Gödel’s Theorem over-interpreted: There is no such thing as de re self-reference
It is surely a commonplace in Philosophy Departments that certain sentences can refer to themselves, particularly when couched in logically-perspicuous language(s).
Such is the assumption undergirding the contextualisation of probably the most important result this century in mathematical logic. I refer to Gödel’s First Incompleteness Theorem (Though I will speak only of its prose contextualisation and interpretation; not of the theorem ‘per se’, considered only 'syntactically', as it were... . The formal proof, as Wittgenstein noted, just proves whatever it proves. Any actual applications of it which succeed in being made, one can have no objections to. It is the informal proof, which might be comprehensible to us in prose, which might allow us to understand how a non-self-contradictory instance of self-reference has been accomplished, that has been most influential, that we can usefully talk about, and that is my 'target' here). This Theorem is taken to be of great logical and even philosophical significance when read as constructing the schema for a proposition that refers to itself. This proposition is supposed to ‘say’ of itself that it is unprovable.
But how can a proposition say something of itself, or self-refer, outside of the external factum of our determination that it should be read so to say? “Surely, by means of as it were pointing to itself in the (logical) language in question. The language, once established, is not vulnerable to human decision as to its meaning and interpretation.” Possibly so; but still, how can a sentence point to itself? “We must imagine metaphysical lines of projection, going from the sentence back to itself; just as we might imagine such lines going from the word ‘Sun’ to the Sun.” Very well; let us endeavour to countenance the imagining of such lines.
But now, whence ‘the sentence itself’? Does the Gödel sentence include the metaphysical lines of (self-)referential projection, or doesn’t it? If it does not, we as yet have no guarantee that the sentence refers to itself (It might be interpreted in a myriad different ways, according to the metaphysical picture (of 'lines of projection') that licenses the concept of de re self-reflexivity). But if it does, then we may ask anew whether it is actually unprovable. For it is different than we initially imagined it to be; it must be read as already containing these “metaphysical lines”. In fact, we may ask anew of the Gödel sentence understood to contain the “lines” whether it refers to itself. In order to guarantee that it does, a fresh set of “lines” will be needed (to be constructed) encompassing the previous version of the sentence. It is evident that we are launched here on an infinite regress. Or alternatively: there is no stability to the concept of “the Gödel sentence”; it has not yet been clearly defined.
The problematic here is illustrated below:
"This sentence is unprovable."
- - - - - - - - - - - - -
"This sentence is unprovable." [Fig.1]
[‘Which sentence?’ (This was asked again because the question could arise as to whether or not the arrow was part of the sentence referred to)]
- - - - - - - - - - - - -
"This sentence is unprovable." [Fig.2]
[‘Which sentence?’ (The demonstrative may it seems be called upon to perform infinitely many tasks even of the same general kind. )]
- - - - - - - - - - - - -
And so on and so forth...
It might be thought relevant (and problematic) that the example -- the version -- considered above involves a demonstrative. But this actually makes no difference. One could equally well substitute some other method of 'self-reference' or 'self-involvement' or 'self-inclusion', such as a numbering system (The proposition could be written as "No.1 is unprovable", and called "No.1"), or the happenstance of a location (e.g. The proposition could be written as "The sentence on this blackboard is unprovable", and be the only sentence on the blackboard). The same problem would apply. One would still be to ask whether the sentence was the same sentence any more 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. For two reasons: (i) Because there could be other blackboards in the room or in nearby classrooms, and there could be other sentences that happened to be 'named' "No.1'. (ii) Because without these 'arrows', or some similar device, the sentence could not be truly claimed to be 'pointing to' or 'hooking up with' itself.
To self-refer, a sentence cannot just sit there; it must do something. It must point to itself, or something similar. But when we say this, do we know what we are talking about? Have we made any sense yet out of what we want to say?
