Thinking Machines: Some Fundamental Confusions

Thinking Machines: Some Fundamental Confusions

JOHN T. KEARNSDepartment of Philosophy and Center for Cognitive Science, State University of New York atBuffalo, Buffalo, N.Y. 14260, U.S.A. (email: kearns @ acsu.buffalo.edu)

Abstract. This paper explores Churchs Thesis and related claims made by Turing. Churchs Thesisconcerns computable numerical functions, while Turings claims concern both procedures for manip-ulating uninterpreted marks and machines that generate the results that these procedures would yield.It is argued that Turings claims are true, and that they support (the truth of) Churchs Thesis. It isfurther argued that the truth of Turings and Churchs Theses has no interesting consequences forhuman cognition or cognitive abilities. The Theses dont even mean that computers can do as muchas people can when it comes to carrying out effective procedures. For carrying out a procedure is apurposive, intentional activity. No actual machine does, or can do, as much.

Key words: Churchs Thesis, Turing machine, effective procedure, effectively computable function.

1. Churchs Thesis

Discussions of the issues of whether computers can, do, or will think are frequentlymarked by unclarity and confusion. When this confusion is removed, there is verylittle left to the issues being discussed. They are then scarcely worth discussing.In this paper I will illustrate (and defend) my position by considering some claimsmade and some results established by Alonzo Church and Alan Turing. Theseclaims and results are all related to Churchs Thesis, which is frequently taken tohave important consequences for our understanding of human cognition. (For anexample of someone claiming such consequences, see Nelson 1987.) The point ofthis paper is to discover just what consequences the work of Church and Turingdoes have for our understanding of human cognition. Does this work either showor suggest that cognition is simply computation of the sort carried out by digitalcomputers? However, while there are connections between Churchs and TuringsTheses and Godels Incompleteness Theorem (one such connection is stated inKleene 1987), the present paper will have nothing to say about Godels Theoremor its consequences.

In school we learn procedures for beginning with certain numerals and, bya series of steps, obtaining others. We learn to add, subtract, multiply, divide,take square roots, etc. The numerals are names of numbers, but we cant get ourhands on the numbers. The procedures that we carry out with numerals give usknowledge of numbers, and of their properties and relations. Numerical proceduresare also the source of our idea of a function. Actual numerical functions may existindependently of us, but they come to our attention through our reflecting on

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numerical procedures. Functions are conceived on the model of procedures, forfunctions do things they take arguments and yield values.

A more sophisticated understanding regards functions as abstract correlations as sets of ordered pairs, perhaps. From this more sophisticated perspective,functions are prior to our numerical procedures; they arent derived from theprocedures. Epistemically, we must proceed from performing procedures to under-standing functions. Ontologically, functions are the objects our procedures aredesigned to explore. Numerical procedures from numerals to numerals merelyreflect the functions which link numbers to numbers.

Let us understand a numerical procedure to be a procedure for coming up withone or more canonical numerical expressions. (Arabic numerals provide the bestexample of canonical numerical expressions. The phrase Johns mothers age inyears on January 1, 1980 is a noncanonical numerical expression.) A procedureis an activity of some kind, but the very idea of a numerical procedure isntvery restrictive. One procedure is, given an input numeral, to write down the firstnumeral that comes into ones head. Another procedure that students sometimesuse for solving numerical problems is to stare at a formulation of the problem untilone simply sees what is the right answer, and then to write this down.

A procedure is goal-directed if it has a goal to which one must attend in carryingit out. Trial and error procedures are typically goal-directed, but not all goal-directedprocedures will allow for false starts. Procedures for solving equations are goal-directed.

Another important kind of numerical procedure takes one or more canonicalnumerical expressions as inputs, and yields a sequence of numerical expressions asoutputs. If the procedure terminates after producing finitely many expressions, thelast expression in the sequence is the procedures result for the given inputs. Oneof these input-output procedures is input governed if each element of the outputsequence is obtained on the basis of the inputs and previous elements of theoutput sequence. Not all input-governed procedures will be goal-directed, but aprocedure could be of both kinds. The input to an input-governed procedure mightspecify the goal to be achieved. (Every input-governed procedure will presumablyhave a goal: at the very least, to get the right output. But some input-governedprocedures can be carried out without attending to the goal.)

If there are rules for carrying out a particular input-governed numerical proce-dure, which rules can be communicated and are such that anyone who correctlycarries out the procedure must produce the same output sequence for the sameinputs (in order), then the input-governed procedure is deterministic. If, further, theprocedure is one which can be carried out in a purely routine matter no additionalinformation must be gathered, no flashes of insight are needed, nor any creativeproblem solving, then the deterministic input-governed procedure is effective.

An effective numerical procedure is one that any competent and knowledgeableperson can simply carry out and, if she exercises due care, get the right answeror answers. But this is in principle, not always in practice. For what we count as

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an effective procedure can be so complicated and time consuming, or demandingso much in the way of pencils and paper that no person can actually carry itout. In considering procedures, we are able without great effort to idealize theseprocedures and consider procedures impossible to perform, which procedures wesimply see to be essentially similar to procedures we really can, and do, perform.We might take this to show that there are ideal, platonic procedures to which wehave cognitive access, some of which are exemplified by our actual calculations.The ideal procedures would have much the same status as numbers, while ourspatiotemporal episodes would be analogues of numerals. However, this platonicaccount raises more questions than it answers. It is sufficient to note that we aresimply able to grasp the principles governing actual procedures, and can then thinkof and investigate procedures never carried out, even procedures that could neverbe carried out, which are governed by those same principles.

