theorem end inferences

Metamathematics and the Philosophy of MindAuthor(s): Judson WebbSource: Philosophy of Science, Vol. 35, No. 2 (Jun., 1968), pp. 156-178Published by: The University of Chicago Press on behalf of the Philosophy of Science AssociationStable URL: http://www.jstor.org/stable/186484 .Accessed: 04/03/2014 19:52

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METAMATHEMATICS AND THE PHILOSOPHY OF MIND*

JUDSON WEBBt

The metamathematical theorems of Godel and Church are frequently applied to the philosophy of mind, typically as rational evidence against mechanism. Using methods of Post and Smullyan, these results are presented as purely mathematical theorems and various such applications are discussed critically. In particular, J. Lucas's use of Godel's theorem to distinguish between conscious and unconscious beings is refuted, while more generally, attempts to extract philosophy from metamathematics are shown to involve only dramatizations of the constructivity problem in foundations. More specifically, philosophical extrapolations from metamathematics are shown to involve premature extensions of Church's thesis.

1. Introduction. In his classic paper Godel [5]1 discovered the recursive incompletability of formal number-theory; but although the relevance of this phenomenon for the philosophy of mathematics cannot be disputed, its consequences for even this domain, indeed for Hilbert's Programme itself, where it prima facie applies, have not yet been fully clarified to the satisfaction of those mathematical logicians working in foundations (see Kreisel [13] and [14] for a full discussion of this matter). Also, if there is difficulty in making the mathematical significance of a theorem explicit and agreed upon, an even greater difficulty ought to be expected when we try to extract the philosophical meaning. Nevertheless, some philosophers have proceeded rather unawares in this direction: for example, Lucas [15] argues that Godel's theorem (i) refutes mechanism, (ii) enables us to distinguish between conscious and unconscious beings, and (iii) allows us to "begin to see how there could be room for morality without its being necessary to abolish or even circum- scribe the province of science." Claim (iii) relieves a tension which "not even Kant could resolve."

In this paper, Lucas's claim (ii) will be critically examined and found to be groundless.2 The more general claim (i) is not discussed, except for the following remark: since Godel has only shown that number-theory is not recursively axio- matizable, any use of his theorem to establish claim (i) would necessarily have to offer a clarification and justification of the general demand of constructivity, and in particular of Church's Thesis, since the assertion that a Turing machine could do anything that any computing machine whatsoever could do is just another form of Church's Thesis. However, the problem of constructivity, so crucial to the foundations of mathematics, is far from being solved (see Kreisel [14]). These

* Received February, 1967. t I wish to thank my teacher Dr. Raymond Nelson for encouragement and many helpful

discussions on the topics treated. He is, of course, not thereby responsible for my errors. 1 A name followed by a number in brackets is a bibliographical reference. 2 The arguments are obviously applicable with appropriate modifications to numerous, more

articulate, expressions of the view that Godel's theorem supports an anti-mechanistic philosophy, e.g. G. Frey's "Sind Bewussteinsonaloge Maschinen moglich ?", Studium Generale, 19, pp. 191-200.

156


METAMATHEMATICS AND THE PHILOSOPHY OF MIND 157

reflections will be amplified briefly in our concluding remarks. Claim (iii) is discussed at the very end of the paper. 2. Lucas's Views. After treading through the usual argument from Gbdel's theorem against mechanism, i.e. since any "machine" cannot solve certain problems which a mind can, the so-called "Godel-problems," therefore no mechanical model of the human mind can be adequate, Lucas remembers that the hypothesis of consistency is essential to Godel's argument, and furthermore that Godel also proved that the consistency of a system (machine) could not be proved in that system (machine). There are serious confusions in his discussion of the consistency problem into which we will not enter here, except to mention that this problem is not what it seems, but is rather another aspect of the constructivity problem: we are not asking whether a particular system of number-theory is consistent, but rather by how constructive a proof it can be shown to be so; for if a system can be proved consistent by such-and-such methods, then every theorem of that system can be proved by using only those methods, and so the proofs of the system which use stronger methods can be replaced in favor of weaker ones. If a given system is thought to characterize just the "constructive" methods of proof in number-theory, then Gbdel's work shows that both its Gbdel-sentences and its consistency require non-constructive proofs (again see Kreisel [13]).3

But Lucas reacts to the consistency problem in the following way: he imagines an analogous consistency problem for human beings, and so to complete his refutation of mechanism he seeks a way for human beings to transcend their G6del theorem on consistency:

Thus in order to fault the machine by producing a formula of which we can say both that it is true and that the machine cannot produce it as true, we have to be able to say that the machine (or, rather, its corresponding formal system) is consistent; and there is no absolute proof of this.... At best we can say that the machine is consistent, provided we are. But by what right can we do this ? Godel's second theorem seems to show that a man cannot assert his own consistency, and so Hartley Rogers argues that we cannot really use Godel's first theorem to counter the mechanist thesis unless we can say that "there are distinctive attributes which enable a human being to transcend this last limitation and assert his own consistency while remaining consistent." (Lucas [15], p. 52)

Lucas then, following Rogers' suggestion, finds the required "attribute" to be man's "self-consciousness":

It therefore seems to be both proper and reasonable for a mind to assert its own consistency: proper, because although machines, as we might have expected, are unable to reflect fully upon their own performance and powers, yet to be self-conscious in this way is just what we expect of minds.... (ibid. p. 56)

Now, insofar as mechanism is understood as eschewing any fundamental difference between self-conscious and non-self-conscious beings, the contention of the

3Still another problem facing the philosopher who would appeal to Godel's second theorem is the circumstance that there are many reasonable syntactical formulations of the consistency of a given theory which are in fact provable in that system; the second theorem applies only to formulations of consistency which appeal to substantially more information about the proof- predicate than the first theorem. It is not obvious how this circumstance is to be reckoned with philosophically. See Solomon Feferman: "Arithmetization of Metamathematics in a General Setting," Fundamnenta Mathematica, XLIX, pp. 35-92. See note 15.


158 JUDSON WEBB

above-quoted passage might fairly be judged to be question-begging, unless some further support is adduced for it; however, in search of such evidence, Lucas is led back to G6del's first theorem, namely, the alleged "self-referring" character of Godel-sentences:

We can now see how we might have almost expected Godel's theorem to distinguish self-conscious beings from inanimate objects. The essence of the Godelian formula is that it is self-referring. It says that "This formula is unprovable in this system." When carried over to a machine, the formula is specified in terms which depend on the particular machine in question. The machine is being asked a question about its own processes. We are asking it to be self-conscious, and say what things it can and cannot do. Such questions notoriously lead to paradox.... The paradoxes of consciousness arise because a conscious being can be aware of itself, as well as of other things, and yet cannot really be construed as being divisible into parts. It means that a conscious being can deal with Godelian questions in a way in which a machine cannot, because a conscious being can both consider itself and its performance and yet not be other than that which did the performance. A machine can be made in a manner of speaking to 'consider' its own performance, but it cannot take this into account without thereby becoming a different machine, namely the old machine with a new part added. But ... a conscious mind can reflect upon itself ... and no extra part is required to do this: it is already complete, and has no Achilles' heel (ibid., pp. 56-57).

Thus the overall strategy of Lucas's argument proceeds as follows: he wants to use G6del's first theorem against mechanism in the well known, if not well founded, way; but noticing the hypothesis of consistency in the first theorem, and then misconstruing the second theorem on the proof of consistency, he imagines a consistency problem for human beings, which he tries to solve by arguing that human minds, being self-conscious, can assert their own consistency; while machines, being inanimate objects, cannot assert their own consistency. This latter distinction, which seems to be just what mechanism denies, is based, in turn, on the special nature of the G6del-sentences arising in the first theorem. Thus the entire argument flirts with circularity. At any rate, we shall direct our specific criticism against Lucas's last step (labeled as claim (ii) in the introduction), i.e. the use of the "self-referring" character of G6del-sentences to distinguish between conscious and non-conscious beings.

In sections 3, 4, and 5 we introduce the requisite machinery and results prepara- tory to the critical discussion beginning with section 6.

3. Machines and Recursively Enumerable Sets.4 We will follow Smullyan in defining a ";machine" (this notion can easily be shown to be equivalent to that of a Turing machine) as any device which generates some set which is formally representable in an elementary formal system. When the objects of the considered set are numbers, e.g. Gbdel-numbers, this is equivalent to defining as a machine any device generating a recursively enumerable5 set of integers, and, conversely, an r.e. set is one

4The ensuing discussion assumes some familiarity with either chapter 1 of Smullyan [201 or Post [17], or equivalent material. The "elementary formal systems" of Smullyan are essentially Post's "bases" for generated sets, or again, with certain modifications, they can be thought of as the "machines" of Turing [21]. An advantage of this approach is that it affords a high degree of intuitive appeal without sacrificing rigor. More importantly, the extraction of the incompleteness results from a purely mathematical context may help to forestall premature imputations of philosophical significance to the results.