Let me make quite clear my point of concern, of attack. The question is not so much whether we can read the Gödel sentence as referring to itself as whether we are compelled to. My claim is that, in prose, a sentence saying of itself simply that it is true, or false, or unprovable, or what-have-you, is not an idea which has yet been made good sense of. As yet, it seems to me at best only poetry, only an attempt at the kind of self-referentiality that Deconstructionist critics (arguably quite mistakenly, indeed incoherently) believe much Modern poetry accomplishes. And therefore we surely do not have to read it as referring to itself! If we do not yet know what it could possibly mean for something to refer unequivocally to itself, it surely follows that we are not compelled so to read it! In short, I am questioning the standard interpretation of the Gödel sentence, that it "says of itself that it is unprovable'. For I do not yet know what this means. I do not yet know how to read the Gödel sentence.
Thus even if on a formal level meta-mathematicians can show something that they choose to call the undecidability of the Gödel sentence (and thus to prove in formal terms, for example, what they choose to call the unprovability of the Continuum hypothesis in Zermelo-Fraenkel set theory), even if they can give a mathematical basis for the claim that the Gödel sentence is correctly interpreted as a statement about its own provability, we are no closer to understanding what the hell they are talking about. I think, that is, we are no closer to understanding what Gödel's Theorem might mean -- for we are no closer to understanding what pure 'self-reference', such as in the above examples, such as in "I am unprovable", might be.
Should any of this surprise us? Not those of us familiar with Wittgenstein’s ‘Philosophical Investigations’. There, less cryptically than at moments in the parts of the ‘Remarks on the Foundations of Mathematics’ explicitly on Gödel and Hilbert, which some readers have found inaccessible or obtuse, are all the resources needed to show that the mainstream conceptions of ‘self-reflexivity’and 'self-referentiality' are in the main incoherent, and to explain in prose just why the purported self-induced/automatic disambiguation of “the Gödel sentence” challenged and dissected in the figures above has not been coherently defined. I am arguing that "the Gödel sentence", and indeed its prose contextualization(s), are simply appealing forms of words that can and will be rendered unappealing, unattractive.
An upshot of PI para.86 (highly-relevant here, since it concerns different possible ways of reading a table), and indeed in a sense of the entire rule-following-considerations and anti-private-language-argument, is that there is no such thing as a self-interpreting item of language, and that seemingly magical or metaphysical connections between (e.g.) objects and designations are just inchoate reflections of our grammar. The railway timetable one visualises does not tell one how to use it (no more than does the timetable on the wall in the station), even if it contains lots of horizontal and vertical arrows. Likewise, a proposition alleged to refer to itself does not do so itself, (there is no pointing to oneself, simpliciter, unless one is an agent (e.g. a human), as I explicate more fully below), no matter what arrows (visible or otherwise) it ‘contains’.
The decisive sequence runs from c. para.s 492-509 of PI. Compare especially para.502:
Asking what the sense is. Compare:
“This sentence makes sense.”-- “What sense?”
“This set of words is a sentence.”-- “What sentence?”
Both ‘assertions’, Wittgenstein implies, are to be “withdrawn from circulation” (like the Gödel sentence, perhaps?...), on grounds of being, in isolation, senseless (and it is not their sense that is senseless -- see para.500).
The questions put to them, in both cases (no gap between syntax and semantics here...), ask what it is that they say, and the answer is: so far, nothing at all (And note that the same point would have been made, if by a somewhat different route, if it had been asked in the second case, “What set of words?”; or indeed if it had been asked in the first case, "What sentence?"). They are cited as apparent attempts at unambiguous 'pure' self-reflexivity, attempted exclusions of the practices of the reader, but both can at best only 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 moment’s reflection will show us that, outside of high Theory, outside the average English or Philosophy Department (and perhaps some philosophically-misled mathematicians), 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, attribute-thing relationships (As in metonymy, ordinary cases of recursion, references to oneself,etc.). 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. Compare (imagining, perhaps, an explanation of certain technical aspects of grammar to a non-native speaker): “This set of words is an English sentence: “English is not a Romance language.”.” “This set of words is not an English sentence.” [Pause] “What set of words do you mean?” “Oh, this set: “English not Romance language is a.”
