Mechanism as a theory of mind would be refuted if it could be shown that a human being (or his mind)* can do what no machine can do, and there is a family of arguments invoking Gödel’s Theorem which purport to prove just that.1 I wish to show that all these arguments must fail because at one point or another they must implicitly deny an obvious truth, namely that the constraints of logic exert their force not on the things in the world directly, but rather on what we are to count as defensible descriptions or interpretations of things. The common skeleton of the anti-mechanistic arguments is this: any computing machine at all can be represented as some Turing machine,† but a man cannot, for suppose Jones over there were a realization of some Turing machine TMj, then (by Gödel) there would be something A that Jones could not do (namely, prove TMj’s Gödel sentence). But watch!—this is the crucial empirical part of the argument—Jones can do A; therefore Jones is not a realization of TMj, and since it can be seen that this will be true whatever Turing machine we choose, Jones transcends, angel-like, the limits of mechanism. The error in this lies, I will argue, in supposing that the determination of Jones’s acts and hence his abilities, and also the determination of the activities of a computing device, can proceed in a neutral way that will not beg the question of the applicability of Gödel’s Theorem.
Gödel’s Theorem says that in any consistent axiom system rich enough to generate the arithmetic of the natural numbers, there are statements we cannot prove in the system, but which can be seen by other means to be true. Gödel’s Theorem, then, is about axiom systems, and the “machines” it governs are as abstract as the axiom systems. A Turing machine can be viewed as nothing more than a finite system of instructions to perform simple operations on strings of symbols which constitute the “input.” The instructions are gathered into “machine states,” each of which is a finite sequence of instructions, and a master instruction, or state-switching function, which prescribes which sequence of instructions is to be followed given the input. Such a specification is obviously entirely neutral about how such operating and switching is to be accomplished, and hence a particular Turing machine can be “realized” in very different ways: by a mechanical tape-reading device, by simulation on a digital computer, or by “hand simulation,” where the operations are performed by a person or persons following written instructions on “state” cards.
The engineer setting out to construct directly a mechanical realization of a Turing machine has the following task: he must exploit the laws of nature in such a way as to achieve the regularities prescribed or presupposed in the Turing machine specification. Thus, for each of the symbols in the alphabet in which the input and output are to be expressed, the engineer must devise some physical feature that can be reliably distinguished mechanically from its brethren (like a hole in a paper tape and unlike a pencil mark or spoken word). These features should also be relatively stable, quickly producible and small scale—for obvious reasons of engineering economy. Paired with the symbol features must be the devices, whatever they are, that react differently to the symbol features, that “read” or “discriminate” them. Then, whatever these different reactions are, they must in turn be so designed to differ from one another that the next rank of effectors, whatever they are, can be caused to react differently to them, and so forth. For each shift of state there must be a corresponding physical shift, which might be a shift of gears, or the sliding of a different cam onto a drive shaft, or the opening and closing of series of electrical or hydraulic relays. The whole machine may exploit only a few simple principles of electronics or it may be a Rube Goldberg contraption, but in either case it must be more or less insulated from the rest of the environment, so that coincidental features of the outside world do not interfere with its operation, e.g., changes in temperature or relative humidity, or sudden accelerations. The better the design, the more immune to interference the machine will be. But what counts as interference, and what counts as a physical change “read” as input by the machine is relative to the designer’s choice of physical laws. A hole in the tape may be a symbol to one machine, a major disruptive event to another (depending usually on whether we are speaking of paper tape or magnetic recording tape). Similarly, what internal changes in the machine are to count as state changes, and what are to count as breakdowns is also relative to the designer’s scheme of realization. If we discover a machine that is drastically affected by accelerations, it may be a computer poorly designed for mobile applications, e.g., in airplanes, but on the other hand it may be an inertial guidance system, and the accelerations may be its input.
Since the choices engineers actually make when designing hardware are a fairly standard and well-known lot, and since the purposes of machines are usually either obvious or suitably announced by the manufacturer, it is easy to overlook this relativity to the designer’s choices and suppose that we can directly observe the input, output, operations and state changes of any device, and hence can settle in an objective fashion which Turing machine, if any, it is. In principle, however, we cannot do this. Suppose Jones and Smith come across a particular bit of machinery churning away on a paper tape. They both study the machine, they each compile a history of its activity, they take it apart and put it back together again, and arrive finally at their pronouncements. What sorts of disagreements might there be between Jones and Smith?
