1 From Neural Circuitry to Mechanistic Model-based Reasoning

Jonathan Waskan

To appear in Springer Handbook of Model-based Science

Model-based reasoning in science is often carried out in an attempt to understand the kinds of

mechanical interactions that might give rise to particular occurrences. One hypothesis regarding

in-the-head reasoning about mechanisms is that scientist rely upon mental models that are like

scale models in crucial respects. Behavioral evidence points to the existence of these mental

models, but questions remain about the neural plausibility of this hypothesis.

This chapter will provide an overview of the psychological literature on mental models of

mechanisms with a specific focus on the question of how representations that share the

distinctive features of scale models might be realized by neural machinations. It is shown how

lessons gleaned from the computational simulation of mechanisms and from neurological

research on mental maps in rats can be applied to make sense of how neurophysiological

processes might realize mental models.

The goal of this chapter is to provide readers with a general introduction to the central

challenge facing those would maintain that in-the-head model based reasoning about

mechanisms in science is achieved through the use of scale-model-like mental representations.

1.1 Overview

A central form of model-based reasoning in science, particularly in the special sciences, is

model-based reasoning about mechanisms. This form of reasoning can be effected with the aid of

external representational aids (e.g., formalisms, diagrams, and computer simulations) and

through the in-the-head manipulation of representations. Philosophers of science have devoted

most of their attention to the former, but the latter is arguably at the heart of most of what passes

for explanatory understanding in science (Sec. 1.2). Psychologists have long theorized that the

humans and other creatures (e.g., rats) reason about spatial, kinematic, and dynamic relationships

through the use of mental representations, often termed mental models, that are structurally

similar to scale models, though clearly the brain does not instantiate the very properties of a

modeled system in the way that scale models do (1.3). A key challenge facing this view is thus to

show that brains are capable of realizing representations that are like scale models in crucial

respects. There have been several failed attempts to show precisely this, but a look at how

computers are utilized to model mechanical interactions offers a useful way of understanding

how brains might realize mental representations of the relevant sort (Sec. 1.4). This approach

meshes well with current research on mental maps in rats. In addition, it has useful ramifications

for research in A.I. and logic (Sec. 1.5), and it offers a promising account of the generative

knowledge that scientists bring to bear when testing mechanistic theories while also shedding

light on the role that external representations of mechanisms play in scientific reasoning (1.6).

1.2 Mechanistic Reasoning in Science

A common reason that scientists engage in model-based reasoning is to derive information

that will enable them to explain or predict the behavior of some target system. Model-based

explanations provide scientists with a way of understanding how or why one or more

explanandum occurrences came about. A good model-based explanation will typically provide

the means for determining what else one ought to expect if that explanation is accurate1 – that is,

it will enable one to formulate predictions so that the explanation may (within widely known

limits) be tested.

1 One must bear in mind, however, that models are often accurate only in certain respects and to certain degrees [2].

Model-based reasoning can, corresponding to the diversity of representational structures

that count as models – including external scale models, biological models, mathematical

formalisms, and computer simulations – take many forms in science. As for what models

represent, it is now widely accepted that mechanisms are one of the principal targets of model-

based reasoning. This is most obviously true in the non-basic sciences (e.g., biology, medicine,

cognitive science, economics, and geology).

In philosophy of science, much of the focus on mechanisms has thus far been on the role

they play in scientific explanation. The idea that all genuine scientific explanation is mechanistic

began to gain traction in contemporary philosophy of science with the work of Peter Railton,

who claimed that “if the world is a machine – a vast arrangement of nomic connections – then

our theory ought to give us some insight into the structure and workings of the mechanism,

above and beyond the capability of predicting and controlling its outcomes…” [1]. Inspired by

Railton, Wesley Salmon abandoned his statistical-relevance model of explanation in favor of the

view that “the underlying causal mechanisms hold the key to our understanding of the world”

[3]. On this view, an “explanation of an event involves exhibiting that event as it is embedded in

its causal network and/or displaying its internal causal structure"[4]. Salmon was working in the

shadow of Carl Hempel’s covering law model of explanation, according to which explanations

involve inferences from statements describing laws and, in some cases, particular conditions.

Salmon tended, in contrast, to favor an ontic account, according to which explanations are out in

the world. He thought that progress in understanding those explanations requires ‘exhibiting’ the

relevant mechanisms. However, even though he rejected representational and inferential

accounts of explanation, he naturally recognized that reasoning about mechanisms, which

requires representations (models), plays a big part in the process of exhibiting those mechanisms.

A more recent formulation of the mechanistic account of explanation is supplied by

Machamer, Darden, & Craver, who claim that “Mechanisms are entities and activities organized

such that they are productive of regular changes from start or set-up to finish or termination

conditions” [5]. A central goal of science, on their view, is to formulate models, which take the

form of descriptions of mechanisms that render target occurrences intelligible:

Mechanism descriptions show how possibly, how plausibly, or how actually things work.

Intelligibility arises...from an elucidative relation between the explanans (the set-up

conditions and intermediate entities and activities) and the explanandum (the termination

condition or the phenomenon to be explained)....

As with ‘exhibiting’ for Salmon, the process of ‘elucidating’ how set-up conditions lead to

termination conditions requires a significant contribution from model-based reasoning.

Bechtel offers a related account of mechanisms. He claims: “A mechanisms is a structure

performing a function in virtue of its component parts. The orchestrated functioning of the

mechanism is responsible for one or more phenomena” [6]. As compared with other mechanists,

Bechtel is much more explicit about the role that model-based reasoning plays in science and

about the diverse forms of representation that may be involved (e.g., descriptions, diagrams,

scale models, and animal models). He is, moreover, among the few to acknowledge the

importance of ‘in-the-head’ model-based reasoning. He suggests that its central form may

involve a kind of mental animation. As Bechtel and Wright put it, “One strategy is to use

imagination to put one’s representation of the mechanism into motion so as to visualize how that

phenomenon is generated” [7]. Bechtel claims that the representations underlying this mental

animation process may have a structure similar to that of the diagrams scientist use in their

thinking and to the animated renderings of computer simulations scientists construct to represent

proposed mechanisms in action [6]. As for prediction, he notes:

what the scientist advances is a representation of a mechanism… She or he then evaluates

the representation by using it to reason about how such a mechanism would be expected to

behave under a variety of circumstances and testing these expectations against the behavior

of the actual mechanism [6].

In other words, once the scientist possesses a model of the mechanisms that may be responsible

for an occurrence, which may take the form of a mental model, he or she may then use it to

formulate predictions in order to test that model.

While external representational artifacts may sometimes be required in order to achieve

explanatory understanding of how a mechanism could produce a given phenomenon, plausibly

those artifacts are not themselves sufficient for explanatory understanding.2 Instead,

representational artifacts may have the important function of facilitating understanding by

enhancing the scientist’s ability to mentally simulate the process by which the proposed

mechanism would produce the target phenomenon.3 Through manipulation of those mental

simulations, scientists may also discover novel predictions of a given model.

1.3 The Psychology of Model-based Reasoning

Given the potentially crucial role that mental models play in the process of mechanistic

explanation and prediction, it may be that we cannot hope to attain a truly adequate, deep

understanding of science without first understanding how the mental modeling process works.

An obvious way of going about making sense of the role mental models play in science is to

inquire into the nature of those models themselves. A good question to ask here is: What form

2 For evidence that there is a crucial psychological component to explanatory understanding, see [8]. 3 As shown in Section 5.2, external representational aids may also enable forms of reasoning that would otherwise

(e.g., due to the complexity of the mechanism) be impossible.

must our mental models take if they are to play the role that they do in science? One increasingly

popular answer has its origins in Craik’s landmark monograph, The Nature of Explanation.

Regarding everyday reasoning, Craik suggests:

If the organism carries a 'small-scale model' of external reality and of its own possible

actions within its head, it is able to try out various alternatives, conclude which is the best

of them, react to future situations before they arise ... and in every way to react in a much

fuller, safer, and more competent manner to the emergencies which face it [9].

On Craik’s view, scientific explanation is just an extension of this everyday reasoning process –

that is, it involves the construction of internal world models that are akin to scale models.4

What may be considered the first attempt to put this view to experimental scrutiny came in

the prelude to the cognitive revolution with Edward Tolman’s seminal studies of spatial

navigation in rats [9]. In his most famous experiment, Tolman’s team placed rats in a simple

alley maze, similar to the one depicted in Figure 1a, and rewarded the animals with food when

they reached the end. After learning to perform the task without hesitation, the maze was

replaced with a radial maze similar to the one in Figure 1b, where the alley that the rats had

previously learned to traverse was blocked. Upon discovering this, the vast preponderance of rats

then chose the alley that led most directly to where the food source had been in previous trials.

On the basis such experiments, Tolman concluded that rats navigate with the aid of cognitive

maps of the relative spatial locations of objects in their environment.

[Insert Figure 1 About Here]

Figure 1. Alley maze (a) and radial maze (b), based on Tolman (1948).

4 Bechtel is explicit in crediting Craik, when he maintains that the use of mental models in scientific reasoning about

mechanisms is to be understood by analogy with the use of external images and scale models [6]. Fellow mechanists

Nancy Nersessian [11] and Paul Thagard [12] also credit Craik.

