Descartes’S Regulae, Mathematics, And Modern Psychology: The Noblest Example of All in Light of Turing’s (1936) On Computable Numbers

History of Psychology Copyright 2000 by Ihc Educational Publishing Foundation2000, Vol. 3, No. 4, 299-325 1093-451DrtJG/$5.00 DOI: 10.1037//1093-4510.3.4.299

DESCARTES'S REGULAR, MATHEMATICS,AND MODERN PSYCHOLOGY:

"The Noblest Example of All" in Lightof Turing's (1936) On Computable Numbers

Geir Kirkeb0enUniversity of Oslo

There are surprisingly strong connections between the philosophy of mind and thephilosophy of mathematics. One particular important example can be seen in theRegulae (1628) of Descartes. In "the noblest example of all," he used his newabstract understanding of numbers to demonstrate how the brain can be consideredas a symbol machine and how the intellect's algebraic reasoning can be mirrored asoperations on this machine. Even though this attempt failed, it is illuminating toexplore it because Descartes launched 2 traditions—mechanistic philosophy of mindand abstract mathematics—that would diverge until A. Turing (1936) approachedsymbolic reasoning in a similar "symbol machine-existence proof" way. Des-cartes's and Turing's thought experiments, which mark the beginning of modernpsychology and cognitive science, respectively, indicate how important the devel-opment of mathematics has been for the constitution of the science of mind.

Cartesianism. . . created an unbridgeable gulf between the domain of natural

science . . . and the domain of the soul.1

Descartes's reflex model of behavior was once considered the modest begin-nings of modern psychology.2 Recently, however, several historians have criti-cized the view of Descartes as one of the field's founding fathers.3 Richards, forexample, in his work on the origin of psychological ideas, claimed that "no singletap-root of Psychology is present in the seventeenth century."4 Danziger, in hisfascinating investigation of the gradual replacement of Aristotelian-Scholasticcategories with modern ones, agreed, and wrote, "there is no reason to begin theseexplorations much before the year 1700."5 Danziger and Richards did indeeddiscuss Descartes's writings, but they both overlooked his most important con-tributions to a non-Aristotelian psychology, which is not his reflex model ofbehavior but his attempts in the Optics (1637) and the Rules for the Direction ofthe Mind (1628, henceforth Regulae)6 to bridge, in Danziger's words, the "un-bridgeable gulf between the domain of natural science . . . and the domain of thesoul."7 These attempts, both of which were made possible by Descartes's new

Geir Kirkeb0en was educated as a psychologist and a computer scientist. He is a professor inthe program Language, Logic, and Information, ILF, University of Oslo. He has published severalarticles on Descartes's psychology and on the influence of computer technology on psychology ina historical perspective.

I thank Stephen Gaukroger, Lars Kristiansen, and Richard Watson, who made detailed criticalcomments on a draft of this article. A short version of this article was presented at the conferenceof Cheiron, the International Society for the History of the Behavioral and Social Sciences, June1997, Richmond, Virginia.

Correspondence concerning this article should be addressed to Geir Kirkeb0en, Language,Logic, and Information, ILF, Pb. 1102, University of Oslo, 0317 Oslo, Norway.

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algebraic geometry, which was developed in the early 1620s, are largely over-looked by other historians of psychology too. As a result, the close link betweenthe early development of modern mathematics and modern psychology is notrecognized. My general aim in this article is to throw light on the important rolethat the development of a formal mathematical language played in the break withAristotelian-Scholastic psychology and the initial development of a non-Aristo-telian or "modern" one.

In the 1630s Descartes explained how a mechanically conceived world,including the human body and brain, can give rise to, respectively, color percep-tion and perception of the external world's metric properties.8 Although Des-cartes, as Danziger emphasized, clearly adhered to several Aristotelian categories,these theories of perception constitute the first fundamental break with theAristotelian-scholastic understanding of the relation between man's "intellect"and the world and can therefore, according to Danziger's own criteria, beconsidered as the very beginnings of the modern science of mind.

Descartes's theory of metric perceptions in the Optics implicitly presupposeshis new algebraic geometry. However, the close connection between his mathe-matics and psychology is far more explicit in his failed attempt to bridge "the gulfbetween the domain of natural science and the soul" in what, in Rule 8 of the earlyRegulae, he called "the noblest example of all." Descartes called it "the noblestexample of all" because it is an application of the method (contained in Rules 3-7)relating to "the problem of investigating every truth for the knowledge of whichhuman reason is adequate" (AT396).9 In Rules 12-21 this "example" is expandedinto an investigation of how our cognitive facilities operate, in particular how theyoperate when we reason in accordance with the algebraic method Descartes putforward in the Regulae. In short, in the "noblest example of all" he tried to showhow the immaterial intellect's algebraic symbols and symbol operations can beconnected to what they are about, that is, the external material world. This attemptis the topic of the present article.

The "noblest example of all" is a strange and isolated thought experiment tofind out how the human mind might operate when engaged in symbolic reasoning.Descartes's main problem was to show how the intellect's algebraic reasoning canhave a material process in the brain as a correlate. He did this in two steps. First,he showed that the corporeal imagination (a part of the brain) can be consideredto be a symbol machine, and then he tried to demonstrate how algebraic opera-tions in general can be regarded as operations on this machine. I argue that no onebefore Descartes, and none of his contemporaries in the 1620s, aspired to doanything similar; neither did Descartes himself in the works he wrote after he gaveup the Regulae in 1628. No one in the following 300 years attempted to accountfor mental phenomena in a way similar to Descartes's Regulae until Alan Turing,in his 1936 article On Computable Numbers, answered "the real question atissue . . .'What are the possible processes which can be carried out [by a humancomputer] in computing a number?' "10

There are obviously several differences in Descartes's and Turing's motiva-tions and in the contexts of their works. There are clearly also fundamentaldifferences in the way they understand both human cognition and mathematics.Nevertheless, my claim is that there are some striking similarities in the way thesetwo young mathematicians approached what today are called mental phenomena.

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In general, they both used their own results in mathematics to demonstrateprecisely how symbolic reasoning can be considered a mechanical brain process.

Although more than 300 years separate Descartes's and Turing's thoughtexperiments, there is a kind of connection between their endeavors. Both arelinked to the development of an algebraic mode of thought in modern mathemat-ics, a mode of thought that has three main characteristics: (a) the use of anoperative symbolism, (b) dealing with mathematical relations rather than objects,and (c) freedom from ontological commitment. To a large extent, Descartes'sthought experiment and his mathematical contributions initiated this development,and Turing's thought experiment, in a sense, completed it. Their thought exper-iments were also pivotal in the development of the modern science of mind. In theRegulae Descartes attempted to show precisely how a mechanistically conceivedworld and brain can give rise to perception and perceptual cognition. His failedendeavor in "the noblest example of all" motivated and determined his modernnaturalistic theories of perception in the 1630s. In On Computable NumbersTuring presented the first precise, abstract mechanical model of how a restrictedkind of mental process might be generated and controlled. Turing's article isconsidered by several authors not only as an inspiration and indirect cause of thecognitive revolution but also as the seminal work in cognitive science andartificial intelligence.11

In a recent book, Lakoff and Johnson expressed the popular view that

Descartes created a theory of mental representation—essentially the view inher-ited by first generation cognitive science. In this theory you can separate theproblem of how we think with ideas from the problem of what the ideas aresupposed to designate.12

This is wrong. Precisely because Descartes did not "separate the problem of howwe think with ideas from the problem of what the ideas are supposed to designate"he considered abstract symbols, like today's "second-generation" cognitive sci-ence, as being "represented in the same system as the perceptual states thatproduced them."13 In Turing's thought experiment and in first-generation cogni-tive science, on the other hand, the symbols are inherently nonperceptual or"amodal." I show that this important difference between Descartes's and Turing'sthought experiments mirrors an important difference between the initial and thefinal stage in the development of an algebraic mode of thought in mathematics.Finally, I argue that it is not until today's second-generation cognitive science thatone finds attempts to unify perception and symbolic cognition similar to Des-cartes's hypothetical thought experiment in the Regulae, that is, attempts to give"existence proof that one can develop a fully functional symbolic system that isinherently perceptual."14

Descartes's main concern in the Regulae was to explain how the intellect'sscientific thinking can be visualized or simulated in a passive corporeal brain. Inthe first part of this article I clarify the nature of this problem and how it isconnected with his new advanced algebra. In the second part, I consider in detailDescartes's approach to this problem in the light of Turing's On ComputableNumbers. In the third part, I discuss some recent interpretations of "the noblestexample of all" in comparison to my own Turing kind of cognitive scienceinterpretation, and I argue that even these interpretations are influenced by today's

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cognitive science. In the last part of the article I consider Descartes's mathemat-ical psychology in the Regulae in a wide historical context, and I defend my claimthat "the noblest example of all" is unique with respect to Descartes's predeces-sors, his own psychology after 1630, and other attempts to materialisticallyaccount for symbolic cognition in the period between 1628 and 1936. I alsoelaborate on the difference between, on the one hand, the amodal (nonperceptual)understanding of symbolic cognition of Turing and the first generation of cogni-tive science, and, on the other hand, the attempts of Descartes and second-generation cognitive science to ground symbolic cognition in perception andimaginations.

Descartes's Problem: The Ingenium in Abstract Reasoning

"The noblest example of all" is Descartes's reflections in the Regulae on thefour cognitive facilities: "the intellect, imagination, sense-perception and mem-ory" (AT411). In Rule 12 Descartes gave a short outline of his understanding ofthe last three of these faculties together with reflex behavior. He simply outlined"as briefly as possible what, for my purposes, is the most useful way of conceivingeverything" (AT412). Descartes's main purpose in "the noblest example of all,"as it is in the Regulae ad directionem ingenii as a whole, is to explain the"direction of the ingenii."

