> 5@ bjbj22 .XX/L&&&^^^r8NT<rzZ((:PP9;<TYYYYYYY$X\R^^Z^>8l9>>Z^^PP/4ZvCvCvC>^P^PYvC>YvCvCJE%X^^YP3?XYJZ0zZX_@b_(Yrr^^^^_^Y <LA=6vCw=,=<<<ZZrrC^rrCan sentences self-refer?: Gdel and the Liar
[Version: July 25, 2005]
Asking what the sense is. Compare:This sentence makes sense. What sense?This set of words is a sentence. What sentence?(Wittgenstein, Philosophical Investigations, 502)
The argument
The argument that we want to develop in this paper is that words (and consequently sentences) do not have any effect in and of themselves (even if understood to operate within a system) but that it is human actions and practices that ensure that words and sentences have effects. It is only by losing sight of human action that one becomes enchanted by words and symbols which seem to be altogether out of the control of humans, and which seem to mean whatever they mean, and do whatever they do, magically, without any reference to a speaker, a thinker, a community, a context, and so on. We feel that such enchantment is particularly likely in fields such as mathematics, where the procedures of the subject are designed to as to eliminate any seemingly subjective elements.
We will try to illustrate this by focussing on the issue of self-reference, which has been the subject of much debate (and controversy) in the philosophy of language and mathematics. The topic of this paper thus is sentences such as I am a liar., I am not provable., or This sentence makes sense.. These sentences seem to lead to all sorts of paradoxes that need to be explained and accounted for by philosophers. Offering a radically Wittgensteinian twist on some seemingly familiar territory, we want to show that these paradoxes themselves are based on a metaphysical picture of how words operate.
Not sentences but humans self refer
We would like to suggest that words in and of themselves cannot do or say anything. Words need something else in order to have any effect. Words without us using them, and taking responsibility for that use, cannot do any of the extraordinary, often magical, things that are attributed to them. A fortiori, they cannot refer to themselves:
it should strike us as absurd to maintain that sentences say something. Sentences dont talk; people do. And so sentences dont say anything, either.
Wittgensteins Philosophical Investigations (PI) and his Remarks on the Foundations of Mathematics (RFM) contain many examples of the paradoxes that may arise by losing sight of the role that human action plays. Our argument in a sense applies Wittgensteins remarks about calculating machines to self-referring sentences:
Does a calculating machine calculate? (RFM, V, 2, p.257)
If calculating looks to us like the action of a machine, it is the human being doing the calculation that is the machine. (RFM, IV, 20, p.234)
It is sometimes argued and we go along with the argument, by and large that words only have sense within a context or system. However, that does not solve our problem, as again the system itself cannot do anything. It is not the system which does but human practice. The success of words depends on us being part of a (larger) context or system.
Following Stanley Cavell, we might want to say then that there is a tendency, especially in philosophy, to avoid responsibility for ones thoughts, words, (as well as other) deeds. One tends not to acknowledge them as truly ones own much as, all-too-often, one tends not to acknowledge other people as fully people. In particular, there is a tendency to put forward the view that it is not humans that say certain things but the words themselves. However, in our view there is no way of understanding language (or any other human practice, including mathematics) that does not take into account that it is not words that do the speaking or doing but people. As a consequence, one cannot police or predict how words might be used in new ways without falling back to the tendency to avoid responsibility.
Ordinary self-reference
There has been a lengthy debate in such disciplines as Literary Studies and Philosophy over the exact nature of the (supposed) self-reflexivity of many things: texts, sentences, utterances, words, or occasionally even objects. In recent years for a while it became almost a commonplace in English Departments that poems can and do (even: always do) refer to themselves, and in Fine Art Departments that artworks refer to themselves. And it is surely commonplace in Philosophy Departments that certain sentences can refer to themselves, in particular when couched in logically perspicuous language(s).
In this section, we will investigate sentences such as I am lying. or This sentence makes sense., in order to get an initial grasp on the notion of self-reference or self-reflexivity.
Metaphysical lines of projection
How can a proposition say something of itself, outside of the external factum of our determination that it should be read so to say? Is there a means in the language in question by which a sentence can (itself) be said to be pointing to itself? Is it possible to establish a language that is not vulnerable to human decision as to its meaning and interpretation? Possibly so, but still: How are we to make sense of the view that a sentence can point to itself? Are we to imagine metaphysical lines of projection, going from the sentence back to itself (just as we might imagine such lines going from (say) the word sun to the Sun)? Well, let us endeavour to countenance the imagining of such lines.
This sentence makes sense.
[Fig.1]
Then we might ask: Which sentence? And the answer would be to follow a (metaphysical) line:
This sentence makes sense.
[Fig.2]
However, we might still want to ask: Which sentence? (because the question could arise as to whether or not the arrow was part of the sentence referred to):
This sentence makes sense.
[Fig.3]
And so on and so forth
We would like to argue that any putative instance of self-reference will involve some kind of perpetual ambiguity as shown in these figures.
It might be thought relevant (and problematic) that the example considered above involves a demonstrative (This). But this actually makes no difference. One could equally well substitute other methods of self-reference or self-involvement (e.g., the proposition could be written as The sentence on the blackboard does make sense., and be the only sentence on the blackboard). The same problem would apply. One would still be able to ask whether the sentence was the same sentence after the lines of projection (ensuring that it was the sentence being referred to) had been added. And one would have to have the lines of projection, in one way or another. For two reasons: (i) because there could be other blackboards in the room or in nearby classrooms, etc.; (ii) because without these arrows (or similar devices) there would be no prima facie plausibility to the claim that the sentence was pointing to or hooking up with itself by itself (i.e., without any involvement from language-users).