An effectively calculable function is one for which there is an effective proce-dure (in the in principle sense) which computes the values of the function. Eventhough functions are mathematical objects, this characterization of effectively cal-culable functions is not a mathematical, or formal, characterization. The functionsare identified on the basis of there being, in principle, a certain way of investigat-ing the functions. Any function which can be identified at all can be identified ininfinitely many different ways (in Shapiro 1980, the author speaks of different pre-sentations of a function). From some characterizations of an effectively calculablefunction, we can immediately come up with an effective procedure for comput-ing the functions values. Other characterizations dont make this possible. Andan effectively calculable function can be identified in a way that doesnt indicatewhether or not the function is effectively calculable.

Although the concept of an effectively calculable function is not a mathematicalconcept, being informal or preformal, this does not mean that the concept is vague,in the logicians sense of vague. A vague concept is one for which there areborderline cases. The criteria of application for the concept are such that someobjects, the borderline cases, cant be determined either to fall under the conceptor not to fall under it. There are reasons for thinking the concept does apply to aborderline case, and reasons for thinking the concept doesnt apply, and the criteriaofficially (conventionally) associated with the concept are insufficient to decidethe matter. A borderline case for a concept isnt a case where it is merely hard totell if the concept applies; it is a case where it is impossible, in principle, to tell.

As I understand concepts, they are a matter of linguistic practice. The criteriaassociated with a concept are determined by the practice (competence as well asperformance) of knowledgeable, or even expert, users of the language. A borderlinecase for a vague concept might be one that competent language users agree to beundecidable on the basis of the criteria associated with the concept. But it can alsohappen that competent speakers disagree about whether certain objects fall underthe concept. The same competent speaker may even reach different conclusions

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about a single object when judging on different occasions. Such disagreement issufficient to mark the disputed objects as borderline cases.

The concept of an effective numerical procedure is informal. Our understandingof the concept depends as much on our mathematical practice as it does on lexicaldefinitions. But there is no reason to think that this concept is vague. We dont haveexamples of functions that we can see to be borderline cases for the concept ofeffective calculability. Nor do we find experts disagreeing about the effective cal-culability of particular functions. The fact that some characterizations of functionsdont allow us to determine whether the functions are effectively calculable is nosign that those functions are borderline cases. No more than my inability to tell ifsome soft yellow metal is gold is a sign that the concept of gold is vague.

I dont believe the concepts of an effective procedure and an effectively calcu-lable function to be vague. The current understanding of what can, in principle,be effectively carried out is sufficient to leave no space for borderline cases. InShapiro 1981, it is suggested that to hold such a view is to adopt a structuralist viewof mathematics and mathematical objects. For the structuralist supposes either thatthere is a mathematical structure common to all mechanical computation devicesor that all minds have a common mathematical structure in virtue of which theyhave the potentiality to grasp and execute algorithms and, therefore, the ability tocompute (p. 355). Even if the denial that the concept of effective calculability isvague doesnt entail a structuralist view of mathematics, Shapiro think structural-ism is the only view currently on the market which accommodates the claim. Imust confess that I dont really understand what a stucturalist view amounts to. Inreading Shapiro, I get the feeling that he isnt using the word structure for thesame things I do. In any case, I dont see that my claim about the nonvagueness ofthe concept of effective calculability has anything to do with the organizations orstructures of either computers or minds.

We develop concepts, and apply them to objects in the world, in order to makesense of things. If a concept is adequate, then it should somehow reflect genuinefeatures of the objects to which we apply it. So our concept of gold should revealfeatures of the actual stuff gold. The gold in the world must possess features whichdistinguish gold from other things, and our concept should reveal at least some ofthose features, enough to allow us to distinguish gold from other stuff. Similarly, ourconcept of an effective procedure must reveal features that an effective proceduremust possess. It provides absolutely no enlightenment to say that the featuresconstitute a mathematical structure. Whatever are the characteristic features of aneffective procedure, they will be features of acts of carrying out the procedure, notfeatures of the agents that carry them out. Even for those procedures which are toodemanding to actually carry out, we think of what it would be like (in idealizedcircumstances) to carry them out. An understanding of the distinctive features ofeffective procedures provides little understanding of what equipment is neededto carry out such procedures.

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Recursive functions can be characterized formally in terms of the expressionsused to present them. The class of recursive functions can also be identified withrespect to a class of effective procedures for obtaining numerical results. The for-mal presentations of the functions are easily construed as directions for carryingout procedures. And recursive functions are known to be the same as Turing-computable function, -definable functions, and the functions definable in H. B.Currys combinatory arithmetic. These last characterizations use effective proce-dures to get at the associated functions. The one class of functions is identifiedin terms of different classes of effective procedures.

Carrying out a procedure is doing something, but functions dont really dothings, though we sometimes speak otherwise. The different characterizations ofrecursive functions in terms of effective procedures can all be transformed intoformal characterizations which represent the relation between the arguments andvalues of these functions. Procedures that people carry out arent mathematicalobjects, and characterizations in terms of procedures arent mathematical charac-terizations. But the formal characterizations of the recursive functions are math-ematical characterizations. The class of recursive functions is a mathematicallywell-defined class of functions.

Churchs Thesis is the claim that this mathematically well-defined class is thesame as the class of effectively calculable functions. Specific concepts of kindsof effective procedures, like the concept of a Turing machine, can be transformedinto formal characterizations/definitions of functions in a straightforward way. Theconcept of an effective-procedure-in-general cannot be similarly transformed. Thiscould be because the concept is inherently vague, but there is no evidence to supportthe claim of vagueness. It could also be because the concept is too general. Thereare more effective procedures than there are effectively calculable functions, inthe sense that each function is captured by infinitely many different procedures.Any formally acceptable definition which characterizes the class of effectivelycomputable functions will correspond to a specific class of effective procedures,not to effective-procedures-in-general.

There are many questions one can raise about Churchs Thesis. Perhaps the mostimportant is whether the Thesis is true. More precisely, what reasons do we haveto think it is true? The second question, which interests me at least as much as thefirst, is a conditional question. If Churchs Thesis is true, what is the importanceof this fact for human cognitive abilities?

