# How is Wittgenstein’s “notorious paragraph” about the Gödel's Theorem not obviously correct?

Timm Lampert quotes from Wittgenstein's "notorious paragraph" (§8 of Remarks on the Foundations of Mathematics, Appendix 3) in http://wab.uib.no/agora/tools/alws/collection-6-issue-1-article-6.annotate

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 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.[…]

When we examine these things using the mathematical formalist approach:

https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics)
In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules.

When analytical truth in language is defined as tautologies between finite strings then:
(1) There is no possible way that truth can diverge from provability.
(2) Truth is always definable.
(3) Incompleteness gaps cannot possibly exist.

Tractatus Logico-Philosophicus, by Ludwig Wittgenstein (1921)
seems to have this same view on tautology.

This is how my view unifies with the view of Wittgenstein:
Analytical truth expressed in language is merely finite strings that have been defined to have the semantic property of Boolean true (axioms) and relational connections to these finite strings (Theorems). Any expression of language that is not defined to have the semantic property of Boolean true (axiom) or has a relational connection to these finite strings (theorem) is necessarily untrue.

The following can be understood from the mathematical formalist perspective of [tautological] relations between finite strings:

Formalizing Wittgenstein's highlighted words:
P ↔ (RS ⊬ P)
∀x (True(RS, x) ↔ (RS ⊢ x))
∀x (False(RS, x) ↔ (RS ⊢ ¬x))

Examples of Provability relations:
(PA ⊢ "2 + 3 = 5") // Provability relation exists----------AKA True
(PA ⊬ "2 + 3 = 7") // Unprovability relation exists------AKA ¬True
(PA ⊢ ¬"2 + 3 = 7") // Refutability relation exists-------AKA False

Within the truth predicates derived from Wittgenstein's specification if Unprovability is not proved within Russell's System then Unprovability does not count as true within Russell's System.

This view (based on Wittgenstein) seems to be irrefutable
Only statements inside the scope of provability within Russell's System derive truth values from Russell's System. The Truth of the statements of a Theory is only relative to that Theory. The formalization of (the Wittgenstein specification of) P relative to RS shows that P is self-contradictory relative to RS thus not true in RS even if it is true elsewhere.

These are Two Different Questions:
(1) Is P true in RS?---NO
It cannot be proved in RS that P cannot be proved in RS because P is self-contradictory in RS thus ill-formed in RS and therefore untrue in RS.

(2) Is P True outside of RS?---YES
It can be proved outside of RS that P cannot be proved in RS.

When the above two questions are conflated together as one question [Is P true?] it really seems that Gödel Incompleteness and Tarski Undefinability have been proven. When this one question is divided so that it is relative to a formal system then the Gödel and Tarski conclusions cease to be sustained.

Actual truth in the world is simply relations that have been defined between finite strings. The relation [is a] has been defined between "cat" and "animal". You can pile layers of extraneous complexity on top of that to confuse the issue, yet the self-evident truism remains, conceptual truth in language is simply defined relations between finite strings. Provability is nothing more than determining whether or not one of these defined relations exists.

This aspect will be continued in a new question:
Are truth and provability inherently inseparable because every means of ascertaining conceptual truth can be construed as a formal proof?

The key issue is not a math error by either Tarksi or Gödel. If Truth and provability are truly inseparable then Tarksi and Gödel are wrong.

Wittgenstein proves that Gödel's G is merely self-contradictory when Truth and provability are inseparable. When truth and provability mutually define each other then Tarski Undefinability ceases to exist.

The actual refutation Tarksi and Gödel requires a proof that Truth and provability be proved inherently inseparable and cannot be defined coherently otherwise.

• Possible duplicate of Was Wittgenstein anticipating Gödel? Jun 1, 2019 at 7:44
• Also from the quoted text: "Actually, Gödel rejected Wittgenstein’s argumentation accusing him to confuse his argumentation with “a kind of logical paradox”" Jun 2, 2019 at 1:36
• Putting this convoluted mess aside, isn't the only real question whether truth indeed is defined as provable in Russell's system? If not, the answer is trivially "No, it's not obvious that he's right, and arguably he's wrong." Jun 12, 2019 at 17:19
• @PhilipKlöcking Actual truth in the world is simply relations that have been defined between finite strings. The relation <is a> has been defined between "cat" and "animal". You can pile layers of extraneous complexity on top of that to confuse the issue, yet the self-evident truism remains, conceptual truth in language is simply defined relations between finite strings. Provability is nothing more than determining whether or not one of these defined relations exists. Actual truth the way it really is supersedes anyone's misconceptions of it. Jun 12, 2019 at 21:33
• @PhilipKlöcking Maybe it's time to lock this post? The edits keep bumping it to the top. Jun 13, 2019 at 17:53

Timm Lampert, cited by the OP, quotes Wittgenstein (§8 of Remarks on the Foundations of Mathematics, Appendix 3):

‘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.

Lampert claims Wittgenstein is assuming what needs to be proven:

Whether P = ΠP and ¬P = Π¬P are valid is just what is in question and the philosophical upshot of Gödel’s proof is to have proven that these assumptions are wrong.