To self-refer a sentence cannot just simply stand there; it must do something. It must point to itself, or something similar. But when we say this, (especially after reading the passage from Wittgenstein quoted below), do we know what we are talking about? Have we any clear sense yet of what we really want to say? In particular, do we know in which sense a sentence can do or say anything?
Our method of questioning here is akin to Wittgensteins, for example at PI 141, where he is discussing how lines of projection might seem to supply meaning, might seem to effectuate an assertion or saying of something; in this case, what the word cube means.
Suppose, however, that not merely the picture of the cube, but also the method of projection comes before our mind? How am I to imagine this? Perhaps I see before me a schema showing the method of projection: say a picture of two cubes connected by lines of projection. But does this really get me any further? Cant I now imagine different applications of the schema, too? Well, yes, but then cant an application come before my mind? It can: only we need to get clearer about the application of this expression. Suppose I explain various methods of projection to someone so that he may go on to apply them; let us ask ourselves when we should say that the method that I intend comes before his mind.
Now clearly we accept two different kinds of criteria for this: on the one hand the picture (of whatever kind) that at some time or other comes before his mind; on the other, the application which in the course of time he makes of what he imagines. (And cant it be clearly seen here that it is absolutely inessential for the picture to exist in his imagination rather than as a drawing or model in front of him; or again as something that he himself constructs as a model?)
Would-be lines of projection cannot do the task one fantasises for them: they cannot take from one the responsibility for using, over time, the expressions concerned. The insistence that a sentence self-refers is just that: an insistence. If one takes a sentence as referring to itself, it is not the sentence that does this, by itself. In other words, the question is not so much whether one can read a sentence as referring to itself as whether one is compelled to. And one is surely not compelled to; for one is actively involved even in reading a sentence as referring to itself; and so, if one withdraws that active involvement, as one surely can, it will be possible to read the sentence another way.
In sum, the claim that a sentence is saying something of itself is not an idea that has been often made very good sense of. It seems nearly always to be a case of (bad?) poetry or attempts of the kind of self-referentiality that Deconstructionist critics (arguably quite mistakenly, indeed incoherently) believe much Modern poetry accomplishes. And therefore one surely does not have to read it as referring to itself.
Wittgensteins take on self-reference
Wittgenstein provides us with the resources to show more thoroughgoingly that and how the mainstream conceptions of self-reflexivity and self-referentiality are in the main incoherent (and based on optional and in the end relatively-unattractive metaphysical pictures of how language operates). Wittgenstein tries to show why the purported self-induced or automatic disambiguation of any self-reference sentence has not been coherently defined. His main argument, the reader will perhaps now be unsurprised to hear, runs roughly as follows: Nothing in a sentence (or alternatively, a rule) itself ensures its application (or sense). It is only because sentences (and rules) are part of human practices that they are meaningful.
Note, however, that this does not mean that it is thus human agreement (in the ordinary sense of that word) that determines the sense of a sentence (no voting is involved here; see PI 240-2). Normally, there simply is no wedge between the sentence and its meaning (the rule and its application) any such wedge can only be inserted afterwards (by the analyst or theorist). As Sharrock and Button in their discussion of rule-following argue: understanding the rule and understanding what to do are the same thing, i.e., learning to follow a rule is learning what to do. Furthermore, as one of us has argued extensively elsewhere, there need be no implicit metaphysics of rules present here.
Wittgenstein, in PI 86, argues that there is no such thing as a self-interpreting item of language, and that seemingly magical or metaphysical connections between, for example, objects and designations, are just inchoate reflections of our grammar. For example, a railway timetable does not literally tell one how to use it, even if it contains lots of horizontal and vertical arrows. Likewise, as already intimated: a sentence alleged to refer to itself does not do so by itself. There is no pointing to oneself, simpliciter, unless one is an agent (e.g., a human), no matter what arrows (visible or otherwise) one contains or exhibits.
Self-reference as pointing to something else
Sentences that are cited as apparent attempts of unambiguous pure self-reflexivity can at best be taken to be almost implying or pointing toward some other string of words, and declaring of it that it is a sentence, or that it makes sense. A moments reflection will show us that outside of high theory, outside the average English or Philosophy Department (and perhaps some philosophically-misled mathematicians see below), this is the way putatively self-referential sentences are taken: not in isolation, but as referring to some other (usually following) proposition.
In short, in common parlance, outside certain very peculiar philosophical/logical contexts (contexts in which the relevant thought-community enforces upon its members the idea that there is such a thing as (de re) self-reference), the only types of self-reference that may relatively unproblematically be said to exist are part-whole or attribute-thing relationships (as in metonymy, ordinary cases of recursion, references to oneself, and so on). Sentences which the philosophically-minded will force to conform to a self-reflexive mould are, in normal contexts, read in effect as ending in colons (or there is a but waiting to happen, as in: I know Im a cad and a liar, but you must believe me this time when I tell you that ). Compare (imagining, perhaps, an explanation of certain technical aspects of grammar to a non-native speaker):
Case One:
A: This set of words is an English sentence. English is not a Romance language..