Ingenii, or ingenium, is often translated as "mind" or "natural intelligence."However, the ingenium does not refer to the mind or intellect as such, "But when[the intellect] forms new ideas in the corporeal imagination (phantasia), orconcentrates on those already formed, it is properly called [ingenium]" (AT416).15

The ingenium, according to this definition, is the power of forming and manip-ulating ideas or images in the corporeal imagination.16

In the Regulae Descartes emphasized that the corporeal imagination has a roleto play in all kinds of reasoning, stating that "we shall not be undertaking anythingwithout the aid of the imagination" (AT443). In Rules 13-21 he attempted toexplain precisely how the ingenium mirrors the immaterial intellect's algebraicreasoning in the passive, corporeal brain. I refer to this as Descartes's problem ofmind in the Regulae.

This psychological problem has its origin in Descartes's new mathematics. Inthe early 1620s he developed a new algebraic approach to geometry, as a resultof which he is often considered the father of analytic geometry. However, theinsight that grounds his mathematical thought is incompatible with the funda-mental notion of analytic geometry, namely, that the equation is the essentialdatum.17 For Descartes, algebraic equations are simply a shorthand way ofperforming time-consuming geometrical operations. He viewed the equationspurely as a useful symbolic language in which geometrical constructions can bestored, and he emphasized that each algebraic manipulation must correspond todefinite geometrical operations. Unless they can be accompanied by geometricaloperations, algebraic manipulations are regarded by Descartes as the result ofmeaningless play with empty symbols.

This understanding of algebra conforms to the dominant view of the lateMiddle Age and the Renaissance. In this period it was common to make adistinction between practical—that is, applied—mathematical disciplines and

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theoretical arithmetic and geometry. Algebra was practiced as an artful procedurein practical calculating and measuring.18 To legitimize an algebraic approach totheoretical geometry, Descartes had to demonstrate the relation between the artfulalgebraic manipulations and scientific geometric constructions. He had to showhow his abstract algebra could be grounded in geometrical operations. This is themathematical problem that Descartes confronted in the Regulae. Descartes's newunderstanding of scientific reasoning and mathematical activity, together with hisunderstanding of the intellect, turned this mathematical problem into his problemof mind.

Scientific reasoning, according to Descartes in the Regulae, is algebraicmathematical reasoning. His method is intended to be a universal problem-solvingmethod; in it, he defined a problem in general terms as "something unknown"(AT430) in a subject matter. In short, Descartes explained how, in a "certain andevident" (AT362) way, we can abstract the quantitative aspects from all kinds ofsubject matters, symbolize the quantities, transform the relations between theminto algebraic equations, and solve the equations, that is, express the unknownquantities in terms of the known ones.

The Aristotelian paradigm of mathematical activity was the dialogue. Thefinal criterion of mathematical truth was negotiations among the community ofmathematical practitioners. In the Regulae Descartes prescribed not only a newscience of nature but also a new "natural" science. The naturalness consists in hisconviction that logic and scientific method must be rooted in how the cognitivefaculties naturally operate. The difference between an Aristotelian discursiveunderstanding of mathematical activity and a Cartesian facultative understandingis exemplified in the different ways that Aristotle and Descartes justified infer-ential principles. Aristotle justified the law of noncontradiction by showing that anopponent who denies it must, in denying it, actually assume its truth. ForDescartes, only the "light of reason" or the "light of nature" could justify basicinferential principles.19 He internalized mathematical activity. The paradigmaticprocedure of scientific or mathematical activity becomes calculation using alge-braic symbols in an individual's intellect.

In the Regulae the intellect is considered to be immaterial and active, and "itis of course only the intellect that is capable of perceiving the truth" (AT411). Thecorporeal brain, on the other hand, is conceived as being inert and passive.Nevertheless, Descartes strongly emphasized the intellect's limitations when itacts on its own. Consequently, it "has to be assisted by imagination, sense-perception, and memory if we are not to omit anything that lies within our power"(AT411). However, the intellect is also sharply divided from the corporeal body;he wrote, "[it] is no less distinct from the whole body than blood is distinct frombone, or the hand from the eye" (AT415). In short, Descartes understood scientificreasoning as algebraic mathematical reasoning in a limited intellect sharplydivided from the corporeal brain and world.

Descartes, then, had at least two reasons to explain how the ingenium canvisualize abstract scientific reasoning in the corporeal brain. First, to legitimatethe new mathematical approach to natural phenomena—that is, his physico-matematica—Descartes had to show how the abstract objects and operations ofhis new mathematics can be identified with "the things which are outside us andquite foreign to us" (AT398).

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The second reason is the intellect's limitations when it is operating on its own.

According to Descartes, "knowledge is certain and evident cognition" (AT362).His method requires certainty at each step in the reasoning process and is,ultimately, based on the intellect's truth-perceiving abilities, that is, what Des-cartes, in the Regulae, calls " 'intuition' . . . the conception of a clear and attentive

mind, which is so easy and distinct that there can be no room for doubt about whatwe are understanding" (AT368). Intuition (or vis cognoscens) is the fundamental

way of knowing. Problem solving, therefore, consists in reducing everything to aform in which we can grasp it in an intuition. The intellect must visualize abstractscientific reasoning in the corporeal brain so that the vis cognoscens or "the lightof nature" (AT440) can verify that every step in the reasoning process is "certain

and evident... and incapable of being doubted" (AT362).

"The Noblest Example of All" in Light of Turing'sOn Computable Numbers

Turing's On Computable Numbers is a classic article in mathematical logic.

However, Turing's approach in this article is also motivated by his attempt toanswer an old problem of mind,20 namely, how to reconcile an understanding ofthinking and free will or mental control with the scientific description of matter.As late as 1933 Turing was something of a Cartesian dualist with regard to this

problem; he wrote, "We have a will which is able to determine the action of the

atoms probably in a small portion of the brain, or possibly all over it."21 Turing

based this view on quantum mechanics. However, within a few years he changedhis view completely. His own mathematical results then made it possible for himto account for apparent mental control in a mechanistic and deterministic way.

Contrary to Descartes in the Regulae, Turing assumed that the mind can be

identified with the activity of organized matter.22 Nevertheless, there is an im-portant similarity in Descartes's and Turing's problems of mind: Each had to

show how symbolic reasoning can be described as a mechanical "brain" process.There is also a resemblance in Turing's and Descartes's mathematical prob-

lems. In On Computable Numbers, Turing answered Hilbert's Entscheidungs-

problem. Hilbert's statement of the problem is as follows: "The Entscheidungs-problem is solved if one knows a[n] [effective] procedure which will permit one

to decide, using a finite number of operations, on the validity, respectively thestatisfiability of a given [first order] logical expression."23 What I consider to beTuring's mathematical problem is limited to the central difficulty of defining thevague intuitive notion of an effective procedure or definite method. Turing had toground this abstract notion in mechanical (symbol) operations, whereas Descartes

had to ground abstract algebra in geometrical operations.Hodges described Turing's problems as follows: "The lack of any simple

connection between mathematical symbols and the world of actual objects fasci-nated Alan .. . the task was to relate the abstract and the physical, the symbolicand the real."24 This characterization also fits Descartes's problems of mind and

mathematics in the Regulae.


Descartes's and Turing \i Symbol Machines

Turing and Descartes approached their problems of mind in a similar way.The key to their solutions is what I call their symbol machines. Turing inventedhis Turing machine to define the vague intuitive notion of a definite method. Afterdescribing his computing machine's simple operations, Turing claimed that "It ismy contention that these operations include all those which are used in thecomputation of a number."25 He showed, then, how complex operations can bedescribed precisely, as a table of behavior,26 with respect to the Turing machine.Turing identified the notion of a definite method with such a table of behavior. Ina sense, he grounded the abstract notion of a definite method in the operations ofthe Turing machine.

Descartes's mathematical problem was to justify the application of algebra togeometry. According to his understanding of mathematics, he had to groundabstract algebra in geometrical operations; that is, he had to show that the basicoperations of algebra have a geometrical interpretation. Descartes's new symbolicunderstanding of numbers made this possible. In ancient mathematics, a numberis always a number of something. Arithmetic is a form of metrical geometry.Numbers are considered as line lengths. Multiplication, for example, thereforeinvolves a dimensional change. The multiplication of three numbers (or linelengths) is a cube, and the product of four numbers is considered to be meaning-less.27 In the Regulae Descartes stressed "For though a magnitude may be termeda cube . . . it should never be represented . . . otherwise than as a line or a surface"(AT457). For Descartes, line lengths and rectangles are not what numbers are;they are just a way of representing them—they are symbols. In Rule 18 of theRegulae he showed how the basic algebraic operations (addition, subtraction,multiplication, and division) can be understood as geometrical operations on suchfigure symbols. I consider Descartes's figure-symbolic representation of numbersand the basic algebraic operations on them to be his symbol machine.

Turing's Abstract-Mechanical Model of Cognition

In On Computable Numbers Turing had to justify that his definition of adefinite method included everything that can possibly be counted as such amethod. One way he did this (in paragraph 9) is by considering what people canpossibly be doing when they think—or, more precisely, when they compute anumber by following a deterministic procedure. Turing argued that human cog-nition in that case can be modeled using the Turing machine.28

To show this, Turing first had to make several simplifying assumptions thatadapt the human computer29 to his Turing machine:

Computation [cognition] is carried out on one-dimensional paper .. . the numberof symbols which may be printed is finite . . . the number of states of mind whichneed to be taken into account is finite... Let us imagine the operationsperformed . . . are so elementary that it is not easy to imagine them further divided.30

The result of Turing's analysis is an abstract model of a human being doingcalculations.