5 Subsequently abbreviated as "r.e."


METAMATHEMATICS AND THE PILOSOPHY OF MIND 159

which can be generated by some machine. If the complement of an r.e. set can also be generated by a machine, it is called a recursive set. Church's Thesis asserts that a set is effectively decidable iff6 it is recursive, and Church's Theorem states the existence of r.e. nonrecursive sets of numbers, and hence, by the thesis, the existence of effectively unsolvable decision problems.

To prove the Post form of Church's theorem, Smullyan [20] constructs a certain "universal system" U in which can be expressed every sentence of form n E A, where n is a number and A is an r.e. set of numbers. The following results for U are basic: (i) the set To7 of Gbdel-numbers of true sentences of form n E A is r.e. (a completeness property for the theory of r.e. sets), (ii) the set To is not recursive, i.e. To is not r.e. (the Post form of Church's theorem), (iii) every recursively generated logic (T, R), where R is any r. e. subset of T, is incomplete: there will be a false sentence X, i.e. X E T, not refutable in (T, R), i.e. X 0 R (the Post form of G6del's theorem). Thus for this domain of what Post [17] calls "well determined propositions," the problem of formalization is encountered not in connection with truth, but rather with falsity and negation: the system Eo of all r.e. sets of numbers is not closed under complementation. Furthermore, since Eo comprises just the mechanically generated sets of numbers, any formal system for number-theory will have to "represent" (in some sense) at least these sets. (It turns out, in fact, that Eo comprises exactly the ranges of the general recursive functions.)

4. Representation Systems.8 We now describe a class of "systems" which are abstractions and generalizations of the usual logistic systems in the literature. Their utility lies just in their generality: they

allow us to study representability in systems of highly diverse syntactical structures ... (and) to treat the mathematically significant aspects of incompleteness and undecidability without getting entangled in the formal peculiarities of any one type of representation system (Smullyan [20], p. 41).

By a Representation System Z is meant a collection of the following things: (1) a denumerable set E of expressions, G6del-numbered 1-1 by a function g onto the set N of natural numbers; (2) a subset S of E called sentences; (3) a subset T of S called valid, true or provable sentences, or theorems of Z; (4) a second subset R of S called contra-valid, false or refutable sentences of Z, (if Z contains a symbol for negation, say '-', R might be the set of all sentences X such that - X E T); (5) a set P called the predicates of Z; (6) a function (D from each pair (X, n), X E E and n E N, into E, i.e. (D(X, n) E E, and such that for every predicate H E P, D(H, n) E S, for every n E N. (D is called the representation function for Z; intuitively, (D is simply the syntactical means by which Z symbolizes the application of a predicate (function) to its argument(s). The simplest (D would be just the operation of con-

6 Abbreviation for "if and only if." 7 We adopt Smullyan's convention of denoting by Xo the Godel-number of the sentence

X, and by To the set of Godel-numbers of the set T of sentences; and similarly for other metamathematical variables for sentences and sets of sentences.

8 We follow Smullyan [20], chapter 3. Often in the discussion that follows I will use metamathematical symbols autonomously. An advantage of the notion of a representation system is that this can be done without serious confusion.


160 JUDSON WEBB

catenation. Thus Z can be regarded as an ordered set Z = (E, S, T, R, P, (D) and by an extension Z' of Z we will mean the ordered set Z' = (E, S, T', R', P, D) where T', R' are supersets of T, R respectively, though still subsets of S; so that Z' contains exactly the same symbols, formulas and sentences as Z, differing only in the distribution of the distinguished subsets of S.

A basic notion for any Z is that of a predicate H representing a set A of numbers. Where W is any subset of E, let Hw be the set of numbers n such that H(n) E W; so HT = n(H(n) E T), and H is said to represent HT. Thus for a set A of numbers, H represents A iff for all n, n E A


incompleteness merely means that there is a Gbdel-sentence for R, whereas Church's theorem for Z, i.e. that To is not recursive, means that for every (r.e.) set W of expressions disjoint from T there is a Gbdel-sentence.11

We also remark that by reasoning essentially similar to that by which we proved the diagonalization lemma, it can be shown that every universal system is normal, which is based on the fact that for every r.e. set of numbers or expressions there is a Gbdel-sentence.

6. The Truth and Self-Reference of Giodel-Sentences. It is essential to such uses as Lucas makes of Gbdel's theorem that we can talk of the "truth" of the Godel- sentences of the given Z, which is usually a system based on the first-order predicate calculus, as in Gbdel's original example. This leads to the locution that we can "see the truth" of a sentence which a machine cannot. However, as we noted in section 4, the underlying difficulty was rather falsity than truth: since the representation of Eo can be taken as an explication of the usual phrase "adequate for elementary number-theory," the limitations on formal systems for number-theory will arise because the set T of U is not r.e., for if Z is formal its R will be r.e. and leave infinitely many of its translations of U's false sentences unrefuted. If Z contains negation, of course, then the negations of these falsehoods will count, on the intended interpretation of the symbols, as truths. Thus the problem of recog- nizing truth in any such Z is based on the non-constructive character offalsity in a fixed system which may be taken to express the basic facts of number-theory that we try to formalize with such Z's. We should also notice in connection with this situation that Tarski's well known theorem on the indefinability of truth holds only for complemented systems Z (see Smullyan [201, p. 45). We now look at this more closely.

Let Z be a formal universal representation system which has a symbol for negation whose application to a sentence we denote metamathematically by - X, and such that for every X E S, X E R +-* - X E T. By formality, Ro is r.e.; by universality Ro is rep. in Z by some H E P; by normality R* is also rep. in Z by some H E P. Let H rep. R*; then by the diagonalization lemma Hh is a Godel-sentence for R, i.e. Hh E R

162 JUDSON WEBB

Hence Hh expresses a falsehood, and so on the intended meaning of the symbol -Hh expresses a truth which is unprovable in Z.

However, one could also argue for the truth of Hh: the intended (or expected) meaning of R is that of the unprovable sentences of Z, and Hh is not only unprovable but expresses the proposition that h E R*, i.e. that Hh E R. In other words: it "expresses its own unprovability," and so Hh expresses a truth when R is in- terpreted as comprising all of the unprovable sentences of Z. The ambiguity of INh is due to the fact that no r.e. R can exhaust T.

This situation illustrates another general metatheorem: if a sentence X is undecidable in a system Z, then there are two models M and M' such that X is true in M and false in M'. Thus when we say that we can see the truth of an Hh which a machine cannot, this means nothing more than our choosing which model M we had in mind when we constructed Z.12 Usually this will be done by truth-definitions

12 This may seem to involve us in an irreducibly mentalistic idiom, and so to support the view that Godel-sentences enable one to distinguish between conscious and mechanical operations. But this would be premature. The notion of a "standard model" can be defined, for Peano arithmetic (as well as for a very general class of theories) in a straightforward syntactical manner, involving no prima facie intentional notions. (See Montague: "Set Theory and Higher-Order Logic," pp. 131-148, in Formal Systems and Recursive Functions, North Holland, 1964.) Goodstein [7] has lucidly criticized the tendency to interpret Gbdel's results as supporting the need to consider some irreducible intuition (of extralinguistic realities) in order to account for the notion of arithmetical truth:

When we say that no formal system can characterise the number concept, we do not mean that the number concept is something which we already have independently of the formal system; I may reject every definition of the meaning of a word, because it fails to characterise what I mean by the word, and maintain, rightly, that I know well what the meaning is, and yet my knowing what the meaning is may consist in nothing more than my rejection of the definitions. Just as I may write a story and be left with the feeling that this is not the story I meant to write, although of course I have not already in mind another story with which I compare it. When we contrast formal mathematics with intuitive mathematics we are not contrasting an image with reality, but a game played according to strict rules with a game with rules which change with the changing situation.... (ibid., p. 215)

The difference between "strict" and "changing" rules is simply that between recursive and non-recursive ones. Interestingly enough, as Goodstein also remarks, no non-standard model of the pure theory of recursive arithmetic can be recursive. At any rate, it is clear that any attempt to introduce mind as a necessary presupposition for explaining this situation will have not only to justify the general demand of recursiveness, but, more problematically, must also apparently find some coherent connection between the mental and the non-recursive. See note 15 for further observations on the possibility of relating Godel's work to the general notion of mind.