Case Two:
A: This set of words is not an English sentence.
((pause))
B: What set of words do you mean?
A: Oh, sorry; this set: English not Romance language is a..
In Case Two, As first remark need not be heard as self-referring, just because it took a while before A made clear what he was doing with the remark!
Wittgenstein supposed that there are two things that one can equally well do with, for example, I am lying. (or: This sentence makes sense.). Either one simply excludes it from the language-game as ill-formed (through making, if you like, a useful and utterly reasonable ad hoc alteration in the ordinary grammatical rules). Or, if not, then one can stress that it is just as little worth attempting to worry about if actually spoken as if simply read as part of an arcane pedagogic discussion or such like:
Is there harm in the contradiction that arises when someone says: I am lying.So I am not lying.So I am lying.etc? I mean: does it make our language less usable if in this case, according to the ordinary rules, a proposition yields its contradictory, and vice versa?the proposition itself is unusable, and these inferences equally; but why should they not be made?It is a profitless performance! Its a language-game with some similarity to the game of thumb-catching. (RFM, I, App. III, 12, p.120)
Such a contradiction is of interest only because it has tormented people, and because this shews both how tormenting problems can grow out of language, and what kind of things can torment us. (RFM, I, App. III, 13, p.120)
It might also be said: his I always lie was not really an assertion. It was rather an exclamation. (RFM, IV, 58, p.255)
One might of course say, for example, I am lying. I didnt really just get off the bus, at the end of a long cock-and-bull story; but in that case I am lying would refer to the preceding remarks (or possibly, in some troublesome cases, to the succeeding remarks as well). In other words I am lying. would not refer to itself.
In sum, we have tried to argue that we have yet to encounter anywhere or anywhen a convincing example of something that we would actually definitely want, all things considered, to refer to as a sentence referring to itself. (As we shall see, this is of some considerably import, when it comes to the philosophy of mathematics, where in effect it is sentences that definitely in and of themselves refer to themselves that is needed, for certain alleged philosophically-inflected results to be deduced.)
So is there any self-reference at all?
We are of course not denying that there is such a thing as, for example, the conclusion of a paper referring back to earlier parts of the same paper. However, we are questioning whether there is such a thing as the conclusion of a paper referring (simply and only) to itself. We have tried to demonstrate that aside from certain, philosophical or pedagogical (but never simply practical) purposes, there seems to be no pure self-reference.
Indeed, we can go further and say this, analogously to the upshot of earlier discussions: even if one were to allow that example (of the conclusion of a paper referring back to itself alone), we would be no nearer to an unambiguous de re instance of self-reflexivity because ambiguity is built into the concept of a conclusion to a paper which conclusion is self-referential. If it were purely self-referential, in virtue of what would it be the conclusion of the paper in question, or even part of the paper at all?
There is nothing in principle to stop one from having a practice of taking some pieces of language (or better: some linguistic actions) as being self-reflexive. Indeed, in some instances it seems obvious that this may be the best way to describe things (e.g., the looking up of dictionary in a dictionary, or the occurrence of the word orthography in the discipline of orthography). For we humans can choose, ceteris paribus, to make language work in all sorts of different and novel ways for us. We can choose, even, to take a sentence such as This sentence has five words. as referring to itself, if it might serve certain purposes to do so (e.g., teaching a child numbers). But that has not yet indicated a sense for a sentence I am a liar. supposed to refer to itself. For without even a potential use, we do not yet have meaning, or sense. That is to say, it is very hard to see what one could do with a sentence such as I am a liar. apart from simply trying to confuse or amuse someone.
In sum, none of the present discussion should lead one to claim that there is a thing called self-referentiality apart from certain specific human practices. Neither has pure self-reference been coherently elucidated.
Guns and people
In an important sense, then, it is only people who refer, not sentences. This might sound like the U.S. National Rifle Association's infamous slogan that Guns don't kill people; people kill people. but the NRA's slogan would actually be quite reasonable, unobjectionable, if it were amended to read: Guns don't kill people, people kill people ... usually with guns.: only then one doubts whether it would have quite the rhetorical and political impact that the NRA hopes for)... The correct analogy then is this: "Sentences don't refer; people refer... usually with sentences.
This is what we are claiming. And it should hardly be controversial at all, once one has thought about it for a moment. Sentences do not refer in and of themselves, any more than guns kill in and of themselves (thus far, at least, the NRA are right!). People refer by means of using sentences. Some of those sentences already have a clear reference, given a certain human practice or language-game. Some do not; and purportedly self-referential sentences are among their number. Far from having an absolutely obvious interpretation, most self-referential sentences are always liable in practice to evoke bizarreness reactions, until a community has instituted such practices, i.e., has institutionalised a particular reading of the kind of strings in question.
To give an example in a different key: one should, to say the least, beware of thought-experiments involving worlds where everything is its own name! To avoid misleading anthropomorphising of objects, be they real or abstract (e.g. linguistic, logical), one should perhaps keep in mind that the best thing to say hereabouts is surely that only humans directly self-refer (e.g., I am almost 2 metres in height., I am not dying.). And there are reasonably strict grammatical limits to even such self-reference, as we saw earlier. In particular, one should not think that isolated, idling self-references involve self-reference de re, that is, outside the context of our sometime voluntary determination that they should do so. Again, such voluntary determination is over and above the sense in which all of language, trivially, is part of human practice. It is rather more akin to the decision required when two rules of a game are taken to clash with one another.