Turing then claimed, "We may now construct a machine to do the work of this[human] computer."31 His idea was simply that each state of mind of the human

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computer can be represented by a configuration of the corresponding Turingmachine. Cognition can then be described as computations on a Turing machine.Turing demonstrated that "The behaviour of the [human] computer at any momentis determined by the symbols which he is observing, and 'his state of mind' at thatmoment."32 He had turned his symbol machine into an appropriate deterministicmodel of symbolic reasoning.

"Somehow," Hodges wrote, "[Turing] perceived a link between what toanyone else would have appeared the quite unrelated questions of the foundationsof mathematics, and the physical description of Mind."33 My contention is thatDescartes, in the Regulae, perceived a similar kind of link between his questionof the foundation of mathematics and his question of how symbolic reasoning canbe described as a process in the corporeal brain.

Descartes's Simplifying Assumptions on Cognition

Descartes, like Turing, first made assumptions that adapted his understandingof the cognitive faculties or cognition to his symbol machine. In Rule 12 of theRegulae Descartes stated that his method is based on considerations of "only twofactors . . . the knowing subjects, and the things that are the objects of knowledge"(AT411). His considerations of these two factors are subordinated to his main aimin the Regulae, that is, to use his figure symbolism to answer the question of howthe ingenium can visualize algebraic reasoning as a mechanical process in thecorporeal brain.

With regard to "the knowing subjects," Descartes first assumed that sense-perception is purely passive and that it "occurs in the same way in which waxtakes on an impression from the seal" (AT412). He considered everything, evencolors, to be perceivable by the senses, as figures; he wrote that "we simply makean abstraction .. . and conceive of the differences of white, blue, and red, etc. asbeing like the differences between . . . figures" (AT413). He further assumed thatthe figure the sense organ receives "is conveyed at one and the same moment" tothe phantasia or the corporeal imagination that is "a genuine part of the body-.. . large enough to ... take on many different figures and . . . retain them for

some time" (AT414). By making these assumptions Descartes achieved twothings. First, he linked the corporeal brain or imagination to the external world,which is necessary to legitimize his new physico-matematica. Second, and moreimportant, his assumptions make it possible to consider the corporeal imaginationas an organ capable of representing abstract mathematical operations.

"The objects of knowledge," according to Descartes, are not the "things thatare outside us" (AT398). In the Regulae he was "concerned . . . with things onlyin so far as they are perceived by the intellect" (AT418). When Descartes stressedin his method that one has to "distinguish the simplest things" (AT381), hetherefore did not speak of the things "in accordance with how they exist in reality"(AT418). He termed " 'simple' only those things that we know so clearly anddistinctly that they cannot be divided by the Mind into others that are moredistinctly known" (AT418). According to Descartes, "the whole of human knowl-edge consists uniquely in our achieving a distinct perception of how all thesesimple natures contribute to the composition of other things" (AT427).

These simple natures are crucial in Descartes's attempt to legitimate his new


physico-matematica in the Regulae. Descartes postulated three kinds of simplenatures: intellectual, material, and common. "Those simple natures that theintellect recognizes by means of a sort of innate light, without the aid of anycorporeal image, are purely intellectual" (AT419). The material simple natures"are recognised to be present only in bodies—such as shape, extension, andmotion, etc." (AT419). They are the simplest thing the intellect perceives clearlyand distinctly in the corporeal organs. The material simple natures give theintellect access to the corporeal brain. I demonstrate above that the brain, bymeans of Descartes's understanding of sense-perception, is mechanically con-nected to the external world. Thus, the material simple natures link the intellect tothe corporeal world. The common simple natures "are ascribed indifferently, nowto corporeal things, now to spirits—for example, existence, unity, duration, andthe like" (AT420). Below I show that both the material and the common simplenatures are necessary assumptions in Descartes's explanation of how the inge-nium can visualize abstract reasoning in the corporeal imagination.

Descartes's Abstract-Mechanical Description of Cognition

In Rule 13, the first rule on problem solving in the Regulae, Descartes showedhow "we can abstract a problem ... and reduce it to ... certain magnitudes ingeneral and the comparison between them" (AT431). According to Descartes,entities conceived in the intellect are indeterminate. To render them determinate,the intellect has to represent them in the corporeal imagination. The heading ofRule 14 summarizes what one has to do: "The problem should be re-expressed interms of the real extension of bodies and should be pictured in our imaginationentirely by means of bare figures. Thus it will be perceived much more distinctlyby our intellect" (AT438). Descartes's new symbolic understanding of numbersmakes it possible to conceive how this can take place.

The analogy Descartes saw between geometry and arithmetic, between alge-braic operations on continuous quantities (line lengths) and discrete quantities(number), legitimates his algebraic approach to geometry. This analogy alsomakes it possible for him to explain how the ingenium can visualize the intellect's"symbol-generating abstractions" as figure symbols in the corporeal brain. Heobviously had his figure symbolism in mind when, in Rule 14, he claimed that"nothing can be ascribed to magnitudes in general that cannot also be ascribed toany species of magnitude [particular instance]" (AT440-1). Descartes's figuresymbols can represent every quantity, and the basic algebraic operations can beperformed on them. It is also possible, then, to imagine how magnitudes in generalcan be represented in the corporeal imagination.

The ingenium does not merely re-express abstract problems in terms of figuresymbols in the imagination. It also is able to turn its focus to "ideas" the corporealimagination offers it and "completely .. . reduce continuous magnitudes to a set"(AT451). The ingenium then transforms the continuous magnitudes into corporealfigure symbols in the imagination. These figure symbols correspond to the lettersin Descartes's new algebra. Precisely because Descartes considered the figures inthe corporeal imagination to be symbols, and not simply geometrical figures, hestated that "We have as much reason to abstract propositions from geometrical

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figures, if the problem has to do with these, as we have from any other subjectmatter" (AT452).

Descartes's assumptions about the simple natures are necessary to explainboth how this figure symbolism—his symbol machine—can be implemented inthe corporeal imagination and how the intellect has access to the figure symbols.The material simple natures are the basic constituents of these figure symbols, andthe intellect, by definition, has clear and distinct access to them. The commonsimple natures are necessary to explain how the ingenium turns continuousmagnitudes or figures received through sense-perception into figure symbols. Todo this, the ingenium must apply a "unity . . . the common nature tha t . . . all thethings that we are comparing must participate in equally" (AT499) to the con-tinuous magnitudes in the imagination. "If no determinate unit is specified in theproblem, we may adopt as a unit either one of the magnitudes already given or anyother magnitude, and this will be the common measure of all the others"(AT450).34

The ingenium, then, approaches the magnitudes in the corporeal imaginationin the same way that Descartes turned geometrical problems into algebraicequations in his new algebra. Descartes presented for the first time the idea thatabstract symbols can be represented in a corporeal brain, and he explainedprecisely how symbols are coded into brain states. His new mathematics made thispossible.

In Descartes's view people do not manipulate concepts according to formallaws; they link simple natures intuitively perceived by means of the commonnatures. They are, in a sense, Descartes's rules of inference, "those commonnotions that are, as it were, links that connect other simple natures together, andwhose self-evidence is the basis for all the rational inferences we make" (AT420).Such an understanding stands in sharp contrast to how Turing comprehended rulesof inference. For Turing, and modern logicians, rules of inference depend only onthe physical forms of the expressions and not at all on their meaning or thelogician's intuition.

Turing considered each state of mind of the human computer as beingrepresented by a configuration of the corresponding Turing machine. For Des-cartes, each perceptual state could be represented by a configuration of linepatterns, a symbolic representation, in the corporeal imagination. His assumptionsabout the knowing subjects and the objects of knowledge turn the corporealimagination into a symbol machine with the symbols built up by rectilinear figuresand line lengths. Descartes attempted to show that scientific reasoning as pre-scribed by his method can be duplicated by the ingenium as a process in thecorporeal imagination, which is considered to be such a symbol machine.

Hodges claimed that Turing machines "offered a bridge, a connection be-tween abstract symbols and the physical world."35 Descartes's symbol machineplayed a similar role in his attempt to anchor the intellect's abstract reasoning inthe corporeal brain. The figures in the imagination are abstract symbols and, aspart of the corporeal imagination, they are also corporeal entities connected to theexternal world. Descartes's figure symbolism together with his mechanistic modelof sense-perception and his notion of the simple natures made it possible for himto show exactly how the corporeal imagination can act as a bridge betweenabstract thinking and the physical world.


Turing's and Descartes's Existence Proofs

Turing claimed that the most complex procedures can be built out of elemen-tary states and positions, reading and writing. He showed that any definite methodthat can be performed by his abstract human computer can also be performed byhis Turing machine, and vice versa.36 He also showed that one single machine, theuniversal Turing machine, can take over the work of any machine. By doing this,he implicitly gave an existence proof for the possibility that anything performedby a human computer can, in principle, also be performed by a Turing machine.