2. Turings Thesis

In this section, I will consider results obtained by Turing that will provide us a wayto approach the questions I have raised with respect to Churchs Thesis.

Let us understand a mark to be a single, repeatable, and readily identifiablecharacter, such as a letter in an alphabet or a circle, a square, a star, etc. A markmust be such that we can produce it without (much) difficulty. A manageable system

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of marks is a finite collection of distinct marks mark types whose memberscan be readily distinguished from one another. If the marks are properly produced,different marks in a manageable system must not be hard to tell apart.

A mark manipulation procedure is a procedure for producing a sequence ofstrings or (one-dimensional) arrays of marks belonging to a single manageablesystem. From the standpoint of the procedure, the only features of the marksthat count are their shapes and arrangements. If any of the marks are mean-ingful expressions of a language, their meanings must be irrelevant to carryingout the procedure. A mark-manipulation procedure is goal-directed if a specifiedgoal guides the person carrying out the procedure where the last element of asuccessful output sequence achieves the goal. A procedure for constructing proofsin a deductive system is typical of a goal-directed procedure (except that proofsare characteristically two-dimensional). The results specified in advance are thegoals to be achieved. An input-governed mark manipulation procedure is one inwhich the output sequence is obtained on the basis of inputs. An input-governedmark manipulation procedure can be deterministic and effective in the same waysas an input-governed numerical procedure. An effective mark-manipulation pro-cedure is governed by a finite system of rules/principles such that the procedure isa routine step-by-step procedure, the rules are definite, and it is always determinedwhich rule, if any, applies. The person carrying out an effective procedure onlyneeds to understand the rules and know the inputs to carry out the procedure. Shedoesnt need to know the meanings of any marks that have them. The require-ment that rule-governed mark manipulation procedures be given by nonsemanticrules/principles does not correspond to a requirement on the numerical proceduresconsidered earlier. In carrying out a numerical procedure, it might be necessary tounderstand the significance of the numerals employed.

For certain inputs, an effective mark manipulation procedure may producean output sequence that doesnt terminate. Of course, for anyone who actuallycarries out the procedure, the output sequence will terminate. No one will, or can,continue the output sequence forever. But, in principle, the output sequence mightnot terminate. If, for given inputs, the output sequence does terminate (because it issupposed to), the last string in that sequence is the procedures result for the giveninputs.

A Turing machine can be regarded either (1) as a mark manipulation procedurefor someone to carry out or (2) as a device which operates independently of us.These two ways of thinking about the machine are not equivalent, except withrespect to outputs.(1) If we regard the machine as a procedure, the description of the machine becomesa (somewhat cryptic) statement of rules for a person to follow. For someone to carryout a procedure is for that person to act intentionally, in the on purpose sense. Eventhe person who carries out an input-governed procedure is acting purposively,and has at least the minimal goal of correctly carrying out the procedure. She

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must understand the rules for the procedure, and intend to implement them. Sheintentionally tries to get the right result.(2) Conceived as an independent device, a Turing machine is a spatiotemporalengine which operates according to causal principles. No one actually builds Turingmachines, because they would be so tiresome to work with. But there is no difficultyin making one, only a (possible) difficulty in keeping it supplied with tape. In aphysically realized Turing machine, physical tokens of marks will be causallyeffective in determining just how the device maneuvers and what strings thedevice produces. The physical Turing machine does not follow rules, for causalprocesses are neither purposive nor intentional.

There are two theses that I will associate with Turing, reflecting the two waysof regarding a Turing machine. Turings Procedural Thesis, which I will oftensimply call Turings Thesis, is the claim that every effective mark manipulationprocedure can be carried out, or modelled, by a Turing machine procedure. Tur-ings Mechanical Thesis is the claim that, in principle, physical Turing machinescould be built which would, if sufficient tape were available, causally producethe outputs of Turing machine procedures, given the inputs to those procedures.Turings Mechanical Thesis should not be confused with Thesis M of Gandy 1980.That thesis does not really distinguish between procedures and mechanical causalprocesses. Instead Gandys paper provides still another characterization of a classof effectively calculable procedures and shows that Turing machines are up to thetask of giving the results of such procedures.

I will begin by focussing on the Procedural Thesis. There are two forms of thisthesis, a strong and a weak form. The weak form claims that for every effectivemark manipulation procedure, there is a corresponding Turing machine procedure,which yields results for exactly those inputs for which the manipulation procedureyields results, and the manipulation and the Turing machine procedures yield thesame results. But for those inputs for which the manipulation procedure grindson forever, the Turing machine procedures output sequence need not match theoriginal output sequence.

The strong form of Turings Procedural Thesis is harder to state, but the idea issimple: For every effective mark manipulation procedure, there is a correspondingTuring machine procedure which yields the same output sequences as the manip-ulation procedure, given the same inputs (and the manipulation procedure and theTuring machine procedure take the same inputs). This formulation isnt quite right,because while a mark manipulation procedure, for certain inputs might yield anoutput sequence 1, 2, 3, . . . , the Turing machine procedure which modelsthe manipulation procedure might need to produce a sequence 1, . . ,. , r, 1,

r+1,...., s, 2, s1,..., t, 3,.... The Turing machine procedure will be gov-erned by different principles than the original mark manipulation procedure, andwont produce exactly the output sequence of the manipulation procedure. TheTuring machine procedure which corresponds to the mark manipulation procedure

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will produce output sequences which contain the manipulation procedures outputsin some suitably defined sense of contain.