First we might find them disagreeing only on the interpretation of the input and output symbols, and hence on the purpose or function of the Turing machine, so that, for instance, Jones treats the symbol-features as numbers (base two or base ten or what have you) and then “discovers” that he can characterize the Turing machine as determining the prime factors of the input numbers, while Smith interprets the symbol features as the terms and operators of some language, and has the Turing machine proving theorems using the input to generate candidates for proof sequences. This would not be a disagreement over which Turing machine had been realized, for this is purely semantic disagreement; a Turing machine specification is in terms of syntactic relationships and functions only, and ex hypothesi Jones and Smith agree on which features are symbols and on the rules governing the production of the output strings. In principle a particular Turing machine could thus serve many purposes, depending on how its users chose to interpret the symbols.
More interesting and radical disagreements are also possible however. Jones may announce that his device is TMj, that its input and output are expressions of binary arithmetic, and that its function is to extract square roots. However, let us suppose, he proves mathematically (that is, on the basis of the machine table he assigns it and not the details of engineering) that the program is faulty, giving good answers for inputs less than a hundred but failing periodically for larger numbers. He adds that the engineering is not all that sound either, since if you tip the machine on its side the tape reader often misreads the punched holes. Smith disagrees. He says the thing is TMs, designed to detect certain sorts of symmetries in the input sequences of holes, and whose output can be read (in a variation of Morse Code) as a finite vocabulary of English words describing these symmetries. He goes on to say that tipping the machine on its side amounts to a shift in input, to which the machine responds quite properly by adjusting its state-switching function. The only defect he sees is that there is one cog in the works that is supposed to be bent at right angles and is not; this causes the machine to miscompute in certain states, with the result that certain symmetries are misdescribed. Here there is disagreement not only about the purpose of the machine, or the semantics of the language it uses, but also about the syntax and alphabet. There is no one-to-one correspondence between their enumerations of symbols or instructions. The two may still agree on the nature of the mechanism, however, although they disagree on what in the mechanism counts as deliberate design and what is sloppiness. That is, given a description of the physical state of the machine and the environment, and a physical description of the tape to be fed in, they will give the same prediction of its subsequent motions, but they will disagree on which features of this biography are to be called malfunctions, and on which parts of the machine’s emissions count as symbols. Other sorts of disagreement over interpretation are possible in principle. For instance, one can treat any feature of the environment as input, even in the absence of any salient and regular reaction to it by any part of the machine, if one is prepared to impute enough stupidity to the designer. There is no clear boundary between entities that count as imperfect or broken or poorly designed realizations of TMx and entities that are not at all realizations of TMx. By the same token, discovering that an entity can be viewed as a highly reliable, well-designed TMa does not preclude its also being viewed as an equally good realization of some other TMb. To give a trivial example, almost any good computer could be construed as a Turing machine yielding as output “p” if and only if it receives as input “q,” where our symbol “p” is realized by a very faint hum, and “q” by turning on the power switch.
Faced with the competing interpretations of the tape reader offered by Jones and Smith, if we decide that one interpretation, say Jones’s, is more plausible all things considered, it will only be because these things we consider include our intuitions and assumptions about the likely intentions and beliefs of the designer of the machine. The quick way to settle the dispute, then, is to ask the designer what his intentions were. Of course he may lie. So it seems we may never find out for sure; only the designer knows for sure, but what is it he knows that we do not? Only what his intentions were, what Turing machine he intended to realize—and he may even discover that his own intentions were confused. In any case, what the designer’s intentions were in his heart of hearts does not determine any objective fact about the device before us. If Smith purchases it on the spot and proceeds to use it as a symmetry classifier, then what he has is just as truly a symmetry classifier as any he could build on his own, under his own intentions, inspired by this prototype. If we find something on the beach and can figure out how to use it as a TMb, then it is a TMb in the fullest possible sense.