The Wittgensteinian, behavioristic flavor of Goodstein's remarks is evident. Thus it may be thought that any decisive objections, if such there be, to these presuppositions would refute Goodstein. But probably not: for he does not need them as a general epistemology and theory of meaning, but merely their fruitful application to mathematics, where they prove immeasur- ably less problematic. But it is not appropriate to go more deeply into this at present. Insofar as number theory is concerned the position is strikingly supported by the new game-theoretic semantics, in terms of "dialogues," developed by Lorenzen (see his Metamathematik, Mannheim, 1962).

Also, it may be mentioned here that there is all too much of an uncritical emphasis on the "limitative" (Fraenkel & Bar Hillel), "grave" (Beth), and "devastating" (Davis) in Godel's work. Thus Kreisel [12] writes as follows in his review of Mostowski's well known book on Godel's work:

In one respect the treatment is one-sided. All the undecidability results are treated as proofs of the inadequacy of the systems considered-a constant plaint throughout. But



in the manner of Tarski and may involve some non-constructive set-theoretic notions, e.g. the notion of an arbitrary infinite set of numbers.

Lucas bases the truth of Hh upon its alleged self-referring nature: it "expresses its own unprovability" and so is true. He also, we recall, uses this feature of Hh to support the distinction between conscious and non-conscious beings. Now although one might agree that Lucas has singled out a property of conscious minds, viz. their ability to answer questions about themselves without becoming other than themselves, unfortunately, when this property is made a sufficient condition for being conscious, we will be forced, when the nature of a Godel- sentence is examined more closely and properly understood, to conclude that either the limitation of a machine has nothing whatsoever to do with whether or not it is "conscious" or else that machines (and formal systems too!) are conscious after all and the refutation of mechanism will collapse on all sides.

We now take a closer look at the theory of Gbdel-sentences. The idea. we re- member, was that the human transcends the machine because, being conscious, he can manage the self-reflection necessary to answer Gbdel-questions about his own processes, whereas a machine, because it is not a conscious being, cannot deal with questions about itself without becoming another machine, i.e. the same machine with a new part. This is totally unconvincing.

In the first place, the self-reference itself is not clearly understood: this self- reference is not any intrinsic property of the Godel-sentence but is relative to the Gddel-numbering, e.g. it may express the proposition that a certain natural number has a certain arithmetical property, but relative to the G6del-numbering g, this number happens to be the one which g assigns to the sentence expressing this proposition. Obviously we cannot change the truth or falsity of an arithmetical proposition by simply taking up a different ordering of the set S of sentences which contains the given sentence. Thus consider the Hh of previous examples, which, relative to g, "refers" to itself. Now suddenly we may decide that g is too cumber- some and switch to a more convenient numbering g' of expressions containing Hh; and now Hh, though still expressing the same arithmetical proposition (because it is still the same formula with the same arithmetical interpretation), clearly has ceased referring to itself, because h, relative to g', is no longer the Godel-number

equally one may point out, that they show this: any proof in a formalized system of arithmetic gives information not only about the integers, but about concepts which are essentially different from the integers (a kind of unexpected or unintended efficiency of the system). These concepts provide the so-called non-standard models of a formalized system, where the individuals are, say, the integers 1, 2, . . . and in addition a.0, a

164 JUDSON WEBB

of H. In other words, under g' our Hh may have a different syntactical interpretation, or none at all. Or does one believe that simply by rearranging a set of sentences one can change their sense and truth-value?

This shows the futility of basing the truth of Godel-sentences on their reflexiveness relative to a particular Godel-numbering: the point is that this reflexiveness, which arises when the diagonal argument is applied to propositional functions (whose appropriate values are sentences) instead of number-theoretic functions (when diagonalization leads to a new set or function), is, in connection with a given enumeration g of predicates with free variables, of considerable heuristic aid in helping one to suspect the truth or falsity (on the intended interpretation of the symbols) of Hh; but the final decision as to its truth has nothing to do with this relative reflexiveness under a numbering. (See Kleene [10], pp. 205-206.) We may also add that Godel has here discovered how to use Cantor's diagonal method to discover new axioms for number-theory.

More specifically, one is confronted with the set S of a system Z for number- theory and, assuming R and T disjoint, one wonders whether there are any elements in S- (T u R), and if so, how to find them. Godel found that by refining the diagonal method they could indeed be constructed: one orders the unary predicates H1(n), H2(n), . . ., Hx(n) with free variables n, considers the predicate IHn(n), and if this is the qth and rep. in Z, then Hq(q) will be a sentence equivalent to its own negation.13 In other words, he found the normality of the usual systems,

13 Kreisel [11] remarks that Godel's work could be thought of as providing a method for converting (diagonal) proofs of non-denumerability (in Cantor's sense) into proofs of undecidability. Thus, for some purposes, we might consider the precise concept "non-recursively enumerable" as an explication of the vague notion "non-denumerable." This is perhaps hinted in Church [1] where it is suggested that, since we cannot give any clear meaning to the notion of a set being denumerable but not effectively so, we may be able to put a non-denumerable set into 1-1 correspondence with a subset of a denumerable set. His original unsolvability theorem took the form (in effect) of showing that there could be no recursive method of determining whether an arbitrary function of positive integers is defined for an arbitrary positive integer. Commenting on (a corollary to) this result, he remarks:

This corollary gives an example of an effectively enumerable set ... which is divided into two non-overlapping subsets of which one is effectively enumerable and the other not. Indeed, in view of the difficulty of attaching any reasonable meaning to the assertion that a set is enumerable but not effectively enumerable, it may even be permissible to go a step further and say that here is an example of an enumerable set which is divided into two non-overlapping subsets of which one is enumerable and the other non-enumerable. (ibid., p. 362)

As both Goodstein [7] and Watson [24] have pointed out, G6del's improvement of the diagonal argument turns on his recognition of the difference between giving a number to each element of an enumerated set, and giving an element to each number. In other words, when all the assumptions packed into Cantor's argument for the non-denumerability of the set of functions of natural numbers are made explicit and unambiguous (including especially those providing the needed enumerations and closure conditions), we get at most only non-r.e. sets out of the diagonal method. For example, the minute we realize that our function concept admits partial functions as a subset of all functions, the diagonal argument no longer yields a "new" function not in the enumerations, but rather one which must merely be undefined for its own index (or indices). Cf. Kleene [12], p. 341 on the enumeration theorem for partial recursive functions. Thus we might regard recursive function theory itself as a rigorous theory of what can significantly and unambiguously be established by diagonalization. See also Wilder [25], pp. 91- 101, especially the "fallacious theorem" on p. 95. Moreover, from the work of Lorenzen, it



i.e. those in which recursive methods can be used for constructing sentences. The Hh we discover will, of course, depend on the g chosen, but obviously Hh will not suddenly become decidable in Z if we take up another Godel-numbering in search of other undecidable sentences. Moreover, we have no general method for finding all of them, since the set of undecidable sentences is not r.e.

For undecidable sentences in first-order theories the following points ought to be made. Let P(x, y) be the "proof predicate" for such a Z, i.e. it means, relative to g, that x is the Godel-number of a proof for the sentence with G6del-number y; and let s(x, y) be Godel's famous substitution function, i.e. s(x, y) = g((X,, N(y))), where 1D is the representation function of Z, X E P, x = g(X), and N(y) is Z's numeral for the number y. Then we consider the familiar predicate (x) -P(x, s(y, y)), call it Hy, which was expected to be a representing predicate for Tt, and if h - g((x) -P(x, s(y, y))), then Hh is our undecidable sentence. However, abbreviating -P(x, s(h, h)) by A(x), we have a predicate A(x) such that each of A(1), A(2), .. ., A(k) are provable in Z for every particular number k, but (x)A(x), which is Hh, is not provable in Z (if Z is consistent). The truth of (x)A(x) is argued from the intended interpretation of Z, in which the variable x in A(x) takes only the (signs for) natural numbers as values; while the unprovability of (x)A(x) results from various deficiencies of different systems Z, of which the most common are the following: (i) the presence in Z of "non-standard integers" is called o- inconsistency if they are eliminable by a stricter definition of natural number, otherwise it is called "numerical insegregativity"; (ii) the unavailability in Z of a suitable predicate for (the induction step of) a proof of (x)A(x) by mathematical induction, i.e. the set of numbers satisfying the needed predicate, is not (completely) rep. in Z.14 But even this sense of the truth of (x)A(x) is heuristic since this sentence appears that Cantor's notion of non-denumerability is not, as often supposed, necessary for the purposes of classical analysis. See his Einfuhrung in die operative Logic und Mathematik, Berlin, 1955

Turing [21], p. 246 declares outright that when we "apply the diagonal process argument correctly," we prove not non-denumerability, but rather, in effect, the non-existence of a positive solution to the Entscheidungsproblem. Of course, it is not here urged that the notion of non- denumerability is useless. On the contrary, it has proved to be a powerful instrument of mathematical discovery; e.g. Finsler made essential use (i.e. using what Kreisel [11] calls its "picturesque" meaning: there are "more" functions of numbers than numbers) of the notion to argue for the incompletability of formal number theory five years before the appearance of G6del [5], which may be regarded as a constructive confirmation of Finsler's result. See his paper "Formale Beweis und die Entscheidbarkeit," Mathematische Zeitschrift, vol. 25, pp. 672- 682. Cf. Kreisel's remark mentioned at the beginning of this note. See also Watson [24].