The strong form of Turings Thesis has been called Churchs Super Thesis byKreisel. In Kreisel 1987, he has awarded credit for the Superthesis to Barendregt.But Kearns 1969, which appeared earlier than the work Kreisel cites in Kreisel1987, makes a claim which is an analogue of the strong Turing Thesis, for a systemof combinatory logic. The claim is not linked to Churchs Thesis in Kearns 1969, butis supported by an argument which employs Turing machines. Turings ProceduralThesis, in either its strong or weak form, is not simply a version of Churchs Thesis,for it concerns effective mark manipulation procedures, not effective numericalprocedures. And it has nothing to say about mathematical functions.

I believe that both forms of Turings Procedural Thesis are true. The reason foraccepting the weak form is that it follows from the strong form, and the strong formis true. But what are the reasons for accepting the strong form? To see what theyare, we must reflect on the concept of an effective mark manipulation procedure.This is probably not a concept that figures in our ordinary thinking and our ordinarypractices, though it is similar to concepts that do figure. In describing a mark manip-ulation procedure, I have taken pains to make clear that the meaningful characterof any marks is irrelevant to the procedure I have carefully avoided using theword symbol in describing mark manipulation procedures. But in our actual prac-tice, the marks we manipulate are ordinarily meaningful expressions, and we takeaccount of their meanings when we calculate with the expressions. However,we have little difficulty in abstracting away from semantic considerations, and weunderstand manipulation procedures where meanings dont matter. Although theconcept of a mark-manipulation procedure is vague in certain respects, our under-standing of the concept together with our familiarity with algorithmic proceduresis sufficient for recognising the truth of the strong form of Turings ProceduralThesis.

The concept of an effective mark manipulation procedure is vague because theconcept of a mark is vague, and so is the concept of a manageable system ofmarks. We can with pens and pencils actually produce shapes which are borderlinecases for the concept of a readily identifiable, easily producible mark. We canalso produce a collection of marks which is a borderline case for the concept of amanageable system of marks. These sources of vagueness dont make the conceptof an effective procedure for manipulating marks vague, though I have relied onour familiarity with calculating and with carrying out algorithms, procedures wherewe do manipulate marks, in identifying effective mark manipulating procedures.Without this familiarity, what I have said would be insufficient to pick out theprocedures. Once we have settled on the marks, our understanding of a procedurefor manipulating them will be somewhat vague, but not our concept of an effectiveprocedure for manipulating them. The concept of a Turing machine procedure isvague in exactly the same places as the concept of an effective mark manipulation

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procedure. Both concepts become much less vague if we require the procedures toemploy only marks which constitute ordinary letters and numerals.

The way we can tell that the strong form of Turings Procedural Thesis is true isby reflecting on our own practices in manipulating marks. What sort of proceduresdo we carry out? What sort can we carry out? We can look at our own practices anddetermine that each step that we might make in carrying out a procedure can berealized by a Turing machine procedure, though what we accomplish in one stepmight take several machine procedure steps. While it may be that all the moves wecan think of now can be realized by a Turing machine procedure, someone mightworry that we could be forgetting, or overlooking, some moves we are capable of.But if we were, this surely would have been noticed by now. An analysis of ourpractice supports the strong form of Turings Thesis and, of course, Turing basedhis description of his machine procedures on an analysis of our practices.

Another, somewhat far-fetched, worry about Turings Thesis might be raised asfollows. Let it be granted that Turing machine procedures are capable of realizing(or modelling) what we now conceive to be effective procedures for manipulatingstrings of marks. In the future, people might adopt new practices, and classify newactivities carried out with marks together with our present procedures. With respectto the reconceived manipulation procedures, it might no longer be true that all suchprocedures that would then be considered to be effective are realizable by Turingmachine procedures. In the unlikely event that something of the sort happened,it wouldnt undermine Turings Thesis. That Thesis (those Theses) should beunderstood with respect to procedures as they are now conceived and carried out.If it is true with respect to such procedures, it is simply true once and for all. Formark manipulation procedure are not a natural kind which can turn out to containnew and unexpected instances. What it is to be a mark manipulation procedureentirely depends on our concept of these procedures. Should we reconceive what isa procedure for manipulating marks, Turings Procedural Thesis will not concernthe new procedures. After such a shift, our present procedures may no longerhold any interest. Turings Procedural Thesis will be equally uninteresting but itwont be untrue. Of course, someone could choose to maintain Turings Thesis as aprediction about the future as well as a claim about the present, or as a claim aboutwhat is humanly possible. These claims are unreasonable. Whether or not we havegone about as far as we can go in operating with marks, Turings Procedural Thesisresults from an analysis of our current practices, and owes its significance to thesepractices.

My argument for the strong form of Turings Procedural Thesis has, in effect,construed this claim as an a priori truth. For a priori truths are those which reflectfeatures of our concepts and conceptual frameworks, no matter how these truthsare discovered or supported. (It is an a priori truth that 4 + 3 = 7 even for thecalculator who counts on his fingers.) It is an empirical fact that a certain person orcommunity has adopted a certain conceptual framework, and that this frameworkhas whatever features it possesses. But statements made from inside a framework,

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which statements merely reflect features of the framework, are a priori. Theyare only a priori with respect to their conceptual framework, but I think this isthe case with all a priori statements. Mark manipulation procedures are activitiesconstituted (in part) by our concepts of them. For such procedures, unlike activitieson the order of breathing and digesting food, a person must have some idea of whatshe is doing in carrying them out. Our manipulation procedures and the conceptswe use to define and characterize them are constitutive of a conceptual framework.The claim that all mark manipulation procedures can be realized by Turing machineprocedures reflects features of this conceptual framework. The claim is a prioriwith respect to this framework.