Later, Shepard & Metzler would show that the time it takes for people to determine if two

3D structures have the same shape is proportional to the relative degree of rotational

displacement of those structures [13]. One neat explanation for this finding is that people engage

in the mental rotation of 3D models of the two structures until they are aligned in such a fashion

as to enable easier comparison. In another landmark study of mental imagery, Kosslyn showed

that reaction times for scanning across mental images of a map was proportional to distance, but

not to the number of intervening objects, suggesting that spatial reasoning is better explained by

a process akin to scanning across a real map than to a process of sentence-based reasoning (e.g.,

working through a list structure) [14].

All of this research points to the existence of mental models of 2D and 3D spatial

relationships, but to support the full range of inferences implicated in mechanistic model-based

scientific reasoning, mental models would need to capture kinematic and dynamic relations as

well. There is some support for the existence of these models as well. For instance, Schwartz &

Black observed similar, proportional reaction times when subjects were asked to determine

whether or not a knob on one gear would, when that gear is rotated, fit into a grove on a

connecting gear (Figure 2a) [15]. Schwartz & Black found, moreover, that subjects were able to

“induce patterns of behavior from the results depicted in their imaginations” [16]. Subjects

might, for instance, infer and remember that the second in a series of gears will, along with every

other even-numbered gear, turn in the opposite direction of the drive gear (Figure 2b). Having

inferred this through simulation, the information becomes stored as explicit knowledge, thereby

eliminating the need to generate the knowledge anew for each new application.

[Insert Figure 2a & 2b About Here]

Figure 2. Knob and groove on connecting gears (a), based on Schwartz & Black [16]. Gears in

series (b), based on Schwartz & Black [17].

In addition, Hegarty [17] and Hegarty, Kriz, & Cate [16] have shown that mental modeling

of dynamic relationships is often effected in piecemeal fashion, a process that is much better

suited for tracing a sequence of interactions through a system than for simulating collections of

dynamic effects all at once. All of this research fits well with Norman’s early assessment of

mental models. He notes:

1. Mental models are incomplete.

2. People’s abilities to “run” their models are severely limited.

3. Mental models are unstable: People forget the details of the system they are using…

4. Mental models do not have firm boundaries: similar devices and operations get confused

with one another [19].

These limitations on the human ability to construct and manipulate mental models surely have a

great deal to do with more general limitations on the capacity of human working memory and

with the high cognitive load associated with creating, maintaining, and manipulating mental

models.

In everyday reasoning with mental models, the behaviors of the components structures in

our models will not typically be tied in any direct way to fundamental physical laws (e.g.,

Newtonian, quantum mechanical, or relativistic). Rather, many of the kinematic and dynamic

principles governing object behavior in our mental simulations will be rooted in early

experiences of collisions, impenetrability, balance and support, projectiles, blocking, and so forth

[20-2]. In addition, in everyday reasoning, and even more so in scientific reasoning about

mechanisms, many of the behaviors of the components of our models will not be the result of

early learning. Some of these will be one-off ‘brute’ events – such a meteor striking the earth, a

gene mutating, or a latch coming undone – for which one does not have or require (i.e., in order

to formulate a satisfactory answer to the question of why the explanandum occurred) any deeper

explanation. Such occurrences might be imposed upon a mental model in much the same way

that one would impose them – that is, through direct intervention – on a scale model. In the same

way, one could also impose newly learned or hypothesized regularities on a model. Some of

these might be discovered through simple induction (e.g., one might notice that one’s car engine

becomes louder in cold weather) or through prior model-based reasoning (e.g., as in Schwartz’

study with gears). However, when formulating mechanical explanations, particularly in science,

one sometimes simply hypothesizes, as a way of making sense of the available data, that a

particular regularity obtains. A good example of this is the way that the hypothesis of periodic

geomagnetic pole flipping was used to make sense of the patterns of magnetization in rocks

found lateral to mid-ocean rifts [2]. Such ideas accord well with recent work regarding

mechanistic explanation in the philosophy of science, where it is generally recognized that our

models of mechanisms typically bottom out at brute ‘activities’ [5] or ‘functions’ [6].

The above empirical research sheds light on the properties of the models we use to reason

about mechanisms in everyday life and in science. There is, in addition, a great deal of research

that simply hypothesizes that we do utilize such models to understand other cognitive processes

such as language comprehension [23-8], concepts [29], or learning [30].

The hypothesis of mental models has also been invoked by Johnson-Laird to explain

deductive reasoning, though here the term ‘mental model’ is used somewhat differently than it is

in the research cited above [31].5 Like many proponents of mental models, Johnson-Laird does

claim to be directly inspired by Craik, an inspiration that shows up in his suggestion that mental

5 Below, I explain, in greater depth, the contrast between deductive reasoning more generally and the mental models

approach to mechanistic reasoning espoused here.

models have “a structure that is remote from verbal assertions, but close to the structure of the

world as humans conceive it” [32]. However, if we look more closely at the way in which

Johnson-Laird employs the mental models hypothesis in accounting for reasoning processes

(viz., deductive, inductive, and abductive), it begins to look as though he has something very

different in mind. For instance, with regard to deductive reasoning – that is, reasoning that

mainly involves the semantic properties of top-neutral logical operators such as ‘if…then…,’

‘and,’ ‘all,’ and ‘some’ – Johnson-Laird proposes that we reason internally through a process not

unlike the formal method of truth table analysis. For instance, on Johnson-Laird’s view, the

conditional, “If the door is pushed, then the bucket will fall,” would be mentally represented as

something like the following spatial array, which lists those scenarios (models) that would be

consistent with the truth of the statement:6

door pushed bucket falls

¬ door pushed bucket falls

¬ door pushed ¬ bucket falls

If presented with the additional premise, “The bucket did not fall,” one could then eliminate all

but the last of these models, enabling a valid deduction to “The door was not pushed.” The

formal, topic-neutral nature of this strategy means that it works in exactly the same way

regardless of what items (e.g., balloons, satellites, or mice) we are reasoning about. To say

nothing of the viability of the approach, Johnson-Laird’s proposals regarding deductive (as well

as inductive and abductive) reasoning thus seem, except insofar as they appeal to such structures

as spatial arrays, at odds with his avowed view that mental models have a structure closer to the

world than to our descriptions of it.

6 ‘¬’ here signals negation.

1.4 Mental Models in the Brain: Attempts at Psycho-neural Reduction

While there has been considerable research on mental models in recent years, what has

been very slow to materialize is a demonstration that brains do or, what is even more worrisome,

that they could harbor mental models that are like scale models in crucial respects. One can see

how this might raise concerns about the mental models hypothesis. After all, if there can be no

such models, then the above explanatory appeals to mental models come out looking misguided

from the outset. At the same time, there is a competing hypothesis which faces no such

difficulties. In its most audacious form, it is the proposal that all of cognition is effected through

formal computational operations – that is, operations that involve the application of syntax-

sensitive inference rules to syntactically structured (viz., sentential) representations.

Proponents of the computational theory of cognition know that they have nothing to fear at

least with regard to the matter of whether or not brains are capable of realizing the relevant kinds

of syntax-crunching operations. McCulloch & Pitts showed, decades ago, that collections of

neuron-like processing units can implement logic gates and, in principle, a universal Turing

machine [33]. Indeed, it was in no small part because von Neumann recognized the functional

similarities between McCulloch-Pitts neurons and electronic switches (e.g., transistors) that he

was inspired to create the first fully programmable computers, ENIAC and EDVAC. More

recently, it has been shown that recurrent neural networks are, memory limitations

notwithstanding, capable of implementing computers that are Turing complete [34]. There is,

then, no longer any doubt that it is possible to bridge the divide between neural machinations and

syntax-crunching operations.

In contrast, a satisfactory demonstration that neural machinations might realize mental

models – that is, non-sentential mental representations that are like scale models in crucial

respects – has proven far more elusive. Indeed, difficulties arise the moment one tries to specify

what the ‘crucial respects’ might be, as is evidenced by the fact that each past attempt at doing

this has been argued, not without justification, to run afoul of one or the other of the following

two desiderata:

(i) an adequate account of mental models must be compatible with basic facts about the

brain.

(ii) an adequate account of mental models must be specific enough to distinguish mental

models from other kinds of representation (viz., sentential representations).

Again, this is no small matter, for given that brains are known to be capable of formal

computational operations, if it cannot be shown that they are also capable of realizing mental

models, this will cast doubt on all those psychological theories mentioned above that advert to

mental models. This is a concern made all-the-more pressing by the fact that proponents of the

computational theory of cognition have no shortage of alternative explanations for the behavioral

data cited in support of mental models. For instance, to the extent that people report having

model-like phenomenology, this might be dismissed as a mere epiphenomenon of the actual,

underlying computational operations. Similarly, to the extent that behavioral data, such as

reaction times, suggests reliance upon model-like mental representations that undergo continuous

transformations, this might be chalked up to demand characteristics (e.g., subjects may feel

compelled to pretend that they are scanning a map). Some of these specific objections could be

vulnerable in that they give rise to their own testable predictions [35], but, as explained below,

proponents of the computational theory have an ace up their sleeve, for computational accounts

are flexible enough to handle virtually any behavioral data. All of this is quite general, so let us

turn to some of the specific attempts to spell out the distinctive features of mental models.