Mathematical self-reference
Having dealt with some ordinary instances of self-reference, let us now deal with perhaps the most famous example of self-reference, namely Gdels Incompleteness Theorem (GIT) and his formally undecidable sentence P that is supposed to say I am not provable.
The foundations of mathematics: mathematics or philosophy?
Before tackling GIT directly, some remarks about the context in which it was developed are necessary. Gdels theorem was a contribution to discussions about the foundations of mathematics, in particular in the form developed by Hilbert, which is often called metamathematics or proof theory.
Hilberts idea was to formalise classical mathematics (e.g., set theory or number theory) as a formal theory, i.e., to restate set theory or number theory in a logically more perspicuous way and to make explicit the notion of formal proof by writing down all admissible inference rules (how to arrive at new propositions out of previously proven ones). The goal was to reduce all of mathematics to a minimal number of axioms (the formalisation of the classical theory) and additional inference rules in order to see more clearly and be able to demonstrate that all inferences made within the theory are valid and true (and therefore do not lead to paradoxes). Gdel in 1933 summarised Hilberts project thus:
a perfectly precise language has been invented by which it is possible to express any mathematical proposition by a formula. Some of these formulas are taken as axioms, and then certain rules of inferences are laid down which allow one to pass from the axioms to new formulas and thus to deduce more and more propositions, the outstanding features of the rules of inference being that they are purely formal, i.e., refer only to the outward structure of the formulas, not to their meaning, so that they can be applied by someone who knew nothing about mathematics or by a machine. [This has the consequence that there can never be any doubt as to what cases the rules of inference apply to, and thus the highest possible degree of exactness is obtained.]
This quote nicely exhibits that at least part of the motivation of metamathematics was philosophical in nature, namely to provide mathematics with secure foundations (of a roughly Cartesian kind). As Shanker puts it:
the whole point of meta-mathematics lay in its deliberate transgression of the boundary between mathematics and philosophy; viz. in Hilbert's conviction that he could use this tool to solve mathematically what were au fond philosophical problems.
Note further that Cavells suggestions about our (human) tendency to avoid responsibility could fruitfully be applied to the development of Hilberts metamathematics at the beginning of the 20th century. By conceiving of mathematical proofs as meaningless symbol manipulation, so that they can be applied by someone who knew nothing about mathematics or by a machine, Hilberts philosophical project was to put the responsibility for mathematical truth on the well-functioning of a machine. In other words: Hilbert wanted to dissolve any sense of rational beings having responsibility for the certainty of mathematics. Hilberts project was motivated by the wish to avoid the kind of (Cavellian) responsibility and acknowledgement that we opened this paper by mentioning. Polemically, we might say that Hilbert wanted humans (or at least, mathematicians) to be so enchanted by the words and symbols of mathematics that they would seem to work their magic without any participation by us at all.
Finally, note that within Hilberts metamathematics (to which Gdels theorem was a significant contribution) we have now two notions of proof: firstly, formal proofs within the formal system itself, and, second, classical proofs (metamathematical proofs), the proofs of the mathematical theory to be formalised and all proofs about the formal system itself. Gdels own proof in his famous 1931 paper was of the latter kind: it was a classical proof about formal systems but not itself a formal proof.
Moving on to Wittgensteins remarks about metamathematics, we think that Wittgensteins position could be summarised by saying that Wittgenstein did not question the sense of theorems and proofs (e.g., by Hilbert or Gdel) with respect to the former (the study of formal proofs) but wondered whether they had shed any light on the latter (the study of classical proofs). In other words, Wittgenstein had no quarrels with the study of a new mathematical structure called formal proofs (which, in a sense, is not in principle different from, say, algebra, number theory, or analysis). However, Wittgenstein did wonder whether the work of Gdel or Hilbert had really helped at all to clarify any philosophical issues in the neighbourhood, such as the question of what a mathematical proof is a question that has troubled philosophers since the time of the Ancient Greeks.
Finally, the same distinction, that between formal and classical proofs, must be made with respect to mathematical truth. Tarski developed a notion of mathematical truth (based on model-theoretic arguments) that is nowadays typically accepted by mathematicians (and by a large number of philosophers) working in proof theory or model theory. However, the same basic question applies: has Tarskis definition of formal truth shed light on the philosophical question of mathematical truth that has troubled philosophers since (at least) Plato?
Gdels Incompleteness Theorem
We now come directly to a discussion of Gdels famous (First) Incompleteness Theorem (GIT) and Gdels construction of a formally undecidable sentence P. We agree with Floyd and Putnam that Wittgenstein at no point questioned the metamathematical (in Hilberts sense) contribution of GIT. In other words, Wittgenstein accepted GIT as a genuine contribution to proof theory (metamathematics). However, Wittgenstein objected to the view that the theorem had an impact on the questions that troubled Wittgenstein himself, namely about mathematical truth and the nature of proofs in general including Gdels own proof of the GIT. The disagreement boils down to the question whether GIT is only a contribution to the mathematics of formal proofs or also a contribution to the philosophy of mathematics. In other words: whether it is a contribution to the relatively recent branch of mathematics called proof theory or whether it is a contribution to philosophical questions regarding mathematical truth.
The view, that Gdels theorem has implications for philosophy has been (partly) due to Gdel himself. Gdel starts his paper with a metaphorical summary a summary that is not necessary for the subsequent technical discussion, but a summary that might have led many philosophers astray.