14 See Smullyan [20], p. 46 for the important distinctions between representability, complete representability and definability. The notion of "numerical insegregativity" was introduced by Quine in his paper "On w-Inconsistency and a So-called Axiom of Infinity," Journal of Symbolic Logic, 18 (1953), pp. 119-124. Perhaps the ultimate source of the incompletability of formalized number theory has been unearthed by Kleene [9]:

It is impossible to confine the intuitive mathematics of elementary propositions about integers to the extent that all true theorems will follow from explicitly stated axioms by explicitly stated rules of inference, simply because the complexity of the predicates soon exceeds the limited form representing the concept of provability in the stated formal system. (ibid., p. 65)

Using (obviously) constructive elementary number theory in conjunction with the diagonal method, Kleene has constructed a "hierarchy" of all arithmetical predicates, which are ordered


166 JUDSON WEBB

fails after all to express universality in Z. If g is a prime-factor numbering, then (x)A(x) expresses the proposition that no natural number can be factored in a certain way. This is true, but its truth has nothing whatsoever to do with our having discovered (x)A(x) by means of the diagonal argument; this, relative to g, produces an heuristic epiphenomenon of "self-reference," which, however, disappears relative to other numberings g'.

Goodstein [7] has observed that even this epiphenomenon is confused in the case of (x)A(x) because, although each A(k) expresses, relative to g, that k is not (the Goidel-number of) a proof of (x)A(x), (x)A(x) does not itself express anything relative to g. His critique goes even further:

For even supposing, as is not in fact the case, that there is a formula of the formal system (let us call it O) such that as a sentence of the code b says something about the formula q of the formal system, we still could not claim that j is an example of a successful self- reference or self-description, for as a sentence of the code k refers not to itself, i.e. not to its meaning, but to the sign by which it is expressed, in the way the sentence 'This is written in chalk' refers to its physical character, not to its sense. (Goodstein [7], p. 219)

These points are hard to see when, as in Godel's original example, the function 4F of Z involves substitution and g is a prime-factor code; but the points are especially clear for the G6del-sentences of systems where d) is just the operation of concatenation and g is chosen isomorphic to 0, e.g. the dyadic numbering used in Smullyan (1962) to prove Church's theorem for the universal system U. Thus suppose that K = (a,, .. ., a,) is the alphabet of primitive symbols (which can be finitized if it is not already finite) of a system Z. We define the dyadic Godel- numbering as follows: for each i, let

g(ai) = 122 ... 2, e.g. (g(a2a1a3) = 122 12 1222, where a2ala3 symbolizes the concatenation of the 2nd, 1st, and 3rd symbols of K. Thus for strings X and Y, (XY)o = Xo Yo, which will also be true for those strings formed from K that serve as numerals, which have also their own dyadic Godel- numbers, e.g. if a, = 1, then g(l) 12, etc. If we now further suppose that the 4( of Z is concatenation and then extend the notion of diagonalization to include numerals as well as predicates, the following situation arises in our numbering: if n = XO, then the diagonalization of n is the Godel-number of the diagonalization of X, i.e. n = Xo -- nno = g(XXo), since we have nno = Xono = g(Xn) g(XX0) according to their increasing structural complexity. For any predicate in the hierarchy there exist more complex ones: and in particular, this will be true of any predicate which happens to be chosen as the proof-predicate of any formal system. This analysis of Kleene's is especially important for anyone interested in applications of incompleteness to the philosophy of mind, since it would appear to lead to a purely structural, i.e. syntactical, account of the limitations of formal systems. It reveals the incompleteness phenomenon to be of exactly the same kind as, e.g. Ackermann's construction of a general recursive function which is not primitive recursive: the constructed function grows faster than any of the kind considered. Moreover, from this point of view, it becomes evident why the diagonal method is so effective in establishing these kinds of results: it is the obvious method for constructing functions which grow faster than pre- assigned ones. See Kleene [10], pp. 271-272; also R. C. Buck's paper, "Mathematical Induction and Recursive Definitions," American Mathematical Monthly, vol. 70, pp. 128-135 contains an instructive example and discussion of the way diagonalization produces functions of accelerated growth rates.



by the isomorphism of the dyadic Godel-numbering to concatenation. Extending yet another notion, we write A* for the set of all numbers n such that nnO E A where A is a given set of numbers. It is easy to prove normality for sets of numbers, i.e. if A is rep. in Z (or r.e.), then A* is rep. in Z (or r.e.). Now assume Z universal and also formal. If H rep. R* and A* = R*, the following will hold:

1. (n)[Hn e T< n e A*] and by definition of A* we have

2. (n)[Hn E T -nno E A] and setting n = h where h = Ho we have

3. Hh E T

168 JUDSON WEBB

any system Z without any "new parts": Mu generates all r.e. sets and its activity is simply an attending now to this, now to that r.e. set of theorems.

But this is still not the weakest part of Lucas's argument, for even if we grant the dubious "new part" notion of extensions, it is easy to show that the original un- extended system Z, and hence the machine Mz, already qualify handsomely as conscious beings if we grant Lucas his interpretation of Godel-sentences. For let a Z be consistent, formal and universal; then Z is also normal. Then Mz has all these properties too. Now assume with Lucas that Godel-sentences represent questions put to Mz about its own processes. By the fundamental result for systems Z, i.e. the diagonalization lemma, it follows that for any set W of expressions of Mz's output, there will be a Godel-sentence for W if W* is rep. by Mz. And whenever H rep. W*, Hh will be a Godel-sentence for W saying that g(Hh) E WO, but reading with Lucas, Hh will say "of itself" that it is in W. Since there are an infinite number of r.e. sets W rep. by Mz, we will have by normality and our lemma, that every one of these W has an Hh saying of itself that it lies in W; indeed, infinitely many of these Hh will be both true and provable by Mz. For example, S is r.e. certainly, hence so is S*, and if H rep. S*, then Hh is a G6del-sentence for S, saying of itself that it lies in S, i.e. saying of itself that it is a sentence; and this Hh will be true and provable in Mz (this is because S is not only r.e. but also recursive). In fact, Kreisel has even constructed Godel-sentences for T, which say of themselves that they are provable, and then shown that in fact they are provable (in the given Z), and hence true; for if Z(Mz) is formal and normal, T* is r.e. and if Mz is further universal (i.e. adequate for number-theory), then T* will be rep. by some H and Hh will say of itself that it is provable, and under appropriate circumstances, may well be true, but not because it "says" so, but because it expresses a truth of arithmetic. (For general conditions for provability of G6del-sentences for T, see Lob, [14a].)

These facts clearly show that the "limitations" and/or "unconsciousness' of Mz, whatever they might mean, have nothing to do with Mz's inability to answer questions about its own processes. We have just seen that, granting the self-referring interpretation on Godel-sentences, Mz can answer truthfully infinitely many questions about itself, and needs no new parts to do it. Thus Mz (and why not Z too!) would seem to be quite conscious after all; indeed, Lucas even says that the questions about itself which it fails to answer are "rather niggling, even trivial, question(s)." To object because Mz does in fact fail for many such questions would be to forget that humans are also unable to answer many questions about themselves-or do some people have complete self-knowledge?

Thus, if Godel-sentences are allowed as literally self-referring, and ability to answer these questions is taken as the criterion of being conscious, then far too much comes out as conscious. However, as we saw, this rests on an utterly uncritical understanding of G6del-sentences: once we see that this self-reference is purely an heuristic epiphenomenon of the application of Cantor's diagonal method in conjunction with a given enumeration of strings of Z, we are forced to conclude that Godel's theorem fails to illumine any such distinction between conscious and non-conscious beings as Lucas envisages. Machines may well be without con-



sciousness, but Godel's theorem, applied the way it is by Lucas, fails to explain this. And, as we saw in the introduction, this was what he needed, by his own lights, to get around the consistency problem. But since he did misconstrue the consistency problem in a way which made his task unnecessarily difficult, we certainly cannot claim to have shown the impossibility of refuting mechanism by appeal to Godel's theorem.15

15 It seems clear that Godel's theorem cannot establish that the human mind is "stronger" than a machine: the machine can prove just as much, probably more, than a mind. More exactly: the machine can enumerate just as large a subset of its unprovable Godel sentences (this set is not r.e.) as any mind. The question is rather: do minds and machines establish their theorems in basically the same way, or do they proceed in essentially different ways? And does Godel's result, assuming they prove essentially different, illuminate this difference?