Three hundred years before Turing's time Descartes attempted to prove thatall operations in algebraic scientific reasoning can be simulated in a clear andevident way in his symbol machine, that is, that they can be described as concreteoperations on figure symbols represented in the corporeal imagination. Descartesfailed. He wrote out only the headings of Rules 19, 20, and 21. The solutions tothe problems he posed in these headings demand complex algebraic calculations,and it is not possible to show that such complex scientific cognition can besimulated in a clear and distinct way in his concrete-logical framework. Heabandoned the Regulae in 1628, possibly for this reason.37

"The Noblest Example of All" and/as Cognitive Science

Canguilhem maintained that "The discipline whose history one is studyingactually changes with each epistemological break."38 With respect to "the noblestexample of all," this claim makes sense. The so-called cognitive revolution in the1950s was an epistemological break between behaviorism and cognitivism. Cog-nitive science, the main result of this "break," has indirectly influenced mostrecent interpretations of Descartes's work.39 In more than 200 years since the firstpublication of the Regulae in 1701, no one read into it what I emphasize aboveand what has been emphasized by several interpreters in recent years. Gibson, forexample, in an extensive review of Descartes's Regulae in Mind in 1898, devotedno attention at all to the aspects of the Regulae that I stress.40

Before the rise of cognitive science, there was no basis for interpretingDescartes's understanding of the corporeal imagination in Rule 14 as, for exam-ple, Schuster did: "The imagination is an ontologically suitable 'screen' uponwhich extension-symbols can be manipulated."41 The Turing machine and its tapeare fundamental to this interpretation. Such an interpretation presupposes the ideaof coding in the nervous system that was—after Descartes—first developed byTuring, Shannon, McCulloch, Pitts, and later cognitive scientists.42

It is a curious fact that the first person to note that Descartes, in the Regulae,attempted to ground his abstract mathematics in a symbolism consisting of real,concrete line lengths depicted in the imagination—namely, the mathematician andhistorian Jacob Klein, in his brilliant work on Greek Mathematical Thought andthe Origin of Algebra—did so the same time—and from a similar stance43—asTuring worked out his On Computable Numbers.

"The Noblest Example of All" as a Turing Kind of Cognitive Science

In most respects, my interpretation of Descartes's "noblest example of all" fitsother recent interpretations. In one important respect, however, "the noblest

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example of all" looks quite different in my Turing kind of cognitive scienceinterpretation. The essence of my interpretation is that Descartes's considerationsof the cognitive faculties in Rule 12 are wholly determined by his main aim in theRegulae, namely, to explain how the ingenium may visualize the immaterialintellect's algebraic reasoning in the corporeal brain. From a diachronistic per-spective, the relation between Rule 12 and Rules 14-18 looks quite different. Inthe late 16th and early 17th centuries, theorists strongly psychologized logic; thatis, they emphasized the basis of logic in the activities of the mind. For example,Sepper wrote that

Late Scholastic logicians took the established distinction between formal andmaterial logic, the one culminating in syllogistic, the other in the analysis of theacts of the mind, and attributed primacy to the latter. Formal logic was no morethan an abstraction from material logic.44

From such a perspective it is natural to consider Rule 12 as being an independent"inquiry into mental function and perception"45 and, further, to consider bothDescartes's method and his explanation of the ingenium in Rules 14-18 as beingbased on his analysis of the acts of the mind in Rule 12.

The way Descartes presented "the noblest example of all" was probablyinfluenced by psychologized Scholastic logic.46 However, I cannot see that Rule12 contains any inquiry into perception and mental functioning at all. Descarteswrote, "I should like to explain . . . what each particular faculty does; but I lackthe space" (AT411). This is a rather direct way of saying that an inquiry into thecognitive faculties is not the issue. As I show above, Descartes assumed in Rule12 exactly what is necessary, and hardly anything else, to explain in Rules 14-18how the ingenium operates and how the intellect has access to the external world.Rule 12 is not therefore an independent analysis of the acts of the mind or our"material logic" from which Descartes then later abstracted his formal logic ormethod. The formal logic (presented in Rules 14-18) determined how Descartesconsidered cognitive faculties or material logic in Rule 12.47

Descartes worked out Rule 12 in 1628. The mathematical ideas he presentedin Rules 14-18 date from several years earlier.48 This is also a reason forconsidering Descartes's investigations into the cognitive faculties or materiallogic as being shaped by his formal logic. Another reason is his introduction of thesimple natures. After giving up the Regulae in 1628 he never explicitly used thenotion of simple natures again. This indicates that he invented the simple naturesin the Regulae explicitly in order to explain the operations of the ingenium andhow the intellect can have access to the figure symbols in the corporealimagination.

Descartes's main purpose in Rule 12 was clearly not to explain how percep-tion and mental functioning take place but only how they might take place. Hestressed again and again the hypothetical character of his suggestions. He toldreaders, for example, "Of course you are not obliged to believe that things are asI suggest" (AT412). The following extract makes this even clearer:

To this end, as before, certain assumptions must be made in this context thatperhaps not everyone will accept. But even if they are thought to be no more realthan the imagery circles that the astronomers use to describe the phenomena they


study, that matters little provided they help us to pick out the kind of apprehensionof any given thing that may be true and to distinguish it from the kind that may befalse. (AT417)

Only his assumption on sense-perception must be thought of in the way Descartessuggested, namely,

It should not be thought that I have a mere analogy in mind here: we must thinkof the external shape of the sentient body as being really changed by the objectsin the same way as the shape of the surface of the wax is altered by the seal.(AT412)49

To summarize, "the noblest example of all" must be understood as a thoughtexperiment to find out how a mechanically conceived world can give rise toperception and abstract reasoning. In Rule 12 Descartes seemed only to set up anew framework that made it possible for him later, in the Regulae, to imagine howa mathematical science of a mechanistic world is possible. Hodges's generalcharacterization of Turing's thought experiment in On Computable Numbers alsofits quite well with Descartes's "noblest example of all":

But it was not only a matter of abstract mathematics, not only a play of symbols,for it involved thinking about what people did in the physical world. It was notexactly science, in the sense of making observations and predictions. All he haddone was to set up a new model, a new framework. It was a play of imagination.. . What he had done was to combine . . . a naive mechanistic picture of the mind

with the precise logic of pure mathematics.50

"The Noblest Example of All" in the History of Psychology

We have seen that, in "the noblest example of all," Descartes attempted todemonstrate in two steps how the immaterial intellect's algebraic reasoning can beconsidered as a material brain process. First, he considered the corporeal imagi-nation as a figure symbol machine. Second, he attempted to demonstrate that theintellect's algebraic reasoning can, in general, be duplicated as a process on thismachine. In the historical sketch below I substantiate my claims that one does notfind a similar two-step "symbol machine-existence proof approach to explainingsymbolic reasoning again until Turing's On Computable Numbers and that it isnot until today's second-generation cognitive science that one finds attempts at amodal two-step symbol machine-existence proof approach to symbolic cognition,as one finds rudiments of in Descartes' s failed project in the Regulae. I first brieflyconsider "the noblest example of all" with respect to Aristotelian-inspired Scho-lastic cognitive psychology and abstraction theory.

"The Noblest Example of All" and Descartes's Ancestors

The way in which Descartes attempted to legitimate his new abstract physico-

mathematica in the Regulae is, to a large extent, in accordance with a theoreticalunderstanding of abstraction and abstract reasoning that was widely accepted inEurope from the 13th to the 17th century. This understanding, based on Aristotle'scognitive psychology, assumes that we grasp external reality through a process ofabstraction from physical sensation.51 Abstract concepts were taken to be images

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abstracted, or drawn out, from sensory impressions, and thinking was understoodas being grounded in such images in accordance with the Aristotelian dictum"There is no thought without phantasms,"52 that is, there is no thought withoutcorporeal images in view of which the power of understanding exercises itsactivity. Descartes's "noblest example of all," in which he stressed the immaterialintellect's dependency on the corporeal imagination, stands in this Aristoteliantradition.53

Descartes's "embodied" and "imagistic" psychology in the Regulae is, inmany respects, traditional. For example, his comparison of external sensation witha stamp in wax is an analogy that was also used by the early Stoics (e.g., Zeno),and his division of the cognitive faculties into sense perception, imagination,memory, and the intellect is a simplification of a conceptualization that dates backto Avicenna (980-1037 AD). Avicenna distinguished between the intellect andthe five external and internal senses. He also gave the internal senses precisebodily locations in the ventricles of the brain.54 With minor modifications,Avicenna's conceptualization, which can be considered an elaboration of theAristotelian distinction between a sensitive and an intellectual soul, dominated theMiddle Ages.

The controversial question was how the immaterial intellect operates withinthe system of cognitive faculties.55 Avicenna himself assumed that the corporeallylocated internal senses serve the rational soul, but he argued that the intellectrequires such help only for the preliminary stages of thought. In contrast toAvicenna, Thomas Aquinas insisted, in the Summa Theologiae (1273), that theintellect remains dependent on sense impression even while thinking, and heexplicitly repeated Aristotle's dictum, which had been ignored by Avicenna, thatis, non contingit intelligere sine phantasmate.™

Aquinas's theory places the intellect in close collaboration with the body.According to Aquinas, the human intellect

abstracts a universal from many particulars, but it does not then take leave of thesensible forms from which it has derived this knowledge, for in thinking of auniversal, man always employs some phantasm, just as the geometer uses adiagram.57

Aquinas's emphasis on the intellect's dependency on the brain, as well as theanalogy he drew between phantasmata and geometrical diagrams, anticipatedcentral ideas in Descartes's "noblest example of all."

Vesalius's anatomical findings, illustrated in detail in his De humani corporisfabrica (1543), were probably the main cause of the gradual decline of theneuropsychological theory of the inner senses. His anatomical discoveries under-mined the physiological basis of the theory.58 Vesalius expressed the problemraised by his anatomical studies as follows: "I cannot understand to my satisfac-tion how the brain performs its office in imagination, reasoning, cogitation andmemory."59

In the Renaissance it was still common to assume that the internal sensesacted to bridge the gap between external sensation and the abstract operation ofthe intellect.60 The specific novelty of "the noblest example of all" is Descartes'sattempt to answer Vesalius's "how" question in respect to algebraic reasoning,


that is, his attempt to explain precisely how the corporeal imagination may act asa bridge between abstract algebraic reasoning and the physical world.