Now how does this affect the possibility of there being living Turing machines? It is not at all far-fetched and is even quite the vogue to suppose that an animal can profitably be viewed as a computer or finite automaton of a special sort, and since any finite automaton can be simulated by a Turing machine, this amounts in a fashion to the supposition that we might want to treat an animal as a Turing machine. (There are difficulties in this, but let us concede to the anti-mechanists the shortest visible route to their goal; it will only hasten their demise.) The question then is, can we settle whether or not we have chosen the correct Turing machine interpretation of a particular animal? First we have to decide which of the impingements on the animal count as input and which as interference, and it is not at all clear what criteria we should use in deciding this. Suppose we ask ourselves if changes in barometric pressure constitute input or interference for a particular animal. Probably in most animals we will find detectable, salient physical changes associated with changes in barometric pressure, but suggestive as this might seem, what would it show? Suppose we learn that the effect of such pressure changes in cows is to make them all lie down in the field. So what? What advantage, one wants to ask, accrues to the cows from this reaction? Our search for a plausible Turing machine specification is guided here, as it was for the paper tape device, by the assumption that a Turing machine always has some point, some purpose. From a strictly mathematical point of view this assumption is unwarranted; a Turing machine may compute a function of no interest, elegance or utility whatever, of no value to anyone, and still meet the formal requirements for a Turing machine. Of course we would not be interested in the notion of a Turing machine at all were it not the case that we can isolate and study those that can be used to serve interesting purposes. The application of the concept to animals will be fruitful just so long as it leads us to mechanizations of apparently purposeful activity observed in the animal. Thus in some animals changes in barometric pressure can be highly significant, and may be responded to in some appropriate way by the animal—by finding shelter before the impending storm, for instance—and in these cases we will have reason to treat the effects on the animals as the receipt of information, and this will set us searching for an information-processing model of this capacity in the animal, or in other words (to take the short route again) to view the animal as a Turing machine for which barometric pressure is input.
At another level in our examination of the living candidate for machinehood, we would have to decide which features of the animal’s physical constitution are working as they were designed to, as they were supposed to, and which are malfunctioning, misdesigned or merely fortuitous. The “response” of mice to the “stimulus” of being dropped in molten lead is no doubt highly uniform, and no doubt we can give sufficient physiological conditions for this uniformity of reaction, but “burning to a crisp” does not describe a sort of behavior to which mice are prone; they are not designed or misdesigned to behave this way when so stimulated.* This does not mean that we cannot treat a mouse as an analogue of the one-state, two-symbol humming Turing machine described above. Of course we can; there just is no point. In one sense an animal—in fact any fairly complicated object—can be a number of different Turing machines at once, depending on our choice of input, state descriptions, and so forth. No one of these can be singled out on purely structural or mechanical grounds as the Turing machine interpretation of the animal. If we want to give sense to that task, we must raise considerations of purpose and design, and then no objectively confirmable answer will be forthcoming, for if Smith and Jones disagree about the ultimate purpose of particular structures or activities of the animal, there is no Designer to interview, no blueprint to consult.
Similar considerations apply pari passu when we ask if a man is a Turing machine. As a complicated chunk of the world he will surely qualify for any number of Turing machine characterizations, and it is even possible that by some marvelous coincidence some of these will match Turing machine interpretations of interest to mathematicians. Thus the wandering patches of light on a baby’s retina, and the subsequent babble and arm-waving, might be given a systematic interpretation as input and output so that the arm-wavings turn out to be proofs of theorems of some non-Euclidean geometry, for example. It is important to recognize that it is solely non-mathematical assumptions that make this suggestion outlandish; it is only because of what we believe about the lack of understanding in babies, the meaninglessness of their babble, the purposes, if you will, for which babies are intended, that we would discard such an interpretation were some mathematician clever enough to devise it. By suitable gerrymandering (e.g., incessant shifting of input vocabulary) it ought to be possible to interpret any man as any Turing machine—indeed as all Turing machines at the same time. So construed, every infant and moron would be engaged (among its other activities) in proving theorems and Gödel sentences (any Gödel sentence you choose), but of course the motions that constituted these feats of proving would not look like feats of proving, but like sleeping, eating, talking about the weather. The antimechanist is not interested in Turing machine interpretations of this sort; the activities and abilities he supposes he has crucial information about are those of mature, sane mathematicians in their professional endeavors.