Evidently our best chance would be to show that Godel's work creates a situation in mathematics which can be fully explained only by making an essential appeal to intentional notions. The leading candidate for such a notion would seem to be that of "meaning." Thus we might pose the problem as follows: (i) can we coherently describe the (or some) difference between reasoning which proceeds according to the meaning of expressions and that which is guided solely by their syntactical structure; and (ii) does Godel's result help to establish that there are significant arguments from which the appeal to meanings cannot be eliminated? From the discussion in GcSdel [6] it appears that the distinction (i) is closely connected with both the consistency problem and the use of higher-order logic. Commenting on consistency proofs known from the literature and their bearing on Hilbert's finitary formalism, he writes:

Da die finite Mathematik als die der anschaulichen Evidenz definiert ist, so bedeutet das..., dass man fiur den Widerspruchsfreiheitbeweis der Zalentheorie gewisse abstrakte Begriffe braucht. Dabei sind unter abstrakten (oder nichtanschaulichen) Begriffen solche zu verstehen, die wesentlich von zweiter oder h6herer Stufe sind, das heisst, die nicht Eigen- schaften oder Relationen konkreter Objekte (z. B. von Zeichenkombinationen) beinhalten, sondern sich auf Denkgebilde (z. B. Beweise, sinnvolle Aussagen usw.) beziehen, wobei in den Beweisen Einsichten uber die letzteren gebraucht werden, die sich nicht aus den kombinatorischen (raumzeitlichen) Eigenschaften der sie darstellenden Zeichenkom- binationen, sondern nur aus deren Sinn ergeben. (ibid., p. 76)

Thus our problem is whether consistency can be established solely on the basis of the combinatorial (space-time) relations between the symbols of a formal system, without any appeal to their intended meanings. As remarked in note 3 however, Feferman has shown that we must further distinguish between formulations of consistency which are extensional and those which are intensional. Roughly speaking, the intensional formulations are those which can be seen to express consistency. And so Godel's second theorem on consistency says, roughly, that if an appropriate Z is consistent and Conz satisfies certain conditions on our being able to see that Conz expresses the consistency of Z, then Conz is not provable in Z. However, this deliberate use of the intentional idiom is eliminable in the sense that Feferman's distinction itself is defined in purely syntactical terms (see his paper referred to in note 3). On the other hand, it must be admitted that the significance, i.e. the explicanda, of the two kinds of theorems requires intentional notions to be fully understood. As Godel mentions in the above quote, the notions of previous consistency proofs which had been used on the basis of their "Sinn" were essentially the abstract concepts of higher-logic. But Godel has lately changed this situation in an essential way with his new consistency proof which employs as its only abstract concept that of a "computable function of finite simple type." Of this notion he writes: "Dieser Begriff ist als un- mittelbar verstandlich zu betrachten, . . ." and adds by way of prolepsis:

Man kann daruber im Zweifel sein, ob wir eine genugend deutliche Vorstellung vom Inhalt dieses Begriffs haben, aber nicht daruber, ob die weiter unten angegebenen Axiome fur ihn gelten . . ."

A. M. Turing hat bekanntlich mit Hilfe des Begriffs einer Rechenmaschine eine Defini- tion des Begriffs einer Berechenbaren Funktion erster Stufe gegeben. Aber wenn dieser Begriff nicht schon voher verstaindlich gewesen waire, hatte die Frage, ob die Turingsche Definition adaquat ist, keinen Sinn. (ibid., p. 79)

Thus the situation now is that we can prove the consistency of classical number theory by using


170 JUDSON WEBB

7. Concluding Remarks. We need to appreciate the danger that such philosophical applications of metamathematics may only be picturesque dramatizations of very special problems in the foundations of mathematics; indeed, we might say that they amount to a dramatization of the diagonal argument. First of all, let us note that the whole complex of limitative theorems usually so appealed to follows already from the following:

(i) There exists an r.e. set of numbers which is not recursive.16 (ii) A set of numbers is effectively decidable iff it is recursive. (iii) A system is formal iff its proof-predicate P(x, y) is recursive.

Now (i) is a formally provable fact, while (ii) and (iii) are just Church's thesis17 and Kleene's thesis respectively, which may be regarded as hypotheses concerning the significance of (i). Thus regarded (ii) says that the significance of (i) is that there exist effectively unsolvable decisions, while (iii) implies that in view of (i) every formal system which is adequate, i.e. universal, will fail to refute certain falsehoods or, if negation is present, to prove certain truths. Since, assuming (ii), (iii) may simply be regarded as an entirely unproblematic definition of a formal system, we will concentrate on (i) and (ii). Since (iii) implies that To, Ro are r.e., we see that it must be assumed in order to satisfy the hypothesis of the form we gave to G8del's theorem in section 5 (for universal systems Z), viz., the hypothesis that R* be representable in Z.

Anyone using Godel's theorem against mechanism must realize that, in effect, he is saying that the existence of a D implies consequences for the "whole of philosophy" (Lucas). Thus G$del's work may be said to consist in the construction of an r.e. non-recursive set of numbers by application of existential quantification to the proof-predicate of his system, i.e. where P(x, y) is the proof-predicate, he considers the set x.(3y)P(x, y) of theorems. But he was concerned only with the Godel-sentence for R. Church (1936) explicitly formulates (CT) and focuses on the

as our only non-combinatorial notion that of a computable function (unexplicated by Church's Thesis). Moreover, it cannot be objected that the notion is too vague to be significantly used, for Godel's new consistency proof uses only axioms for the notion which are evidently valid. But the overall situation is far from being completely understood. We can only say with Kreisel [13] that the notions of "constructive" and "finitist" are "ripe for systematic study."

We have already remarked on the connection of Gbdel's work with the notion of non- denumerability (note 13). We may also remark here that this notion proves on close inspection to be intentional itself. Thus Bernays observes in his discussion of the relativity of non-denumerability suggested by Skolem's paradox:

Freilich muss zugestanden werden, dass durch diese Relativitat der Umstand uns starker zum Bewusstein kommt, dass die hoheren Machtigkeiten in der Mengenlehre sozusagen nur intendiert, nicht eigentlich aufgebaut sind. In diesem Sinne kommt den Abstufungen der Machtigkeiten eine gewisse Uneigentlichkeit zu. (Betrachtungen zum Paradoxon von Thoralf Skolem, Oslo, 1957)

So from this point of view Godel's result seems to suggest that we can replace a notion which can only be "intended" with one which can be constructed, giving, so to speak, a reduction of the purely intentional, to something pretty much syntactical.

16 See Smullyan [20], p. 55, Theorem 16(a). In subsequent discussion the letter "D" will stand for any fixed r.e. nonrecursive set.

17 Subsequently abbreviated as "(CT)."



fact that there is a Godel-sentence for every r.e. subset of x -(3y)P(x, y), i.e. T. (See the end of section 7 and the reference to Smullyan [20].) What wondrous consequences, we might ask, are waiting to be extracted from the existence of continuous non-differentiable functions in analysis? Since (CT) is already a hypothesis concerning the meaning of (i),18 we must ask ourselves whether there is any room or justification for a further extension of the significance of (i) which is not simply a pleonastic, picturesque equivalent of (CT) itself. For example:

Suppose that a mechanist asserts that he can build a machine which does anything (in arithmetic) a human can do. Lucas (and others) want to reply here that the machine falls prey to Godel's theorem, but there is a step here which has not been made explicit: in order to apply the theorem to the proposed machine it is essential that some Turing machine be assumed equivalent to it; but to say that this is the case for any machine which a mechanist proposes to build (or asserts to exist) is simply to repeat (CT) in another form (since Turing-computability is equivalent to general recursiveness). So when the issue is stated without fanfare in exact terms, it boils down to an argument about (CT): the mechanist is denying it and Lucas is upholding it.19 Presumably one would then point out the heuristic evidence for (CT) (see Kleene [10], sec. 62); however Kalmar [8] argued against (CT) by showing that (CT) together with classical logic (i.e. the law of excluded middle used non-constructively) implies the existence of absolutely (not just relative to a formal system) unsolvable propositions which can yet be known to be false, a serious anomaly indeed. In a review of Kalmar's argument Kreisel can defend (CT) only by employing principles of intuitionistic logic and arguing that it is the logic relevant to the context. Not that we are accepting Kalmar's argument, or rejecting it: we are only emphasizing that the mind-machine-Godel controversy, on closer examination, appears (to paraphrase a quip of Wittgenstein's about Russell's logic) to consist only of frills of mind-talk tacked onto recursive function theory plus (CT). (See Wang [23], pp. 87-92.) At any rate, nothing in Lucas's discussion (and his is one of the most thorough of its kind in the literature) indicates how we are to meet the mechanist position when it is supplemented by a denial of (CT). Equally unclear is Lucas's position against the mechanist who would accept (CT), and then, on the basis of (CT) itself, assert that a human mind was just as ineffectual

18 For (CT) would be pointless if all sets proved to be recursive. '9 I lhave recently learned by verbal communication from my teacher Dr. Raymond Nelson

that his student D. R. Daykin has shown in his doctoral dissertation that there are non- recursive relations definable by machines whose computations are in a certain exact sense "hyperbolic" rather than "euclidean" in character. The method used in obtaining the result is to represent Turing-like computations by mosaics (generalized dominoes) and to consider spaces of mosaics Hx E', H a hyperbolic plane. I have not examined this example, but it could conceivably lead to counterexamples to (CT).