This appears to be the case.

In Ernest Nagel and James R. Newman's summary of Gödel's proof Gödel was able to establish the truth of the undecidable statement G without using the proof system, but through Gödel numbering of meta-mathematical arguments: (page 93)

Third, we recall that metamathematical statements have been mapped onto the arithmetical formulas. (Indeed, the setting up of such a correspondence is the raison d'etre of the mapping; as, for example, in analytic geometry where, by virtue of this process, true geometric statements always correspond to true algebraic statements.) It follows that the formula G, which corresponds to a true meta-mathematical statement, must be true. It should be noted, however, that we have established an arithmetical truth, not by deducing it formally from the axioms of arithmetic, but by a meta-mathematical argument.

The OP asks how Wittgenstein is not obviously correct.

One way to see why Wittgenstein is not obviously correct is to see that Gödel established a way with his Gödel numbering to decide truth without having a derivation using the inference rules of the proof system. Wittgenstein assumed he must use a derivation. This is similar to using a truth table in propositional logic to establish truth rather than a proof using inference rules.

Unlike the case for propositional logic, Gödel found that a consistent arithmetical system cannot be complete. There are true arithmetical statements that cannot be derived using the consistent axioms of the proof system.

Nagel, Ernest, and James R. Newman. Gödel's proof. New York University Press, 1986.

Lampert, T. "Wittgenstein’s “notorious paragraph” about the Gödel Theorem" Retrieved on June 1, 2019 from http://wab.uib.no/agora/tools/alws/collection-6-issue-1-article-6.annotate

• "Wittgenstein assumed he must use a derivation." Actually Wittgenstein defined that a derivation is required. He did not assume he stipulated. The key question seems to be what is the correct specification of analytical truth? Tarski answered: "It can't exist", yet the Wittgenstein specification seems to be consistent. It simply rejects previously undecidable sentences as untrue. Jun 9, 2019 at 3:58
• @PL_OLCOTT The main question is whether truth transcends provability. In certain systems such as arithmetic it does. In propositional logic it does not. Wittgenstein also did not allow natural numbers to exist as a concrete class in his logic while Russell and Frege did. Numbers for him were defined formally as exponents. Wittgenstein's logic, without the arithmetic, may be simple enough so that truth does not transcendent provability. But that is not the case in Russell's arithmetic. Jun 9, 2019 at 11:03
• Truth has to be evaluated relative to a formal system and not outside of it. liarparadox.org/Haskell_Curry_45.pdf We cannot say that a textbook about algebra is incomplete because it says nothing about physics. Relative to Russell's System the above expression is not provable because it is self-contradictory so we cannot use that as a basis for deciding that Russell's System is incomplete. The fact that the logic sentence is neither provable nor refutable within Russell's System proves that it is self-contradictory, thus ill-formed relative to Russell's System. Jun 9, 2019 at 14:12
• @PL_OLCOTT In propositional logic there is a means to assign truth values without provability using truth tables. Truth tables are outside the inference rules associated with provability. That the set of true propositions are the same as the set of provable propositions is the consequence of arguments for soundness and completeness of propositional logic. Russell's system is more complicated and involves arithmetic. This arithmetic was used to find a way to determine truth independent of the proof system. That they give the same results is what Godel showed was not the case. Jun 9, 2019 at 14:42
• "Truth tables are outside the inference rules associated with provability." Only those things inside the scope of provability within Russell's System derive the truth values of statements of Russell's System, because the Truth of the statements of a Theory is only relative to that Theory: (last highlighted sentence) liarparadox.org/Haskell_Curry_45.pdf The formalization of P relative to RS shows that P is self-contradictory relative to RS thus not true in RS even if it is true elsewhere. Jun 9, 2019 at 17:43

Wittgensteins early prose was notoriously obscure and difficult to parse. Thus to call it ‘obviously’ anything seems very wrong-headed.

This is possibly why he gave up on his former philosophy calling it not philosophy at all and completely wrong.

His later philosophy is far more human in that it’s concerned with human values.

• I added this clarification: ALL conceptual truth is simply relations between expressions of language that are stipulated to be true and these same stipulations make these expressions of language provable. Jul 19, 2019 at 15:27

Philosophy aims at the logical clarification of thoughts. Philosophy is not a body of doctrine but an activity

If Godel shows that "P = ΠP and ¬P = Π¬P" are not valid then any use of his ideas that also use "P = ΠP and ¬P = Π¬P" are, assuming we're talking philosophy, surely a misuse of "P = ΠP and ¬P = Π¬P".

Any activity that misuses logic (or language) surely generates nonsense that says nothing, whatever it shows

“What can be shown cannot be said,” that is, what cannot be formulated in sayable (sensical) propositions can only be shown... Even the unsayable (metaphysical, ethical, aesthetic) propositions of philosophy belong in this group—which Wittgenstein finally describes as “things that cannot be put into words. They make themselves manifest. They are what is mystical” (TLP 6.522).