Symbolic Reasoning and the Brain in Descartes's Modern

Naturalistic Psychology

In the Regulae Descartes assumed that the intellect does not have direct accessto the external world. The intellect can only inspect material figures, built up fromsimple natures, in the brain. This assumption constitutes a fundamental break withthe Aristotelian tradition.61 However, Descartes maintained another central Aris-totelian-Scholastic assumption, namely that there is a perceptual relation betweenthe intellect and what the intellect perceives, that is, the figures in the corporealimagination. In his writings after abandoning "the noblest example of all," thisassumption is also abandoned, and he replaced it with the assumption that therelation between brain activity and the intellect is a "causal" (or occasional)one. Probably motivated by his failure in "the noblest example of all," henow considered ideas or imaginations as consequences of mechanical brainmovements.62

Descartes's naturalistic psychology as developed after 1629 is founded on thisradical new epistemological hypothesis. In the Regulae he showed how it ispossible to consider reflex behavior and sense-perception, memory and perceptualcognition as a result of mechanical processes in corporeal brain organs. However,his descriptions were abstract and purely functionalistic. He only sketched thestructure of a causal path that could realize them. In his naturalistic psychologyafter 1629 he speculated on the particular realization of these causal paths.

Historians of psychology have focused almost exclusively on Descartes'sdualistic philosophy in the Meditations (1640) and his purely physiologicaltheories in I'Homme (ca. 1633), in particular his theory of reflex behavior.However, Descartes also discussed what today one would consider to be psycho-logical phenomena in ways clearly distinct from his concern with the status ofsensory knowledge, on the one hand, and from his purely mechanistic physiology,on the other. An example of particular interest to my discussion is his theory ofmetric perceptions in the Optics. I have discussed this theory in another article,63

and here I mention only the crucial role of Descartes's new algebraic geometryand the lack of any attempt to specify the bodily correlate of the symbolicinferences (or "natural geometry") that he attributed to the intellect in his theoryof metric perception.

For Descartes, explaining perception was of decisive importance, partlybecause "the principle argument which induced philosophers to posit real acci-dents was that they thought that sense-perception could not be explained withoutthem."64 In particular, it was important to explain metric perception, that is, theperception of objects' "primary qualities," and the distance to them, to "bridge thegulf between natural science and the domain of the soul" (i.e., of the scientist whomakes the observations). Descartes clearly recognized that a new natural sciencedemanded, in particular, a new psychology of metric perception.

In the Optics Descartes assumed that when a person looks at an object, lightbeams reflected on its surface hit the eye and (mechanically) activate a set of nerveendings on the retina. Descartes then had to explain how a set of nerve endings

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set into motion can have sufficient representative content to represent the geo-metrical aspects of the perceived object. He answered this question in his analyticgeometry.65 In Greek mathematics, shapes or figures cannot be reduced to mag-nitudes. Consequently, from the perspective of Euclidean geometry it is notpossible to imagine how a set of activated points on the retina can representgeometrical figures. Seen from the standpoint of Descartes's algebraic geometry,a transformation from activated points on the retina to perceived figures is,however, conceivable. Tn his geometry he demonstrated how all geometricalfigures can be understood as being generated from geometrical points. He con-sidered a line to be the result of a point's movement, a plane to be the result ofa line's movement, and so on.66 As a result, his new algebraic geometry served asan existence proof for the possibility of bridging "the gulf between the domain ofnatural science . . . and the domain of the soul" in his theory of metric perceptionin the Optics, as it did (in a quite different way) in the "noblest example of all"in the Regulae.

In the Regulae Descartes maintained a "critique of pure reason." The inge-nium, and the brain, have an important role to play in that work in all kinds ofthinking, as well as when one is engaged in abstract scientific reasoning. Forexample, Descartes argued in the Regulae that "Those who attribute wonderfuland mysterious properties to numbers" (AT445) do not understand the intellect'slimitations when it operates independently of the corporeal imagination. In the1620s he required that it must always be possible to imagine how certain kinds ofbrain processes may correlate with, or mirror, the intellect's abstract reasoning.After abandoning the Regulae, Descartes departed from this requirement.67 In hislater writings he never speculated on the bodily correlate of the inferences heattributed to the intellect, for example, the algebraic inferences (or "naturalgeometry") in his theory of metric perception. In short, the radical change inDescartes's understanding of the relation between the intellect and the materialbrain in 1628-1629 meant that a project such as the "noblest example of all" wasno longer of interest to him.

The Absence of a Two-Step "Symbol Machine-Existence Proof

Approach to Cognition Until 1936

Descartes's contemporary, Hobbes, went a step further in explaining cogni-tion materialistically. Hobbes reduced physics to matter in motion and equatedmental activity with the latter. Contrary to both Aristotelian and Cartesian theoriesof cognition, Hobbes denied that human beings, unlike animals, have an imma-terial intellect; he wrote: "The Imagination that is raised in man (or any othercreature induced with the faculty of imagining) by words, or other voluntarysignes, is that we generally call Understanding; and is common to Man andBeast."68 Hobbes also formulated general principles of association in order toexplain the regularity in people's thought or chains of imaginations. However,Hobbes did not approach symbolic cognition in the two-step "symbol machine-existence proof kind of way that Descartes used in the Regulae. Hobbes, like LaMettrie, Hartley, Pavlov, Hull, and others who put forward materialistic theoriesof symbolic cognition in the 300 years after Descartes's Regulae, assimilated

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symbolic reasoning directly to the brain mainly by simply postulating thatparticular physiological or mechanistic principles underlie such reasoning.

Leibniz, in his logic, came far closer than Descartes to the modern notion ofa formal system (or symbol system). In contrast to Descartes, Leibniz maintainedboth a mechanistic (syntactic) concept of inference and a related idea aboutmechanizing proofs.69 His "universal characteristic" (or "artificial language") isbased on these ideas. However, Leibniz did not combine his understanding of aformal system with a mechanistic understanding of the human mind.70 In general,it was not common among 1 Sth-century thinkers working on machinery that couldsimulate thought processes to draw "the disparaging analogy between mind andmachine."71 One reason for this was simply that "the actual calculating machineshad proved such dismal practical failures."72 Neither did Babbage, early in the19th century, try to explain cognition on the basis of his impressive AnalyticEngine.73

It is common to attribute a disembodied understanding of cognition to thedualist Descartes. However, Descartes never explained (what is now called)cognition or cognitive processes without taking the brain and body into account.Whether cognition was beyond the powers of a corporeal machine was, for him,an empirical question.74 Not until the period after Descartes's death did a disem-bodied analysis of mental phenomena attain a prominent position, in particularwithin British empiricism.75 Locke, for example, explicitly delimited himselfaway from the corporeal correlate of mental processes.76 Even if Locke and (mostof) his followers within British empiricism have as their aim to "mechanize themind," they do not attempt to relate mental functioning to the material brain.

The British empiricists' disembodied approach to mental phenomena waspartly influenced by the Newtonian scientific ideals that characterized the 18thcentury. Particularly influential was Newton's explicit dissociation from theCartesians' speculative mechanical hypotheses in science.77 In contrast to Des-cartes, Newton accepted the use of (mechanically) "unexplainable" concepts(gravitation, mass, etc.) in scientific explanations. The success of Newtonianphysics discredited speculative mechanical models and legitimized the use of"unexplainable" concepts within epistemology, psychology, and physiology, too.For example, Hume, who obviously assumed that cognition is a result of materialprocesses, did not speculate on the mechanisms that associate ideas (as Descartes,in fact, did). Hume legitimated the fundamental concept of association by drawinga parallel with the "unexplained" concept of gravitation or attraction in physics;he wrote, "here is a kind of ATTRACTION, which in the mental world will befound to have as extraordinary effects as in the natural, and to show itself in asmany and as varied forms."78

One of Descartes's main motivations in the 1620s was to undermine vitalisticand animistic theories that flourished in the Renaissance.79 Mainly for that reason,he postulated that matter is "inert," a view that dominated mechanistic theoriesuntil Newton reintroduced "forces" into matter. Newton's success legitimated theintroduction of vitalistic forces within physiology as well.80 It was partly for thisreason that vitalism dominated science and philosophy in the 18th century and thefirst decades of the 19th century.81 Major philosophers in this period, such asLeibniz, Kant, and Hegel, also argued against the assumption that organisms andlife processes can be considered to be processes in "inert matter." In general,

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Newtonianism did not encourage attempts at considering symbolic reasoning asrepresented (or realized) as processes in a brain considered to be a (hypothetical)symbol machine.

In the middle of the 19th century there was a move from vitalism tomechanism in physiology, and the second part of the century was dominated bya strong mechanical conception of man. However, no one at that time seems tohave any idea of how the material brain may perform symbolic reasoning orinferences. This is indicated by Pavlov's reaction to Helmholtz's concept ofunconscious inference: "Evidently, what the genius of Helmholtz referred to as'unconscious conclusion' corresponds to the mechanism of the conditioned re-flex."82 As late as 1942 Boring characterized "the concept of unconscious infer-ence . . . [as] a negative explanation . . . essentially a confession of ignorance."83

Boring's statement is representative of the attitude prevalent in the first half of thiscentury toward the idea that unconscious inferences are taking place in the brain,an idea that most psychologists today consider to be unproblematic.

Pavlov himself did not make any attempt to explain precisely how particularsymbolic inferences can be performed by conditioned reflexes. Hull's attempts inthe 1920s to demonstrate that machinery can do what mental processes do wereprobably the most ambitious ones within psychology until the 1940s. Hullconstructed, for example, a mechanical machine to calculate correlation coeffi-cients.84 However, Hull did not seem to have any idea as to how (symbolic)inferences can be realized in the nervous system, and he did not give anyconsideration at all to the brain as a (universal) symbol machine.

