our antecedent knowledge of the grammar andatomic constituents (e.g., beer, Munich). More specif-ically, thought is productive: in principle, one canentertain and produce an infinite number of distinctrepresentations because the mind has a combinatorialsyntax. Relatedly, thought is systematic. A represen-tational system is systematic when the ability of thesystem to entertain and produce certain representa-tions is intrinsically related to its ability to entertainand produce other representations.6 For example, onedoes not find normal adult speakers who entertainand produce ‘David likes Mitsuko’ without also beingable to entertain and produce ‘Mitsuko likes David’.As with productivity, systematicity arises from the factthat thoughts display combinatorial structure.1,4,6,13

It is commonly agreed that LOT excels inexplaining these language-like features of thought.From the connectionist vantage point, however,thought is not primarily language-like; instead, cog-nition is a species of pattern recognition consisting inassociative relationships between units. The language-like and combinatorial character of cognition hastherefore been a thorn in the side of the connection-ist. Whether any connectionist models, and especially,any models that are genuinely non-symbolic (moreon this shortly), can explain these important fea-tures of thought are currently a source of intensecontroversy.6,13–18 Relatedly, it has been argued thatthe symbol processing view best explains how mindsdistinguish between representations of individualsand of kinds, encode rules or abstract relationshipsbetween variables, and have a system of recursivelystructured representations.16 As with systematicityand productivity, which are closely related to thelanguage-like structure of higher thought, these fea-tures of thought are more naturally accommodated bythe symbolic paradigm.

Not only is it controversial whether purely con-nectionist models can fully explain higher thought, butit is also unclear how very simple models of isolatedneural circuits are supposed to ‘come together’ to giverise to a larger picture of how the mind works.19

Existing ‘big picture’ accounts are fascinating, yetpatchy and speculative. Furthermore, higher cognition

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is the domain in which we would see validation ofthe symbol processing approach, if validation is tocome at all.4 The discrete, combinatorial representa-tions of the prefrontal cortex are distinct from themore distributed modality-specific representation ofthe posterior cortex; prima facie, this latter repre-sentation seems more straightforwardly amenable totraditional connectionist explanation.c

Furthermore, the relationship between LOTand connectionism is subtle. The advocate of LOThas an important rejoinder to the connectionistattempt to do without mental symbols: to the extentthat the connectionist can explain the language-like,combinatorial nature of thought, the connectionistmodel would, at best, merely be a model inwhich symbols are implemented in the mind. Soconnectionism is not really a genuine alternativeto the LOT picture, for the networks would justbe the lower-level implementations of symbolicprocesses. This position is called ‘implementationalconnectionism’.6,16 As Gary Marcus explains:

The term connectionism turns out to beambiguous. Most people associate the term withthe researchers who have most directly challengedthe symbol-manipulation hypothesis, but the field ofconnectionism also encompasses models that havesought to explain how symbol-manipulation can beimplemented in a neural substrate. . . The problem isthat discussions of the relation between connectionismand symbol manipulation often assume that evidencefor connectionism automatically counts as evidenceagainst symbol-manipulation (Ref 16, p. 2).

If connectionism and symbolicism represent gen-uine alternatives, a view called ‘Radical Connection-ism’ must be correct. But existing connectionist modelsof higher cognitive function are few, and there arepersuasive arguments that putative radical connec-tionist models in fact make covert use of symbolicrepresentations.16

Furthermore, the mind may be a sort of ‘hybrid’system, consisting in both neural circuits that satisfysymbolic operations and other neural circuits thatcompute according to connectionist principles, anddo not satisfy symbolic operations at all. (These otherneural circuits may lack representations that combinein language-like ways, and in this case, the proponentof LOT would not even claim that they are imple-mentations of symbolic systems.) Indeed, certain con-nectionists currently adopt hybrid positions in whichsymbol processing plays an important role.20Many ofthese models employ connectionist networks to modelsensory processes and then rely on symbol processingmodels for the case of cognition, but perhaps even the

cognitive mind will be a mix of different formats, apoint we consider in the final section of this piece.

The upshot is that connectionist success stories inthe domain of pattern recognition do not suggest thatLOT is false. LOT is compatible with both the successand failure of connectionism. Instead, the proponentof LOT should look to connectionism with greatinterest, as it seeks to uncover the neurocomputationalbasis of thought. Pursuing a better understanding ofLOT is thus crucial to contemporary cognitive science.