We can imagine some beings who employ concepts and use language, but areintellectually much less well endowed than we are. These beings are either inca-pable of carrying out mark manipulation procedures or else they can carry out onlya limited class of the procedures we are capable of carrying out. (It is hard to makethis example concrete for what, exactly, might it be that they couldnt do? ) Thoseless gifted beings wouldnt have our concept of an effective mark manipulationprocedure. They wouldnt be in a position to assert, or even to understand, Tur-ings Thesis. Since the whole range of mark manipulation procedures would notbe accessible to them, Turings Thesis wouldnt be relevant to their practice, but itwould still be true.

Similarly, we can imagine intelligent agents vastly more gifted than we are.These agents still have some use for operating with uninterpreted marks, but,except in their capacity as anthropologists of us, they have no interest in our classof effective mark manipulation procedures. The Turing Thesis would have nointerest or importance with respect to their own practices, though it would be true.Turings Procedural Thesis is not about our abilities. It concerns our practices andconcepts. Although these certainly have important connections to our abilities, wecan investigate our practices and concepts without raising (or being able to answer)the question whether these are the most powerful practices and concepts anyonecould ever employ.

However, this last example of the more intelligent agents seems too far in therealm of science fiction to be credible. It strikes me as unreasonable to suggest thathighly intelligent agents might be capable of carrying out operations with marksthat are essentially different, and much more powerful, than our mark manipulationprocedures. Unless those procedures are literally inconceivable to us, they mustrequire abilities which we can conceive but do not possess. It wouldnt be enoughfor the highly intelligent agents to simply do more things at a time than we can,or work faster. It seems that those agents must possess some infinite abilities. Butthere is no telling what agents with infinite abilities would be able to carry out,or would want to carry out. As far as finite agents go, agents who have someinterest in manipulating marks in algorithmic fashion, effective mark manipulationprocedures are just the ticket. And Turing has provided a convenient and adequatecharacterization of these procedures.

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Turings Procedural Thesis is an a priori truth, but Turings Mechanical Thesis(which also has weak and strong versions) is not. However, Turings MechanicalThesis is true. The Mechanical Thesis claims that for every effective mark manip-ulation procedure, we can, in principle, build a physical device which, by causalprocesses, produces the outputs that we would get (in principle again) by carryingout the procedure. It is our knowledge of causal processes, and of our manufactur-ing abilities that allows us to determine that Turings Mechanical Thesis is true. Itis this knowledge together with the knowledge that Turings Procedural Thesis istrue.

Machines that operate causally dont act intentionally, they dont strive to realizepurposes and goals of their own. But Turings Mechanical Thesis assures us that wecan exploit causal processes to realize calculational purposes and goals of our own.We can (we do) build machines that save us the time and trouble of actually carryingout lengthy and involved mark manipulation procedures. The machines dont carryout the procedures for us, because to carry out a procedure is to act intentionally.But the machines produce the right answers for us. However the answers are onlyright with respect to the procedures; they are the answers we would get if we didcarry out the procedures. Whatever a machine turns out belongs to the causalorder; the causal order has no answers at all, certainly no right or wrong answers.

3. The ChurchTuring Thesis

Turings Procedural Thesis is not a numerical or arithmetic claim. It concernsprocedures for manipulating arbitrary, uninterpreted marks. A mark manipulationprocedure can manipulate numerals as well as other marks, but the significance ofthe numerals is not a factor in carrying out the procedure. A mark manipulationprocedure, whether or not it manipulates numerals, can be modelled mathemat-ically. The marks and strings of marks can be assigned numbers, and functionsfrom numbers to numbers will correspond to procedures from mark strings to markstrings. That mark manipulation procedures can be modelled mathematically doesnot show them to be mathematical, no more than the fact that there are mathematicalmodels of weather shows the weather to be mathematical.

Churchs Thesis and Turings Procedural Thesis are distinct. Neither entailsthe other, at least, neither obviously entails the other. Turings Thesis concernsprocedures for operating with uninterpreted marks, and can be determined to be trueby reflection on our concepts and practices. Churchs Thesis deals with numericalprocedures, with procedures for beginning with certain numerals and coming upwith sequences of numerals. But for numerical procedures, there is no requirementthat the rules/principles for carrying out the procedures be nonsemantic. A numeralis a kind of mark, but it is a meaningful mark, and a numerical procedure mightrequire the agent who carries out the procedure to understand the numerals as wellas understanding the rules/principles of the procedure. Churchs Thesis is not a

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claim in mathematics, but it is a claim about mathematics and those numericalfunctions we can effectively calculate.

Churchs Thesis and Turings Procedural Thesis together entail what I will callthe ChurchTuring Thesis: For every effective numerical procedure, there is acorresponding effective mark manipulation procedure. The effective manipulationprocedure, for the same numerical inputs, yields exactly the same results as thenumerical procedures. This is the weak form of the ChurchTuring Thesis. Thestrong form claims that for every effective numerical procedure, there is a corre-sponding effective mark manipulation procedure whose output sequences contain,in some suitably defined sense of contain, the output sequences of the numericalprocedure (for the same inputs). The strong form of the ChurchTuring Thesis doesnot claim that every effective numerical procedure is an effective mark manipula-tion procedure, for it may be that the effective numerical procedure requires theperson carrying it out to take account of features of the numbers designated bynumerals which figure in the procedure. But every effective numerical proceduremight as well be an effective mark manipulation procedure. The weak form claimsthat whenever we use an effective numerical procedure to obtain results, we couldas well use an effective mark manipulation procedure to get those results.

Given that both the strong and weak forms of Turings Procedural Thesis are truea priori, the strong and weak forms of the ChurchTuring Thesis are consequencesof Churchs Thesis. The weak form is an obvious consequence, for it is well-known that Turing machine procedures are adequate for calculating the values ofrecursive functions. For every effectively calculable numerical function, there isa mark manipulation procedure in the form of a Turing machine which does thecalculating. An argument showing that the strong form also follows is found inthe appendix to this paper. Turings Thesis together with Churchs Thesis entailsthe ChurchTuring Thesis. And Turings Thesis together with either form of theChurchTuring Thesis entails Churchs Thesis. So, having determined that TuringsThesis is true, we conclude that Churchs Thesis is equivalent to the ChurchTuringThesis. (A list of the various theses is found at the end of the present paper.)