Neither Descartes (1628) nor Turing (1936) took neuropsychology into con-sideration in their thought experiments, and neither of them provided realisticideas as to how symbolic inferences are represented in the material brain. The firstconcrete hypothesis of how the brain may work as a symbol machine was putforward by McCulloch and Pitts in 1943.85 They established the logical principlesof the brain as a computer by combining the insights of Turing and others in logicand automata theory with the view of the nervous system as a structurally andfunctionally coordinated network of neurons or nerve cells.86 The acknowledgedfact that the brain was first considered as a symbol machine in 1943 is in itself agood reason for doubting that anyone in the period between Descartes and Turinggave the same kind of consideration to symbolic reasoning as they did in theirthought experiments.

Descartes's and Turing's Thought Experiments and the "AlgebraicMode of Thought"

Descartes's mathematical contributions were a major factor in the initial stageof the development of the algebraic mode of thought in mathematics.87 Inparticular, his new abstract understanding of numbers was an important breakwith the intuitive geometrical mode of thought that characterized mathematicsfrom antiquity until the early 17th century. Descartes's thought experiment in theRegulae was motivated by this geometrical mode of thinking in mathematics.However, it was his main contribution to an algebraic mode of thought—that is,his abstract understanding of numbers—that made it possible for him to considerthe imagination as a figure (perceptual) symbol machine.

DESCARTES'S REGU1AE AND MODERN PSYCHOLOGY 317

With the gradual development of an algebraic mode of thought during the 200years after Descartes, mathematicians moved further and further away from theintuitive physical world and entered an abstract world of structures. Hilbert, whoconsidered mathematics to be a completely abstract enterprise, is the central figurein the final stage of this development. In the sense that Turing in On ComputableNumbers gave a final answer to the last one of Hilbert's three general questions,he concluded the development of the algebraic mode of thought in mathematics.

The fully developed algebraic mode of thought in mathematics contributed,by means of Turing (1936), the development of formal languages, the computermetaphor, and so on, to the cognitive revolution and a new kind of cognitivepsychology in the late 1950s. This psychology considers the mind to be a symbolsystem and cognition to be symbol manipulations. Compared to earlier theories ofcognition, the symbol system theories incorporate three important novelties. First,they have the ability to represent abstract concepts, to combine symbols produc-tively, to represent propositions, and so on—that is, they are fully functionalconceptual systems. Second, the theories themselves have the potential to dupli-cate or simulate the phenomena they are intended to explain. In a sense, they arethemselves existence proof for the possibility of their own correctness. Third, thetheories are amodal in the sense that the symbols' "internal structures bear nocorrespondence to the perceptual states that produced them."88

In particular, the distinction that this new symbol system psychology makesbetween conception and perception is an important novelty in the history ofpsychology. Throughout recorded history, from Aristotle's dictum "no thoughtswithout phantasms" until well into the present century, cognition had beenconsidered and understood as being grounded in perception and imagination.89 Itwas the completion of a fully algebraic (and amodal) mode of thought inmathematics that both directly and indirectly made possible and legitimizedabstract, amodal symbol system theories of cognition in psychology as well.

The "Noblest Example of All" and Second-Generation Cognitive Science

In the first decades after the cognitive revolution, cognitive science wasdominated by amodal symbol system theories of cognition. I mention above thestrengths of these theories. However, they also have obvious shortcomings. Tworelated problems are directly linked to their amodality: the symbol transductionproblem and the symbol grounding problem. The first problem is that the symbolsystem theories do not provide a satisfactory account of the process that mapsperceptual states into amodal symbols, and the second is that the theories do notexplain how symbol meaning is grounded in something other than just moremeaningless symbols.

Contrary to first-generation cognitive science, Descartes attempted in the"noblest example of all" to provide a solution to both the symbol transductionproblem and the symbol grounding problem. He attempted, in particular, toanswer a question similar to the one raised by the symbol grounding problem,namely, "How can the semantic interpretation of a formal symbol system be madeintrinsic to the system, rather than just parasitic on the meanings in our heads?"90

In fact, his main motivation in the Regulae was to demonstrate that it is possible

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to imagine how the immaterial intellect's algebraic reasoning or symbol opera-tions are connected to what they are about, that is, the external material world.

A central topic in today's second-generation cognitive science is how toreunite conception and perception. Among others, Barsalou and his colleagueshave attempted in several articles to develop "modal" perceptual symbol systemtheories that have the same strengths as amodal theories but without theirshortcomings. They attempt to do this by proceeding in a way which, on a generallevel, has similarities to the two-step symbol machine-existence proof approachthat Descartes followed when he united algebraic conception and perception in the"noblest example of all."

Compare how Descartes considered the corporeal imagination as a figuresymbol machine above with the "first step" in Barsalou's most recent attempt toconsider the brain as a "perceptual symbol machine":

A perceptual state can contain two components: an unconscious neural represen-tation of physical input, and an optional conscious experience. Once a perceptualstate arises, a subset of it is extracted via selective attention and stored permanentlyin long term memory. On later retrievals, this perceptual memory can functionsymbolically, standing for referents in the world, and entering into symbol ma-nipulation. As collections of perceptual symbols develop, they constitute therepresentations that underlie cognition.91

Descartes's intended "second step" was to provide an existence proof for thepossibility that the brain, considered as a figure (perceptual) symbol machine, canrepresent and visualize algebraic concepts and operations in general. The "secondstep" taken by Barsalou and his colleagues was "to establish an existence proofthat a completely perceptual approach is sufficient for establishing a fully func-tional symbolic system."92 So far, no one has managed to demonstrate how aperceptual symbol system can be considered as a fully conceptual system. Thereis still a long way to go.

Clearly, theories of perceptual symbols and the evidence for them remain to bedeveloped considerably in many regards. At this time, this approach primarilyattempts to provide an existence proof that, in principle, perception and conceptioncan be united in a way that does not require amodal symbols.93

Moreover, Descartes's failed attempt in the Regulae was limited to demonstratingthe possibility that conception (of a certain kind) can be united or grounded inperception and imagination. The fact that it is still hard to see how a modal orperceptual symbol system can function as a conceptual system is probably the bestargument for my claim that there is no evidence of a two-step symbol machine-existence proof approach to symbolic cognition in the period between 1628 and1936, when cognition was understood imagistically or modally.

Conclusion

It was not, as Danziger claimed, Cartesian dualism that created "an unbridge-able gulf between the domain of natural science . . . and the domain of the soul."It was the new natural science (of Galileo, Descartes, and others) itself that createdthis gulf. In fact, Descartes' s problem of bridging the gulf between the immaterial


intellect's algebraic reasoning and the material world in the Regulae was theproblem of dualism as he first encountered it. Jacob Klein epitomized theimportance to natural science of Descartes's attempt at solving this problem inthe Regulae:

Descartes' great idea now consists of identifying . . . the "general" object of thistnathesis universalis—which can be represented and conceived only symbolically—with the "substance" of the world, with corporeality as "extensio." Only by virtueof this identification did symbolic mathematics gain that fundamental positionwhich it has never since lost.94

My main point is that Descartes's attempt at legitimizing a new mathematicalphysics also entailed a first fundamental break with Aristotelian-Scholastic psy-chology and that his failure in the "the noblest example of all" was (probably) themain reason why, in the late 1620s, he assumed that ideas must be understood as(mysterious) consequences of brain movements. On the basis of this radical newepistemological hypothesis, Descartes developed his "modern" naturalistic psy-chology in the 1630s.95

Descartes's two-step symbol machine-existence proof approach to bridgingthe gulf between abstract reasoning and the material brain in the Regulae is basedon his most important contribution to the algebraic mode of thought in mathe-matics, namely, his new abstract understanding of numbers. No one, until Turingin 1936 completed the development of an algebraic mode of thought in mathe-matics, approached symbolic reasoning in a similar way. His thought experimentinspires, directly and indirectly, a new view of cognition as abstract manipulationsof amodal symbols.

Thus, the beginnings of both modern psychology and cognitive science wereinfluenced by a young mathematician's new symbolic formalisms and calculationtechniques. It is well known that the creation of a formal mathematical languagewas necessary for the constitution of modern mathematical physics. Descartes'sand Turing's thought experiments indicate that the development of a formalmathematical language has also been important to the constitution of the modern

science of mind.

Notes

1. Kurt Danziger, Naming the Mind: How Psychology Found Its Language (Lon-don: Sage, 1997), 55.

2. See, for example, Edwin G. Boring, A History of Experimental Psychology (NewYork: Appleton Century Crofts, 1950).

3. See, for example, Danziger, Naming the Mind; Roger Smith, "Does the Historyof Psychology Have a Subject?" History of the Human Sciences \ (1988): 147-177;Graham Richards, Mental Machinery: The Origins and Consequences of PsychologicalIdeas. Pan I. 1650-1850 (Baltimore: John Hopkins University Press).

4. Richards, Mental Machinery, 92.5. Danziger, Naming the Mind, 15.6. Rene Descartes, "Rules for the Direction of the Mind" and "Optics," in The

Philosophical Writings of Descartes, vol. 1, Trans. John Cottingham, Robert Stoothoff,and Dugald Murdoch (Cambridge, England: Cambridge University Press, 1985), 9-78and 152-175.

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7. Several of Descartes's contemporaries claimed that his reflex model of behaviorwas a plagiarism of a theory put forward by the Spanish doctor Gomez Pereira in 1554.See Javier Bandres and Rafael Llavona, "Minds and Machines in Renaissance Spain:Gomez Pereira's Theory of Animal Behavior," Journal of the History of the Behavioral

sciences 28 (1992): 158-168. Dcscartes's psychology in the Regulae and the Optics, onthe other hand, was based on his new algebraic geometry and, for that reason, clearly notanticipated by anyone; see below.