He is interested in those motions of a man the purpose or interpretation of which is natural and manifest—his actions in short—but once we turn to the question of which Turing machine interpretation fits these actions, and hence might deserve to be called the description of the man, we come up against the relativities encountered by Smith and Jones: the ultimate function and design of every part of a man is not in the end to be decided by any objective test. Moreover, since the Turing machine interpretation of a man (if there is one) is picked out as the one best capturing the biological design of a man, and since man the biological entity has more ulterior goals than mere theorem-proving, no plausible candidate for the Turing machine interpretation of any man will be of the right sort to give the anti-mechanist the premise he needs. In addition to whatever computing a man may do (in school, in business, for fun) he also eats, acquires shelter, makes friends, protects himself and so forth; we do not need Gödel to demonstrate that man is not just a computer in this sense—that is, a device whose sole purpose is to compute functions or prove theorems. Suppose, to illustrate this, we have a particular hardware TMk churning out theorems in some system, and a mathematician, Brown, sitting next to the computer churning out the same theorems in the same order. If we tentatively adopt the hypothesis that we have two realizations of TMk, then we can go on to apply the Gödelian limitations to them both, but we have an easy way of disproving the hypothesis with respect to Brown. Once we have fixed, for our hypothesis, the list of Brown-motions that count as the issuing of output symbols (and these will appear to be in one-to-one correspondence with some symbol-printing motions of the hardware model), we merely ask Brown to pause in his calculations for a moment and give forth with a few of these symbols “out of order.” His doing this is enough to establish not that Brown is not a machine, but that Brown is not (just) a (good) TMk. Brown is not a TMk because here we see output symbols being emitted contrary to the hypothesized instructions for TMk, so either our request broke him (is a mathematician to be viewed as a sequence of exquisitely fragile self-repairing mechanical theorem-provers?) or he was not a TMk in the first place.
The fact that Brown was producing the same proofs as the hardware TMk does not imply that if Brown is a mechanism he is a hardware TMk, for producing the same proofs is not a sufficient condition for being, in this sense, the same Turing machine. Perhaps we were fooled into thinking Brown was a TMk because, for a while, he had “hand simulated ” a TMk. Simulating a TMk is in one sense being a TMk (a simulation is a realization), but of course it is not the sense the antimechanist needs, for in the sense in which simulating is being, the anti-mechanist claim is just false: it is not true that if a man were a TMk he could not prove S, where S is TMk’s Gödel sentence. If a man’s being a TMk is a matter of simulating a TMk then in all likelihood he can prove S; all he has to do is cease for a while following the instructions that amount to simulating a TMk, and Gödel says nothing about this role-changing being impossible. What the man cannot do (and this regardless of whether he is a machine, organism or angel) is prove S while following the instructions of TMk, but this is no more a limitation on his powers than is his inability to square the circle.
Gödel’s Theorem has its application to machines via the notion of a Turing machine specification, but Turing machine specifications say very little about the machines they specify. Characterizing something as a TMk ascribes certain capacities to it, and puts certain limitations on these capacities, but says nothing about other features or capacities of the thing. From the fact that something A is a realization of a TMk we cannot deduce that A is made of steel or has rubber tires, nor can we deduce that it cannot fly, for although the specification of a TMk does not stipulate that it can fly, it does not and cannot rule out this possibility. We also cannot deduce that A cannot speak English. Perhaps A can, and perhaps while speaking English, A may issue forth with a proof of TMk’s Gödel sentence. A could not do this if A were just a TMk, but that is precisely the point: nothing concrete could be just a particular Turing machine, and any concrete realization of any Turing machine can in principle have capacities under one interpretation denied it under another.
The fundamental error behind attempts to apply Gödel’s Theorem to philosophy of mind is supposing that objective and exclusive determinations of the activities and capacities of concrete objects are possible which would determine uniquely which Turing machine specification (if any) is the specification for the object. Once we acknowledge this error, this apparent application of Gödel’s Theorem to the philosophy of mind reveals its vacuity: if a man were (a realization of) a particular theorem-proving Turing machine with Gödel sentence S, then in his role as that Turing machine he could not prove S, but this says nothing about his capacities in other roles on the one hand, and on the other we surely have no evidence—and could have no evidence—that a man while playing the role of a Turing machine can do what Gödel says he cannot.
We can put the point of this paper in a form that should be mildly astonishing to the anti-mechanists who hope to use Gödel: a realization of the Universal Turing machine can, in principle, do the one thing Gödel’s theorem says the Universal Turing machine cannot do: prove the Gödel sentence of the Universal Turing machine. How could this be? A thought experiment will explain.
Children can be taught to hand simulate simple Turing machines. There are primers and textbook chapters designed to do just that. Suppose a graduate student in Artificial Intelligence took as his dissertation task, as his “toy problem,” writing a program that could learn to “hand simulate” a Turing machine just as children do—by reading the instruction book, doing the practice exercises, etc. Call this imaginary program WUNDERKIND. The rules of this project are that WUNDERKIND, after debugging but before being given its “lessons,” should be unable to hand simulate a Turing machine, but after being fed exactly the same instruction (in English) as the school children—no more, no less—should be able to hand simulate a Turing machine as well as, or even better than, the school children. (This proviso has nothing to do with the applicability of Gödel’s theorem to the case; it is added to give the project a non-trivial task, and hence to give WUNDERKIND something like a human set of abilities and interests.)
