A Note on Wittgenstein's "Notorious Paragraph" about the Gödel Theorem Author(s): Juliet Floyd and Hilary Putnam
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Source: The Journal of Philosophy, Vol. 97, No. 11 (Nov., 2000), pp. 624-632 Published by: Journal of Philosophy, Inc.
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A NOTE ON WITTGENSTEIN'S "NOTORIOUS PARAGRAPH" ABOUT THE GODEL THEOREM*
I imagine someone asking my advice; he says: "I have constructed a proposition (I will use 'P' to designate it) in Russell's symbolism, and by means of certain definitions and transformations it can be so interpreted that it says: 'P is not provable in Russell's system'. Must I not say that this proposition on the one hand is true, and on the other hand is unprovable? For suppose it were false; then it is true that it is provable. And that surely cannot be! And if it is proved, then it is proved that it is not provable. Thus it can only be true, but unprovable."
Just as we ask, "'Provable' in what system?," so we must also ask, "'True' in what system?" "True in Russell's system" means, as was said, proved in Russell's system, and "false in Russell's system" means the opposite has been proved in Russell's system.-Now what does your "suppose it is false" mean? In the Russell sense it means, "suppose the opposite is proved in Russell's system"; if that is your assumption you will now presumably give up the interpretation that it is unprovable. And by "this interpretation" I understand the translation into this English sentence. -If you assume that the proposition is provable in Russell's system, that means it is true in the Russell sense, and the interpretation "P is not provable" again has to be given up. If you assume that the proposition is true in the Russell sense, the same thing follows. Further: if the proposition is supposed to be false in some other than the Russell sense, then it does not contradict this for it to be proved in Russell's system. (What is called "losing" in chess may constitute winning in another game.)'
W A te believe that this "notorious paragraph"2 contains a phil- osophical claim of great interest which has been almost entirely missed in the brouhaha about whether Ludwig Wittgenstein "misunderstood" Gddel's first Incompleteness Theo- rem. Our purpose here is to detach that claim, so to speak, from that disputed question (although the fact that Wittgenstein's critics seem to have missed it must surely be relevant to the dispute).
*Floyd is grateful to Jan Harald Alnes, Akihiro Kanamori, and Rohit Parikh for conversations on points related to the issues discussed in this paper.
1 Ludwig Wittgenstein, Remarks on the Foundations of Mathematics, G.H. von Wright, R. Rhees, and G.E.M. Anscombe, eds.; Anscombe, trans. (Cambridge: MIT, 1978, revised edition), i, Appendix iII, ?8.
2 So-called by Floyd in her "Prose versus Proof: Wittgenstein on G6del, Tarski and Truth," Philosophia Mathematica (forthcoming). Floyd gives a detailed reading of this paragraph in "On Saying What You Really Want to Say: Wittgenstein, G6del and the Trisection of the Angle," inJaakko Hintikka, ed., From Dedekind to Gddel: Essays on the Development of the Foundations of Mathematics (Boston: Kluwer, 1995), pp. 373-425.
The claim is simply this: if one assumes (and, a fortiori if one actually finds out) that -UP is provable in Russell's system one should (or, as Wittgenstein actually writes, one "will now presumably") give up the "translation" of P by the English sentence 'P is not provable'. To see that Wittgenstein is on to something here, let us imagine that a proof of -UP has actually been discovered. Assume, for the time being, that Russell's system (henceforth PM) has not actually turned out to be inconsistent, however. Then, by the first Incompleteness Theorem, we know that PM is w-inconsistent. But what does w-incon- sistency show? w-inconsistency shows that a system has no model in which the predicate we have been interpreting as 'x is a natural number' possesses an extension that is isomorphic to the natural numbers.
But why did we "translate" P as 'P is not provable in PM'? Well, P has the form: (3 x) (NaturalNo. (x).Proof(x, t)), where 't' abbreviates a numerical expression whose value calculates out to be the Godel number of P itself, 'Proof' abbreviates a predicate that is supposed to define an effectively calculable relation that holds between two nat- ural numbers nmjust in case n is the Godel number of a proof whose last line is the formula with Godel number m, and 'NaturalNo. (x)' is the predicate of PM we interpret as 'x is a natural number'.