For other recent results pertaining to (CT), see Kreisel [14], pp. 143-147. Kreisel points out that (CT) is, in intuitionistic mathematics, a purely mathematical statement: "Evidently there is no reason why the question (of the validity of (CT)) should not be decided by means of evident axioms about constructive functions.... The discovery of axioms about constructions which are inconsistent with Church's thesis is certainly one of the really important open problems." (ibid., p. 147) See also Kreisel's review of Kalmar [8] in the Mathematical Reviews, 1960, vol. 21, #5567, p. 1029.


172 JUDSON WEBB

as a machine in the face of a non-recursive set (of problems).20 Thus Wang [23] writes:

It is not entirely clear what bearings the well-known results on unsolvable problems have on the theoretical limitations of the machine. For example, it is not denied that machines could be designed to prove that such and such problems are recursively unsolvable. If one contends that we could imagine a man solving a recursively unsolvable set of problems, then it is not easy to give a definite meaning to this imagined situation. It is of the same kind as imagining a human mind so constituted that given any proposition in number theory he is able to tell whether it is a theorem. It is a little indefinite to say that it is logically possible to have such a mind but logically impossible to have such a machine. (Wang [23], p. 107)

Note, however, that without (CT) we cannot say that it is logically impossible to build an actual machine, or that it is impossible that such a machine could exist.

20 The point here is that there is an ambiguity in what resources human minds are to be regarded as enjoying, and that whatever we decide and/or discover about this question must necessarily affect in a decisive way our interpretation of the Godel result. For on the one hand Post [17] declares: "Like the classical unsolvability proofs, these proofs are of unsolvability by means of given instruments. What is new is that in the present case these instruments, in effect, seem to be the only instruments at man's disposal." And on this assumption, D thwarts a man as well as a machine. But on the other hand, there is really nothing to prevent us from assuming at the outset that man's resources comprise more than the machine's. But then Godel's theorem is no longer needed, since we have now assumed what we wanted to prove! See Smart [19].

We recall that negation and complementation difficulties were responsible for D; also that there was difficulty in first-order theories with universal quantification over (just) the integers: the quantifier may catch other notions unexpectedly. These difficulties have prompted Myhill to reject these notions: "These represent a region which the human central nervous system, being subject to all the limitations of a Turing machine, is incapable of dealing with, and may therefore be rejected as meaningless." (Journal of Symbolic Logic, 15, p. 195.) Now this is tantamount to Post's assertion that man is essentially limited to constructive methods. From this viewpoint, Godel's work has revealed a "profound limitation on what man can accomplish ... essentially that he cannot eliminate the necessity of using his intelligence." (Rosenbloom [18], p. 163) The Post-Myhill thesis that man is limited to constructive methods is evidently equivalent to the mechanist thesis itself, if we assume (CT).

But the fact remains that, prima facie, men have used non-constructive methods-or at least, that is what some methods have been called. Does mechanism mean that these methods were only regulative ideas of reason in the Kantian sense (Hilbert)? or simply meaningless (Myhill) ? or motivated by naive platonic metaphysics (Brouwer)? It is significant that we must here refer to the names of mathematicians rather than declared mechanist philosophers (e.g. Smart), since the mechanists themselves have not troubled over the problem of constructivity, despite the fact that the basic conceptual tool of their recent formulations, the Turing machine, was introduced to clarify just this problem of constructivity for number-theoretic functions; or alternatively, we may say that Turing-machines were intended as an analysis of machine-like behavior, i.e. to explicate the notion of algorithm. Now the intended significance of algorithms for (infinite) classes of problems is that they replace ingenuity by systematic procedure in dealing with such a class; in other words, they bring us relief from having to consider the meaning and/or reference of (the notation for) each individual member of the class-in short, they save us the trouble of having to think about each member of the class. This they achieve by character- izing the set syntactically by some structural property common to (the given notation for) each problem. Thus if we succeed in constructing an algorithm for a class we then say that the class can be dealt with "mechanically." From this viewpoint, it would seem to be rather incongruous to argue the question whether or not machines can think: their very purpose being to relieve us of this burden wherever possible! Mechanism now begins to appear as a belief in some future utopia in which the necessity of thought will be completely eliminated. Thus to say that a machine could do something is to say that we could do that same something without thinking. And to say that we could not eliminate thinking from some problem class is to say that no machine could solve it. (Still untouched, though, is intentionality of perception.)



And so our philosophical argument plunges into very technical questions currently studied in the foundations of mathematics.

Or again: suppose someone asserts that number-theory is completely formalizable, by that system Z whose axioms are sentences which are true under Hilbert- Bernay's truth-definition for number-theory, and whose rule of inference is the identity relation. Or alternatively, one may simply apply Lindenbaum's well known lemma stating that every consistent system can be extended to a consistent and complete system. The reply is that Z's axioms do not form an r.e. set. This reply is as true as it is exact: but what is its force? Well, we say, assuming (CT), this means that they are not constructively characterized. But what does this mean? And so what?

These last questions are as unavoidable as they are difficult and motivate a large part of current study in foundations. Without an answer to them the present writer does not see how any application of metamathematics to the philosophy of mind can be more than hand-waving.

Copi [3] has argued that Godel's theorem is manifest evidence for the existence of synthetic a priori propositions. Turquette [22] objected on various grounds, ultimately questioning "the wisdom of associating a highly-refined logical language as is used in Godel's theorems with a philosophical language which is as richly colored with historical meaning as the phrase 'synthetic a priori."' Copi [4] rejoins by, among other things, appealing to Kleene's example of a number-theoretic proposition which is true classically, but not constructively provable. He criticizes Turquette's fear of "drawing philosophical conclusions from logical results," but admits that "in a sense this is a moral issue." I want to conclude now by relating my discussions in this paper to these three papers.

First of all, these three papers are worthy of very close study, especially by anyone interested in the relation of (modern) logic to philosophy.21 It is hard not to feel sympathy with both writers. Nevertheless, it is clear that my arguments are closely akin, spiritually at least, to those of Turquette. On the other hand, I have the pro- foundest respect for Copi's suggestion that such issues are ultimately "moral." Specifically, I want to make two comments.

1. My overall position in the present paper may be stated by saying that the mind-machine-Godel problem cannot be coherently treated until the constructivity problem in the foundations of mathematics is clarified. Hence I note with particular interest that Copi [4] defends himself at a critical point by appealing to Kleene's

21 Of course, they consider a different specific problem than does the present paper, at least prima facie; but the underlying question of the relation of logic to philosophy is the same. For a still different specific problem, still against the background of the same underlying problem, see the three papers, Myhill [16]. Myhill tries to interpret Godel's (and Church's) results as having the significance of psychological laws, viz. to the effect that certain organisms cannot learn to recognize certain properties of sentences in certain languages. Benes objects that a law having empirical content cannot be extracted from a purely analytic theorem. Myhill rejoins, in effect, that although the existence of D may be considered analytic, the conjunction of D with (CT) cannot. This amounts to interpreting (CT) as a psychological hypothesis (cf. Wang [23], pp. 87 ff.). So again, the attempt to state the philosophical implications of these limitative theorems consists actually of an attempt to read into D more significance than Church himself had intended with his own formulation of (CT).


