-
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
Your use of the JSTOR archive indicates your acceptance of the
Terms & Conditions of Use, available at
.http://www.jstor.org/page/info/about/policies/terms.jsp
.
JSTOR is a not-for-profit service that helps scholars,
researchers, and students discover, use, and build upon a wide
range ofcontent in a trusted digital archive. We use information
technology and tools to increase productivity and facilitate new
formsof scholarship. For more information about JSTOR, please
contact [email protected].
.
The University of Chicago Press and Philosophy of Science
Association are collaborating with JSTOR todigitize, preserve and
extend access to Philosophy of Science.
http://www.jstor.org
This content downloaded from 181.118.153.139 on Tue, 4 Mar 2014
19:52:51 PMAll use subject to JSTOR Terms and Conditions
-
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 meta- mathematics 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 incom- pletability 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 phil- osophy, e.g. G. Frey's "Sind
Bewussteinsonaloge Maschinen moglich ?", Studium Generale, 19, pp.
191-200.
156
This content downloaded from 181.118.153.139 on Tue, 4 Mar 2014
19:52:51 PMAll use subject to JSTOR Terms and Conditions
-
METAMATHEMATICS AND THE PHILOSOPHY OF MIND 157
reflections will be amplified briefly in our concluding remarks.
Claim (iii) is dis- cussed 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 prob- lems 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 con- sistent 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 re- futation 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 con- sistency: 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 diff- erence 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.
This content downloaded from 181.118.153.139 on Tue, 4 Mar 2014
19:52:51 PMAll use subject to JSTOR Terms and Conditions
-
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 generat- ing 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 essen- tially
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."
This content downloaded from 181.118.153.139 on Tue, 4 Mar 2014
19:52:51 PMAll use subject to JSTOR Terms and Conditions
-
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 meta- mathematical variables for sentences
and sets of sentences.
8 We follow Smullyan [20], chapter 3. Often in the discussion
that follows I will use meta- mathematical symbols autonomously. An
advantage of the notion of a representation system is that this can
be done without serious confusion.
This content downloaded from 181.118.153.139 on Tue, 4 Mar 2014
19:52:51 PMAll use subject to JSTOR Terms and Conditions
-
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
-
METAMATHEMATICS AND THE PHILOSOPHY OF MIND 161
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 sec- tion 4, the underlying
difficulty was rather falsity than truth: since the representa-
tion 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 unprov- able 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 un- decidable 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
This content downloaded from 181.118.153.139 on Tue, 4 Mar 2014
19:52:51 PMAll use subject to JSTOR Terms and Conditions
-
METAMATHEMATICS AND THE PHILOSOPHY OF MIND 163
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 interpreta- tion, 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 reflexive- ness 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 undecid-
ability. 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 assump- tions 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 sig- nificantly 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
This content downloaded from 181.118.153.139 on Tue, 4 Mar 2014
19:52:51 PMAll use subject to JSTOR Terms and Conditions
-
METAMATHEMATICS AND THE PHILOSOPHY OF MIND 165
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 mathe-
matical 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
This content downloaded from 181.118.153.139 on Tue, 4 Mar 2014
19:52:51 PMAll use subject to JSTOR Terms and Conditions
-
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 cer- tain 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 es- pecially 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.
This content downloaded from 181.118.153.139 on Tue, 4 Mar 2014
19:52:51 PMAll use subject to JSTOR Terms and Conditions
-
METAMATHEMATICS AND THE PHILOSOPHY OF MIND 167
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 arith- metic. (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-re- ferring
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 them- selves-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 un-
critical 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-
This content downloaded from 181.118.153.139 on Tue, 4 Mar 2014
19:52:51 PMAll use subject to JSTOR Terms and Conditions
-
METAMATHEMATICS AND THE PHILOSOPHY OF MIND 169
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 cer- tainly
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 mathe- matics 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 com- binatorial (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 in- tentional 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 "com- putable 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
This content downloaded from 181.118.153.139 on Tue, 4 Mar 2014
19:52:51 PMAll use subject to JSTOR Terms and Conditions
-
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 phil- osophy" (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-denumer- ability 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)."
This content downloaded from 181.118.153.139 on Tue, 4 Mar 2014
19:52:51 PMAll use subject to JSTOR Terms and Conditions
-
METAMATHEMATICS AND THE PHILOSOPHY OF MIND 171
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 any- thing (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
prob- lems." (ibid., p. 147) See also Kreisel's review of Kalmar
[8] in the Mathematical Reviews, 1960, vol. 21, #5567, p. 1029.
This content downloaded from 181.118.153.139 on Tue, 4 Mar 2014
19:52:51 PMAll use subject to JSTOR Terms and Conditions
-
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 in- tegers:
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.)
This content downloaded from 181.118.153.139 on Tue, 4 Mar 2014
19:52:51 PMAll use subject to JSTOR Terms and Conditions
-
METAMATHEMATICS AND THE PHILOSOPHY OF MIND 173
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 formaliz- able, 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-theor- etic 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 prob- lem, 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).
This content downloaded from 181.118.153.139 on Tue, 4 Mar 2014
19:52:51 PMAll use subject to JSTOR Terms and Conditions
-
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 construc-
tion 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 conse- quences 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 combin- atorial 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
This content downloaded from 181.118.153.139 on Tue, 4 Mar 2014
19:52:51 PMAll use subject to JSTOR Terms and Conditions
-
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 exis- tence 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 in- duction.... 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
This content downloaded from 181.118.153.139 on Tue, 4 Mar 2014
19:52:51 PMAll use subject to JSTOR Terms and Conditions
-
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 recur- sive
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
conceiv- ably 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
This content downloaded from 181.118.153.139 on Tue, 4 Mar 2014
19:52:51 PMAll use subject to JSTOR Terms and Conditions
-
METAMATHEMATICS AND THE PHILOSOPHY OF MIND 177
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 con- structed 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:
This content downloaded from 181.118.153.139 on Tue, 4 Mar 2014
19:52:51 PMAll use subject to JSTOR Terms and Conditions
-
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
konstruk- tiven 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.
This content downloaded from 181.118.153.139 on Tue, 4 Mar 2014
19:52:51 PMAll use subject to JSTOR Terms and Conditions
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