But in discovering that PM is w-inconsistent we have discovered that:
(1) 'NaturalNo. (x)' cannot be so interpreted. In all admissible interpre- tations of PM (all interpretations that fit at least one model of PM), there are entities that are not natural numbers (and, a fortiori, not G6del numbers of proofs).
(2) Those predicates of PM (for example, 'Proof(xt)') whose exten- sions are provably infinite, and which we believed to be infinite subsets of N (the set of all natural numbers), do not have such extensions in any model. Instead, they have extensions that invari- ably also contain elements that are not natural numbers.
In short, the "translation" of P as 'P is not provable in PM' is untenable in this case-just as Wittgenstein asserted! This does not, however, affect the correctness of Gddel's proof, for nothing in that proof turns on any such translation into ordinary prose.3
Wittgenstein's aim is not to refute the Godel theorem but to "by-pass" it.4
In addition, we may point out that, if PM is actually inconsistent, and not merely w-inconsistent, then it has no "admissible interpreta- tions"-which is not to deny that in various contexts, and for various reasons, we may want to correlate its sentences with sentences in English.
I. BUT SURELY WITTGENSTEIN COULDN'T HAVE KNOWN ABOUT THAT STUFF!
But is it believable that Wittgenstein could have known about the "numerical insegregativity"5 of w-inconsistent systems? Are we not being overly charitable in giving this as Wittgenstein's reason for saying that, if one assumes that --Pis provable in Russell's system, one "will now presumably give up the interpretation" of P by the English sentence 'P is unprovable'?
The answer is that we have testimony (and not from a particularly sympathetic source!) that Wittgenstein thought about what are now called "nonstandard" models of the natural numbers, and connected them with the G6del theorem. In 1957, R.L. Goodstein6 wrote:
Wittgenstein with remarkable insight said in the early thirties that G6del's results showed that the notion of a finite cardinal could not be expressed in an axiomatic system and that formal number variables must necessarily take values other than natural numbers; a view which, fol- lowing Skolem's 1934 publication, of which Wittgenstein was unaware, is now generally accepted (ibid., p. 551).
And in 1972, Goodstein7 wrote:
I do not think Wittgenstein heard of G6del's discovery before 1935; on hearing about it his immediate reaction, with I think truly remarkable insight, was to observe that it showed that the formalization of arithmetic with mathematical induction and the substitution of numerals for vari-
4 See Remarks on the Foundations of Mathematics, VII ?19: "My task is, not to talk
about (e.g.) G6del's proof, but to by-pass it."
5 A term introduced by Quine to refer to the model-theoretic fact that W-incon-
sistent systems have no model in which the "integers" of the model are (isomorphic to) the natural numbers. See his "On w-inconsistency and the So-called Axiom of Infinity," in Selected Logic Papers (Cambridge: Harvard, 1995, enlarged edition), pp. 114-20; compare Set Theory and Its Logic (Cambridge: Harvard, 1969, revised edi- tion), pp. 305-06.
6"Critical Notice of Remarks on the Foundations of Mathematics," Mind, LXVI
(1957): 549-53.
7 "Wittgenstein's Philosophy of Mathematics," in A. Ambrose and M. Lazerowitz,
ables fails to capture the concept of natural number, and the variables must admit values which are not natural numbers. For if, in a system A, all the sentences G(n) with n a natural number are provable, but the universal sentence (n)G(n) is not, then there must be an interpretation of A in which n takes values other than natural numbers for which G(n) is not true (in fact in 1934, Th. Skolem had shown that this was the case, independently of G6del's work) (ibid., p. 279).