8. It is arguable that Descartes's theory of color perception (in the eighth discourseof the Meteorology, 1637) worked out in 1629 is the first theory in "modern" biologicalpsychology and that his theory of metric perception in the Optics (1637) is the firstmechanical—mathematical (cognitive science kind of) explanation of a mental phenome-non; see, for example, Geir Kirkeb0en, "Descartes' Psychology of Vision and CognitiveScience: The Optics (1637) in the Light of Marr's (1982) Vision," Philosophical Psychol-ogy I I (1998): 161-182.

9. The quotations from the Regulae are adapted from Descartes, "Rules for Direc-tion of the Mind." However, I give only the Adam and Tannary (AT, volume X) numbersin Oeuvres de Descartes, eds. C. Adam and P. Tannery, (rev. ed., Paris: Vrin/C.N.R.S.,1964-1976). The AT numbers are also given in the translation of the Regulae byCottingham et al.

10. Alan Turing, "On Computable Numbers, With an Application to the Entschei-dungproblem," in The Undecidable, ed. Martin Davis (1936; reprint, New York: RavenPress, 1965), 135.

11. See, for example, Andrew Hodges, Alan Turing: The Enigma. (London: Bur-nett, 1983); Justin Leiber, An Invitation to Cognitive Science (Oxford, England: Black-well, 1991); Stuart Shanker, "Turing and the Origins of AI," Philosophia Mathematica 3(1995): 52-85. See Diane Proudfoot and B. Jack Copeland, "Turing, Wittgenstein and theScience of the Mind," Australasian Journal of Philosophy 72 (1994): 497-519, for a

critique of such an interpretation of Turing's article On Computable Numbers,12. George Lakoff and Mark Johnson, Philosophy in the Flesh (New York: Basic

Books, 1999), 407.13. Lawrence W. Barsalou, "Perceptual Symbol Systems," Behavioral and Brain

Sciences 22 (1999): 523.14. Robert L. Goldstone and Lawrence W. Barsalou, "Reuniting Perception and

Conception," Cognition 65 (1998): 236.15. Cottingham, Stoothoff, and Murdoch, in The Philosophical Writings of Des-

cartes, translate the last part of the quoted sentence as: "The proper term for it is 'nativeintelligence' " (AT416). See Dennis Sepper, Descartes' Imagination: Proportion, Images,

and the Activity of Thinking (Berkeley: University of California Press, 1996), for athorough discussion of Descartes's use of the term ingenium in his early writings.

16. When Descartes referred to the imagination as a physical organ in the brain hecommonly named it phantasia. Sometimes he also used the term corporeal imagination.The term imagination in the Regulae usually refers to the result of the knowing force orintellect applying itself to the phantasia.

17. See, for example, Timothy Lenoir, "Descartes and the Geometrization ofThought: The Methodological Background of Descartes' 'Geometric', Historia Math-

ematica 6 (1979): 355-379.18. See, for example, Jacob Klein, Greek Mathematical Thought and the Origin of

Algebra, trans, by Eva Brann (1934; reprint, Cambridge, MA: MIT Press, 1968).19. See Steven Gaukroger, Cartesian Logic: An Essay on Descartes' Conception on

Inference (Oxford, England: Clarendon Press, 1989), for a full treatment of what Gauk-roger described as a shift from "a discursive to a facultative conception of inference"(p. 4).


20. Hodges, Alan Turing, showed that Turing "was profoundly concerned to find ascientific description of Mind long before his encounter with mathematical logic" (p. 63).My interpretation of Turing's On Computable Numbers is inspired by Hodges's biographyof Alan Turing.

21. Turing, quoted in Hodges, Alan Turing, 66.22. In his later articles Turing explicitly made such an assumption about the human

mind. See, for example, Alan Turing, "Intelligent Machinery," in Machine Intelligence 5,

eds. B. Meltzer and D. Michie (1947; reprint, Edinburgh, Scotland: Edinburgh UniversityPress, 1969), 3-23; and Alan Turing, "Computing Machinery and Intelligence," Mind 59(1950): 433-460. In his 1936 article, Turing only discussed a human being performingcalculations. However, his approach in this article too was, as Hodges convincinglydemonstrated in Alan Turing, inspired by his concern with finding a scientific descriptionof mind as a whole.

23. Hilbert and Ackerman, translated and quoted in Robin Gandy, "The Confluenceof Ideas in 1936," in The Universal Turing Machine: A Half-Century Survey, ed. RolfHerken (Vienna: Springer, 1994), 58. Related to his program for an investigation into thefoundations of mathematics, Hilbert posed three questions at a congress in 1928. First, ismathematics complete; that is, can every statement either be proved or disproved in aformal calculus? Second, is mathematics consistent; that is, can untrue statements neverbe arrived at by a sequence of valid steps of proof in the formal calculus? Third is thequestion that Turing answered negatively in On Computable Numbers: Is Mathematicsdecidable ?

24. Hodges, Alan Turing, 86.25. Turing, On Computable Numbers, 118.26. A table of behavior is simply a set of quintuples (of the form: old state, symbol

scanned, new state, symbol written, direction of motion) that describes the relation amongthe machine's states, inputs, and outputs.

27. See, for example, Steven Gaukroger, "Aristotle on Intelligible Matter," Phro-nesis 25 (1980): 187-197.

28. In fact, as Gandy pointed out in The Confluence of Ideas, Turing machinesappear as a result, as a codification, of Turing's analysis of calculations by humans.

29. Hodges, in Alan Turing, observed that "the word 'computer' .. . meant onlywhat that word meant in 1936: a person doing calculations" (p. 105).

30. Turing, On Computable Numbers, 135-137.

31. Ibid., 137.32. Ibid., 136.33. Andrew Hodges, "Alan Turing and the Turing Machine," in The Universal

Turing Machine: A Half-Century Survey, ed. Rolf Herken (Vienna: Springer, 1994), 6.34. In more detail; "We should realise that, with the aid of the unit we have adopted,

it is sometimes possible completely to reduce continuous magnitudes to a set and that thiscan always be done partially at least. The set of units can then be arranged in such an orderthat the difficulty involved in discerning a measure becomes simply one of scrutinising theorder. The greatest advantage of our method lies in this progressive ordering" (AT452).

35. Hodges, Alan Turing, 107.36. Gandy, in The Confluence of Ideas, expressed what he named "Turing's theo-

rem" thus: "Any function which is effectively calculable by an abstract human beingfollowing a fixed routine is effectively calculable by a Turing machine... and con-

versely" (p. 77).37. Why Descartes abandoned the Regulae is controversial. Several commentators

believe the reason was his inability to show how complex algebraic calculations can besimulated in a clear and distinct way in the corporeal brain, for example, John A. Schuster,"Descartes' Mathesis Universalis: 1619—28," in Descartes: Philosophy, Mathematics and

322 KIRKEB0EN

Physics, ed. Steven Gaukroger (Sussex, England: Harvester Press, 1980): 41-96; StevenGaukroger, Descartes: An Intellectual Biography (Oxford, England: Clarendon Press,1995). Sepper, in Descartes' Imagination, speculated that Descartes put the Regulae asidebecause of the results of his own anatomical studies, which he began in 1628-1629.Jean-Luc Marion, in Sur I'ontologie Grise de Descartes, 2nd ed. (Paris: LibrairePhilosophique J. Vrin, 1981), argued that Descartes's main reason for giving up theRegulae was that he was not able to make clear in an absolute sense the crucial distinctionin the Regulae between what is "simple" and what is "complex." Edwin M. Curley,Descartes Against the Sceptics (Cambridge, MA: Harvard University Press, 1978), sug-gested that Descartes abandoned the Regulae because he became convinced that he "didnot go deeply enough into the problem of knowledge" (p. 38) and therefore did not giveanswers to skeptical (pyrrhonian) arguments which Descartes, according to Curley, beganto consider as the main threat around 1628.

38. Georges Canguilhem, Ideology and Rationality in the History of the Life

Sciences, (Cambridge, MA: MIT Press, 1988), 16. In the Bachelardian history of science,an epistemological break is the transition between two scientific conceptualizations.

39. See, for example, John A. Schuster, "Descartes' Mathesis Universalis": StevenGaukroger, Descartes; William R. Shea, The Magic of Numbers and Motion (Canton,

MA: Watson, 1991).40. A. Boyce Gibson, "The Regulae of Descartes," Mind 1 (1898): 145-158,

332-363.41. Schuster, "Descartes' Mathesis, Universalis," 67.42. For example, Roy Lachman, Janet L. Lachman, and Earl C. Butterfield, Cog-

nitive Psychology and Information Processing: An Introduction (Hillsdale, NJ: Erlbaum,1979), 68, claimed that the concept of coding was not used in psychology before Shannonpublished his information theory in 1948. They overlooked the fact that Descartes, in theRegulae, showed how the ingenium codes and represents symbols in the corporeal brain.In his explanation of sense perception in the Optics, Part V, Descartes also assumed thata kind of (temporal) coding has to take place in the corporeal brain; see Kirkeb0en,"Descartes' Psychology of Vision." Warren S. McCulloch, "A Historical Introduction tothe Postulational Foundations of Experimental Epistemology," in Embodiments of Mind,

ed. W. S. McCulloch (Cambridge, MA: MIT Press, 1965), 359-372, argued, wrongly, thatthis is the first theory of coding in the nervous system. McCulloch also ignored Rule 14of the Regulae.

43. Klein, like Turing, took the symbolic mathematics (and mathematical physics)of the 1930s as his starting point; he wrote, "this study . . . will confine itself to the limitedtask of recovering to some degree the sources . . . of our modern symbolic mathemat-ics . . . . However far afield it may run, its formulation will throughout be determined bythis as its ultimate theme" (p. 4).