Before considering whether Churchs Thesis or the ChurchTuring Thesis istrue, before I argue that it is true (that they are true), it will be helpful to considerwhat its truth would show about us and our cognitive abilities. That Turings Thesisis true affords us very little insight into the workings of human cognition. TheThesis does not reveal a limitation of human abilities. It does not even suggest that,when carrying out effective mark manipulation procedures, people are operatingthe way computers do. Computers dont, and cant, carry out procedures, becausecomputers dont act purposively. To carry out a procedure is to act intentionally,in order to achieve a purpose, trying to get things right. The causal processeswhich completely determine the machines operation are not processes in whichthe machine tries to do anything. It simply isnt possible to program intentional orpurposive activity.

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However, while causal processes dont have goals, it is possible to intentionallyexploit causal processes to achieve goals of our own. The character of effective markmanipulation procedures is such that, given our knowledge of causal processes,we are able to devise machines which produce, for the same inputs, the outputs wewould obtain by carrying out the procedures. This is Turings Mechanical Thesis.Because it is true, such machines will be valuable to us, to the extent that we findmark manipulation procedures either useful or important.

Now suppose that the ChurchTuring Thesis is true. It is clear that the Thesisdoesnt entail anything about the way we operate in carrying out numerical pro-cedures. But if the ChurchTuring Thesis is true, its truth should be something ofa surprise. For one would think that a numerical procedure requiring us to under-stand the numbers designated by numerals is inherently more "demanding" thana procedure specified in terms of marks and their arrangement. The truth of theChurchTuring Thesis calls for some explanation. Why is it that we could alwaysget the right numerical answers without thinking about numbers? The simplestanswer would be that effective numerical procedures simply are mark manipula-tion procedures. That when someone carries out a numerical procedure and thinksabout the meanings of the symbols, this thinking about meanings is either inci-dental to the mark manipulation procedure she actually carries out or else thinkingabout the meanings of the symbols is nothing other than paying attention to thesymbols themselves while following the rules for a mark manipulation procedure.This proposal has the consequence that a person who carries out a numerical pro-cedure and thinks that she is thinking of the objects these symbols denote doesntreally know what she is doing. The absurdity of this consequence overthrows theproposal. Who is in a better position than the person who carries out the procedureto know what she is doing?

One reason why someone might find this first proposal plausible is that theidea of grasping a meaning attached to the expression is obscure. The idea ofmanipulating uninterpreted marks according to definite rules is much clearer, and somakes an attractive substitute for dealing with marks plus meanings. But meaningsarent stuck on to words, just waiting to be digested by minds. It is rather thatpeople use expressions to perform various kinds of meaningful acts. A simple kindof meaningful act is performed by using an expression to think about, or identify,a particular object. I find that I do this when adding a column of numerals, andwhen multiplying with numerals. Of course, numerical notation is important. Icant easily add a column of Roman numerals. But when I add a column of Arabicnumerals, I first run my eyes down the right-most digits, keeping a running total inmy head. Then I write down the rightmost digit of the numeral for the right-columntotal, and etc. I use the tokens of numerals to attend to, to think of, the quantitiesinvolved. I am not simply running my eyes over shapes which I notice. My addingis a numerical procedure but not a mark manipulation procedure.

The first attempt to explain the truth of the Church-Turing Thesis would reduceone kind of procedure to another, so that numerical procedures become a special

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282 JOHN T. KEARNS

class of mark manipulation procedures. A second attempt is much more radical.This explanation for the truth of the Thesis is that whenever we carry out aneffective numerical procedure, the nonsemantic features of the expressions weemploy are causally effective in producing our behavior, even though it doesntseem that way to us. The procedure one thinks she is carrying out is so muchwindow dressing for the underlying causal processes. This proposal explains thecoincidence involved in the Church-Turing Thesis by appealing to a third factorcommon to both numerical procedures and mark manipulation procedures.

This second attempt at an explanation is more preposterous than the first. It hasthe consequence that none of us knows what she is doing in carrying out procedures the consequence that we are all wrong about what we are doing. When we thinkwe are acting in order to obtain either a result or an output sequence, being carefulto follow the rules, rules which we understand, we are actually being drivenby causal processes which involve neither rules nor understanding, and whichhave no goals to realize. This explanation isnt even suggested by the truth of theChurchTuring Thesis. Whatever plausibility it might seem to have must be due tomisunderstanding the similarities between effective mark-manipulation proceduresand mechanical causal processes.

The word cause is commonly used in connection with explanations of variouskinds. I will speak of the mechanical causal order when I wish to restrict attention tothe causality that figures in the natural sciences. Historians limit the application ofmechanical to outdated views which make causality a matter of pushes and pulls.But our own understanding of the world has developed from those beginnings, andI use the adjective mechanical more broadly, to include the understanding of theway the world works that has been developed from the sixteenth century to thepresent. A broadly mechanical understanding of causality can be contrasted withAristotelian views, among others. It is characteristic of this mechanical understand-ing that nature operates according to uniform principles, that the past drives thepresent and the future, and that the causal order pursues no goals, and achieves nopurposes proper to that order.

Our conception of the mechanical causal order is derived from our understandingof causal explanations. The explainers in a mechanical causal explanation providelogical support to the description of the event to be explained. If the causalityis deterministic, the explainers entail the description. Less conclusive support isprovided in cases of statistical causality. Given objects of certain kinds in specifiedsorts of circumstances, it is essential or probable that they behave in certain ways.They wouldnt be of those kinds if they behaved otherwise. But the objects arenttrying to behave like that, they just do behave that way. Our conception of deter-ministic causality appears to be further informed by our understanding of effectivenumerical procedures. Any procedure is a purposive activity which can be donecorrectly or incorrectly. But if we abstract from its purposive character, an effec-tive procedure provides a nice appreciation of the inevitability of a deterministicprocess.

