[FOM] Understanding Wittgenstein's Notoriou s Paragraph based on Intuitionism

[庄朝晖]

Wittgenstein made the remark on Goedel's Incompleteness 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 (or clarified) 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.) (RFM, I, Appendix III, §8)

The above paragraph was universally regarded as "notorious paragraph". In 2000, Floyd and Putnam argued that Wittgenstein's remark should be understood based on the non-standard models of PM. In 2004, Bays argued that, if Floyd and Putnam were true, then PM should be modified to adapt to the standard model. 
But the problem is that the standard model of PM is still debatable, especially for Intuitionists. Now we would re-consider Wittgenstein's remark from an Intuitionistic perspective, especially in the sense of Brouwer's Intuitionism.
Challenged by Brouwer's Intuitionism, Hilbert put forward Hilbert's Program to attempt to persuade Intuitionists to accept classical mathematics. But Goedel's Incompleteness Theorem showed that most of the goals of Hilbert's Program were impossible to achieve. Thus Intuitionists could insist on their philosophy. 
Brouwer said: From the present point of view of intuitionism therefore all mathematical sets of units which are entitled to that name can be developed out of the basal intuition, and this can only be done by combining a finite number of times the two operators: "to create a finite ordinal number" and "to create the infinite ordinal number w"; here it is to be understood that for the latter purpose any previously constructed set or any previously performed constructive operation may be taken as a unit. Consequently the intuitionist recognizes only the existence of denumerable sets, i.e., sets whose elements may be brought into one-to-on correspondence either with the elements of a finite ordinal number or with those of the infinite ordinal number w. And in the construction of these sets neither the ordinary language nor any symbolic language can have any other role than that of serving as a non-mathematical auxiliary, to assist the mathematical memory or to enable different individuals to build up the same set. 
In the Cambridge lecture on Intuitionism, Brouwer said: Now every construction of a bounded finite nature in a finite mathematical system can only be attempted in a finite number of ways, and each attempt proves to be successful or abortive in a finite number of steps. 
For Intuitionists, they could accept expressibility in a formal system and recursive functions, but would deny any concept of truth outside provability in first order arithmetic. Based on Intuitionism, Wittgenstein's remark could be better understood. For Wittgenstein, if a proposition is provable in some other than the Russell's system, then this proposition is true in some other than the Russell sense. For Wittgenstein, we had to focus on the scope of Russell's system when we talk about the properties of Russell's system. Therefore, Wittgenstein said: "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. (RFM, I, Appendix III, §8)
Thus Wittgenstein said: "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." (RFM, I, Appendix III, §8)

>From the perspective of Intuitionism, Wittgenstein's remark could be better understood. 

Chaohui Zhuang
Xiamen University, P.R.China