Wittgenstein held in fact that this point applies just as effectively to the classic logico-linguistic paradoxes, even if one imagines them as actually uttered by a human being. That is, he supposed that there are two things we can equally well do with (e.g.) "I am lying" (or "This sentence is false"). Either we simply exclude it from the language-game, as ill-formed, through making if you like a useful and utterly-reasonable ad hoc alteration in the ordinary 'syntactic' rules (We will return to this mode of response -- it is worth noting immediately that given that the sentences in question were not expressly considered in the course of the setting up of the rules, and were only designed by people with the purpose of being awkward, it's arguably not even worth calling an alteration / adjustment.). Or, if not, then we 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!... 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. 
One might of course say (e.g.): "I am lying. I didn't 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!...). I.e. "I am lying" would not 'refer to 'itself''. I hope, in fact, by now to have made plausible to you that we have yet to encounter anywhere or anywhen a convincing example of something that we would actually want, all things considered, to refer to as 'a sentence referring to itself'.
There is however a critical objection to deal with. It might be put thus:
"It may have been effectually shown above that simple direct statements (e.g. "This sentence is false") do not simply self-refer. Alternatively: we might even be on the point of conceding that and why only humans self-refer directly. But even if one or other of these points were admitted, it would not affect Gödel's proof. For its genius lies in the fact that the Gödel sentence self-refers only indirectly. Once one understands aright the distinction between syntax and semantics, one sees that there is self-reference in and of itself here, not through direct assertion, but only through some particularly clever working within the previously-defined rules of syntax of certain logico-mathematical systems."
This is a crucial objection to my argument, in part because it is a close cousin to Gödel's own interpretation of this central device of his proof:
We therefore have before us a proposition that says about itself that it is not provable [in the Principia Mathematicasystem].*
*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.
Now, if it is a proposition that we have before us that is in question, then Gödel cannot escape a potential Wittgensteinian critique through his sometime resort to propositional Platonism. So, the proposition we have before us (in Gödel's proof) 'asserts' that a certain formula is unprovable, and it 'so happens' that this formula is ... that formula which expresses the proposition (Purportedly, much as it ‘so happens’ that the Morning Star is the Evening Star, one presumes). But the proposition has already to be understood before it is even so much as the proposition that it is (In Gödelian / Philosophy of Language terms: the formula is a 'syntactic' entity, the proposition a 'semantic' entity). So now what is to stop someone from simply and immediately disallowing this formula -- as soon as they figure out that the syntax 'points' toward a contradictory semantics -- as not a real proposition, in much the same way as we can disallow "I am lying", if isolated from anterior or posterior sentences? To put it the other way around: it is only an (unforced) determination to treat the syntax uniformly across diverse cases, i.e. conflating language-games, and thus to treat the Gödel sentence as well-formed despite any and all appearances to the contrary, which enables the proof of unprovability to convince (An analogy might be: a proof of a contradiction within the system of the positive integers based upon the principle that the order of these integers ought not to be affected by dividing through each by any positive integer -- and the ('obvious, allowed for by the arithmetic syntax') inclusion among the positive integers of zero... ). And that 'determination', that 'treatment', is a factor of meta-mathematicians' (and others') motivations and practices, not of the nature of numbers (or of logic) themselves. If the Gödel sentence self-refers 'indirectly', it is this fairly extreme and unwanted (by believers in the proof) kind of 'indirectness' which is in play.