174 JUDSON WEBB

example mentioned above, of a number-theoretic proposition (KC) which is "'unprovable ... for all constructive methods of reasoning" (Kleene [9], p. 70). The situation here is parallel in two important respects to the work of Church [1]. First, (KC) is constructed by the diagonal method. Secondly, the formal construction is given significance (in the sense of being related to familiar notions well entrenched in mathematical contexts) by a thesis, which can be regarded as a natural extension of (CT):

(KT): A proposition of form (x)(3y)A(x, y) containing no free variables is provable constructively iff there is a general recursive function +(x) such that (x)A(x, +(x)).

It is clear that, just as with the previous case of Church's theorem, where D alone without (CT) will not help Lucas against the mechanist, (KC) alone without (KT) would be of no help to Copi. Needless to say, this strengthens my belief that, if we limit ourselves to notions taken from modern logic, the mind-machine problem is only a dramatization of the constructivity problem. Note also, that such a (KC), given such a meaning by (KT), still depends on the existence of a D. Without such a D, (KT) would be as pointless as (CT). Thus in Copi's argument, as well as Lucas's, we are in the position of extracting from a D, by sheer analysis, consequences for the whole of philosophy. (On this constructivity issue see Church, [2].)

As a worthy companion piece, consider Craig's well known construction. He observes that if C is the closure of an r.e. set B under some relation R (and a certain further trivial condition is satisfied), then we can construct a primitive recursive set A such that C is still the closure of A under R. By applying this construction to (the Godel-numbers of) formal theories in science (assuming that there could be such), we are led to the conclusion that, as far as deductive observable consequences of theories are concerned, theoretical terms are unnecessary in science! What is interesting here is not the conclusion per se, but rather the character of the argument: one infers the dispensability of theoretical concepts from mere combinatorial properties of calculi ideally envisaged for their expression.

2. There is indeed a good possibility for moral conflict on the issue. On the one hand, one can experience a definite feeling of moral commitment to (and respon- sibility for) the consequences of one's actions and premisses, just because they are consequences, of which experience can be particularly sharp if one has cultivated the practice of Socratic elenchus. Thus a Lucas, or a Copi, or a reader can experience commitment to sufficiently tight chains of argumentation, regardless of their subject matter. On the other hand, we have a strong belief in the proposition that our comforts should be earned: we believe with Spinoza, that "all good things are as difficult as they are rare."

And so, although we can make the argument, say, from the existence of a D to our moral freedom, none of us can really, I think, believe that we have earned the right to say we are free on the basis of D. Thus, contrary to Lucas, it seems to me that Godel's work has led to an even greater tension than the one that "not even Kant could resolve." It is the tension between our belief in the honest truth stated


METAMATHEMATICS AND THE PHLOSOPHY OF MIND 175

by Russell that the more logic progresses, the less one can prove on its basis, and our sharing with Kemeny the inherent fascination of Godel's work.

The thinking of (the historical) Plato is characterized by his willingness to pro- ject the philosophical consequences of a purely mathematical result, e.g. the existence of irrational numbers. This example is interesting since it is intimately connected with problems which led naturally to Godel's discovery (see Watson [24] for details). In fact, the existence of irrationals can also be proved by the diagonal argument! Is there no end to what we can get at with this argument? Yes:

Each of the theorems of the present section was proved by means of the diagonal method. Each of the above constructions amounted to a definiti6n of a function by induction.... Godel's construction of an undecidable arithmetical predicate and Kleene's construction of a nonrecursive, one-quantifier form made similar use of the diagonal method. In sections 4-7, 9, and 11 we will see that the diagonal method lacks the power needed to obtain results about degrees deeper than those of sections 2 and 3. (G. Sacks: Degrees of Unsolvability, p. 20)

It is comforting to know that although the method peters out when we are probing the deeper degrees of recursive unsolvability, when it must be replaced by the more powerful category and measure-theoretic arguments, it still can help us to discover such simple things as our moral freedom and the synthetic a priori! Surely to conclude thus is a pitiful degradation of philosophical argument. I have tried in this paper to short-circuit some of the degrading arguments.

APPENDIX After submitting this paper my attention was called to the interesting paper of

P. Benaceraff, "God, the Devil, and Godel," Monist, Vol. 51, No. 1, pp. 9-33, which is also a critical study of Lucas's argument, although our readings of Lucas diverge at places. The remarks below attempt to bring my own reading into better focus.

1. I take Lucas to be arguing not that minds are stronger than Turing-machines, but rather that, on the evidence of Godel's first theorem, they are essentially different. Answering an objection of Turing (the human), he seemed explicit:

We are not discussing whether machines or minds are superior, but whether they are the same. In some respects machines are undoubtedly superior to human minds; and the question on which they are stumped is admittedly a rather niggling, even trivial question. But it is enough, . . . to show that the machine is not the same as the mind. (Lucas [15], p. 49)

(cf. note 15 above.) Professor Benaceraff, however, takes him otherwise: Lucas argues that the mind is not a Turing-machine on the grounds that I can prove more than any Turing-machine. .. If his argument for the non-machine- hood of the mind based on the supposition that the mind can prove more than any machine should fail, he might like to avail himself of the view that minds are limited to proving what turn out to be non-recursively enumerable subsets of what perfectly sound machines can prove. (loc. cit. p. 26)

I have trouble reconciling the first sentence with Lucas's words, but the real problem is to make sure phrases like "stronger than" and "prove more than" are


176 JUDSON WEBB

appropriately explicated. The second sentence causes trouble in this respect. If Lucas had produced an argument showing the deductive output of the mind to be non-r.e., I should have hailed him as thereby showing the mind as indeed stronger than any Turing-machine: for assuming that one can legitimately be said to generate one's deductive output (and-mind or machine-why not?), this would have been to show the mind capable of generating non-r.e. sets, something, by (CT), no machine can do. Surely this would justify a superiority claim: degrees of recursive unsolvability measure, in a cognate sense, the complexity of number sets, and it would be natural to regard them as measures of the "intelligence" of beings who could generate exactly their members, but, say, not those of higher degree. But I am fantasizing: Lucas produced no such argument. Worse yet, neither Lucas nor Benaceraff give any serious reasons for supposing that questions like "Is the mind's output non-r.e. ?" are even meaningful (cf. Wang's remarks). It is a question of significantly attributing certain mathematical properties to Benaceraff's set S = {x I I can prove x}, which he so resourcefully tries to fit into the framework of metamathematics. In one literal sense S is finite (hence trivially recursive), and so we may not be able even to get the problem airborne. But assume S is infinite, and suppose someone offers us what they claim to be an inductive definition b of S. We may imagine it even looks like one, appearing to have both direct and conditional clauses. Now what principles of evidence (of classical mathematics) could conceivably give any meaning to a dispute over whether S really does satisfy q uniquely? But S could conceivably be a meaningful mathematical object of intuitionistic mathematics. (See Kreisel's paper "Informal Rigour and Completeness Proofs," in Philosophy of Mathematics, ed. by Lakatos, North Holland.)

2. Lucas, on my reading, is primarily concerned with machines for number theory. Thus it puzzled me to see him entertain the inconsistency of number theory, since he quite obviously contemplates no counterexamples to Gentzen's proof (cf. Rosenbloom [18], p. 64, 72 for what he calls the "put up or shut up" criterion). I find the result so obviously constructive and informative I wouldn't know what one would look like, hence Lucas's doubts seem pointless to me. The Devil cannot survive long in the atmosphere of number theory.

Lucas also mentions "naive set theory.... deeply embedded in common sense ways of thinking". Perhaps. But inconsistent? I have never really seen a demon- stration of this, nor could I say what would count as one. I see that (3y)(x)(R(x,y) *-4

R(x,x)) is an inconsistent schema of elementary logic, but not that any one "naive" definition (least of all Cantor's) of set was ever plausibly translated as asserting it for the case where R is the membership relation. Here we meet with Quine's problem of distinguishing contradiction from bad translation. It seems more sensible simply to note that Russell's paradox has many uses, one of which is checking comprehension principles.

As for the consistency of human beings, I can't begin to make sense of this problem. Suppose, for example, it were a question about the formal consistency of our "internal programs." Would then their consistency imply, in accordance with Gbdel-Henkin completeness theorems, that they possess realizations ? Standard ones? And perhaps their inconsistency would then prevent them from



having-what? Bodies? Having sacrificed its control over the existential idiom via completeness, what role would this consistency notion be left to play other than as a synonym for conflict?