Interestingly, Goodstein entirely missed the connection between this "remarkable insight" and the claim we are discussing-which may not only be taken to be contained in Appendix I, ?8 of Remarks on the Foundations of Mathematics, but also as constituting virtually the whole of ?10 of the same Appendix!8 But, as is well known, the remarks on G6del's theorem were written as notes for Wittgenstein himself, and there was no reason for their author to state explicitly everything that he knew in connection with them. The fact is that Wittgenstein did understand-"with remarkable insight"-that "variables must neces- sarily take on values other than the natural numbers." (Indeed, this must be the case whether PM is w-inconsistent or not; the point about w-inconsistency is that, if PM is w-inconsistent, then there is no interpretation under which PM's theorems come out true in which the formal number variables take only the natural numbers as values.) We know this not only from Goodstein, but also from remarks made by Wittgenstein's student Alister Watson9 in a 1938 paper in Mind. Watson explicitly states that his interpretation of the Godel incom- pleteness result "owes much to lengthy discussions with a number of people, especially Mr. Turing and Dr. Wittgenstein" (ibid., p. 445). He then gives an argument very similar to the one we sketched above
8 Remarks on the Foundations of Mathematics, i, Appendix iii, ?10: " 'But surely P cannot be provable, for, supposing it were proved, then the proposition that it is not provable would be proved'. But if this were now proved, or if I believed-perhaps through an error-that I had proved it, why should I not let the proof stand and say I must withdraw my interpretation 'unprovable'?"
Compare Goodstein's dismissal of Wittgenstein's remarks on G6del, "Critical Notice of Remarks on the Foundations of Mathematics," p. 551, and "Wittgenstein's Philosophy of Mathematics," p. 279.
9 "Mathematics and Its Foundations," Mind, XLVII (1938): 440-51. According to
Andrew Hodges, Turing's biographer, discussions were held between Watson, Turing, and Wittgenstein in the summer of 1937, when Turing returned for a
season to Cambridge between his years at Princeton-Alan Turing: The Enigma (New
York: Touchstone, 1983), pp. 109, 136. The "notorious paragraph" of Remarks on the Foundations of Mathematics, i, Appendix iII, ?8 was penned on September 23, 1937,
about the effect of w-inconsistency on the power of any recursively axiomatized system of arithmetic to express the notion of natural number. He does this, he says, in order to show that the intuitive argument about the true but unprovable proposition P "obscures rather than illuminates the point" of G6del's theorem.'0 This, as we have just argued, may be seen to have been Wittgenstein's point as well in the "notorious paragraph."
II. BUT EVEN SO...
Even though, as we explained, our purpose here is to examine the valuable and still relatively unappreciated point that Wittgenstein makes in ?8 (and also in ?10) of Appendix I of Remarks on the Foundations of Mathematics-that is, that the "translation" of the fa- mous Godel sentence Pas 'Pis unprovable in PM' is not cast in stone, but is something that we have to give up in certain contexts-rather
10 Watson, "Mathematics and Its Foundations," pp. 446-47:
"If we assume for the moment that this axiomatic system is indeed a good basis for arithmetic, we shall have to conclude that the formula is not provable, and therefore, since this is just what it says, that it is true. For if it were provable, it would be false, and the system would be incorrect.
This method of putting the argument, however, obscures rather than illuminates the point. Suppose we assume the falsity of the formula, we cannot, of course, derive a contradiction, for this would amount to a proof of the formula. Instead, we reach the following peculiar situation, which is called by G6del an n0-contradiction (c, is the "ordinal number" of a sequence). We find that there is a function of a cardinal variable, say f(n), such that (all on the basis of the falsehood of G6del's formula) (n)f(n) can be disproved, and yet we can convince ourselves that we can prove in turn f(O),f(1),f(2) and so on. In other words, we apply mathematical induction to the proofs of the system, and obtain f(O), and from a proof of f(n) for any particular value of n, a proof for n+1.
Why should we object to an no-contradiction? Why should we not still say that G6del's formula may be false? The answer is that if we do this we shall feel compelled to say that the cardinal numbers cannot be all the values of the variable n, if f(n) can be true for each particular value of n, and yet (n)f(n) be false....
Thus the notion of a cardinal variable, i.e. of a number in the everyday sense, is something that cannot be completely expressed in the axiomatic system."
