Logic and Language

Copyright   James R Meyer    2012 - 2023 https://www.jamesrmeyer.com

# Jean-Yves Girard on Incompleteness

Page last updated 06 Oct 2021

Jean-Yves Girard is a research director at the mathematical institute of the University of Aix-Marseille, his specialties are logic and proof theory. One of his books, The Blind Spot: lectures on logic, is a collection of lectures on various topics. One lecture is on the subject of incompleteness, and which is replete with fallacious arguments that should not be allowed to pass without notice. (Footnote: Jean-Yves Girard, The Blind Spot: Lectures on Logic, European Mathematical Society, 2011.) A large part of the lecture is a tirade against objections to the notion that incompleteness has been proved, and much of this tirade is presented in rather flowery language, but we shall concentrate here on the parts of the lecture that have some substance.

At the outset of this lecture on incompleteness, Girard states:

For several reasons, it is out of the question to enter into the technical arcana of Gödel’s theorem:

1. This result, like the late paintings of Claude Monet, is easy to perceive, but from a certain distance. A close look reveals only fastidious details that one perhaps does not want to know.
2. Neither is there a need to know, since this theorem is a scientific cul-de-sac: in fact it exposes a way in with no way out. Since it is without exit, there is nothing to seek and it is of no use to be expert in Gödel’s theorem. (Footnote: Emphasis not in the original.)

This is an inane remark. If one is not an expert on Gödel’s theorem, then how is one to know anything about:

1. how it achieves its result?
and
2. how that result and the method of proof might be applicable to other areas?

So if you are never to attempt to become an expert on Gödel’s theorem, that leaves you with the only option that you must simply accept another person’s claim that he is an expert on it, and so you must simply accept that what he tells you about it must necessarily be correct. It’s worth noticing that this was the same line that was pedaled by the Christian church long ago, where the priests told the people that they didn’t need to know every detail of their religion, they didn’t need to read the bible - only experts were able to interpret it correctly, and the priests were the experts. Today this viewpoint is unacceptable, and in the same way no-one should accept Girard’s self-proclaiming grandiosity.

He continues:

It is however important to know the general sense and the structure of the proof. Further, since the theorem is a genuine paradox,* one is naturally tempted to get around it - which is indeed the only way to understand it. The examination of various objections which have been raised to the theorem, all of them wrong, requires more than a mere detailed knowledge of the proof. Rather than attempting to tease out those tedious details which ‘hide the forest’, we shall spend time examining objections, from the most ridiculous to the less stupid (none of which, in the long run, prove themselves to be respectable).
* Footnote: In the literal sense, ‘exterior to the dogma’.

He then talks about the Gödel numbering system:

In particular (and this is the ‘reflexive’ aspect of the theorem), a system of arithmetic can represent itself, ‘speak’ about itself, just as a programming language can be represented as a specific program of the same language.

Here he simply repeats without any logical argument the old lie that such coding enables a formal system to ‘speak’ about itself, a claim which is patently false, since both the Gödel numbering system and the formal system are required to generate statements that refer to expressions of the formal system, and the Gödel numbering system cannot be expressed within the formal system. The statements of the formal system that include numbers that are Gödel numbers state nothing about expressions of the formal system. (Footnote: For more on this, see for example Gödel’s Incompleteness theorem, A Simplified Explanation of Gödel’s proof and A Step by Step Guide to Gödel’s Incompleteness Proof and the formal paper PDF The Impossibility of Representation of a Gödel Numbering Function by a Formula of the Formal System.)

We will not go into any further detail on Girard’s outline proof, since the fallacious assumption renders it completely invalid. After presenting this outline proof, Girard states his intention to find errors in any objections to the notion that incompleteness has been proved:

Incompleteness is not a deficiency: … Incompleteness is, similarly, a disease that cannot be healed and it is ridiculous to look for the “missing piece of the puzzle”. We shall now take a critical glimpse at various proposals, starting with the most indigent ones. This is not so much to indulge in teratology, but rather an amused exploration of the various facets of incompleteness.

He then states that many objections are lacking in rigor:

The most radical want is plainly that of rigour. This is why, from time to time, there is a blossoming of refutations of incompleteness; they come in periodic waves (for instance the year 2000 was very productive). There is little to say about them. They are all made from the same mould: one affirms that, since G is true, it must be provable. The problem is not that a few morons keep on repeating over and over the same mistakes, it is the guilty complacency these amateurs find in certain milieus. Artificial Intelligence advertises them insidiously, Emails of the style “we didn’t dare to say it, now we know”, “hmmm, hmmm, hmmm…”, invitations to speak in the name of the “one can exclude nothing”… It is futile to try to repress such baloney, especially when their authors are half-wits.

It’s a cheap trick to pick out objections that are clearly deficient and then lump all objections into that category - but the fact is that some objections are rigorously stated. For those objections, since Girard is complaining about a lack of rigor, then he should also impose the same need for rigor in his own dismissal of all such objections. But by dismissing all objections as all of a kind, he provides no rigorous argument against a detailed logical demonstration that there is a flaw in Gödel’s proof, such as that given by The Flaw in Gödel’s Proof of his Incompleteness Theorem.