INTEGRATING THE PHILOSOPHICALLOT WITH COGNITIVE SCIENCE:BEYOND FODORIAN PESSIMISM

Yet there is an additional reason why connectionistsand other critics of LOT reject LOT: the mind is notanalogous to a digital computer. It does not have aCPU through which every mental operation is shut-tled sequentially, and more generally, the brain hasa complex and unique functional organization thatis not like the functional organization of a standardcomputer. We agree with both of these observations,but we believe that they do not speak against LOT.While the idea that the mind has a CPU was heldby early Classicists, most contemporary advocates ofthe symbol processing approach regard the mind ascomputational because they believe cognition is thealgorithmic computation of discrete symbols, wherethe algorithm that the brain runs is to be discovered bya completed cognitive science.4,7,10 These algorithmscompute on symbols in a manner that is sensitive tothe constituent structure of symbolic strings, enablingthe compositionality, productivity, and systematicityof thought. This position, rather than the outdatedclaim that the brain has the functional organization ofa standard computer, best captures the sense in whichthe brain is said to compute in LOT.

But LOT’s chief philosophical architect, JerryFodor, has argued the cognitive mind is likelynon-computational, for the brain’s ‘central system’will likely defy computational explanation.8,9,21

Fodor calls the system responsible for our abilityto integrate material across sensory divides andgenerate complex, creative thoughts ‘the centralsystem’. The central system is ‘informationallyunencapsulated’—its operations can draw frominformation from outside of the system, in additionto its inputs. And it is domain general, with inputsranging over diverse subjects. Fodor urges thatthe central system’s unencapsulation gives rise toinsuperable obstacles. One longstanding worry is thatthe computations in the central system are not feasiblycomputed within real time. For if the mind truly is

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computational in a classical sense, when one makes adecision one would never be able to determine whatis relevant to what. For the central system would needto walk through every belief in its database, askingif each item was relevant. Fodor concludes from thisthat the central system is likely non-computational.Shockingly, he recommends that cognitive science stopworking on cognition.

Fodor’s pessimism has been influential withinphilosophical circles, situating LOT in a tenuousdialectical position: since LOT itself was originallyconceived of as a computational theory of delib-erative, language-like thought (herein, ‘conceptualthought’),1,3 and the central system is supposed tobe the domain in which conceptual thought occurs,it is unclear how, assuming that Fodor is correct,LOT could even be true. In any case, advocates ofLOT in cognitive science proper reject Fodor’s pes-simism about computation in the central systems.10,16

And some recent philosophical discussions of LOThave responded to Fodor’s anti-computationalism.For instance, Peter Carruthers responds to Fodor’s rel-evance challenge by granting that if the central systemis amodular, as Fodor contends, the problem would beinsurmountable. According to Carruthers, the centralsystem must be massively modular, that is, the cen-tral system features special purpose, innate informa-tion processing modules.7,22,23 Carruthers summonsresearch in areas such as evolutionary psychology,animal cognition, and dual systems theory to developsuch a view.7

Schneider urges that an amodular centralsystem can compute what is relevant. She drawsfrom the work of Murray Shanahan and BernardBaars, who sketch the beginnings of a solutionto the Relevance Problem that is based upon theGlobal Workspace (GW) theory.24–27 According tothe GW theory, a pancortical system (a ‘GW’)facilitates information exchange among multipleparallel, specialized unconscious processes in thebrain. When information is conscious there is astate of global activation in which information inthe workspace is ‘broadcast’ back to the rest ofthe system. At any given moment, there are multipleparallel processes going on in the brain that receivethe broadcast. Access to the GW is granted byan attentional mechanism and the material in theworkspace is then under the ‘spotlight’ of attention.When in the GW the material is processed in a serialmanner, but this is the result of the contributionsof parallel processes that compete for access to theworkspace. (This view is intuitively appealing, asour conscious, deliberative, thoughts introspectivelyappear to be serial.) Schneider summons the GW

theory as the basis for a computational account of thecentral system.

It is also noteworthy that Schneider’s develop-ment of the central system requires that LOT turnits attention to cognitive and computational neuro-science, despite Fodor’s well-known repudiation ofthese fields. First, she urges that integration withneuroscience will enable a richer explanation of thestructure of the central system. In her book, TheLanguage of Thought: A New Philosophical Direc-tion, she illustrates how neuroscience can provide adeeper understanding of the central system by appeal-ing to certain recent work on the central system inneuroscience and psychology, such as, especially, theaforementioned GW theory.4 Second, on her view,cognitive and computational neuroscience are key todetermining what kinds of mental symbols a givenindividual has (e.g., whether one has a symbol forbulldogs, jazz, or Chianti), for these fields detail thealgorithms that the brain computes, and on her view,the type that a given mental symbol falls under isdetermined by the role it plays in the algorithms thatthe brain computes. Third, she stresses that given thatLOT is a naturalistic theory, that is, one that seeksto explain mental phenomena within the domain ofscience, it depends on integration with neurosciencefor its own success.