If it appears that the spirit of the objection still has not been answered, perhaps because I have yet fully effectively to assimilate the actual procedure in Gödel’s proof to that interrogated diagrammatically earlier in this paper, let me make two points. Firstly, again, the present paper concerns prose, not either unintentional poetry or so-called ‘uninterpreted (purely syntactic) symbolisms’. Secondly, and tellingly, the key point I think is that all the technical tricks in the Gödelian arsenal come down to this. Let us by all means allow that the arrow from “This” back to the sentence ‘itself’ (in the diagram earlier) passes through several intermediate stages, and follows a long and devious trajectory, before it returns to its target. But why should this convince us of anything? Why should it make any difference to the logic of the situation if the arrows we draw are long or short? We could easily adapt the diagrams to suit; the very same ambiguities will still arise.
In short; indirectness/indirection buys you nothing.
The ‘critical objection’ to my argument just considered therefore does not escape the argument given earlier. Imposing an over-idealized syntax vs. semantics dichotomy on the Gödel sentence cannot help the usual interpretation of its 'self-referentiality'; for syntax alone does not even generate referentiality, while semantics gives me the infinite regress (sketched earlier) that I need to succeed in my argument in the present paper. (This point is immanent in the Wittgensteinian sense of 'grammar' -- according to which a syntax not coterminous and simultaneous with a semantics is closer to being a nothing than to being a something waiting for an 'interpretation' to be imposed on it -- though no such understanding should be needed, I think, to see the point I have been making.)
None of this means that there is no such thing as language in an exposed state, where its form as language rather than its being part of a working/functional grammar is the most noticeable feature of it, is displayed. Arguably, such is the state of much language that literary critics misleadingly or misguidedly refer to as “self-referential”, or “self-reflexive”.
Nor again, of course, should it be denied that there is such a thing as (e.g.) the conclusion of a paper referring back to earlier parts of the paper... . But is it clear that there is or could be such a thing as the conclusion of a paper referring (simply and only) to itself? It is intuitively reasonable, I think, as well as following from the line of reasoning pursued thus far in the present paper, to say loud and clear that there is not, aside I suppose from a decision that we can choose to make in order to allow for there being a class of such linguistic phenomena (Just as, I suppose, we could try, for certain -- notably always philosophical or (ideologically-motivated) pedagogical, never simply practical -- purposes, deliberately to allow that sentences such as “This sentence is false”, in isolation, shall count as grammatically well-formed, even in the Wittgensteinian sense of ‘grammar’. It is just very hard to see what one could do with such a sentence, other than simply try to confuse or amuse someone.). Indeed, we can go further and say this, analogously to the upshot of our earlier discussions: even if one were to allow or purportedly to presuppose such a decision, we should have got no nearer to an unambiguous de re instance of self-reflexivity. This, 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 a part of the paper at all? And: Even if we concede that a string of words could have such a ‘double-function’ -- conclusion and instance of self-reference -- it is clear that there would be metaphysical strings or arrows attached here just as destabilisingly problematic in their implications as in cases we have already detailed.
Further to the purpose, we can note at this point the (further) analogous double-duty that the Gödel sentence, paradigmatically of any item of ‘meta-mathematics’, is supposed to perform: it ought to be both necessarily -- mathematically -- true, and yet about something. How could this possibly be? But enough; by this stage, one is either preaching to the converted or has not succeeded in effecting a conversion.
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. This is because, of any such case, we can always ask, “Has the remark (sentence, text) ended yet; isn’t there something still to follow? Is this the whole of it, that that we can see?” In short, to Gödel’s sentence, we can ask (as diagrammatically shown earlier): “Does this string include its pointing to itself (independent of any reader with an agenda)?Which sentence is being asserted to be unprovable?!” Only a reply by us can avoid directly changing the Gödel sentence into a different sentence. But the very necessity of giving such a reply is enough to prove that some quite different informal manoeuvre is needed to prove the Incompleteness Theorem, in its philosophic/prosaic interpretation! For the Gödel sentence, in isolation, simply cannot truly be said simply and definitely to refer to itself; though it must, to do the job that has been assigned it.