From Structural to Functional Isomorphism

As we have seen, from the outset, the claim made on behalf of putative mental models is

that they are like scale models in one or more crucial respects. Of course, scale models are

themselves like the actual systems they represent in a very obvious respect: They instantiate the

very same properties as what they represent. It is thus no surprise that the dominant theme in

attempts to specify what makes mental models models is the invocation of one form or another of

isomorphism between mental models, scale models, and the modeled world.

Mere Isomorphism

The most straightforward form of isomorphism invoked in this literature is what might be

termed bare isomorphism, or isomorphism simpliciter, which is a purported relationship between

mental models and what they represent. Despite initial appearances, this is the form of

isomorphism that Craik seems to have had in mind. He claims, for instance: “By a model we thus

mean any physical or chemical system which has a similar relation-structure to that of the

process it imitates” [9]. Latter-day proponents of this proposal include Cummins [36] and

Hegarty, who, in an attempt to summarize the dominant view of mental models in psychology,

notes: “a mental model (or situation model) is a representation that is isomorphic to the physical

situation that it represents and the inference processes simulate the physical processes being

reasoned about” [18].

[Insert Figure 3 About Here]

Figure 3. Kelvin’s first tide predicting device. From:

http://www.sciencemuseum.org.uk/images/ManualSSPL/10300041.aspx

One serious concern about this approach is that it is too liberal, which is to say that it leads

one to classify too wide a range of representations as models. Consider, for instance, that one of

Craik's favored examples of a representation with a ‘similar relation structure’ to what it

represents is Kelvin’s Tide Predictor, a device that consists of an ingenious system of gears and

pulleys arranged so as to support truth-preserving inferences regarding the tides (Figure 3). Says

Craik, “My hypothesis then is that thought models, or parallels, reality—that its essential feature

is...symbolism, and that this symbolism is largely of the same kind as that which is familiar to us

in mechanical devices which aid thought and calculation" [9]. This, of course, is no different

from what proponents of the computational theory of cognition currently maintain. After all, any

syntax-crunching system capable of supporting truth-preserving inferences with respect to a

given physical system will have to be isomorphic with it – that is, there will have to be

correspondences between the parts and relations in the system and the components of the

representation – in ways that get preserved over the course of computation. To that extent, one

might even say that the inference process ‘simulates,’ or even ‘pictures’ [37], the process being

reasoned about. In short, then, the proposal that mental models are merely isomorphic with what

they represent is thus far too vague to satisfy desideratum (ii) above. Indeed, it is for this very

reason that researchers have tried to find a more restrictive notion of isomorphism, one that can

distinguish models from sentential representations.

Physical isomorphism

Perhaps the most restrictive such notion is that of structural [38] or physical [39]

isomorphism, which involves instantiating the very same properties, and arrangements thereof,

as the represented system. This appears to be the kind of isomorphism that Thagard has in mind

when he claims, “Demonstrating that neural representation can constitute mental models requires

showing how they can have the same relational structure as what they represent, both statically

and dynamically” [12; also see 40]. Thagard cites Kosslyn’s research as indicative of how this

demand might be met, and in Kosslyn too, we do find frequent appeals to structural

isomorphisms. For instance, noting the retinotopic organization of areas of visual cortex that are

implicated in mental imagery, Kosslyn claims, "[t]hese areas represent depictively in the most

literal sense . . ." [41].

Unfortunately, the postulation of physically isomorphic mental representations is highly

suspect for several reasons. To start with, the kind of retinotopy that one finds in areas such as

V1 is highly distorted relative to the world due to the disproportionate amount of cortex devoted

to the central portion of the retina (i.e., the fovea). A square in the visual field is thus not

represented in the cortex by sets of neurons that lie in straight, let alone in parallel, lines.

Moreover, visual representation seems not to be carried out through the activity of any single

retinotopically organized neural ensemble. Rather, vision involves the combined activity of a

variety of systems that are, to a considerable extent, anatomically and functionally distinct [42-

4]. Lastly, the kind of retinotopy pointed out by Kosslyn is restricted to two spatial dimensions,

and a 2D representational medium cannot realize representations that are physically isomorphic

with what they represent in three dimensions.7 Nor, a fortiori, can such a medium realize

representations that are physically isomorphic in both 3D and causal respects. Crudely put, there

are no literal buckets, balls, or doors in the brain.

Functional Isomorphism

The main problem with the appeal to physical isomorphism, one that has long been

appreciated, is that it fails to satisfy desideratum (i). As Shepard and Chipman note, "With about

7 Perhaps it is worth noting, as well, how inessential structural isomorphism is to information processing in neural

networks, even in the case of 2D retinotopic maps. The relative physical locations of neural cell bodies seems

irrelevant when compared to the patterns of connectivity between neurons, the strengths and valences of

connections, and the schemes of temporal coding the neurons employ. One would expect then that, so long as all of

this is preserved, cell bodies might be tangled up in arbitrary ways without affecting processing.

as much logic, one might as well argue that the neurons that signal that the square is green

should themselves be green!" [38]. Recognizing this, and recognizing the weakness of appeals to

mere isomorphism, Shepard and Chipman push for the following moderate notion of

isomorphism:

isomorphism should be sought-not in the first-order relation between (a) an individual

object, and (b) its corresponding internal representation-but in the second-order relation

between (a) the relations among alternative external objects, and (b) the relations among

their corresponding internal representations. Thus, although the internal representation for a

square need not itself be square, it should . . . at least have a closer functional relation to

the internal representation for a rectangle than to that, say, for a green flash or the taste of

persimmon [38, italics mine].

The appeal to second-order isomorphism would, they hoped, provide an alternative to physical

isomorphism that is both consistent with basic brain facts (desideratum (i)) and distinct from

sentential accounts (desideratum (ii)).

Another moderate account of isomorphism was put forward at the same time by

Huttenlocher, Higgins, & Clark [45]. They had a particular interest in how subjects make

ordering inferences (viz., those involving the ordering of three items along such dimensions as

size, weight, and height) like this one:

Linus is taller than Prior.

Prior is taller than Mabel.

Linus is taller than Mabel.

Huttenlocher, Higgins, & Clark suggested that subjects might use representations that “are

isomorphic with the physically realized representations they use in solving analogous problems

(graphs, maps, etc.). ...” [45]. The essence of their proposal was that the mental representations

that subjects form in order to solve such problems might function like spatial arrays rather than

like sentences. For instance, what seems distinctive about external sentential representations of

three-term ordering syllogisms like the one above is that, because each premise is represented in

terms of a distinct expression, terms that denote particular individuals must be repeated. On the

other hand, when such inferences are made with the aid of external spatial arrays, the terms need

not be repeated. For instance, one can make inferences about the taller-than relation on the basis

of the left-of relation with the help of marks on a paper like these:

L P M

In fact, the introspective reports obtained by Huttenlocher, Higgins, & Clark did support the idea

that subjects were constructing the functional equivalents of spatial arrays – for instance, subjects

reported that symbols representing individuals were not repeated – and on this basis they claimed

that subjects might be carrying out three-term ordering inferences using mental representations

that function like actual spatial arrays and unlike lists of sentences (also see [40]). This kind of

isomorphism is thus sometimes termed ‘functional’ isomorphism [39].

Shepard & Chipman [38] and Huttenlocher, Higgins, & Clark [45] were clearly after a

notion of isomorphism that satisfies desideratum (i). Unfortunately, the solutions they offer

appears, at least at first glance, to run afoul of desideratum (ii) – that is, appeals to functional

isomorphism, of either the first or second-order variety, seem not to distinguish between

computational representations and model-like representations. Huttonlocher et al. were among

the first to suspect this. They note, “It is not obvious at present whether any theory which

postulates imagery as a mechanism for solving problems can or cannot, in general, be

reformulated in an abstract logical fashion that, nevertheless makes the same behavioral

predictions” [45]. Anderson is generally credited with confirming this suspicion by pointing out

the possible tradeoffs that can be made between assumptions about representational structure and

those concerning the processes that operate over the representations [46]. He showed that the

possible structure-process tradeoffs render computational accounts flexible enough to handle

virtually any behavioral finding. Most have since endorsed his thesis that it is, at least after the

fact, always possible to “generate a propositional [i.e., sentential] model to mimic an imaginal

model" [46]. Alternatively, as Palmer puts it, if you create the right sentential model it will be

functionally isomorphic to what it represents in just the sense that a non-sentential model is

supposed to be [40].