We note that although Watson is right that, if we assume the falsity of G6del's formula-that is, if we assume "(3 x) (NaturalNo. (x).Proof(x,t)) "-then (n) f(n) can
than to discuss in detail the dispute (s) about (1) whether Wittgen- stein "understood" the Godel theorem, and (2) whether his philo- sophical remarks about it have any value, we would be remiss if we did not at least comment on the question of the bearing of the unappre- ciated point on at least the second of those disputes. (The reader who has followed us this far should have no doubts about whether Witt- genstein understood the G6del theorem, and hence about the posi- tion to take with respect to the first dispute!)
The principal source of the question as to the cogency of Wittgen- stein's discussion are the words in the notorious ?8: "'True in Rus- sell's system' means, as was said: proved in Russell's system, and 'false in Russell's system' means: the opposite has been proved in Russell's system" (emphasis added). Is Wittgenstein not aware that arithmeti- cal propositions have a meaning and truth value independent of what system they are formalized in? Is he just identifying "truth" and "provability" out of some misguided combination of formalist and constructivist/finitist motives? And so on.
The answer is that one cannot simply ignore the direction (implicit in the words we italicized) to look at what was said before about 'true in Russell's system'.
Just one paragraph before (?7), there appear the remarks which clearly set the stage for the notorious remark:
7. "But may there not be true propositions which are written in this symbolism, but are not provable in Russell's system?"-'True proposi- tions', hence propositions which are true in another system, i.e. can rightly be asserted in another game. Certainly; why should there not be such propositions; or rather: why should not propositions-of physics, e.g.-be written in Russell's symbolism? The question is quite analogous to: Can there be true propositions in the language of Euclid, which are not provable in his system, but are true?-Why, there are even proposi- tions which are provable in Euclid's system, but are false in another system. May not triangles be-in another system-similar (very similar) which do not have equal angles?-"But that's just ajoke! For in that case they are not 'similar' to one another in the same sense! "-Of course not; and a proposition which cannot be proved in Russell's system is "true" or "false" in a different sense from a proposition of Principia Mathematica.11
If one reads this paragraph with care, one will observe two things: first, that Wittgenstein is telling us that we should look on PM as a "system" in the sense in which a system of non-Euclidean geometry is a "system" of geometry-a sense in which the same sentence (Satz)
can be true in one system and false in another. And, second, that this paragraph does not deny that a proposition that cannot be proved in Russell's system (Wittgenstein obviously means one that cannot be decided, that is, proved or disproved) can, in some sense be "true" or "false" (outside the system)-he only asserts that this is a "different sense" from the sense in which it is true or false as a "proposition of Principia Mathematica." We shall now comment on each of these points in turn.
(1) Wittgenstein's targets in much of Part I of Remarks on the Foundations of Mathematics are Gottlob Frege and Bertrand Russell, not as mathematical logicians but as philosophers of mathematics and logic. These philosophers emphatically did not see themselves as providing a mere notation into which one could transcribe the prop- ositions that mathematicians actually utter, write, and publish in ordinary "mathematical prose," that is, in English or French or Ger- man or.... They saw themselves as providing a freestanding "ideal language" or "concept-language," what W. V. Quine12 has called a first-grade conceptual scheme, which in some sense supercedes ordi- nary language. Moreover, in providing such a scheme, they saw themselves as providing mathematics with a foundation. Ordinary language might be necessary to "lead someone into" the ideal lan- guage, but the "elucidations" we offer for this purpose in ordinary language are, so to speak, a ladder that we can throw away. Frege explicitly argues that ordinary language sentences that we use to explain the ideal notation do not and cannot capture the precise content of the ideal notion.13 It would be utterly foreign to this spirit to explain the truth of a formula of Principia Mathematica by merely writing down an English sentence, and saying this is what it means for P to be true. To confess that this is what one has to do would be to abandon the claim for the foundational status of a system such as Principia Mathematica entirely. (So that when Wittgenstein writes in ?8: "And by 'this interpretation' I understand the translation into this English sentence," he is already denying that this notion of an "inter- pretation" which can only be indicated in English by helpful "hints"'4
12 "Speaking of Objects," Ontological Relativity and Other Essays (New York: Colum-
bia, 1969), p. 24.