Now let us turn to yet a further problem thatLOT faces: LOT and CTM purport to be theories ofthe symbolic nature of thought, but as we shall see,their notion of a mental symbol is poorly understood.

TOWARD AN UNDERSTANDINGOF SYMBOLIC MENTAL STATES

LOT’s commitment to the idea that thinkinginvolves the manipulation of mental symbols is oftenmisunderstood. What are mental symbols? Clearly,there is no orthography or ‘brain writing’ in whichstrings of symbols are encoded. Rather, LOT claimsthat at a high level of abstraction, the brain can beaccurately described as employing a representationalsystem that has a language-like structure. A secondsource of misunderstanding is that LOT is frequentlywedded to a form of radical nativism in which allsymbols are said to be innate. While certain symbolsmay be innate, it is hard to believe that cavemen hadinnate symbols such as [photon] and [internet]. Wesuspect that this implausible view has strengthenedthe sense of mystery surrounding LOT’s notion of amental state, and it has led many to reject LOT asimplausible. However, LOT does not require radicalnativism: indeed, even Fodor no longer endorsesradical nativism,28 and other proponents of LOT

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have for years taken a more moderate approach toconcept nativism. A third source of misunderstandingis more serious, however. Proponents of LOT havenot specified whether LOT’s symbols are typed withrespect to their semantic properties, their neuralproperties, their computational roles, or with respectto some other feature entirely.4,29–32

Without an understanding of how symbols aretyped it is difficult to see how neural activity can, ata high level of abstraction, even be described as themanipulation of mental symbols. It is also difficult todetermine whether a given connectionist model trulyimplements symbolic manipulations. Furthermore,the proponent of LOT summons symbols to doimportant philosophical work, and this work cannotbe done without a plausible conception of a symbol.Symbols are supposed to be neo-Fregean ‘modes ofpresentation’: roughly, one’s way of conceiving things.As such, LOT looks to symbols to explain how anindividual can have different ways of conceiving of thesame object, and to explain the causation of thoughtd

and behavior.4,33,34

Without a clear notion of a LOT symbol,none of this work can be accomplished. However,Schneider has recently developed a theory of symbolsthat is designed to play the aforementioned roles forLOT. She begins by ruling out competing theoriesof the nature of symbols. For instance, she rules outapproaches that identify symbols by what they referto because they are not finely grained enough todifferentiate between different ways one can conceiveof the same object (e.g., consider thinking of ClarkKent as being Superman, versus merely being areporter for the Daily Planet). She then argues thatLOT requires a theory of symbols that assigns asymbol token to its type in virtue of the role thesymbol plays in the algorithms employed by the centralsystem (she calls this ‘the algorithmic conception’).This position is not new: in the past both Jerry Fodorand Stephen Stich have appealed to it.34,35 However,neither philosopher developed the idea in any detail,and Fodor came to repudiate it. Schneider providesthree arguments for the algorithmic conception. Thefirst is that the classical view of computation, to whichLOT is wedded, requires that primitive symbols betyped in this manner. The second is that withoutthis manner of individuating symbols, cognitiveprocessing could not be symbolic. The third is thatcognitive science needs a natural kind that is definedby its total computational role. Otherwise, eitherexplanation in cognitive science will be incomplete, orits generalizations will have counterexamples. Finally,she explores how symbols, thus understood, figurein explanations of thought and behavior in cognitive

science.4,31,32 Her theory of symbols gives rise to atheory of the nature of concepts, to which we shallnow turn.