Imagery and Perception

One last way in which one might try to satisfy the above desiderata, at least with regard to

spatial models, is to point out that visual mental imagery involves the utilization of visual

processing resources. Brooks [47] and Segal & Fusella [48], for instance, discovered that

performance on visual imagery tasks is diminished when subjects must perform a concurrent

visual processing task but not when they perform an auditory task – that is, they found that there

is interference between mental imagery and auditory perception but not between mental imagery

and visual perception (also see [36]). However, if these findings are meant to provide a model-

based alternative to computational theories, the attempt would appear to have the same

fundamental flaw as the appeal to functional isomorphism. As Block notes, because perceptual

processing can, in principle, also be explained in terms of computational processes, “the claim

that the representations of imagery and perception are of the same kind is irrelevant to the

controversy over pictorialist vs. descriptionalist interpretation of experiments like the image

scanning and rotation experiments ...” [49] (also see [46, 50]). That is, the claim that imagery

utilizes visual processing resources fails to satisfy desideratum (ii).

Distinctive Features of Scale Models

The overall realization problem facing putative mental models, then, is just that it has

proven exceedingly difficult to specify what sorts of representational structures mental models

are in a way that is consistent with basic brain facts but that also distinguishes models from

sentential representations. In order to finally see our way past these concerns, it will helpful if we

first take stock of a handful of features that are widely taken, even by proponents of the

computational theory of mind, to distinguish external images and scale models from sentential

representations. Three such features concern the sorts of entities, properties, and processes that

each form of representation is naturally suited for representing:

(1) Images and scale models are not naturally suited for representing abstract entities,

properties, and processes (e.g., war criminal, ownership, or economic inflation). They

are much better suited for representing concrete entities, properties, and processes

(e.g., a bucket, one object being over another, or compression).

(2) Images and scale models are not naturally suited for representing general categories

(e.g., triangles or automobiles). They are better suited for representing specific

instances of categories.8

(3) Images and scale models are not naturally suited for singling out specific properties

of specific objects [37, 50]. For instance, if would be difficult, using a scale model, to

represent just the fact that Fred's car is green, for any such model will simultaneously

represent many other properties, such as the number of doors and wheels, the body

type, and so on.

8 Note: Genera differ from abstracta in that the former can be concrete (e.g., rocks) and the latter can be specific

(e.g., the Enlightenment).

In contrast, sentential representations (e.g., those constructed using natural and artificial

languages) have little trouble representing abstracta (e.g., ‘war criminal’), genera (‘triangle’), and

specific properties of specific objects (e.g., ‘Fred’s car is green’).

While images and scale models are relatively disadvantaged in the above respects, they

are much better suited for supporting inferences regarding the consequences of alterations to

specific, concrete systems. The fact that syntax-crunching systems are quite limited in this regard

first came to light as a consequence of early work in formal-logic-inspired, sentence-and-rule-

based artificial intelligence (AI). The general problem confronting syntax-crunching approaches

came to be known as the frame problem [51].

In its original formulation, the frame problem had much to do with the challenge of

endowing a sentence-and-rule-based representational system with the ability to anticipate what

will not change following an alteration to the world (e.g., tipping over a bottle changes its

orientation but not its color). Today, however, the frame problem is regarded as something more

general – namely, the problem of endowing computational systems (and other artifacts) with the

kind of commonsense knowledge that the average human possesses about what will change and

what will stay the same following alterations to the objects in the world. As Hayes puts it:

The frame problem arises in attempts to formalise problem-solving processes involving

interactions with a complex world. It concerns the difficulty of keeping track of the

consequences of the performance of an action in, or more generally of the making of some

alteration to, a representation of the world [52].

The frame problem can actually be broken down into at least two component problems, the

prediction problem [53] and the qualification problem [54].

[Insert Figure 4 About Here]

Figure 4. A ‘toy’ world: A doorway, a bucket, and a ball. From [55].

As it confronts computational devices, the prediction problem can be summed up as

follows: In order to support inferences about the consequences of alterations to even simple

physical systems, a sentence-and-rule system would have to contain innumerable rules that

explicitly specify how objects will behave relative to one another following each of innumerable

possible alterations. For a simple illustration, consider what we all know about the consequences

of different ways of altering the items in Figure 4. We know, for example, what would happen

were we to use the bucket to throw the ball through the open doorway, were we to place the

bucket over the ball and slide the bucket through the doorway, were we to set the bucket

containing the ball atop the slightly ajar door and then shove the door open, and so on

indefinitely. To endow a sentence-and-rule system with the ability to predict the consequences of

these various alterations, one would have to build in, corresponding to each one, a separate data

structure specifying the starting conditions, the alteration, and the consequences of that

alteration. If these take the form of conditional statements, the system could then make

inferences utilizing domain-general (e.g., topic-neutral, deductive) machinery. Alternatively, the

information could be encoded directly as domain-specific inference rules (e.g., production-

system operators). Either way, from an engineering standpoint, the problem that quickly arises is

that no matter how many of these statements or rules one builds into the knowledge base of the

system, there will generally be countless other bits of commonsense knowledge that one has

overlooked. Notice, moreover, that scaling the scenario up even slightly (e.g., such that it now

includes a board) has an exponential effect on the number of potential alterations and, as such, on

the number of new data structures that one would have to incorporate into one's model [53]. As

Hayes says, “One does not want to be obliged to give a law of motion for every aspect of the

new situation…especially as the number of frame axioms increases rapidly with the complexity

of the problem” [52]. Moreover, as explained in the manual for a past incarnation of the

production system Soar, “when working on large (realistic) problems, the number of operators

[i.e., domain-specific rules] that may be used in problem solving and the number of possible state

descriptions will be very large and probably infinite” [56].

As if the prediction problem were not problem enough, it is actually compounded by the

other facet of the frame problem, the qualification problem [54]. This is because, in order to

capture what the average human knows about the consequences of alterations to a physical

system, not only would innumerable distinct conditionals or inference rules be required, but each

would have to be qualified in an indefinite number of ways. Notice, for instance, that placing the

bucket over the ball and sliding it through the doorway will result in the ball being transferred to

the other side of the wall, but only if it is not the case that there is a hole in the floor into which

the ball might fall, there is a hole in the bucket through which it might escape, the ball is fastened

securely to the floor, and so on indefinitely. To once again quote Hayes, “Almost any general

belief about the result of his own actions may be contradicted by the robot’s observations….there

are no end to the different things that can go wrong, and he cannot be expected to hedge his

conclusions round with thousands of qualifications” [52]. Thus, to capture what the average

human knows, if only implicitly, about the consequences of this one alteration, all of the relevant

qualifications would have to be added to the relevant sentence or rule. Once again, in realistic

situations, the challenge of specifying all of the qualifications is magnified exponentially.

The general failing of sentence-and-rule-based representations that the frame problem

brings to light is that they only support predictions concerning the consequences of alterations

and the defeaters of those consequences if those alterations, consequences, and defeaters have

been spelled out, antecedently and explicitly, as distinct data structures. Representations of this

sort – that is, representations that require distinct structures to support predictions regarding the

consequences of each type of alteration to the represented system – are sometimes termed

extrinsic representations.9

It is worth a quick digression to note that, while the terminology has changed, these general

concerns about the limitations of extrinsic representations antedate work in contemporary AI by

over three-hundred years. They show up, for instance, in Descartes’ best-explanation arguments

for dualism in his Discourse on the Method. Descartes there despairs of there ever being a

mechanical explanation for, or an artifact that can duplicate, the average human’s boundless

knowledge of the consequences of interventions on the world:

If there were machines which bore a resemblance to our bodies and imitated our actions…

we should still have two very certain means of recognizing that they were not real men…

[Firstly, humans have the ability to converse.] Secondly, even though some machines might

do some things as well as we do them…they would inevitably fail in others, which would

reveal that they are acting not from understanding, but only from the disposition of their

organs. For whereas reason is a universal instrument, which can be used in all kinds of

situations, these organs need some particular action; hence it is for all practical purposes

impossible for a machine to have enough different organs to make it act in all the

contingencies of life in the way in which our reason makes us act.

Descartes thought that to match wits with even a ‘dull-witted’ human, any natural or artificial

device would need, per impossibile, to rely upon an infinite number of specific sensory-motor

routines – which bear a striking resemblance to production-system operators – for each new

9 The intrinsic-extrinsic distinction discussed here was introduced by Palmer [39] but modified by Waskan [57-8].

situation the device might confront. What Descartes could not imagine, because he thought that

all such knowledge had to be represented explicitly, was the possibility of (to use Chomsky’s

term) a generative inference mechanism – that is, one that embodies boundless knowledge of

implications through finite means.

What Descartes failed to notice was that there were already artifacts (i.e., scale models) that

exhibited the requisite generativity. Indeed, in contemporary AI, the benefits of an appeal to

scale-model-like representations are now well known. Starting with the prediction problem, one

can use a reasonably faithful scale model of the setup depicted in Figure 4 in order to predict

what would happen were one to use the bucket to throw the ball through the open doorway, were

one to place the bucket over the ball and slide the bucket through the doorway, were one to set

the bucket containing the ball atop the slightly ajar door and then shove the door open, and so on

indefinitely. To use Haugeland's terms, the side effects of alterations to such representations

mirror the side effects of alterations to the represented system automatically [59] – which is to

say, without requiring their explicit specification.10 Notice also that incremental additions to the

represented system will only have an incremental effect on what needs to be built into the

representation. The addition of a board to the system above, for instance, can be handled by the

simple addition of a scale model of the board to the representation.