There are true but unprovable propositions in mathematics is misleading prose for the philosopher, according to Wittgenstein. It fools people into thinking that they understand Gdel's theorem simply in virtue of their grasp of the notions of mathematical proof and mathematical truth. And it fools them into thinking that Gdel's theorem supports or requires a particular metaphysical view.
In other words, Gdels summary does not specify what kind of truth and what nature of proof we are talking about (classical or formal). Some philosophers take this summary to be the only mistake in Gdels paper.
Puzzles of the infinite
Let us quickly comment on the purely technical content of the Gdel proof. In Gdels proof it is necessary to talk about all the (infinitely many) sentences in the formal system. In a way, Gdel extends Cantors diagonalization procedure (to show that the real numbers have a greater cardinality than the natural numbers) into a procedure to reason about formal proofs (hence Gdels proof idea is sometimes called a double-diagonalization procedure). We are thus in the realm of mathematical infinity and should expect that results may be surprising to the non-mathematical reader.
As Watson, in an article that (according to Floyd and Putnam) was close to Wittgensteins position, reminds us: all of the problems or questions that arose in the so-called foundation crisis of mathematics were connected to questions of mathematical infinity, especially when treating a process as a totality:
all the remarkable problems and discoveries of the Foundations of Mathematics, the paradoxes of the theory of aggregates, Russell's theory of types, with its axiom of reducibility, Cantor's arithmetic of transfinite numbers, with its insoluble problems such as the continuum problem, the problems connected with functions in extension and the multiplicative axiom all these merely express in one way or another the well-known difficulties which arise when we attempt to treat an infinite process as completed.
Again, we have no quarrels with mathematical developments in this area, e.g., extending the idea of number from the finite to the infinite. However, one needs to be very careful as to bear in mind that we are now in a new mathematical realm, which may be very far removed from our lives.
I am not provable
To progress now the arguments advanced in this paper towards their conclusion, let us take a closer look at the formally undecidable sentence P. Again, the question is not whether the sentence does the mathematical work assigned to it within Gdels proof (it does), but what one can say about the sentence or what it can say.
A first and critical question might be whether I am not provable is the only translation or rendering of the sentence P. That is to say, the question is not whether it can be translated this way (it can) but whether this is the only possible translation. In other words: What ensures that this is the only possible translation? Something in the sentence itself? Again, the question is not so much whether one can read the Gdel sentence as referring to itself as whether one is compelled to.
Secondly, in line with earlier arguments, we might wonder the sentence itself can say anything? Or, put less strongly, in which sense (if any?) the sentence P says I am not provable.. Does the sentence itself say this? How?
Do not forget that the proposition asserting of itself that it is unprovable is to be conceived as a mathematical assertion for that is not a matter of course. It is not a matter of course that the proposition that such-and-such a structure cannot be constructed is to be conceived as a mathematical proposition.
That is to say: when we said: it asserts of itself' this has to be understood in a special way. For here it is easy for confusion to occur through the variegated use of the expression this proposition asserts something of ....
In this sense the proposition 625 = 25 ( 25 also asserts something about itself: namely that the left-hand number is got by the multiplication of the numbers on the right.
Gdel's proposition, which asserts something about itself, does not mention itself. (RFM, VII 21, pp. 385-386)
Wittgenstein here is questioning who or what is doing the mentioning in mentioning itself. P, in a sense, is only a string of symbols (on an imagined piece of paper). P of itself does not know anything about the natural numbers or of anything. Neither can P do or say anything.
We might ironically sum up the appropriate conclusion to be drawn, then, as follows: isolated cases of purported self-reflexivity are undecidable, aside from an (always finally arbitrary) decision to decide them, one way or another. Only a decision by us a decision for which we as rational agents and as language-users take responsibility allows us to see that, for example, a certain number represents a certain formula, which could be taken to represent a certain statement, and so on. Only we can decide that a certain formula may be taken to say that it is unprovable. But the very necessity of making a decision is enough to prove that some quite different informal manoeuvre is needed to prove GIT (in its philosophical/prose) interpretation. For the Gdel sentence, in isolation, simply cannot truly be said simply and definitely to refer to itself (though it must, to do the philosophical job it has had assigned it).
Indirect self-reference
The Gdel sentence does of course not simply self-refer. The trick in Gdels proof lies in the distinction between syntax and semantics. Thus one can see that there is apparently no self-reference in and of itself here, but only through some particularly cleverly working within the previously-defined rules of syntax and interpretation of certain logico-mathematical systems. Gdel mentioned this fact in a very striking way in the course of his own published proof:
We therefore have before us a proposition that says about itself that it is not provable [in PM].*
*Contrary to appearances, such a proposition involves no faulty circularity, for initially it [only] asserts that a certain well-defined formula (namely, the one obtained from the qth formula in the lexicographic order by a certain substitution) is unprovable. Only subsequently (and, so to speak, by chance) does it turn out that this formula is precisely the one by which the proposition itself was expressed.
The key point in all the technical tricks in the Gdelian arsenal comes down to this. Let us allow that the arrow from This back to the sentence itself (in the diagram on page PAGEREF _Ref109710333 \h 5) passes through several intermediate stages and follows a long and devious trajectory before it returns to its target. Why should this convince one of anything? Why should it change anything? Why should it make any difference to the logic of the situation if the arrows are long or short? We could easily adapt the diagrams earlier to suit; the very same ambiguities will still arise. In short: indirectness or indirection buys you nothing.