A ‘PRAGMATIST’ THEORYOF CONCEPTS

One outgrowth of the philosophical LOT programis a theory of the nature of concepts, conceptualatomism, a view of the nature of concepts devisedby Jerry Fodor, Eric Margolis, Stephen Laurence, andothers.4,28,36 Conceptual atomism holds that lexicalconcepts lack semantic structure, being in this sense‘atomic’. It further holds that a concept has a twofoldnature, consisting in both its broad content (roughly,what it refers to in the world) and its symbol type.e

Much of Fodor’s work on concepts situates conceptualatomism, and indeed, LOT itself, in opposition topositions that he labels ‘pragmatist’—positions onthe nature of concepts that hold one’s psychologicalabilities (e.g., classificatory or inferential capacities)determine the nature of concepts (Ref 37, p. 34).Fodor urges that pragmatism is a ‘. . .catastrophe ofanalytic philosophy of language and philosophy ofmind in the last half of the twentieth century’.38

Schneider claims that both LOT and conceptualatomism must embrace pragmatism. For as noted, sheargues that the nature of symbols is determined by therole they play in computations in the central system.So a symbol type is determined by role a symbolplays in an individual’s mental life, including the roleit plays in inference and classification. Furthermore,even on Fodor’s view, symbols are elements of aconcept’s nature.4,39 Clearly, Fodor did not anticipatethat conceptual atomism is pragmatist; this is likelybecause symbol natures were left underspecified. Todistinguish her position from Fodor’s she calls thisbrand of conceptual atomism Pragmatic Atomism.

Schneider contends that pragmatic atomism is asuperior version of conceptual atomism. Critics havelong charged that conceptual atomism is too skele-tal, merely being a semantic theory in which conceptsare a matter of the information they convey, andsaying next to nothing about the role concepts playin our mental lives. Indeed, the standard conceptualatomist cannot even distinguish between coreferingconcepts that have the same grammatical form (e.g.,Cicero/Tully). This defect stands in stark contrast topsychological theories of concepts, such as the pro-totype theory and the theory theory, which focus onhow we reason, learn, categorize, and so on. However,pragmatic atomism can say that the features of con-cepts that psychologists are traditionally interested inare built into concepts’ very natures. Symbols, being

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individuative of concepts, capture the role the conceptplays in thought. For example, symbols have com-putational roles that distinguish corefering concepts,characterize the role the concept plays in categoriza-tion, determining whether, and how rapidly, a personcan verbally identify a visually presented object, con-firm a categorization judgment, identify features anobject possesses if it is indeed a member of a givencategory, and so on. Here, pragmatic atomism candraw from research from any of the leading psycho-logical theories of concepts. Consider the prototypeview, for instance. In the eyes of pragmatic atomism,the experimental results in the literature on prototypesare indications of features of certain symbols’ under-lying computational roles, and these roles determinethe relevant concept’s natures.4,39

Our discussion has isolated certain longstandingproblems arising LOT; and we have urged that indealing with these problems one must rethink key ele-ments of the standard LOT program, namely, LOT’sstance on concepts, symbols, and computation. Now,we shall conclude by asking: where should the LOTprogram go from here? We believe that the follow-ing topics deserve future attention: (1), understandingthe relationship between symbols, typed in the man-ner outlined herein, and accounts of higher cognitivefunction in cognitive and computational neuroscience.This will facilitate a better understanding of howsymbols and concepts are grounded in the brain,and determine whether the brain indeed computessymbolically. (2) Understanding the relation betweenlinguistically formatted and non-linguistically format-ted representations in light of an emerging challengepresented by the literature on numerical representa-tion. We close by commenting on this latter issue.

CONCLUDING REMARKS: FUTUREINVESTIGATION ON NON-LINGUISTICMATHEMATICAL COGNITION

An interesting development has emerged fromresearch on the representation and processing ofnumerical information. A wealth of experimentationf

suggests that humans as well as certain non-humananimals possess an innate cognitive system that rep-resents cardinality in terms of mental magnitudes.This system represents the cardinality of groups ofobjects as quantities that are proportional in size tothe cardinalities they represent. For example, if a sub-ject counts a set of objects, the system tokens oneincrement for each object counted. The more objectsin the group, the more increments are tokened, andthe larger the resulting representation. However, theincrements themselves are variable in size, and as the

increments are combined to form larger and largerrepresentations, the variability compounds. At a cer-tain point, the system is unable to distinguish a givennumber from its nearby neighbors, where what countsas ‘nearby’ is proportional to the number represented.This indicates that the system cannot ‘recall’ the num-ber of increments that were used to form a compoundrepresentation. Rather, it only has access to the finalproduct. In other words, the system forms compoundrepresentations by combining increments and not byconcatenating them. So the components of compoundrepresentations are not discrete. This is a fundamentaldifference between mental magnitudes and linguisticrepresentations.g