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There are important similarities between effective procedures in general andmechanical causal processes. There are even more similarities between effec-tive mark manipulation procedures and causal processes, for the procedures takeaccount of the same kind of mark features that are causally effective in marktokens. However, in carrying out the procedure, one must recognize the marks andfollow the rules of the procedure, in order to produce the proper outputs. Causalprocesses dont recognize anything, follow rules, or act in order to achieve a pur-pose. An effective mark-manipulation procedure is input-governed, and the inputsprecede outputs like causes precede effects. But causes produce their effects, whileinputs require cooperation from people performing intentional actions to yield theiroutputs.

In a causal process, what happened in the past drives, and determines, the presentand the future. Although a causal process is one in which the past unfoldsinto the present, the process has no purpose or goal. Purposive acts introducegoals, and the distinction between success and failure. Intentional acts are self-conscious purposive acts in which the agent has some conception of what she isdoing. With intentional acts we get the distinction between doing something rightor doing it wrong. Causal processes dont succeed or fail, nor are they corrector incorrect. It is sometimes claimed that the causal and purposive orders canbe reconciled in such a way that a single event can correctly be given both acausal and a purposive explanation. But many attempts at reconciliation reduceapparently purposive activity to causal transactions, leaving nothing to be explainedpurposively. This issue calls for a fuller discussion than can be accommodated inthe present paper. For now I will simply record my view that while a personcan intentionally exploit causal processes, with respect to a single event, causaland intentional explanations are incompatible. Some events and activities haveirreducibly purposive explanations. I have provided a longer treatment of this issuein Kearns 1996. However, I think it is intuitively plausible that causal and purposiveexplanations are incompatible alternatives. A causal explanation makes sense ofan event in terms of what has gone before, while an intentional explanation makessense of an act in terms of what is supposed to come after.

If the ChurchTuring Thesis is true, a sufficient explanation of this fact canbe obtained by appealing to the complexity of strings of marks (by appealing tostring theory). Strings of marks can be used to exemplify, or model, arithmetic.We can, for example, pick a first string and then devise a mark manipulationprocedure that, for the nth string as input, yields the n+1st string as its result. Forevery effective numerical procedure, there is an exactly corresponding effectivestring manipulation procedure. If the ChurchTuring Thesis is true, this doesntreveal a limitation of our abilities, nor does it reflect the secret causal source of ourcalculated results. It simply shows that the complexity of numbers and numericalrelations is reflected in the complexity of strings and their relations.

This explanation for the possible truth of the ChurchTuring Thesis can beturned around to constitute an argument for that Thesis. We use an effective numer-

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284 JOHN T. KEARNS

ical procedure to move from numerical inputs to numerical output sequences. Andwe know that strings of marks are adequate to model arithmetic. So for everyfeature of numbers that is considered in carrying out the numerical procedure,there will be a corresponding nonsemantic feature of numerical expressions. Anynumerical procedure can be transformed to obtain a mark manipulation procedurewhich comes up with the same numerals. The ChurchTuring Thesis is true. Butthen, since Turings Thesis is true, Churchs Thesis is true. This is not so much afact about us, pure and simple, as it is a fact about our procedures. These proceduresmight be the only ones we can use, or the best ones we can use, but knowing thatwe do use them isnt sufficient to support the stronger claims.

4. Conclusions

Any attempt to use the theses discussed in this paper to support claims that peoplefunction like machines operate, or claims that computers can do much the samething that people do in calculating or solving problems must be thoroughly con-fused. Two confusions are prominent in this respect. The first is associated with thenonsemantic character of mark manipulation procedures with the fact that suchprocedures take no account of the meanings of the marks they work with. Someonein the grip of this confusion is likely to claim that there is nothing to meanings overand above syntactic considerations. Understanding meaning is simply knowinghow to manipulate uninterpreted marks and sounds according to various rules. Butthe fact that effective numerical procedures can be captured by nonsemantic markmanipulation procedures tells us nothing about the use of language in general. It iscompletely implausible that everything we say and write results from our carryingout effective procedures. If we did use effective procedures to come up with theutterances we produce, we would need to have a genuine understanding of therules for these procedures, and we would be acting purposively and intentionallyin carrying them out.

The second, and more serious, confusion identifies effective procedures withcausal processes. Effective numerical procedures are an important part of ourrepertoire for understanding causal processes. Our mathematical knowledge andmathematical techniques are essential to our conception of the world and thecausal order. Numerical procedures figure both in giving causal explanations and inmaking precise predictions. The truth of the ChurchTuring Theses shows that notonly can we use numerical procedures to understand the world, but we can exploitcausal processes to give us the results of numerical procedures. (We can also usecausal processes to give us the results of other kinds of effective procedures.) Butit makes no sense to think that the link between causal processes and effectiveprocedures shows or suggests that they are really the same thing. Once werecognize the differences between causal and purposive/intentional explanations,or the differences between the causal and purposive/intentional orders, there shouldbe no temptation to identify one with the other.

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To make such an identification, one must either dismiss the idea that peopleperform intentional actions, or else must claim that the same events can (sometimes)be (correctly) given both causal and intentional explanations. But to claim thatpeople dont act intentionally is itself to act intentionally: it is to intentionallydismiss what is intentional. And claiming that an event can properly be given bothkinds of explanation overlooks the conflicting characters of causal and purposiveexplanations. Aristotle thought a single event could have both an efficient and afinal cause, but he didnt conceive of efficient causes in the modern sense. Giventhe modern conception of causal processes and the causal order, a completelysuccessful causal explanation preempts purposive explanations, leaving them noroom to operate.