Syntax alone does not generate referentiality (as syntax cannot do anything), while semantics gives an infinite regress (as we move from mathematics to meta-mathematics, from meta-mathematics to meta-meta-mathematics, and so on compare our three figures on page PAGEREF _Ref109710333 \h 5 above). This point is immanent in the Wittgensteinian sense of grammar, according to which a syntax not conterminous and simultaneous with a semantics is closer to being a nothing than even to being a something waiting for an interpretation to be imposed on it.
Again, the point we are trying to make is: there is nothing in the sentence itself that ensures self-reference. Note that Gdel in the above quotations says Only subsequently [] does it turn out. Gdel thereby implicitly refers to a human for whom it turns out (because sentences do not notice or see anything).
Conclusion
With respect to the debate between Floyd-Putnam and Bays, what is at stake is the question of whether the mathematical contribution to proof theory has settled or even impacted upon the philosophical questions about truth and proof that rightly concerned Wittgenstein. We believe that our arguments provide independent support for the correct conclusion of Floyd-Putnam, in their first J.Phil piece on Gdel, that it is not a mathematical result, but rather a metaphysical claim, to say that, if PM is consistent, then some mathematical truths are undecidable in PM. Wittgenstein, they write, does not want simply to deny the metaphysical claim; rather, he wants us to see how little sense we have succeeded in giving it.
And we believe that the independent support that we have provided for their conclusion comes from the more general good reason we have given for believing that the mysteries, paradoxes, and logical results that are thought to follow from the consideration of isolated self-referential language or self-reflexive phenomena are in nearly all cases quite illusory. They are only not illusory when we take responsibility for them and then, it is we language-users, not the sentences in themselves, that are the cause of any resulting confusion.
In Remarks on the Philosophy of Psychology I, 65, Wittgenstein makes the following remark:
In the language of a tribe there might be a pronoun, such as we do not possess and for which we have no practical use, which refers to the prepositional sign in which it occurs. I will write it like this: I. The proposition I am ten centimetres long will then be tested for truth by measuring the written sign. The proposition I contain four words for example is true, and so in I do not contain four words. I am false corresponds to the paradox of the Cretan Liar. The question is: What do people use this pronoun for? Well, the proposition I am ten centimetres long might serve as a ruler, the proposition I am beautifully written as a paradigm of beautiful script.
What interests us is: How does the word I get used in a language-game? For the proposition is a paradox only when we abstract from its use. Thus I might imagine that the proposition I am false was used in the kindergarten. When the children read it, they begin to infer If thats false, its true, so it is false, etc. etc. People have perhaps discovered that this inferring is a useful exercise for children.
What interests us is: how this pronoun gets used in a language-game. It is possible, though not quite easy, to fill out a picture of a language-game with this word. A proposition like I contain four words might, for example, be used as a paradigm for the number four, and in another sense so might the proposition I do not contain four words. A proposition is a paradox only if we abstract from its use.
Evidently, Wittgenstein was not operating on the basis of a dogmatic prejudice about uses or about self-reference: one can talk perfectly intelligibly of words (I, this, etc.) or sentences referring to themselves. But such use is never forced upon one. It follows only from contexts of use that one consents to. And this is what is fatal for Gdels Theorem, under its usual philosophic interpretation: for Gdels Theorem to yield, as a philosophy, that there are undecidable truths, that in particular there are mathematical truths that are undecidable in PM if PM is consistent, the standard (Gdel/Bays) interpretation of the Gdel sentence would have to be forced upon one. But it just never is.
Bays concludes his critical response to Floyd-Putnam thus:
There is a perfectly good and indeed, a perfectly canonical interpretation of arithmetic under which Wittgensteins P really does say P is not provable. Given this interpretation, Gdels Theorem helps to show that there are true but unprovable sentences of ordinary number theory. Nothing in Wittgensteins remarks or in Floyd and Putnams analysis of those remarks should lead us to think otherwise.
Bayss argument is based on the view that definitions of formal proofs and formal truth have settled questions about classical proofs and classical truth. And his conclusion is based therefore on the assumption that proof theorists have made a purely mathematical decision on how to interpret P. But has this decision has, in fact, the central work of Hilbert or Gdel shed any light on ones intuitive notion of self-reference? (Does one for instance understand the liar paradox better having gone through Gdels proof?) We have suggested not. And so, for philosophers qua philosophers, a decision to interpret P as saying P is not provable remains a decision, a metaphysically-freighted decision: A decision not to give a clear sense to something that is (so we are told) mathematically clear.
To think as a philosopher that one has to accept that P means P is not provable is to think that one cannot accept mathematics without accepting that mathematics cancels out the human. That it cancels out all that Wittgenstein was drawing to our attention, for instance, in his rule-following considerations. Such an avoidance of responsibility is exactly what the best Wittgensteinian philosophers, following Stanley Cavell, have been warning against for some years now. It is time to apply the lesson that they have been trying to teach to hard problems in the philosophy of mathematics. Sure, sentences can self-refer if that is what one wants them to do. But that wanting and that surety one cannot evade responsibility for. (No matter what expositors of Gdel in the philosophy of mathematics might think that they want, or think that they think.)