13 For the case of what Frege takes to be his primitive or undefinable notions (for
example, function, concept), see his Collected Papers on Mathematics, Logic, and Philos- ophy, Brian McGuinness, ed., Max Black et alia, trans. (New York: Blackwell, 1984), pp. 193-94, 300, 302; Posthumous Writings, pp. 207, 214, 235; Grundgesetze der Arith- metik, I (Hildesheim: Olms, 1966), Appendix 2, n. 1 (p. 240).
and which does not in principle require any dependence on informal mathematical language, makes any real sense.)
Instead, Wittgenstein is suggesting, there is a sense in which a formalism can be free-standing, but it is not the sense of a Begriffs- schrift or an ideal language. It can be free-standing regarded simply as a formal system. But then the only sense of "truth" we shall have available is: being a theorem of the system.15 And in that case, why should it not be the case that one and the same "proposition" should be "true in Russell's system" and false in a different system?
Indeed, something of this kind does happen. If there are only finitely many individuals, then for some natural number x, x= Sx (imagine this written out in the formal notation) is a theorem of Principia Math- ematica, while There are only finitely many individuals andfor every natural number x, x 0 Sx (imagine this likewise written out in formal notation) is a consistent proposition (model theoretically as well as proof theoretically) in Zermelo set theory. Or to change the example, the formula '(3x) (xE x)' ("some set belongs to itself") is "true in Quine's system" (Mathematical Logic) and "false in Zermelo set theory"!
Today, of course, few if any philosophers think that a formal system provides a foundation for either the content or the truth of mathe- matical propositions. But one of us (Putnam) remembers a delightful philosophical conversation between C.G. Hempel and one of Hans Reichenbach's graduate students in Reichenbach's living room in 1950, at which the older attitude and the newer attitude memorably clashed. Hempel was defending Quine's skepticism with respect to the analytic-synthetic distinction, and the graduate student said plain- tively: "Quine's arguments may show that the analytic-synthetic dis- tinction makes no sense in natural language. But why doesn't it make clear sense in a formalized language?"; and Hempel replied: "Every formalized language is ultimately explained in some natural lan- guage. The disease [Hempel meant the unclarity of the analytic- synthetic distinction] is hereditary." Here, Hempel-like Wittgenstein in Remarks on the Foundations of Mathematics-was denying that a
15 After the appearance of semantics with Alfred Tarski, one can add another
formal system could provide us with a standard of truth or clarity that is, in principle, inaccessible to a natural language.
(2) What if someone were to have said to Wittgenstein: "When I say that P is true in Russell's system, what I mean is simply that its translation into English-any one of its mathematically equivalent translations, including 'P is unprovable in PM'-is true?" We believe that Wittgenstein would have pointed out that the notion of truth is eliminable here.16 To understand 'P is true in PM' as meaning "The English sentence 'P is unprovable in PM' is true" (or more colloqui- ally, as meaning "It is true that P is unprovable in PM"), would amount (as we see by "disquoting") just to understanding 'Pis true in PM' as short for 'Pis unprovable in PM', or understanding P itself (in PM) as short for 'Pis unprovable in PM'. In short this just is to accept what Wittgenstein calls "the translation into this English sentence." And this is something, we have shown, Wittgenstein did discuss in the "notorious paragraph"-"with remarkable insight."
That the G6del theorem shows that (1) there is a well-defined notion of "mathematical truth" applicable to every formula of PM; and (2) that, if PM is consistent, then some "mathematical truths" in that sense are undecidable in PM, is nota mathematical result but a metaphysical claim. But that if P is provable in PM then PM is inconsistent and if -P is provable in PM then PM is w -inconsistent is precisely the mathematical claim that G6del proved. What Wittgenstein is criticizing is the philo- sophical naivet6 involved in confusing the two, or thinking that the former follows from the latter. But not because Wittgenstein wants simply to deny the metaphysical claim; rather, he wants us to see how little sense we have succeeded in giving it.
JULIET FLOYD Boston University
HILARY PUTNAM Harvard University
16 For a discussion of the senses in which Wittgenstein was and was not a