The various theses show that computers will be useful for obtaining the resultsof calculation, without carrying out the calculations. This consequence tells usvirtually nothing about human cognition or cognitive abilities. One might wonderif there isnt more to be learned from the theses. For effective procedures are animportant part of our repertoire for understanding and manipulating the world. Thetheses identify an upper limit to these procedures, and show that causal processes areadequate to give us the answers at which these procedures aim. An understandingof the connection between Turings Procedural and Mechanical Thesis can at leastsuggest hypotheses about the way we work. Not about operations taking place whenwe consciously carry out effective procedures, for those operations are accessible toour scrutiny. But it is certainly the case that we exploit causal processes, processesof which we arent consciously aware, in functioning and acting. Machines areartifacts designed to serve our purposes, so we arent machines. But our use ofmachines to give us the answers that would be obtained by calculation may resembleour unconscious use of causal processes to give us answers we use in carryingout routine activities. However, even when causal processes take place inside usrather than in some external device, they remain the very opposite of intentionalprocedures.

5. The Various Theses

CHURCHS THESISEvery function whose value can be calculated by an effective numerical procedureis a recursive function.

TURINGS PROCEDURAL THESISWeak Form For every effective mark manipulation procedure, there is a TuringMachine procedure which, for the same inputs, yields the same results.Strong Form For every effective mark manipulation procedure, there is a TuringMachine procedure which, for the same inputs, yields output sequences whichcontain the output sequences obtained by the mark manipulation procedure.

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286 JOHN T. KEARNS

TURINGS MECHANICAL THESISWeak Form For every effective mark manipulation procedure, one can (in principle)construct a physical Turing Machine such that tokens of the inputs to the procedurecause the machine to produce tokens of the procedures results.Strong Form For every effective mark manipulation procedure, one can (in prin-ciple) construct a physical Turing machine such that tokens of the inputs to theprocedure cause the machine to produce sequences of tokens, which sequencescontain tokens of the procedures output sequences.

THE CHURCH-TURING THESISWeak Form For every effective numerical procedure, there is an effective markmanipulation procedure which yields the same results for the same inputs.Strong Form For every effective numerical procedure, there is an effective markmanipulation procedure which, for the same inputs, yields output sequences whichcontain the output sequences obtained by the numerical procedure.

6. Appendix

Given that Turings Procedural Thesis is true, to show that Churchs Thesis impliesthe strong form of the ChurchTuring Thesis to show that if is an effectivenumerical procedure, then there is an effective mark manipulation procedure which, for the same inputs, yields output sequences which contain the outputsequences obtained by (carrying out) .

Proof Let take inputs i1, i2, . . . and, for these inputs, yield output sequencesS1, S2, . . . These sequences may be finite or infinite.

Let be a numerical procedure which, for inputs ij

, k, where k is a numeral 1, yields a result r

k

which is either one greater than the kth element of Sj

, or is 0 ifSj

has fewer than k elements. is carried out by carrying out for ij

, as far as thekth member of S

j

, then adding 1 and stopping; and supplying a 0 if Sj

terminatesbefore a kth element is reached. So is an effective numerical procedure.

By Churchs Thesis, presents a recursive function F. But then there is aTuring machine M which calculates F. M is an effective mark manipulationprocedure. Since we have an effective mark manipulation procedure which yields

s results for inputs ij

, k, we can devise another mark manipulation procedure which causes M to obtain the sequences S1, S2,... for inputs i1, i2, . . . For agiven input i

j

, will first generate a numeral 1, and use M to get the result forinput i

j

, 1. After getting the result, subtracts 1. Then generates 2 and usesM again. Etc. stops whenever M first yields a 0 for an input i

j

, k.

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ReferencesChurch, Alonzo (1936), An Unsolvable Problem of Elementary Number Theory, American Journal

of Mathematics 58, pp. 345363. Reprinted in Davis 1965.Curry, Haskell B. and Robert Feys (1958), Combinatory Logic, vol 1. Amsterdam: North-Holland

Publishing Company.Davis, Martin (1965), The Undecidable. Hewlett, New York: Raven Press.Gandy, Robin (1980), Churchs Thesis and Principles for Mechanisms, The Kleene Symposium,

edited by Jon Barwise, H. Jerome Keisler, Kenneth Kunen. Amsterdam: North-Holland PublishingCompany, 123148.

Kearns, John T. (1969), Combinatory Logic with Discriminators, The Journal of Symbolic Logic34, pp. 561575.

Kearns, John T. (1996), Reconceiving Experience, A Solution to a Problem Inherited from Descartes.Albany, N.Y.: State University of New York Press.

Kleene, Stephen C. (1987), Reflection on Churchs Thesis, Notre Dame Journal of Formal Logic28, pp. 490498.

Kreisel, Georg (1987), Churchs Thesis and the Ideal of Informal Rigor, Notre Dame Journal ofFormal Logic 28 pp. 499519.

Nelson, R. J. (1987), Churchs Thesis and Cognitive Science, Notre Dame Journal of Formal Logic28, pp. 581614.

Shanker, S. G. (1987), Wittgenstein versus Turing on the Nature of Churchs Thesis, Notre DameJournal of Formal Logic 28, pp. 615649.

Shapiro, Stewart (1980), On the Notion of Effectiveness, History and Philosophy of Logic I, pp.209230.

Shapiro, Stewart (1981), Understanding Churchs Thesis, Journal of Philosophical Logic 10, pp.353-365.

Turing, Alan (1937), On Computable Numbers, with an Application to the Entscheidungsproblem,Proceedings of the London Mathematical Society s. 2, vol 42, 230265. Reprinted in Davis(1965), 115154.

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