As in Stanley Cavells work, especially his Must we mean what we say? (Oxford: OUP, 1976); see also Denis McManuss impressive forthcoming book, The Enchantment of Words.
J. W. Cook, p. 23, Whorfs linguistic relativism: Part 2. Philosophical Investigations 1:2 (1978), 137.
McManuss book is a powerful critique of this tendency. Compare also here the cautions of the New Wittgensteinians, e.g. Ed Witherspoon.
For more detail on what we mean here, see Read and Hutchinsons Whose Wittgenstein?, a review essay of four recent works on Wittgenstein, in Philosophy July 2005, and Read and P. Hutchinsons Towards a perspicuous presentation of perspicuous presentation, forthcoming in D. Moyal-Sharrocks Essays on Wittgensteins Philosophy of Psychology.
An example at the intersection of all these is Michel Foucaults intriguing and highly amusing (but problematic) short work, This is not a pipe (Berkeley: University of California Press, 1983). The problem is that Foucault at times (e.g., p.27, p.30) appears to assume that statement really can refer of themselves (de re) to themselves. A happier way forward that avoids the problems in Foucaults account would be the work of ethnomethodologists such as Lena Jayyusi (The reflexive nexus: photo-practice and natural history. Continuum: The Australian Journal of Media & Culture 6:2 (1993), 2552). Jayyusi makes the following point about scenes depicted by photographs: It is not that the scene speaks for itself; but that it speaks itself in such a way that it can provide the ground and object (the topic and resource) of our speaking for and about it, in the various ways that we do. (p.45)
See the Conclusion below, for a possible exception or two to this generalisation.
W. Sharrock and G. Button, Do the right thing! Rule finitism, rule scepticism and rule following in Human Studies 22:24 (1999), 193210.
Read and W. Sharrocks Kripkes conjuring trick, The Journal of Thought 37:3 (2002), 65-96, and Reads brace of joint papers with James Guetti: Meaningful Consequences, The Philosophical Forum XXX:4 (1999), 289-314, and Acting from Rules, International Studies in Philosophy XXVIII: 2 (1996), 43-62.
A reminder: it is humans that tell each other things. Other uses of the term are typically metaphors. W. Sharrock and W. Coleman (It dont mean a thing: On what computers have to say. Communication & Cognition 33:1/2 (2000), 8395) remind us of the difference between two kinds of telling the time: it is obvious that a calendar tells the date and a clock tells the time in a different way from a person does (p.88). We might want to add that in these discussions it is easy to conflate these two senses.
Cf. the quotation by Jayyusi in Footnote NOTEREF _Ref109874592 \h 5.
K. Gdel, The present situation in the foundations of mathematics. In Collected Works, Volume III: Unpublished Essays and Lectures, pp. 4553 (Oxford: Oxford University Press, 1995). Lecture delivered at the meeting of the Mathematical Association of America in Cambridge, MA, December 2930, 1933.
S. G. Shanker, p. 224, Wittgenstein and the Turning-Point in the Philosophy of Mathematics (London: Croom Helm, 1987).
Note that we do not want to attack Hilberts mathematical contribution, but are only questioning the impact of Hilberts innovations on philosophical questions of what mathematical truth and certainty are.
We are not denying that Gdels proof could be formalised (e.g., see N. Shankars Metamathematics, Machines, and Gdels Proof, Cambridge: Cambridge University Press, 1994), but want to emphasize that proofs about formal proofs are not themselves formal proofs.
Livingston pursues his sociological project in The Ethnomethodological Foundations of Mathematics (London: Routledge, 1986) from a similar starting point. He notes that [t]he early Greeks were both amazed and perplexed by mathematical proofs and by the fact that [t]he mathematical theorem was proved as something necessarily true, a fact anonymous as to its authorship, available for endless inspection, established for all time and this as a required feature of the actual demonstration itself (p. ix). Livingston then discusses the great interest in questions about mathematical truth and the nature of proofs at the turn of the 20th century. However, for Livingston this new interest in the foundations of mathematics was of a different kind, namely motivated by attempts of demonstrating the incorrigibility of mathematical proofs, i.e., to attempt to construct indubitable foundations for mathematics practice (p. x). For Livingston (and, so we would claim, Wittgenstein) these new developments were not a direct contribution to the questions that had troubled the early Greeks, which, so Livingston, remained untouched and unexamined (p. x).
For example, Bays, On Floyd and Putnam on Wittgenstein on Gdel, The Journal of Philosophy 101:4 (2004), 197210.
Floyd and Putnam (Bays, Steiner, and Wittgensteins notorious paragraph about the Gdel theorem, forthcoming The Journal of Philosophy) note that the technical-mathematical definition of truth, as truth-in-the-model N (based on Tarski) is often substituted for the philosophical question of truth. However, whether mathematical model theory is able to solve or in any way contribute toward solving the philosophical question of truth is precisely the issue.
J. Floyd and H. Putnam, p. 628, in A Note on Wittgenstein's Notorious Paragraph about the Gdel Theorem, Journal of Philosophy 97:11 (2000), 624-632.
Livingston (op. cit., p.31) seems to be arguing along similar lines: The argument that will be made is not that a proof of Gdels theorem does not prove what others have claimed it to prove; instead the origins of the rigor of a proof of Gdels theorem will itself be examined and the claim advanced that that rigor of its local work. Thus this argument points to the primordial character of the activity of doing mathematics over some conception of mathematics-in-itself. We take Livingston to be asking: Where does the truth of Gdels proof of the GIT, a proof written in a classical not a formal style, come from?