Nor do scale models suffer from the qualification problem. To see why, notice that much of

what is true of a modeled domain will be true of a scale model of that domain. For instance, with

regard to a scale model of the setup in Figure 4, it is true that the scale model of the ball will fall

out of the scale model of the bucket when it is tipped over, but only if the ball is not wedged into

10 This only holds, of course, to the extent that the model is a faithful reproduction. Unless the model is a perfect

replica, which includes being to scale, there will be some limits on inferential fidelity, though this does not

undermine the claim that scale models are generative.

the bucket, there is no glue in the bucket, and so on indefinitely. Just like our own predictions,

the predictions generated using scale models are implicitly qualified in an open-ended number of

ways. With scale models, all of the relevant information is implicit in the models and so there is

no need to represent it all explicitly using innumerable distinct data structures. Representations

of this sort are termed intrinsic representations. Summing up, scale models are immune to the

frame problem, for one can use them to determine, on an as-needed basis, both the consequences

of countless alterations to the modeled system and the countless possible defeaters of those

consequences – that is, one simply manipulates the model in the relevant ways and reads off the

consequences.

Does Computational Realization Entail Sentential Representation?

The above distinguishing features can help us to know better whether we are dealing with

model-like or sentence-like representations and, ultimately, to appreciate how one might bridge

the gap from neurophysiology to mental models. As noted above, a similar bridge was

constructed from neurophysiology to computational processes by showing that artifacts (e.g.,

collections of McCulloch-Pitts neurons or wires and transistors) characterized by a complex

circuitry not unlike that of real brains can be configured so as to implement, at a higher level of

abstraction, processes that exhibit the hallmarks of traditional syntax-crunching. Because

neurons have similar information-processing capabilities as these artifacts, implementing a set of

formal operations on an electronic computer is already very nearly an existence proof that brain-

like systems can realize the same set of operations.

Might this strategy offer a template for constructing a similar bridge to high-level models?

There is surely no shortage of computer simulations of mechanical systems, and at least as they

are depicted on a computer’s display, these simulations look for all the world like images and

scale models. Many would argue, however, that this approach to bridging the neuron-model

divide is a non-starter. The worry, in short, is that it fails to satisfy desideratum (ii) above. To see

why, it will be helpful to look at the kinds of computational models of mental imagery offered up

by researchers such as Kosslyn [14] and Glasgow and Papadias [60].

Kosslyn's model of mental imagery has several components [14]. One is a long-term store

that contains sentential representations of the shape and orientation of objects. These descriptive

representations are utilized for the construction of representations in another component, the

visual buffer, which encodes the same information in terms of the filled and empty cells of a

computation matrix. The cells of the matrix are indexed by x, y coordinates, and the descriptions

in long-term memory take the form of polar coordinate specifications (i.e., specifications of the

angle and distance from a point of origin) of the locations of filled cells. Control processes

operate over the co-ordinate specifications in order to perform such functions as panning in and

out, scanning across, and mental rotation.

One distinctive feature of actual (e.g., paper-and-ink) spatial matrix representations is that

they embody some of the very same properties and relationships (viz., spatial ones) as – which is

just to say that they are physically isomorphic with – the things they represent. But Kosslyn's

computational matrix representations (CMRs) are clearly not physically isomorphic with what

they represent. After all, Kosslyn’s visual buffer representations are not 'real' matrix

representations that utilize cells arranged in Euclidean space; they are computational matrix

representations. To be sure, modelers may sometimes see literal pictures on the output displays

of their computers, but the representations of interest are located in the CPU (viz., in RAM) of

the computer running the model. Accordingly, the control operations responsible for executing

representational transformations like rotation do not make use of inherent spatial constraints, but

rather they operate over the coordinate specifications that are stored in the computer's memory.

Details aside, at a certain level of description, there can be no doubt that the computer is

implementing a set of syntax-sensitive rules for manipulating syntactically structured

representations; this is what computers do. As Block puts it, “Once we see what the computer

does, we realize that the representation of the line is descriptional” [49]. The received view, then,

a view that has gone nearly unchallenged, is that if a representation of spatial, kinematic, or

dynamic properties is implemented using a high-level computer program, then the resulting

representations must be sentential in character [49, 61-2].11

It would thus seem that the strongest claim that can possibly be supported with regard to

CMRs is that they function like images. Yet, as Anderson notes, it is always possible, through

clever structure-process trade-offs, to create a sentential system that mimics an imagistic one

[46]. Indeed, rather than supporting the mental models framework, one might well take computer

simulations of mental modeling as concrete evidence for Anderson’s claim. Likewise, there is a

case to be made that CMRs and their brethren are, unlike scale models, extrinsic representations

[62]. After all, the computers that run them implement syntax-sensitive rules that provide explicit

specifications of the consequences of alterations. This is no small matter. From the standpoint of

cognitive science, one of the most important virtues of the hypothesis that we utilize mental

representations akin to scale models was that scale models constitute intrinsic representations of

interacting worldly constraints and are thus immune to the frame problem. One could, then, be

forgiven for thinking that any attempt to build a bridge from neurons to models by following the

template set by computational theories – that is, by noting that certain computational artifacts

11 That Fodor shares this sentiment is suggested by his claim that "if ... you propose to co-opt Turing's account of the

nature of computation for use in a cognitive psychology of thought, you will have to assume that thoughts

themselves have syntactic structure" [63].

instantiate the relevant kind of processing – will be doomed to failure from the outset.

What About POPI?

Consider, however, that upon gazing directly at a vast collection of electrical or

electrochemical circuits, one will see no evidence of the harboring or manipulation of sentential

representations. In Monadology, Leibniz turned an analogous observation about perceptual

experience into an objection to materialism:

It must be confessed, moreover, that perception, and that which depends on it, are

inexplicable by mechanical causes, that is, by figures and motions, And, supposing that

there were a mechanism so constructed as to think, feel and have perception, we might

enter it as into a mill. And this granted, we should only find on visiting it, pieces which

push one against another, but never anything by which to explain a perception.

A similar objection might be leveled regarding computational processes. Again, one sees no

evidence of this kind of processing when one looks at electronic or electrochemical circuitry.

Clearly something has gone wrong.

What Leibniz overlooked – and this may be because he lacked the conceptual tools made

available by the information age – was a grasp of the principle of property independence (POPI).

The basic idea of POPI is that properties characterizing a system when it is studied at a relatively

low level of abstraction are often absent when it is studied at a higher level, and vice versa. It is

POPI that allows computer scientists to say that a system which is characterized by electronic

switches and relays at level n may nevertheless be best described in terms of the storing of bits of

information in numerically addressable memory registers at level n+1 and in terms of the

application of syntax-sensitive rules to syntactically structured representations at level n+2. It is

also the very thing that enables proponents of computational theories of cognition to say that

brains and computational artifacts are, despite superficial appearances, capable of implementing

the application of syntax-sensitive rules to syntactically structured representations.

However, when proponents of computational theories of cognition insist that computational

implementation (e.g., of CMRs) entails sentential representation, they are turning their backs on

the very principle that enabled them to bridge divide between low-level circuitry and high-level

computational operations; they are turning their back on POPI. Indeed, nothing about POPI

entails that all syntax-crunching systems must be characterized in terms of sentences and

inference rules at the highest level of abstraction. POPI thus opens up at least logical space for

systems that engage in syntax-crunching operations at one level but that harbor and manipulate

non-sentential models at a higher level.

In point of fact, in this logical space reside actual systems, including finite element models

(FEMs). These were first developed in the physical (e.g., civil and mechanical) engineering

disciplines for testing designs, but they have since become a staple tool in the sciences for

exploring the ramifications of theories, generating novel predictions, and facilitating

understanding. For our current purposes, what matters most about FEMs is that they provide an

existence proof that computational processes can realize non-sentential representations that are

like scale models and unlike sentential representations in all of the crucial respects listed above.

To see why, notice first that there are (among others) two important levels of abstraction at

which a given FEM may be understood. As with scale models, one may understand FEMs at the

relatively low level of the principles that govern their implementing medium. What one finds at

this level are sentential specifications of coordinates (e.g., for polygon vertices) along with rules,

akin to the fundamental laws of nature12, which constrain how those coordinates may change

12 For a close analogy, think the basic rules of Conway’s Game of Life.

(e.g., due to collisions and loads) (see Figure 5). When a given model is ‘run,’ at this low level

one finds a massive number of iterative number crunching operations. Not unlike Leibniz,

enemies of the idea of computationally realized non-sentential models have seized upon this low

level with their suggestion that computational systems harbor only sentential representations. At

this level, however, it is not even obvious that we are dealing with representations (i.e., of

worldly objects and properties) at all, any more than we are, for instance, when we fixate upon

the constraints governing the behaviors of individual Lego blocks.

One only finds representations of objects when one turns to the higher level of the models

that are realized, and multiply realizable, by the aforementioned modeling media. And when we

take a close look at the properties of these high-level FEMs, we find that they share several

characteristics that have long been taken, including by those who suggest that computational

implementation entails sentential representation, to distinguish sentential representations from

scale models.