He then claims that people who assert that there is something wrong with proofs of incompleteness need to be taken aside and told three of Girard’s dogmas. The first is:

Like it or not, there is no bug in the proof of incompleteness of mathematics, a contention that has even been checked by computer.

Girard repeats here the lie that simply because a computer has been used to perform a certain series of operations on some computer code, then that demonstrates that no error has been made anywhere in that system. In fact some of these computer-checked proofs of incompleteness have been analyzed and fundamental errors have been found in each one examined, and in each case the error renders the proof totally invalid. The paper PDF An Error in a Computer Verified Proof of Incompleteness by Russell O’Connor shows that the computer proof by Russell O’Connor, although it was supposedly fully checked by computer, actually relies on untenable assumptions regarding “representability” that are implicit in O’Connor’s definitions in his proof. The paper PDF An Error in a Computer Verified Proof of Incompleteness by John Harrison shows that Harrison’s code fails to observe elementary mathematical principles regarding the domains of variables. And the paper PDF An Error in a Computer Verified Proof of Incompleteness By Natarajan Shankar shows that the proof introduces a statement which includes a fixed non-variable value as one of its values, whereas the value should in fact be a variable value.

The second of the three dogmas is:

A refutation of the theorem (which cannot be a priori excluded, though very improbable) would produce an inconsistency in mathematics.

Here Girard simply asserts this but he doesn’t even begin to explain how that might be the case. Although Girard complains about a lack of rigor in others, he makes sweeping assertions without providing anything to back them up. He provides no explanation at all as to why a refutation, or even a proof that some formal system could be complete, would produce an inconsistency.

And although he glibly notes that a refutation cannot be said to be absolutely impossible, he contradicts himself later on because he asserts: “we have already observed that the theorem cannot be refuted or falsified ” and further on he makes his stance very clear when he states that incompleteness is “one of the few results of the human mind to be absolutely irrefragable”.

And the third of the three dogmas is:

But the hypothesis of the theorem is the consistency of mathematics. A refutation would indeed produce even stronger evidence.

It’s not at all clear what he means by this, he provides no rigorous argument to support it. An incompleteness theorem is supposedly a proof of incompleteness. The conventional notion that a formal system must be either incomplete or inconsistent arises from the same erroneous notion that the formal system can actually make statements about its own expressions. But, as noted above, that is impossible, since it is only the combination of the Gödel numbering system and the formal system that can generate statements that refer to expressions of the formal system, and the Gödel numbering system cannot be expressed within the formal system.

He talks about consistency in relation to incompleteness:

Here, we contend with the second theorem, tamper with the definition of consistency - be creative or perish ! Remember that consistency is the fact that we cannot prove the absurdity ⊥, but this relies on the fact that absurdity implies no matter what formula: “ex falsum quod libet”. A system is consistent when there is something it cannot prove.

This is nonsense, and Girard displays a lack of rigorous logical thinking. It is possible to have a system that cannot prove some statements that can be made in that system and yet it is inconsistent. Simply leave out certain axioms of a system and add in some inconsistent statement as an axiom - since some critical axioms were removed, many of the statements which were previously provable cannot now be proved in the modified system.

#### The extinction of Popperism

The philosopher Popper proposed an epistemology of “falsifiable” properties that we can as well call “recessive”. It is about general laws, subject to partial verifications, tests, for instance, specific experiments, verification up to a given decimal… We easily recognise Hilbert’s influence; it looks like an enhancement of Hilbert’s ontology to apply to the complete field of scientific discourse. The pleasant side of the story is an anti-essentialist positioning; laws do not fall from the sky. Now, since Gödel refutes Hilbert, he refutes a fortiori Hilbert’s epigone Popper.

We can observe that Popper gives no status to Gödel’s theorem, since we have already observed that the theorem cannot be refuted or falsified.

This is a complete misunderstanding of Popper’s concept of falsifiability (or else a misrepresentation). Girard states that the theorem cannot be refuted or falsified, and by that he is asserting that he cannot possibly be incorrect in his belief that Gödel’s proof and other proofs of incompleteness might have a logical flaw. But the whole point of Popper’s falsifiability is that, however unlikely that it may seem, you must always be prepared for the possibility that something may be proved incorrect at some point in time - regardless of how long it has been accepted as correct.

While a proof can appear to be correct, and on casual examination everything may appear to be well defined with clear axioms and rules of inference, that reading depends on the assumption that nothing has been missed, that no errors have been made, that no invalid assumptions have been made. But Gödel’s proof and most other incompleteness proofs are not formal. And those that claim that they are - such as the ‘computer verified’ proofs, have, as noted above, been shown to be flawed.

In summary, it is rather ironic that the title of the book is “The Blind Spot” while Girard himself seems to have foisted upon himself a blind spot to the possibility that proofs of incompleteness are flawed, instead relying on the erroneous assumption that formal systems can express the Gödel numbering system.

Footnotes:

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