44. Sepper, Descartes' Imagination, 129.45. Schuster, "Descartes' Mathesis Universalis," 59.46. According to Gaukroger, Cartesian Logic, the strongest forms of this psychol-

ogized logic are in the texts that Descartes used at La Heche.47. Descartes explicitly defended his mathematical or formal approach to the

cognitive faculties: "But what is to prevent you from following these suppositions if it isobvious that they detract not a jot from things, but simply make everything much clearer.This is just what you do in geometry" (AT412).

48. According to Chikara Sasaki, "Descartes' Mathematical Thought" (Ph.D. diss.,Princeton University, 1989), "the principal part of his mathematical thought had crystal-lised before about 1623" (pp. 4-5), and everybody seems to agree that Descartes workedout Rule 12 in 1626-1628.

49. It is likely that by 1628 Descartes had already worked out his mechanistic

DESCARTES'S REGUIAE AND MODERN PSYCHOLOGY 323

understanding of light (e.g., Schuster, "Descartes' Mathesis Universalis"). Thus, it makessense that he stressed that his mechanical metaphor of sense-perception is not a mereanalogy.

50. Hodges, Alan Turing, 107.51. For a general account of this theory, see A. Mark Smith, "Getting the Big

Picture in Perspectivist Optics," 7m 79 (1990): 69-89.52. Aristotle in De anima, 3.7, 431a 16, translated in Sepper, Descartes' Imagina-

tion, 6.53. See, for example, Sepper, Descartes' Imagination.

54. Avicenna's neuropsychological theory of the inner senses was based on thepsychology of Aristotle and the anatomical discoveries of Galen. See, for example, SimonKemp and Garth J. O. Fletcher, 'The Medieval Theory of the Inner Senses," American

Journal of Psychology 106 (1993): 559-576.

55. See, for example, E. Ruth Harvey, The Inward Wits: Psychological Theory inthe Middle Ages and Renaissance (London: Warburg Institute, 1975).

56. Aquinas in Summa Theologiae la, 84, 6.57. Harvey, The Inward Wits, Aquinas paraphrased on p. 58.58. In particular, the theory of the inner senses assumed that there was a direct

connection between the anterior ventricles and the sensory nerves. Vesalius did not findany such connections. See, for example, Kemp and Fletcher, The Medieval Theory.

59. Harvey, The Inward Wits, Vesalius quoted on p. 30.60. See, for example, Katharine Park, "The Organic Soul," in The Cambridge

History of Renaissance Philosophy, eds. Charles B. Schmitt and Quentin Skinner (Cam-bridge, England: Cambridge University Press, 1988), 464-484.

61. According to Jean-Luc Marion, "Cartesian Metaphysics and the Role of theSimple Natures," in The Cambridge Companion to Descartes, ed. John Cottingham

(Cambridge, England: Cambridge University Press, 1992) 115-139: "[the simple nature]desposes traditional ousia or essence, and banishes it once and for all from modernmetaphysics (despite Leibniz's attempts to bring them back)" (p. 115). Heidegger con-sidered the introduction of the "simple natures" to be the very beginning of modernthinking; see Martin Heidegger, What Is a Thing, trans, by W. B. Barton Jr. and VeraDeutsch (Lanham, MD: University Press of America, 1967).

62. Another probable motivation for this change is his increased interest andknowledge of anatomy around 1628-1629, and his ambition, after 1629, of developing apsychology that was in accordance with this understanding. See also note 37.

63. See Kirkeb0en, "Descartes' Psychology of Vision."64. Ren6 Descartes, "Meditations, Sixth Replies," in The Philosophical Writings of

Descartes, vol. 2, trans. John Cottingham, Robert Stoothoff, and Dugald Murdock(Cambridge, England: Cambridge University Press, 1984), 293.

65. Consult P. J. Olscamp, Discourse on Method, Optics, Geometry and Meteorol-

ogy (Indianapolis, IN: Bobbs-Merrill, 1965) for a translation of the Geometry in English.66. Louis Liard, Descartes (Paris, Bailliere, 1882) was the first to point out that the

possibility of replacing the image-copy notion of sensation by representations mathemat-ically conceived depended on Descartes' specific mathematical achievements. See Kirke-b0en, "Descartes' Psychology of Vision."

67. Descartes explicitly stated that he has changed his critical view on the "pureintellect" when, in 1638, he commented on his theory of the creation of the universe thus:"Only 10 years ago if someone else had written it I would not have been willing to believemyself that the human mind could attain to such knowledge" (Letter to Vatier, 22.2 1638),in The Philosophical Writings of Descartes, vol. 2, trans. John Cottingham, RobertStoothoff, Dugald Murdock, and Anthony Kenny (Cambridge, England: CambridgeUniversity Press, 1991), 86.

324 KIRKEB0EN

68. Thomas Hobbes, Leviathan (1651, reprint, London: Dent, 1973), 8.69. See, for example, Ian Hacking, "Proof and Eternal Truths: Descartes and

Leibniz," in Descartes: Philosophy, Mathematics and Physics, ed. Stephen Gaukroger

(Sussex, England: Harvester Press, 1980), 169-180.70. See, for example, R. McDonough, "Leibniz' Opposition to Mechanistic Cog-

nitive Science," Idealistic Studies 25 (1995): 175-194.71. Lorraine Daston, "Enlightenment Calculations," Critical Inquiry 21 (1994):

192.72. Ibid.73. However, Babbage often stressed the analogy between his Analytic Engine and

new ways of social and work organization. See, for example, Simon Shaffer, "Babbage'sIntelligence: Calculating Engines and the Factory System," Critical Inquiry 2 (1994):203-227.

74. See, for example, Kirkeb0en, "Descartes' Psychology of Vision."75. See, for example, Gary Hatfield, "Remaking the Science of Mind. Psychology

as Natural Science," in Inventing Human Science. Eighteenth-Century Domains, eds.

Christopher Fox, Roy Porter, & Robert Wokler (Berkeley: University of California Press,1995), 184-231.

76. Locke wrote: "I shall not at the present meddle with the physical considerationsof the mind . .. whether those ideas do in their formation, any or all of them, depend onmatter or no. These are speculations which, however curious or entertaining, I shalldecline, as lying out of my way in the design I am upon now." John Locke, An Essay

Concerning Human Understanding (1689; reprint, New York: Meridian, 1964), 63.77. Descartes's rationalism is often described as advocating an a priori and "cer-

tain" scientific methodology. However, the Cartesians in the 17th century were criticizedfor precisely the opposite, that is, the overhypothetical character of their speculativemechanistic theories. See, for example, Larry Laudan, Science and Hypothesis (Dordrecht,the Netherlands: KJuwer Academic Publishers: 1981)

78. David Hume, A Treatise of Human Nature (1740; reprint, London: Penguin,1984), 60.

79. To defend a sharp distinction between the natural, on the one hand, and thehuman soul/intellect and the supernatural, on the other hand, it was necessary forDescartes to demonstrate that all natural phenomena can be accounted for by mechanicaltheories. Thus, the demonstration in "the noblest example of all" that an inert and passivebody and brain might give rise to perception and perceptual cognition probably also hasa religious motivation; see, for example, Gaukroger, Descartes, 147-152.

80. Thomas S. Hall, "On Biological Analogs of Newton Paradigms," Philosophy of

Science 35 (1968): 6-27.81. See, for example, S. Moravia, "From Homme Machine to Homme Sensible:

Changing Eighteenth-Century Models of Man's Image," Journal of the History of Ideas

39 (1978): 45-60.82. Richard M. Warren and Roslyn P. Warren, Helmholtz on Perception: Its

Physiology and Development (New York, Wiley, 1968), Pavlov quoted on p. 18.83. Edwin G. Boring, Sensation and Perception in the History of Experimental

Psychology (New York: Appleton-Century-Crofts, 1942), 167-168.84. Hull and his colleagues wrote several articles on this topic; for example, Clark

L. Hull, "Knowledge and Purpose as Habit Mechanisms," Psychological Review 37(1930): 511-525.

85. Warren S. McCulloch and Walter H. Pitts, "A Logical Calculus of the IdeasImmanent in Nervous Activity," Bulletin of Mathematical Biophysics 5 (1943): 115-133.

86. This is an understanding that goes back to Charles S. Sherrington, The Inte-grative Action of the Nervous System (New York: Scribner's, 1906).


87. See Michael S. Mahoney, "The Beginning of Algebraic Thought in the Seven-teenth Century," in Descartes: Philosophy, Mathematics and Physics, ed. Steven Gauk-roger (Sussex, England: Harvester Press, 1980), 141-155, for a stimulating account ofDescartes's contribution to the change from a geometrical mode of thought to an algebraicmode of thought in mathematics.

88. Barsalou, Perceptual Symbol Systems, p. 524.89. For example, Eva T.H. Brann, The world of the imagination (Savage, MD:

Rowman, 1991); Goldstein and Barsalou, Reuniting Perception and Cognition.90. Stevan Harnad, "The Symbol Grounding Problem," Physica D 42 (1990): p.

335.91. Barsalou, "Perceptual Symbol Systems," 523.92. Goldstone and Barsalou, "Reuniting Perception and Conception," 235.93. Ibid., 236.94. Klein, Greek Mathematical Thought, 197-198.95. Edward S. Reed, "Descartes' Corporeal Ideas Hypothesis and the Origin of

Scientific Psychology," Review of Metaphysics 35 (1982): 731-752, demonstrated thehistorical continuity between Descartes's epistemological hypothesis—or "the corporealideas hypothesis," as he called it—and the rise of a scientific psychology in the 19th

century.

Received May 14, 1998Revision received November 12, 1999

Accepted March 22, 2000 •

Descartes’S Regulae, Mathematics, And Modern Psychology: The Noblest Example of All in Light of Turing’s (1936) On Computable Numbers

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