[Insert Figure 5 About Here.]

Figure 5. Polymesh representation of a blunt impact to a semi-rigid sheet of material. From [57].

To start with, like scale models and unlike sentential representations, FEMs are not (by

themselves) naturally suited to representing abstract entities, properties, and processes (e.g., war

criminal, ownership, economic inflation). They are much better suited for representing concrete

entities, properties, and processes (e.g., a bucket, one object being over another, and

compression). Nor are FEMs naturally suited to representing general categories (e.g., triangles or

automobiles). They are far better suited for representing specific instances of those categories.

Lastly, FEMs are not naturally suited to singling out specific properties of specific objects. For

instance, using an FEM, if would be difficult to represent just the fact that Fred's car is green, for

any such model will simultaneously represent many other properties, such as the number of

doors and wheels, the body type, and so on. In short, just like scale models, FEMs are always

representations of specific, concrete systems. By these reasonable standards, FEMs ought to be

considered computationally-realized non-sentential models that are the close kin of scale models.

The case for this claim looks even stronger once we consider whether or not FEMs

constitute intrinsic representations. As we have seen, the received view is that FEMs and their

brethren (e.g., CMRs) are extrinsic representations, for the constraints governing how the

coordinates of primitive modeling elements may change must be encoded antecedently and

explicitly. Indeed, at the level of coordinates and transformation rules, one gets nothing ‘for

free.’ However, once a modeling medium has been used to construct a suitable FEM of a

collection of objects, the model can then be altered in any of countless ways in order to

determine the possible consequences of the corresponding alterations to the represented objects.

One can, for instance, use a high-fidelity FEM of the door, bucket, ball system to infer, among

other things, what would happen were we to place the bucket over the ball and slide the bucket

through the doorway, what would happen were the bucket used to throw the ball at the open

doorway, what would happen were the air pressure dramatically decreased, and so on

indefinitely [57]. The consequences of these alterations need not be anticipated or explicitly

incorporated into the system. Indeed, as with scale models, much of the point of constructing

FEMs is to find out how a system will behave in light of whichever alterations an engineer or

scientist can dream up.

It bears repeating that it is not at the level of the primitive operations of an implementation

base that we find intrinsic representations, but at the level of the representations realized by a

given, primitively constrained implementation base. Part of what justifies this claim is the fact

that certain constraints will be inviolable at the level of the model, and thus a great deal of

information will be implicit in the model, because it has been implemented using a particular

kind of medium. As Pylyshyn notes:

the greater number of formal properties built into a notation in advance, the weaker the

notational system’s expressive power (though the system may be more efficient for cases to

which it is applicable). This follows from the possibility that the system may no longer be

capable of expressing certain states of affairs that violate assumptions built into the

notation. For example, if Euclidean assumptions are built into a notation, the notation

cannot be used to describe non-Euclidean properties. . . . [61].

This, in fact, is very close to an apt characterization of what is going on in the case of FEMs.

Given that a particular model has been realized through the use of a primitively constrained

medium, certain constraints will be inviolable at the representational level and a great deal of

information will be implicit [57]. As Mark Bickhard (in correspondence) summarizes the point:

“Properties and regularities are only going to be 'intrinsic' at one level of description if they are

built-in in the realizing level – or else they are ontologically 'built-in' as in the case of strictly

spatial relationships in physical scale models.” While scale models are intrinsic for the latter

reason, FEMs are intrinsic for the former. This shows up in the fact that FEMs exhibit a

comparable degree of generativity to scale models and a correlative immunity to the frame

problem. Like scale models, FEMs provide a finite embodiment of boundless tacit knowledge,

which can be made explicit at any time, of the consequences of innumerable alterations to the

systems they represent.

So how does all of this square with Anderson’s [46] contention that it is always possible to

construct a sentential system to mimic an imagistic or model-based system or Palmer’s [39]

claim that if you create the right sentential model it will be functionally isomorphic to what it

represents in just the sense that a non-sentential model is supposed to be? Anderson and Palmer

are surely right that, post hoc, one can always constrain a sentential representational system so

that it mimics the output of a model-based system, but the post hoc character of the strategy is

precisely what gets sentential approaches into trouble vis-à-vis the frame problem. Consider, for

instance, that the traditional AI approach is to take any physical implication of which humans

express knowledge and, after the fact, to build it into the knowledge base of one’s system as a

sentence or inference rule.13 But to solve, or rather to avoid, the frame problem, one must rely

upon representations that embody all of this boundless information as tacit knowledge – that is,

the information cannot be explicitly encoded at the outset, but it can later be generated, and

thereby become explicit knowledge, on an as-needed basis. Put simply, to exhibit anything

approaching true functional isomorphism with scale models, what is needed are high-level,

intrinsic, non-sentential models.

To sum up, those who would contend that FEMs (or even CMRs of 2D spatial properties)

are, qua computational, necessarily extrinsic and sentential have overlooked the fact that there

are multiple levels of abstraction at which a given computational model can be understood. At

the relatively low level of the modeling medium, there are unquestionably extrinsic

representations of the principles governing the permissible transformation of primitive modeling

elements. At a higher level, one finds models that share many distinguishing features, including

immunity to the frame problem, with the scale models they were in large part invented to

replace. Thus, we find once again that FEMs are like scale models and unlike paradigmatic

sentential representations.

13 Despite its shortcomings, this strategy is alive and well, as is evidenced by Lenat’s massive ongoing Cyc project.

Bridging the Divide

All of this is bears directly on the longstanding concern that there is no way to bridge the

divide between neural machinations and the non-sentential models hypothesized by proponents

of mental models. What the foregoing makes clear is that computational processes can realize

non-sentential models that share with scale models the main characteristics that distinguish non-

sentential models from sentential representations. Given that the brain is capable, at least in

principle, of realizing any such computational processes, then one must also agree that brains can

realize non-sentential models. Thus, by appealing to the above distinguishing features of scale

models, we see that there is an account of mental models that (i) distinguishes them (on multiple

grounds) from sentential representations and (ii) is compatible with basic facts about how brains

operate. All of this provides a much-needed foundation for all of that psychological work cited

above that adverts to mental models.

One advantage of showing that low-level computations can realize higher-level mental

models is that it renders the mental models hypothesis robust enough to withstand the discovery

that the brain is a computational system at some level of description. Even if the brain is not a

computational system (i.e., in the syntax-crunching sense), the manner in which computational

systems realize intrinsic, non-sentential models will nevertheless remain quite instructive. It

suggests a general recipe for the creation of intrinsic models that can be followed even without

the computational intermediary: Start by creating a representational medium such that a large

number of primitive elements a constrained to obey a handful of simple behavioral principles.

Next constructs models from this highly productive medium.14 What emerges are generative

structures capable of supporting an open-ended number of mechanical inferences. At the level of

14 ‘Productive’ is here used in Fodor’s sense – that is, to denote a medium capable of representing an open-ended

number of distinct states of affairs [64].

the medium, ‘running’ such a model involves the recursive application of the basic constraints on

the modeling-element behaviors. This will typically be a massive, parallel, constraint-satisfaction

process. Given that this form of processing is the forte of neural networks, there should be little

doubt that neural machinations are up to the task.15

Bottom-up Approaches

Thus far, we have largely approached the question of the neural realizability of mental

models in the abstract, and from the top down. This is partly because there has been a relative

dearth of work that moves in the opposite direction, from the bottom up. One exception is

Thagard’s [12] recent work on the topic, which appeals to such biologically plausible simulations

of neural networks as those of Eliasmith and Anderson [65]. Unfortunately, Thagard has yet to

offer evidence that the neural encoding strategies he discusses exhibit any of the central features,

discussed here, that distinguish modeling from syntax-crunching. Most notably, the neural

representations he cites have not yet been shown to exhibit a significant degree of spatial,

kinematic, of causal generativity. The proof of the pudding here is in the eating.

To the extent that there have been significant advances in the bottom-up endeavor, they

mostly issue from research – such as that of Nobel laureates John O’Keefe, May-Britt Moser,

and Edward Moser – on the biological neural networks that underwrite spatial reasoning abilities

in rats. As you will recall, Tolman’s pioneering work on maze navigation suggested that rats

have an onboard medium for the construction of generative spatial maps of their location relative

to barriers and important items such as food and drink. O’Keefe is famous for showing that the

rat’s hippocampus contains ‘place’ cells which fire preferentially when an animal reaches a

15 At the same time, one should not overestimate inherent immunity to the frame problem of neural networks [58]. It

is only by implementing a primitively constrained modeling medium that neural networks can be expected to realize

intrinsic representations of complex, interacting worldly constraints.

particular location in its environment, cells that fire in sequence as a rat moves from one location

to another [66]. Moser & Moser subsequently showed that the rat’s uncanny spatial navigation

abilities also depend upon ‘grid’ cells in the nearby entorhinal cortex [67]. Individual grid cells

fire when an animal is in one any of several, roughly evenly spaced locations. When lines are

drawn to connect these points, they collectively form what (purely by coincidence) looks a great

deal like the kind of 2D polymesh shown in Figure 5. While each grid cell is tuned to a collection

of locations, different grid cells have sparser or denser coverage of the same region of space.