As I have laid out, we should be clear that there is nothing in principle to stop us from having a practice of taking some pieces of language (or, better: some linguistic actions) as being ‘self-reflexive’, and 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). For we 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 (e.g. teaching a child the numbers, possibly) to do so. But that hasn't yet indicated a use for a sentence such as "This sentence is unprovable", supposed to refer to itself. And without a use, we don't yet have a meaning. Thus none of the present discussion should lead us to claim that there is such a thing as self-referentiality, certain specific human practices aside. Nor to claim that a notion of 'pure' self-reference has yet been coherently elucidated.
It might be worth putting this point in the following way: In an important sense, 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 I am claiming. I think it can hardly be controversial, once one has thought about it for even 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 I count purportedly 'self-referential' sentences among their number. Far from having an absolutely obvious interpretation, most 'self-referential sentences' are always liable in practice to evoke bizarreness reactions, until we have instituted such practices as institutionalise a particular reading of them.
To give a different example: We 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), we should perhaps keep in mind that the best thing to say here 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. And, in particular, we 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 (de dicto, or de se). Again, such ‘voluntary determination’ is over and above the sense in which all of language, trivially, is part of / supervenient upon human practices; it is rather more akin to the decision required when two rules of a game are taken to clash with one another. It cannot legitimately be simply read into the grammar, or presumed. The problem is that Gödel and nearly all other interpreters of his Theorem and of other logical devices structurally analogous to the Gödel sentence (if usually simpler than it, and thus still more obviously vulnerable to the present critique) appear to have pre-supposed otherwise; for understandable reasons. For again, if they did not do so, then their proofs and paradoxes would stand there alone, bereft of most of their beauty: they would then evidently simply be artifices we can unmake as easily as we made them...
In short, the mysteries, paradoxes, and logical results that are thought to follow from the consideration of isolated 'self-referential language' or 'self-reflexive phenomena' are, if my arguments are sound, in nearly all cases quite illusory.
Abstract: "Gödel's theorem over-interpreted"
I argue that if Gödel's theorem depends for our understanding of it upon the notion that there are propostions that can refer in and of themselves only to themselves, then it is fatally flawed -- we have yet to understand it; because it is not clear that there is anything to understand. To show this, I draw on Wittgenstein and others to suggest that 'pure' self-reference such as is envisaged in Gödel (and also in the classic semantic paradoxes etc.) is not possible, because it could only occur consequently upon a decision we make to resolve an unavoidable ambiguity in interpretation -- but then it would not be 'pure', it would not not involve us.
 An example at the intersection of all of these is Michel Foucault's
intriguing and highly amusing but problematic short work, This
is not a pipe (
 E.g. Wittgenstein's Remarks on the foundations of mathematics
('RFM'; posthumous, transl. Anscombe --
 "[W]hat does it mean to say that P and "P is unprovable" are the same proposition? It means that these two English sentences have a single expression in such-and-such a notation." Wittgenstein, (RFM Part I, App. III, No.9; p.119). Cf. also Tractatus Logico-Philosophicus (London: Routledge, 1922) 3.332-3.
 An alternative way of putting this point (with which I am less comfortable for Wittgensteinian reasons, but which others might prefer) would be as follows: We have, one might say, covertly introduced a second interpretation of the sentence (considered now as an uninterpreted strng of characters). There could then be argued to be an indeterminacy in the formal language, allowing us to use two or more, mutually inconsistent interpretations (the 'standard interpretation' and the what we might call the 'coding interpretation(s)'), while supposing that only one, consistent, interpretation is being used. Again, the result would be that the appearance of automatic self-reference has only been illusorily achieved.
 Compare Wittgenstein (ibid.): "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"?"