J. Floyd, p.299, Prose versus Proof: Wittgenstein on Gdel, Tarski and Truth in Philosophia Mathematica 9:3 (2001), 280-307.
O. Helmer, Perelman versus Gdel, Mind 46:181 (1937), 5860, writes: As a matter of fact, Gdel did make one mistake, namely that of writing an introduction to his paper, in which he sketches the main idea of the proof, without of course making any claims for precision. It is the actual lack of precision in these introductory explanations that has misled Perelman and that may mislead others. It can easily be seen that Perelmans objections are applicable only to these inexact explanatory remarks, and not to the exact formal demonstration given later in Gdels paper. (p.59)
Take as an example Hilberts hotel: Hilberts hotel is a thought experiment to demonstrate the bizarre notion of infinity within mathematics. Imagine a hotel with an infinite number of rooms, which are all occupied. One day, a train arrives with an infinite number of new guests who all want a room. The hotel manager is horrified: what shall he do? No problem, says the helpful mathematician, here is what you do: you let all your current guests pack and stand in front of their doors. And then they all go to the room with the double number of the room that they are in. That is to say, if you are in room 3 you go into room 6, if you are in room 452 you go into room 904. Generally speaking: n ( 2n. This means that all rooms with an even number are now occupied, whereas all rooms with an odd room number are free. So we can now put the new guests into the rooms with odd numbers. The example shows the oddity of the idea of an infinite totality.
A. G. D. Watson, p. 450, Mathematics and its foundations, Mind 47:188 (1937, 440451.
Cf., S. G. Shanker, op. cit., Chapter 5.
Floyd and Putnam (op. cit., p. 628) remark that: the translation of the famous Gdel sentence P as P is unprovable in PM is not cast in stone. Rodych (Wittgensteins inversion of Gdels theorem, Erkenntnis 51 (1999), 173-206), also notes: We do not need to assume a natural language meaning for P (e.g., an English meaning) to obtain the threatened contradiction, for it is just a number-theoretic fact that an actual proof of P would enable us to calculate the relevant Gdel numbers and thereby arrive at ~P by existential generalization. (p. 182)
Bays (Op cit., pp. 206-7) seems to acknowledge this: there is nothing in the formal structure of P that is, in Ps very syntax which forces us to interpret P as P is not provable. However, he goes on to say that [t]here is a perfectly good and a perfectly mathematically respectable interpretation of the language of arithmetic under which P expresses the fact that P is not provable. (p. 208) For Bays, this is the canonical interpretation (p. 204). We do not question the status of this as a canonical interpretation within mathematics (in particular, proof theory), but wonder whether it has or should have this status within philosophy (i.e., for questions of the kind that troubled Wittgenstein). Again, Floyd and Putnam (op. cit., p.11): Why Bays thinks that we should assume that the notion of interpretation is reducible to the model-theorist's notion of N he does not say. Nor does he explain what he means by a perfectly canonical interpretation.
Parsons and Kohl (Self-reference, truth and provability, Mind 69:273 (1960), 6973), also note that Gdels sentence, in contrast to the liar paradox, constitutes only indirect self-reference: it is shown, by a correlation of formulae with numbers, that the statement of arithmetic which Gdels formula may be interpreted to express is true if and only if the formula is itself not provable in the system S. It is only in this derivative sense that the formula asserts anything about itself. It is clear that this weak sense of self-reference is quite different from the first two senses and does not, of itself, give rise to paradoxes. (p.70)
K. Gdel, p. 19 and p. 43 in On formally undecidable propositions of Principia Mathematica and related systems I (1931). In S. G. Shanker (Ed.), Gdels Theorem in focus (London: Croom Helm, 1988).
Op.cit., p.632; emphasis ours. This aspect of our line of thought on Gdel, distinguishing rigorously, as do Putnam and Floyd, between Gdels mathematical innovations (with which we have no quarrel) and the philosophical consequences alleged to flow from diagonalization, is also close to Stuart Shanker Wittgensteins remarks on the significance of Gdels theorem, in S. G. Shanker (Ed.), Gdels Theorem in focus, pp. 155256 (London: Croom Helm, 1988), as well as to Charles Sayward, in his On some much maligned remarks of Wittgenstein on Gdel, Philosophical Investigations 24:3 (July 2001), 262-270. Sayward remarks (pp. 265-266): Wittgenstein considers a familiar argument (not Gdels!) that the theorem [GIT] has an important philosophical result: there are arithmetical truths which are unprovable. It would be at odds with Wittgensteins general attitude towards mathematics to question some mathematical result. So why suppose he is doing it here? // The argument Wittgenstein considers purports to establish that there are arithmetical sentences which are true though unprovable. // But Gdels proof does not go so far, for his incompleteness theorem has only syntactic content. It asserts no more than that there is no effective and consistent set of axioms sufficient for the arithmetic of the natural numbers which provides, for each sentence of arithmetic, a derivation of either it or its negation. In the course of establishing this claim it shows as well that there are arithmetic generalizations not themselves derivable from the axioms at hand yet such that each of their numerical instances is so derivable. But on the topic of truth it is silent.
Op. cit., p. 210.
Can sentences self-refer?: Page PAGE 1 of NUMPAGES 30
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