Some years ago, the magazine Inference invited me to respond to David Berlinski’s essay The Director’s Cut. I wrote the response in Spanish, and they kindly translated it into English. However, using Grammarly, I have now translated it according to my own taste, so I am publishing the English translation here on my blog.

The text discusses Gödel’s first incompleteness theorem (FIT). Berlinski spends nearly two-thirds of the text— which is not intended to be formally mathematical— explaining the idea Gödel used for the proof.¹ Then, suddenly, he begins to talk about the implications of the theorem in the philosophical discussion about whether the mind is merely a computer or something more, the so-called «Gödelian argument.»²

In fact, the writing has an engaging and rather formal flow until the explanation of Gödel and Tarski. When it turns to the mind-machine problem, the mathematical depth is relegated (because the topic is no longer as mathematical but more philosophical), and it is almost as if a new writing begins, perhaps also engaging, but in a different direction.

It was the composition that struck me as most strange in Berlinski’s essay. First, after reading it several times, it remains unclear to me whether the mention of the mind-machine problem was merely a parenthetical comment or if it was the point the author intended to make. Second, the final section of the essay, also called «The Director’s Cut,» can be read in both ways, as a conclusion that encompasses the mind-machine problem or as a general reflection on what the FIT provokes, that has little or nothing to do with the Gödelian argument.

Thus, although I immensely enjoyed Berlinski’s explanation of the FIT demonstration, since I have little to add, I will focus on his insights regarding the mind-machine problem and his conclusion.

Unease for the modern man

It must be conceded that, in the most formal sense, it is difficult, based on Gödel’s incompleteness theorems or one of their corollaries, to conclude that the mind is more than a machine. From the statement «A formal system is consistent if and only if it has undecidable propositions,» as Hillary Putnam stated, not much more can be said.

However, it cannot be denied that there is something provocative in the incompleteness theorems that leads one to think that the human mind is more than a machine. Whether this point can (or cannot) be deduced from the theorems is not as interesting to me as how evocative they are in this direction. There is something in the incompleteness theorems that at least invites us to question whether the mind is merely a machine.³ Berlinski cites Gödel on this twice, which demonstrates that even Gödel could not escape the question. And even though Gödel’s response seems more like an application of his own result, thus giving the problem a certain air of undecidability, what is important here is that, even by his own questions, it is clear that there is something in the theorems that leads us to consider the matter.⁴

Is it that, as John Lucas and Roger Penrose say, we can detect truths that the formal system cannot decide? Is it that we can see symbols and go beyond the signs, something that computers (as we define them today) cannot do? It does not matter much; the point is that, with something in the theorems that leads to going beyond mechanistic views, the modern ideal is already the big loser.

Tekel

It is a common belief that mathematics is devoid of paradoxes and possible contradictions. Nevertheless, paradoxes have been discovered for centuries. When Bertrand Russell and Alfred North Whitehead published their Principia Mathematica, they aimed to free mathematics from them.⁵ The spirit of the times had everything to do with it. Modernity exalted reason above all else, so logical reasoning had to be reliable. The illuminated Enlightenment was contrasted with the obscurantist Middle Ages, and this had to be transparent to everyone. Comte, Russell, Wittgenstein, the Vienna Circle, and the Hilbert program are all points that sought to converge toward logical positivism.

Knowledge, to be regarded as such, must be subject to reason, and this must be done with strictly logical criteria. Gödel, with his theorems, undermines this ideal: there are true propositions that we are aware are true but that are beyond the reach of the formal system. Lucas and Penrose are correct about this. While it is true, as Putnam said, that the FIT viewed from the inside says nothing, it is also true that, when viewed from the shore, we see true propositions that the system cannot find. Perhaps this was Berlinski’s intention when he explained in detail the path of the proof up to Gödel’s definition 46, Bew, because, as he says: «Looking at his own film, the director is now able to see himself watching his own film.»

Gödel eliminates the modern ideal. While his FIT does not strictly prove that the mind cannot be reduced to a machine, it throws open the door to consider it as something more, and it also ends any pretense of subjecting all knowledge to a series of logical steps. The big loser is modern positivism in all its forms. Modern science was supposed to demonstrate beyond any doubt that the mind was reducible to a mechanism. Not only has it failed to do so, but Gödel leads us to question this postulate. The FIT leaves a the mouth of the modern man with a uncomfortable sense of tastelessness: it asserts that a formal system is either consistent or complete, but not both. It would be desirable to have both, but if one cannot have both, at least it is preferable to sacrifice completeness rather than consistency. So, we still hope that the system will be consistent.

Of Terence Tao and Edward Nelson

In 2011, the late Edward Nelson, a renowned mathematician from Princeton University and a former member of the Institute for Advanced Study, announced that he had demonstrated the inconsistency of arithmetic. The matter gained attention on social media when John Baez posted it on the blog The n-Category Café. Though Baez stated in the original blog that Nelson’s result was too technical for him to follow, Terence Tao and Nelson had a fascinating conversation in the blog’s comments on the same day.

Perhaps what caught the attention of all the mathematical nerds following the news at the time was that Tao, the mathematician who is considered by many to be the Carl Friedrich Gauss of our times, had to intervene on social media to clarify the matter. This demonstrated the importance of the discussion.

A not-so-small part of me, I confess, began to shift from fascination to morbid curiosity. What would happen if Nelson’s result held? What would be the mathematical implications? What part of arithmetic would remain intact? Which of our rational arguments would still stand?