Collectively they provide effective coverage of the entire region of space in which the animal

finds itself.

Importantly, O’Keefe et al. note regarding place cells that “there does not appear to be any

obvious topographical relation between the field locations [i.e., the places to which cells become

temporarily tuned] and the anatomical locations of the cells relative to each other within the

hippocampus” [68]. Nor do grid cells in the entorhinal cortex exploit any obvious structural

isomorphisms between their respective anatomical locations and the spatial layout of the

environment. However, acting in concert, the two types of cells enable effective navigation, as if

the organism had an internal map that preserves relative locations (place cells) and distances

(grid cells). In other words, the two systems encode maps that are functionally isomorphic with

real maps of the environment. Moreover, they provide a productive modeling medium, one

which, not unlike a collection of Lego blocks, can be used and reused, through a process called

‘re-mapping,’ to encode information about an open-ended number of new environments [69].

The maps constructed in this medium are generative with regard to 2D spatial properties in the

aforementioned sense, as is shown by their role in enabling rats to find efficient new ways to a

destination when familiar routes are blocked. More recent research suggests that the rat’s place

cells are also somewhat sensitive to vertical displacement from a reference plane, perhaps

enabling 3D mapping capabilities [70]. Nor are the lessons learned here applicable only to rats,

for a large body of research suggests that the same anatomical systems may be implicated in

human spatial navigation [71].

Our deepest understanding of how real neural networks create spatial mental models thus

suggests that brains implement a reusable modeling medium and, by exploiting the kinds

functional, rather than physical, isomorphisms that make neural realizability feasible, nothing is

lost in the way of generativity. It also bears mentioning that this modeling medium is well suited

for producing models of the organism’s location relative to its specific, concrete environment. As

such, it may (taken in isolation) be ill suited for representing abstracta or genera. As for the

singling out of specific properties of specific objects, it may be that models that are realized by

neurophysiological processes have a natural advantage over scale models in that populations

representing specific properties may ‘cry out’ for attention (e.g., by oscillating at the appropriate

frequency). There is, moreover, no reason why these lessons could not scale up, so to speak, to

account for the human ability to run off-line models of spatial, kinematic, and dynamic

relationships. Of course, in humans, the neocortex is likely to play a much more prominent role.

As of yet, however, there is little understanding of the precise manner in which the neocortex

does, or might, realize mental models.

1.5 Realization Story Applied

Though we clearly have a long way to go, the above hypothesis about what mental models

are such that neural systems might realize them looks to have important ramifications for work in

several fields, ranging from AI to the philosophy of science.

AI and Psychology: Towards and Intuitive Physics Engine

One obvious ramification of the above is what it suggests about how one might go about

endowing computational artifacts with the kind of boundless commonsense knowledge of the

consequences of alterations to world that humans seem to possess. FEMs prove that there is a

determinate computational solution to the prediction and qualification problems. FEMs are

generative in that they can be manipulated in any of countless ways in order to make inferences

about how alterations to the environment might play out and, by the same token, about the ways

in which those consequences might be defeated. It would thus behoove AI researchers to

incorporate media for the construction of intrinsic models within the core inference machinery of

their devices. Indeed, there has been some movement in this direction in recent years. For

instance, though past manuals for the Soar production-system architecture evidence a certain

degree of exasperation when it comes to the frame problem, more recent manuals indicate that

Soar’s designers have begun to offload mechanical reasoning to non-sentential models. Laird

notes, for instance:

With the addition of visual imagery, we have demonstrated that it is possible to solve

spatial reasoning problems orders of magnitude faster than without it, and using

significantly less procedural knowledge. Visual imagery also enables processing that is not

possible with only symbolic reasoning, such as determining which letters in the

alphabet are symmetric along the vertical axis (A, H, I, M, O, T, U, V, W, X, Y) [72].

While Soar’s imagery module still only supports simple spatial reasoning, it is clearly a step in

the direction of richer, intrinsic models of 3D kinematics and dynamics.

There has also been some movement in the direction of using computationally realized

intrinsic models as a way of making sense of behavioral findings regarding how humans engage

in commonsense reasoning about the world. For instance, after paying homage to Craik, MIT

researchers Battaglia, Hamrick, & Tenenbaum describe their innovative approach to

commonsense reasoning as follows:

We posit that human judgments are driven by an “ intuitive physics engine ” (IPE), akin to

the computer physics engines used for quantitative but approximate simulation of rigid

body dynamics and collisions, soft body and fluid dynamics in computer graphics, and

interactive video games [73].

They simulate the IPE with FEMs of full-blown 3D kinematic and dynamic relationships. They

note that a similar IPE in humans might allow us to read off from our simulations the answers to

questions of ‘What will happen?’ regarding to innumerable novel scenarios. Their pioneering

work also breaks new ground in that it begins to account for probabilistic reasoning by building a

bit of uncertainty into models and treating multiple runs of a model as a statistical sample.

All of this work is very much in the spirit of Schwartz’ claim that “inferences can emerge

through imagined actions even though people may not know the answer explicitly” [74, italics

mine]. It also fits with the following suggestion of Moulton & Kosslyn:

the primary function of mental imagery is to allow us to generate specific predictions based

upon past experience. Imagery allows us to answer ‘what if’ questions by making explicit

and accessible the likely consequences of being in a specific situation or performing a

specific action [35, italics mine].

Exduction

Another important lesson to be learned from computationally realized intrinsic models is

that they support a form of mechanistic reasoning that has found its way into few, if any,

standard reasoning taxonomies. As Glasgow & Papadias claim: “The spatial structure of images

has properties not possessed by deductive sentential representations ... spatial image

representations ... support nondeductive inference using built-in constraints on the processes that

construct and access them” [60]. Of course, there is more to be said about the process of model-

based mechanistic reasoning than that it is not deductive. In fact, the process shares with (valid)

deductive reasoning the property of being monotonic. What makes deduction a monotonic (i.e.,

indefeasible) reasoning process is that the conclusion of a valid argument cannot be overturned

simply by adding premises; it can only be overturned by rejecting one or more of the premises

from which the conclusion was deduced. Other forms of reasoning (e.g., inductive

generalization, analogical reasoning, abduction) are defeasible in that one can overturn their

conclusions simply by adding relevant premises. For instance, if I hear a meowing noise

emanating from my daughter’s closet door, I may infer that the cat is trapped inside. But if I then

see the cat walking through the kitchen and am told that my daughter was given a new electronic

cat toy, my conclusion would be undermined while at the same time leaving the original premise

(i.e., that there is meowing coming from the closet) intact.

One diagnosis for why deduction is monotonic is that, in a certain sense, the premises of a

valid deduction already ‘contain’ the information stated in the conclusion, so adding information

takes nothing away from the support that those premises lend to the conclusion. That means that

insofar as the original premises are true, the conclusion must be as well, and insofar as the

conclusion is false, there must be something wrong with the premises used to derive it. But

deduction is formal, in that topic-neutral logical particles are what bear the entirety of the

inferential load – that is, the specific contents (consistently) connected and quantified over drop

out as irrelevant.

The use of scale models and FEMs makes evident that there is another form of monotonic

reasoning in addition to deduction. As explained above, information derived regarding the

consequences of interventions on a modeled system are to a significant extent already

‘contained’ (i.e., they are implicit) in the models themselves. The only way to overturn a model-

based inference is to call into question some aspect or other of the model from which it was

derived. By the same token, if the conclusion is incorrect, there must be something wrong with

the model. But unlike deduction, model-based reasoning is not effected by abstracting away from

specific contents and allowing logical particles to bear the inferential load. Instead, it is the

specific, concrete contents of the models that do all of the work. As yet, this form of monotonic

reasoning lacks a name. Let us thus call it exduction (ex- out + duce- lead). Like deduction,

exduction may be implemented externally through the use of representational artifacts, but the

hypothesis being explored here is just that we also sometimes engage in internal exductive

reasoning through the use of mental models. If this hypothesis is correct, then exduction must be

added to our standard taxonomy of internal reasoning processes and placed alongside deduction

under the broader heading of monotonic reasoning.

1.6 Mechanistic Explanation Revisited

It was noted at the end of Section 1 that mental models may well play a crucial role in the

process of mechanistic explanation and prediction. If that is so, then we can only hope to attain a

deep understanding of science if we first account for how the mental modeling process works.

Now that we have a clearer conception of the distinctive features of mental models and of the

way in which they might be realized by neurophysiological processes, we can begin to see what

the payoff might be in terms of our understanding of model-based reasoning about mechanisms

in science.

The Prediction and Ceteris Paribus Problems

To give some of the flavor of where this might lead, consider that one benefit of the

foregoing realization story regarding mental models for the philosophy of science is that it offers

a solution to two longstanding problems: the surplus-meaning problem and the ceteris paribus

problem (a.k.a., the problem of provisos). Both problems arose as a consequence of attempts to

apply the methods of formal, mostly deductive methods in an attempt to provide a logical

reconstruction of scientific reasoning.