The existence of mental magnitudes may indeedprovide evidence that LOT is not true of all of cog-nition. But if so, the issue is limited to those domainsthat employ magnitude representations, and evenwithin those domains the research does not necessar-ily challenge symbolic representation. For instance,the presence of magnitude representations of car-dinality does not call into question the view thatmathematical operations are also symbolic, for mostauthors hold that higher thought about numbers ismediated by linguistically formatted representations.For instance, Dehaene has argued for a ‘triple code’theory, according to which numerical information isprocessed by three separate cognitive systems.40 Oneof these systems employs mental magnitudes, oneemploys a visual number form (such as the Arabicnumerals), and one employs auditory verbal repre-sentations. These latter two kinds of representationsare discrete, and indeed, can be individuated by theircomputational roles. Thus, research on mental mag-nitudes (and other non-linguistic forms of mentalrepresentation) can be viewed as being complimen-tary to the LOT program, such that these variousavenues of investigation together seek to answer ques-tions regarding the scope of and relationship betweenlinguistic and non-linguistic representational systems.

Indeed, because mental magnitudes are presentin human infants and non-human animals, mostresearchers have been concerned with whether andhow mental magnitudes play a role in the formationof higher, conceptual, thought about numbers. Forexample, Spelke has argued that mental magnitudesmust combine with another innate representationalsystem—the system of object files—in order toachieve the precision that is characteristic of matureconceptual thought about numbers.41 Carey hasargued that natural number concepts first arise onlyas small number concepts, provided by object-filerepresentations that are ‘enriched’ by quantificationalmarkers in natural language. She argues that mental

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magnitudes only later play a role, as they becomelinked with the enriched object-file representationsto thus provide large number concepts.42,43 In con-trast, Leslie, Gelman, and Gallistel and Laurence andMargolis have argued that even combining mentalmagnitudes with other systems is insufficient to sup-ply mature numerical concepts, and that humansmust in fact possess innate representations of the firstfew natural numbers.44,45 Whichever, if any, of theseaccounts turns out to be accurate, the question beingaddressed can be thought of in terms of whether andhow symbolic language-like mental representationsdevelop from non-symbolic non-language-like mentalrepresentations.h

NOTESaA proponent of LOT need not claim that LOT hasa semantics, for one could reject meanings. But inpractice, an appeal to meaning has generally beena key facet of LOT. For an important exception, seeStephen Stich (1983).35 Given that this section followsthe standard elaborations of LOT, which appeal toa semantics, LOT is basically indistinguishable froma view called ‘CTM’ or the ‘Classical ComputationalTheory of Mind’, a position we discuss later in thissection.bFor another overview of LOT, see Ref 46.cFor a discussion of the distinct processing in the pre-frontal cortex and posterior cortex, see O’Reilly andMunakata (Ref 47, p. 214–219).dFregeans consider mode of presentations (MOPs) tobe semantic. Many advocates of LOT appeal to areferential semantics and do not consider MOPs to besemantic in nature.eBecause symbol natures have been neglected, only thesemantic dimension of the theory has been developed.

Indeed, the conceptual atomists’ concepts are oftenmistakenly taken as being equivalent to broad con-tents, despite the fact that they are individuated bytheir symbol types as well.4,39

f For useful reviews, see for example, Dehaene(1997), Feigenson, Dehaene, and Spelke (2004), Gal-listel, Gelman, and Cordes (2006), and Cantlon, Pratt,and Brannon (2009).48–51

gIn the above description we have omitted sev-eral areas of disagreement about mental magnitudes.First, there is debate about whether the system ofmental magnitudes functions serially or in parallel(see Refs 52 and 53). Second, there is debate aboutwhether variability is present in the magnitudes them-selves or in memory (see Ref 50). Third, there is debateabout whether the system exhibits scalar variabilityor logarithmic compression (see Refs 54 and 55).Despite these disagreements, there is wide consensusthat positing magnitude representations explains thefact that human and non-human discrimination of setsbased on cardinality obeys Weber’s Law, which statesthat � I/I = k, where I is the intensity of a stimulus,�I is the minimal change in intensity required fordiscrimination of the stimulus from another, and k isa constant. In other words, whether a subject is ableto discriminate between two stimuli depends on thedifference in intensity of the two, not their absolutevalues. Positing linguistically formatted representa-tions would not explain why discrimination of setsbased on cardinality obeys Weber’s Law, and hencethe argument that there is a fundamental differencebetween magnitude representations and representa-tions in LOT (see also Ref 56).hFor an overview of recent work documenting mag-nitude representations and more on their relationshipto LOT (see Ref 57).

ACKNOWLEDGMENTS

We are grateful to Carlos Montemayor, two anonymous reviewers, and the editors for their helpful commentson this piece.

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