 This connects up with Wittgenstein's remark: "The proposition "P is unprovable" has a different sense afterwards -- from before it was proved. If it is proved, then it is the terminal pattern in the proof of unprovability. -- If it is unproved, then what is to count as a criterion of its truth is not yet clear, and -- we can say -- its sense is still veiled." (RFM Part I, App. III, No.16; p.121)
 It could be argued that Foucault is engaged in a similar exercise to me here around p.25 ff. of his (ibid.). Certainly, I think that Magritte entertainingly presents us with the opportunity to experience various mental conundrums and head-cramps as a result of his art -- arguably, more entertainingly than Russell or Gödel ever did...
There is such a thing as pragmatic self-refutation too (e.g. A sophomore seriously placing a bumper-sticker on their car which reads "They can send me to college, but they can't make me think"; this example is drawn from Bachman and Hintikka's What If?: Toward Excellence in Reasoning (London: Mayfield, 1991)), though it is not properly a variety of self-reference, certainly not 'de re' self-reference (for explication of why, see below, and my "The Unstatability of Kripkean scepticisms", in Philosophical Papers XXIV: 1 (Ap.1995)).
 RFM, Part I, App. III, para.s 12 -13; p.120.
 The difference may in fact be alleged to lie in this: that "I am true" and "I am false" are not true or false in English; whereas (e.g.) roughly speaking "I am provable" is (allegedly) a theorem in certain formal languages/systems. But this may beg the question against Wittgenstein's remark, quoted earlier, that we need to be on our guard against presuming that what it is for something to be true, or indeed provable, in a certain formal system, is always clear ahead of time, ahead of (e.g.) the assimilation of the proof of a surprising new theorem. Or so, in any case, I argue in the text below.
 Quotation from p.19 of, and footnote from p.42 of "On formally undecidable propositions of 'Principia Mathematica' and related systems I" (1931; reprinted in Gödel's Theorem in focus (ed. Shanker; Beckenham: Croom Helm, 1988)).
 Cf. also pp.227-230 of Shanker's (pp.155-256 of Shanker (op.cit.)) "Wittgenstein's remarks on the significance of Gödel's Theorem".
 There may also be new arguments available, independently, to cast doubts on the semantic effeicaciousness even of Gödel's formal proof. See important forthcoming work by David Clemenson.
 I.e. We could of course choose to treat certain unusual and arguably aberrant forms of words as if (vital proviso!) they were de re self-referential, as it there really were such a thing. Similarly, we could choose to say these words, as many meta-mathematicians do: that the Gödel sentence is self-referential in and of itself -- but we would do better not to. Because we would be fomenting great confusion; because we would be deceiving ourselves if we pretended that this had already been decided for us, ahead of time. And yet Gödel's theorem rendered into prose loses all point, surely, if all it says is that we can choose to regard Logicism etc. as closed off to us, if we choose to regard the Gödel sentence as self-referential...
 For detail on this point, and on the broader issues of mathemtical truth, proof, etc. that it raises, the reader is advised to consult Wittgenstein, Shanker, and Juliet Floyd's important paper, "The trisection of the angle and Wittgenstein's remarks on Gödel's Theorem", in Crary and Read (eds), The New Wittgenstein (London: Routledge, 1999).
 There are further ironies here. As
intimated by J.Guetti in his Wittgenstein and the Grammar
of Literary Experience (
 In fact, all this has of course long been known. Compare for instance Plutarch, writing a little after Socrates's time: "[I]t is as if a man should say that the arrow wounded him, and not the archer with the arrow ... for the act does not belong to the instrument, but to the person to whom the instrument itself belongs, who uses it for the act; and the sign used by the power that signals is an instrument like any other." (Quotation from his The sign of Socrates; found in I.Leudar et al, The sign of Socrates (forthcoming)).
For explication, of this term see J.Guetti’s “Idling Rules”, Philosophical Investigations 16:3 (1993).
 Thanks to an anonymous referee, to Laurence Goldstein, Chrys Gitsoulis, James Guetti, the late Bob Weingard, and especially to David Clemenson, for comments.