The surplus-meaning problem has to do with the fact that explanatory hypotheses have, and

are known to have, countless implications beyond the happenings they explain. To keep things

simple, consider the non-scientific case of what a mechanic knows about the operation of an

automobile engine. Imagine, in particular, that someone has brought an automobile into the

mechanic’s shop complaining that the engine has suffered a drop-off in power. Listening to the

engine, the mechanic might decide that the engine has blown a ring. Setting aside the question of

creativity, one might provide the following formal, deductive reconstruction of his explanatory

model:

If one of the cylinders has lost a ring, then the engine will have lost power.

One of the cylinders has lost a ring.

The engine has lost power.

Consider, however, that the mechanic knows not only that the faulty ring will result in a loss of

power (the explananandum occurrence); he also knows many other implications of this

explanatory hypothesis, such as that oil will leak into the combustion chamber, the exhaust will

look smoky, the end of the tailpipe will become oily, and the sparkplugs will turn dark. He also

knows what interventions will and will not alleviate the problem – for instance, he knows that

replacing the rings will restore power but replacing the air filter will not, and so on indefinitely.

Any suitable deductive reconstruction of the mechanic’s mental model of the source of the

problem must thus account not just for the fact that it implies the explanandum; it must account

for its countless further implications. The general problem with deductive reconstructions of

explanations – a problem that will surely beset any attempt to reconstruct mechanistic

explanations using extrinsic representations – is that they fail to capture the full complexity of

what anyone who possesses that explanation must know, if only implicitly. The problem is that

there is too much surplus meaning to express it all explicitly [75], and these added implications

are essential to how we assess the adequacy of explanations, whether in everyday life of in

science. As Greenwood explains:

Where this “surplus meaning” comes from…is a matter of some dispute, but that genuine

theories poses [sic.] such surplus meaning is not—for this is precisely what accounts for

their explanatory power and creative predictive potential [76].

Notice also that the mechanic’s model of why the car lost power not only contains

information about the various other things he should expect to find if that model is correct; it also

contains information about the countless ways in which each of these expectations might,

consistent with the truth of the explanation, be defeated. The mechanic knows, for instance, that

replacing the rings will restore power, but only if it is not the case that one of the spark plug

wires was damaged in the process, the air filter has become clogged with dust from a nearby

construction project, and so on indefinitely.

Whether we are dealing with commonsense or scientific reasoning about mechanisms, the

problem with attempts at formalizing our knowledge of the ways in which a given implication is

qualified is that what we know far outstrips what can be expressed explicitly in the form of, say,

a conditional generalization. In philosophy of science, the qualifications that would have to be

added are termed ceteris paribus clauses and provisos. As Fodor claims, the general problem is

that “as soon as you try to make these generalizations explicit, you see that they have to be

hedged about with ceteris paribus clauses” [77]. And as Giere claims, “the number of provisos

implicit in any law is indefinitely large” [78, italics mine]. Deductive models, and others that rely

upon extrinsic representations, are thus unable to capture the full breadth of what we know when

we possess a mechanistic explanation for an occurrence. And all of this dark information plays a

crucial role in the testing and retention of mechanistic hypotheses [79]. Accounting for it must

thus be viewed as a central goal for the philosophy of science.

If we abandon deductive (or other extrinsic) schemes for making sense of scientific

reasoning about mechanisms and instead adopt an exductive (and intrinsic) model-based account

of reasoning, the surplus-meaning and ceteris paribus problems dissolve, and the source of the

dark information comes into focus. After all, these two problems are just variants on the

prediction and qualifications problems of AI. This is not surprising given that both sets of

problems were discovered through early attempts to deductively reconstruct everyday and

scientific reasoning about mechanisms. Naturally, the same solution applies in both cases:

Eschew the appeal to extrinsic representations and formal inferences in favor of an appeal to

intrinisic models and exductive inferences. The promising idea that emerges is that scientists

may be utilizing intrinsic mental models to understand the mechanisms that are (or might be)

responsible for particular phenomena. Such models would endow the scientist with boundless

tacit knowledge of the further implications of a given mechanistic hypothesis and of the

countless ways in which those implications are qualified.

Beyond Mental Models

We must not forget, however, that our mental models are limited by working memory

capacity and by the heavy cognitive load associated with mental modeling. Even so, scientists

are somehow able to formulate and comprehend some remarkably complex mechanical

explanations. Seen in this light, it is no surprise that, in their reasoning about mechanisms,

humans rely heavily upon external representational artifacts such as diagrams. These can act as

aids to memory, both short- and long-term, enabling us to off-load some of the cognitive burden

to the world and thereby compensating for our otherwise limited ability to keep track of the

simultaneous influences of many mechanical components (see [80]). Indeed, when aided by

external diagrams, Hegarty, Kriz, & Cate found that high- and low-imagery subjects performed

about equally well on a task that required model-based reasoning about mechanisms [19]. The

compensatory influence of external representations is strong indeed (also see Bechtel, this

volume).

Of course we do not just utilize static pictures to make sense of natural phenomena; we also

sometimes use scale models [81]. On the present view, the reason so many have come to view

these models as an apt metaphor for in-the-head reasoning may be that scale models recapitulate,

albeit in a way that overcomes many of our cognitive frailties, the structure of our internal

models of mechanisms. However, with the advent of sophisticated computer models, we now

have an even better tool for investigating the implications of mechanical hypotheses. As we have

seen, certain computer models (e.g., FEMs) are like scale models in that they constitute intrinsic

non-sentential representations of actual or hypothetical mechanisms. However, these models

have the added virtue that one can easily freeze the action, zoom in or out, slow things down, and

even watch things play out in reverse. These models thus constitute what Churchland &

Sejnowski term a “fortunate preparation” [82]. Such models provide an even more apt analogy

for understanding our own native mental models, for both sorts of models are realized at a low

level by complicated circuitry, and both tend to bottom-out well above the level of nature’s

fundamental laws.16

As noted in Section 3.2, over and above the quantitative limits imposed by working

memory capacity, scale models and FEMs are, and our own mental models may well be, limited

in certain qualitative respects, such as their ability to represent abstracta. But surely thoughts

about abstracta play a big role in scientific reasoning. One way of accounting for this is to say

that our deficiencies in regards to modeling abstracta are the precise problem that analogy and

metaphor were created to solve. This would make sense of why the language we use to represent

abstracta (e.g., ‘economic inflation’) is so shot through with analogies and metaphors rooted in

concrete domains [84-6].

[Insert Figure 6 About Here.]

Figure 6. Various configurations of circuitry, batteries, and resistors.

Gentner and Gentner’s study of human reasoning about electricity lends some credence to

this view [87]. Gentner and Gentner found that in conversation, non-expert subjects commonly

likened the flow of electricity to water moving through pipes or crowds moving through

corridors. Each analogy happens to yield its own unique set of inaccurate predictions about how

electrical current moves through particular configurations of electrical components (Figure 6).

Gentner and Gentner found that subjects’ errors in reasoning about these components tracked the

analogies they invoked when discussing electricity. This suggests that these analogies run deeper

16 One way of describing the interplay between external models (in their various forms) and internal mental models

would be to say that the latter are part and parcel of scientific cognition, whereas the former are representational

artifacts created to aid cognition. An alternative, somewhat speculative, proposal is that our external models are no

less part of fabric of scientific cognition than are our internal mental models [83].

than the surface features of language and penetrate right into subjects’ mental models of the flow

of electricity. Analogies and metaphors are perhaps not the whole story of how we think about

abstracta, but they may well be an important part of it.

1.7 Conclusion

It was noted at the end of Section 1 that mental models might play a crucial role in the

process of mechanistic explanation and prediction in science. As such, we can only attain a deep

understanding of science itself if we first understand that nature of this mental, model-based,

reasoning process. We then saw that experimental psychologists have long maintained that

mental models are distinct from sentential representations in much the way that scale models are.

Insofar as this hypothesis is viable, we can expect that experimental psychology will provide

crucial insight into both the nature and limits of our onboard mental models. At the same time, it

is important to recognize that the many appeals to distinctively model-like mental representations

in psychology will be considered suspect so long as we lack a reasonable way of spelling out

what sorts of representational structures mental models are supposed to be in a way that (i)

shows these models to be distinct from sentential representations while (ii) allowing for their

realization by neurophysiological processes. We can see the way forward, however, if we first

pay attention to some of the distinctive features of scale models that distinguish them from

sentential representations. If we then turn to the computational realm, we see that these very

features (including immunity to the notorious frame problem) are exhibited by certain

computational models of mechanisms such as FEMs. An appeal to the principle that sustains the

computational theory of cognition (i.e., POPI) enables us to understand how this could be so and

how high-level, non-sentential, intrinsic models of mechanisms could in principle be realized by

neurophysiological processes. The broader viability of this realization story for mental models is

suggested by recent work in both AI and experimental psychology and by the elegant solution it

offers to the surplus-meaning and ceteris paribus problems in the philosophy of science. Going

forward, the idea that our scientific reasoning about mechanisms might, to a large extent, involve

the manipulation of representations that are like scale models in crucial respects can be regarded

as at least one, sturdy pillar of a promising hypothesis regarding the nature of model-based

reasoning in science.

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