But the excitement lasted little for me. In the back-and-forth of the conversation between Tao and Nelson, the former found a flaw in the latter’s proof, and Nelson ended up retracting. Nevertheless, something became clear to me: despite the vast majority believing, without having read the proof, that Nelson was wrong, the possibility that he was right remained open, and the confusion witnessed that day spoke louder than words. Baez:

Most logicians don’t think the problem is “making a consistent arithmetic” – unlike Nelson, they believe the arithmetic we have now is already consistent. The problem is making a consistent system of arithmetic that can prove itself consistent… Nelson doubts the principle of mathematical induction, for reasons he explains in his book, so I’m sure his new system will eliminate or modify this principle… Needless to say, this is a radical step. But vastly more radical is his claim that he can prove ordinary arithmetic is inconsistent. Almost no mathematicians believe that. I bet he’s making a mistake somewhere, but if he’s right he’ll achieve eternal glory.

Baez was right: Edward Nelson made a mistake in the demonstration and retracted it. However, despite the fact that Nelson’s first attempt at proof fell through in 2011, he continued to work on the matter until he died in 2014. During this period, he produced two works entitled Inconsistency of Primitive Recursive Arithmetic and Elements, which were uploaded to arXiv posthumously. Both contained an identical epilogue written by Sam Buss and Terence Tao. «Of course, we believe that Peano arithmetic is consistent; therefore, we do not expect Nelson’s project to be completed according to his plans,» the two mathematicians wrote at some point in the epilogue.

In the two parts where Baez uses the verb to believe in his citation, the emphasis is my own, as in the citation of Buss and Tao. In both comments, the use of the verb could not be more appropriate. Given the impossibility of showing that a formal system satisfying the conditions of the incompleteness theorems can be both complete and consistent, the only solution is to accept whatever we accept about consistency by faith. Faith is the most important of the mathematician’s and logician’s tools. It makes no sense for the mathematician to develop mathematics if he believes that the system is not consistent, but this consistency is something that he cannot know, only believe to be true at most.

The writer’s conclusion

Modern science was supposed to move from religion to logical knowledge. Still, Gödel’s second incompleteness theorem (SIT)—which shows that even if a system is consistent, it cannot prove its consistency—leads us to the conclusion that mathematics, our most formal form of knowledge, cannot be based on logic, so continuing to accept it requires faith. Poor Comte.

“No formal system can explain itself. It cannot state anything, and we cannot say everything,” says David Berlinski in the conclusion of his writing. Berlinski stops at the FIT, at undecidability.⁶ It’s fine; there was no need to go further to the SIT, a result he only mentions in passing once. However, it would have been useful for him to reinforce the fact that we cannot say everything. The hope for consistency has turned into uncertainty. The tastelessness of the FIT has turned into bitterness with the SIT. The best we can aspire to is to have no certainty of consistency because once we prove that the system is consistent (or that it is not), we will have merely demonstrated its inconsistency.

Indeed, the formal system cannot explain itself. Furthermore, as Berlinski said, it cannot say anything. Put another way, from the perspective of SIT once again, it can say many things, but none will be definitive. How do we know that some Edward Nelson does not emerge with a working demonstration that arithmetic is inconsistent in the future?

—

A few days ago, the writer Arturo Pérez-Reverte, a fellow of the Royal Spanish Academy, published the following thread of three tweets, which I translate here in full:

Before going to sleep (I just returned from a trip), I leave you, or I propose, an idea that has been in my head for a long time: the perfect and impossible novel, in case any of you is truly gifted (as there must be someone) and feels inspired to write it.

Write a novel whose last page is identical to the first and compels readers to return to the first page so that the new reading of the book, in light of what has already been read, provides a different experience. And dedicate the novel to Borges.

Good night.

The perfect novel that Pérez-Reverte dreams of will have to be based on Gödel’s incompleteness theorems. After all, the crux of Modernity has left us as we began: it is not just that we cannot decide; it is that even our most formal systems require faith. Just as it was before Modernity. The last page of history does not differ from the first, but it does compel us to read anew, “[the incompleteness theorems] have changed the way we see things.”⁷ Before Modernity, we sensed that we had no way to ground reason outside of faith. Now we know.

There is only one novel. All the others are just derivations of Don Quixote. A tribute to Borges

Notes

1. Dan Gusfield, from the University of California, Davis, has a test in the ‘Goldilocks zone’ that is neither too formal to become incomprehensible to the average person nor too relaxed to become superficial. It is suitable for second-year undergraduate students. The written version is here, and the video version is here.
2. Philosophers refer to this discussion as the «Gödelian argument in the mechanistic conception.» It is a long and tedious name.
3. And that something, by the way, is not questioned by computers.
4. In a writing by Jack Copeland, also cited by Berlinski, it states that Gödel seemed to lean more towards immaterialism. The Spanish Wikipedia entry on Gödel’s incompleteness theorems also affirms this: «[Marvin] Minsky has reported that Kurt Gödel told him in person that he believed humans have an intuitive, not just computational, way of reaching the truth and therefore his theorem does not limit what can be known as true by humans.» Unfortunately, Wikipedia does not provide any references.
5. For example, see this talk by Douglas Hofstadter.
6. And in the concept of truth in formalized languages of Tarski.
7. Berlinski, The Director’s Cut.