Imagine that WUNDERKIND was designed, and that it worked—it “learned” to hand simulate a Turing machine, TMk. Suppose we witness a demonstration of the program, actual hardware producing actual symbols. Now just what is this hardware object not supposed to be able to do? Prove “its” Gödel sentence. But which sentence is that? Which Turing machine is in front of us? If the hardware is a standard, commercial, programmable computer, it is—given enough time and storage—a realization of the Universal Turing machine, which has a Gödel number and a Gödel sentence, call it SU. But it is currently running the WUNDERKIND program (let’s ignore the fact that probably today it is time-sharing, and running many bits of many programs in quick succession), which is (mathematically equivalent to) a Turing machine with a different Gödel number and a different Gödel sentence, call it SW. Then there is the Turing machine, TMk, that WUNDERKIND is hand simulating, and it too has a Gödel sentence, call it Sk. Now we know that WUNDERKIND, while hand simulating TMk, cannot as part of that hand simulation produce Sk. But WUNDERKIND does other things as well; it asks questions, reads, practices, corrects errors in its exercises, and who knows what else. Perhaps it plays chess or writes fairy tales. What it does while not simulating TMk is an independent matter. It might well, for all we know, start offering proofs of sentences, and one of them might be Sk. Another might be SU! Or SW! There is nothing in Gödel’s theorem to prevent this. The computer, the actual hardware device, is a realization, let us grant, of the Universal Turing machine, and in that guise it cannot offer a proof of SU. In that guise what it is doing is imitating TMW—WUNDERKIND. When we shift perspective and view the object before us as WUNDERKIND (WUNDERKIND the child-simulation, not WUNDERKIND the algorithm represented by the machine table) we see it not as a theorem-prover-in-the-vocabulary-of-the-Universal-Turing-machine, but as a rather childlike converser in English, who asks questions, says things true and false on a variety of topics, and in that guise might well come up with a string of symbols that constituted a proof of SU or any other sentence.
The idea that WUNDERKIND itself can be viewed from more than one perspective will benefit from further specification. Consider what is wrong with the following argument: all computer programs are algorithms; there is no feasible algorithm for checkmate in chess; therefore checkmate by computer is impossible. The first premise is true, and so is the second. Chess is a finite game, so there is a brute force algorithm that gives the best line of play by simply enumerating all possible games in a tree structure and then working back from the myriad last moves to the line or lines of play that guarantee checkmate or draw for white or black, but this algorithm is impractical to say the least. But of course chess programs are not that bad at achieving checkmate, so the conclusion is false. What is true is that good chess programs are not algorithms for checkmate, but rather just algorithms for playing legal chess. Some are better than others, which means that some terminate in checkmate more often against strong players than others, but they are not guaranteed to end in checkmate, or even in a draw. In addition to the rather unilluminating perspective from which such a program can be viewed as a mere algorithm, there is the perspective from which it can be viewed as heuristic—taking chances, jumping to conclusions, deciding to ignore possibilities, searching for solutions to problems. If you want to design a good chess algorithm, you must look at the task from this perspective. Similarly, WUNDERKIND can be viewed as a mere algorithm for taking symbols as input and issuing symbols as output. That it is guaranteed to do, but from the other perspective it is an English-understanding learner of hand simulation, who follows hunches, decides on semantic interpretations of the sentences it reads, ignores possible lines of interpretation or activity, and so on. From the fact that something can be viewed as proceeding heuristically, it does not follow that it is heuristic “all the way down”—whatever that might mean. And similarly, limitations (Gödelian and other) on what can be done by algorithms are not limitations on what can be done—without guarantee but with a high degree of reliability—heuristically by an algorithm.
This is no refutation of anti-mechanism, no proof that a human being and a computer are in the relevant respects alike—for of course it all depends on my asking you to imagine the success of WUNDERKIND. Perhaps WUNDERKIND is “impossible”—though it looks modest enough, by current AI standards. If it is “impossible,” this is something Gödel’s theorem is powerless to show. That is, if all the versions of WUNDERKIND that are clearly possible fall far short of our intuitive ideal (by being too narrow, too keyed to the particular wording in the instruction book, etc.) and if no one can seem to devise a satisfactory version, this failure of Artificial Intelligence will be independent of the mathematical limits on algorithms.