Perhaps axiomatic set theory will finally precipitate traffic with the Devil: here at last the consistency question is meaningful, Godel's second theorem applies and apparently nothing like Gentzen's proof is available. But the minute we try to take consistency at face value, incoherence threatens: it is just the old and still un- answered (from the face-value standpoint) question how, knowing that every formula is provable in an inconsistent system, we are going to motivate our worry about the unprovability therein of any formula which happens to express consistency. If we could formalize all our consistency proofs therein, we would still have to know the consistency of the system, independently of that proof, in order to know that the proof were correct: at face value, its provability therein could give no assurance whatever. So the second theorem still leaves open the optimal possibility of having our cake and eating it too. Gentzen's proof did this for number theory. It is constructive, we understand why it isn't formalizable internally (by an analysis using constructive ordinals) and yet it is not circular: it is arbitrary (and irrelevant) to say that transfinite induction < c0 applied only to prime formulas "already presupposes" ordinary induction applied to arbitrarily complicated formulas.

3. Feferman has removed the last temptation to take the second theorem at face value. Very roughly, this is why. Let Z be consistent, formal and universal, let k = g(- 0 = 0), let P(x, y) be a formula of Z representing its proof-relation which is constructed in a canonic manner from what Feferman calls RE-formulas and derivatively, call a formula ConpZ expressing the consistency of Z which is constructed from such a P(x, y), say -(]x)P(x, k), an RE-expression of consistency. Under our conditions (1) not F,, ConpZ But the condition of RE-expression is essential: define in Z, P'(x, y) P(x, y) A rP(x, E). Under the other assumptions on Z we have the extensional equivalence P'(x, y) +- P(x, y) provable in Z for all x, y. But now obviously we have also: (2) FI Conp Z, i.e. Fz(x) - P(x, E) What could we have replied to Lucas had he tried to get around (1) by appeal to (2)? We could point out that Conp Z is not an RE-expression of Z's consistency. But our by now familiar problem is that this must be motivated by a thesis about its significance. Recalling remarks of note 15 as well as discussion of (CT) and (KT), noting that Feferman's conditions are generalizations of Lob's for the provability of Godel-sentences for T which in turn go back to derivability conditions of Bernays, and keeping in mind our "semantic ascent" from talk of (r.e.) number sets to talk of their ("best") representing formulas, we might suggest that he could "constructively" or "demonstrably" see that aformula Conz expresses the consistency of Z iff Conz is an RE-expression of Z's consistency. The second theorem then, like the first, appears to derive its significance only from the constructive demand, not from theology. No one has summed up the whole situation as eloquently as Lorenzen:


178 JIUDSON WEBB

Dieser satz, der zuerst-nach Bernays-ein "Fiasko" der Metamathematik zu bedeuten shien, ist Anlass zu vielen Er6rterungen geworden. Am originellsten ist, was der Mathe- matiker A. Weil dazu gesagt haben soll: "Gott existiert, weil die Mathematik widers- pruchsfrei ist, und der Teufel existiert, weil wir es nicht beweisen konnen."

Kritisch betrachtet sagt allerdings der Godelsche Unableitbarkeitssatz nichts uiber Gott, die Welt oder das menschliche Erkenntnisverm6gen aus: er ist nur ein Satz der konstruktiven Mathematik. Die "Berechtigung" der konstruktiven Mathematik wird durch ihn nicht beeintrachtigt, er setzt diese ja zu seinem Beweis voraus. Und die "Berechtigung" der formalisierten Arithmetik wird durch konstruktive Widerspruchsfreiheitsbeweise erbracht. Der Godelsche Unableitbarkeitssatz erklart, warum man sich dazu mehr anstrengen musste, als Hilbert ursprtinglich vermutete. (Metamathematik, p. 132-3).

REFERENCES

[1] Church, A., "An Unsolvable Problem of Elementary Number Theory," American Journal of Mathematics, vol. 58, 1936, pp. 345-363.

[2] Church, A., Review of Copi (1949) and Turquette (1950), Journal of Symbolic Logic, vol. 16, 1951, pp. 221-222.

[3] Copi, I. "Modern Logic and the Synthetic A Priori," Journal of Philosophy, vol. 46, 1949, pp. 243-245.

[4] Copi, I., "Godel and the Synthetic A Priori: a Rejoinder," ibid., vol. 47,1951, pp. 633-636. [5] G6del, K., On Formally Undecidable Propositions of Principia Mathematica and Related

Systems, Oliver and Boyd, 1962. [6] Godel, K. "Iber eine bischer noch nicht beniutzte Erweiterung des finiten Standpunktes,"

Dialectica, vol. 12, 1958, pp. 280-287. [7] Goodstein, R. "The Significance of Incompleteness Theorems," British Journal for the

Philosophy of Science, vol. 14, 1963, pp. 208-220. [8] Kalmar, L., "An Argument Against the Plausibility of Church's Thesis," in Constructivity

in Mathematics (ed. A. Heyting), North Holland, 1959, pp. 72-80. [9] Kleene, S., "Recursive Predicates and Quantifiers," Transactions of the American Mathe-

matical Society, vol. 53, 1943, pp. 41-73. [10] Kleene, S., Introduction to Metamathematics, North Holland and Van Nostrand, 1952;

second printing 1957. [11] Kreisel, G., "Note on Arithmetic Models for Consistent Formulae of the Predicate

Calculus," Fundamenta Mathematica, vol. 37, 1950, pp. 265-285. [12] Kreisel, G., "The Diagonal Method in Formalized Arithmetic," British Journal for the

Philosophy of Science, vol. 3, 1952, pp. 364-374. [13] Kreisel, G., "Hilbert's Programme," Dialectica, vol. 12, 1958, pp. 346-372. [14] Kreisel, G., "Mathematical Logic," in Lectures on Modern Mathematics (ed. T. L. Saaty),

Wiley, 1965, pp. 95-195. [14a] Lob, M. H., in Journal of Symbolic Logic, vol. 20, p. 115. [15] Lucas, J., "Minds, Machines and Godel," Philosophy, Vol. XXXVI, References herein

are to the reprint in Minds and Machines (ed. A. Andersoln), Prentice-Hall, 1964, pp. 43-60. [16] Myhill, J., "Some Philosophical Implications of Mathematical Logic," Review of Meta-

physics, vol. 6, 1952, pp. 169-198. (See also Benes, Philosophical Studies, vol. 4, 1953, pp. 56-88; Myhill, ibid., vol. 5, 1954, pp. 47-48.)

[17] Post, E. L., "Recursively Enumerable Sets of Positive Integers and Their Decision Problems," Bulletin of the American Mathematical Society, vol. 50, 1944, pp. 284-316.

[18] Rosenbloom, P., Elements of Mathematical Logic, Dover, 1950. [19] Smart, J., "G5del's Theorem, Church's Theorem and Mechanism," Synthese, vol. 13,

1961, pp. 105-110. [20] Smullyan, R., Theory of Formal Systems, Annals of Mathematics Studies, 47, Princeton

University Press, 1962, second printing. [21] Turing, A. M., "On Computable Numbers, with an application to the Entscheidungs-

problem," Proceedings of the London Mathematical Society, vol. 42, 1937, pp. 230-265. (Also ibid., vol. 43, pp. 544-546.)

[22] Turquette, A., "Godel and the Synthetic A Priori," Journal of Philosophy, vol. 47, 1950, pp. 125-129.

[23] Wang, H., Survey of Mathematical Logic, North Holland, 1964. [24] Watson, A., "Mathematics and its Foundations," Mind, vol. 7, 1938, pp. 440-451. [25] Wilder, R., Introduction to the Foundationis of Mathematics, Wiley, 1965, second edition.


Article Contentsp. 156p. 157p. 158p. 159p. 160p. 161p. 162p. 163p. 164p. 165p. 166p. 167p. 168p. 169p. 170p. 171p. 172p. 173p. 174p. 175p. 176p. 177p. 178

Issue Table of ContentsPhilosophy of Science, Vol. 35, No. 2 (Jun., 1968), pp. 101-204Front MatterLogic, Probability, and Quantum Theory [pp. 101-111]DiscussionStatistical Ambiguity and Maximal Specificity [pp. 112-115]

Maximal Specificity and Lawlikeness in Probabilistic Explanation [pp. 116-133]New Dimensions of Confirmation Theory [pp. 134-155]Metamathematics and the Philosophy of Mind [pp. 156-178]Phonons--The Quantization of Sound (And Kant's Second Antinomy [pp. 179-184]An Alternative "Fundamental" Axiomatization of Multiplicative Power Relations among Three Variables [pp. 185-186]DiscussionStrawson, Particulars and Space [pp. 187-189]Polarity in the Social Sciences and in Physics [pp. 190-194]

Book ReviewsReview: untitled [pp. 195-197]Review: untitled [pp. 197-198]Review: untitled [p. 198]Review: untitled [pp. 198-199]

Abstracts from Synthese [pp. 200-201]Abstracts from British Journal for the Philosophy of Science [pp. 202-203]Back Matter

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