To the editors:
It is difficult to answer a piece of scholarship with which one mostly agrees, as is the case with David Berlinski’s new essay, so I will instead use it as an excuse to delve into certain ideas. The essay is about Kurt Gödel’s first incompleteness theorem (FIT). Berlinski invests almost two-thirds of the text—which does not pretend to be formally mathematical—in explaining the idea that Gödel used for the proof.1 He then suddenly begins talking about the implications of the theorem as part of a philosophical discussion about whether the mind is just a computer or whether it is something else, the so-called Godelian argument.2
The essay is absorbing and quite formal while explaining the thoughts of Gödel and Alfred Tarski. When it turns to the mind–machine problem, the mathematical depth is decreased, since the subject is more philosophical than mathematical. It seems almost as if a new essay begins at this point. This configuration that feels strange to me. It remains unclear whether the mention of the mind–machine problem is just a parenthetical comment or whether it is really the main point. The final section of the essay, also entitled “The Director’s Cut,” can be read in two ways, either as a conclusion that encompasses the mind–machine problem, or as a general reflection on the results of the FIT, having little or nothing to do with the Godelian argument.
I enjoyed Berlinski’s explanation of the demonstration of the FIT, and I have little to add on that point. Instead, I will concentrate on his assessment of the mind–machine problem and conclusion.
Contemporary Restlessness
In any formal sense, based on Gödel’s incompleteness theorems or corollaries, it is difficult to conclude that the mind is more than a machine. From the statement that a formal system is consistent if and only if it has undecidable propositions, as Hilary Putnam has put it, not much can be said.
But neither can it be denied that there is something in the incompleteness theorems that leads one to think that the human mind might be more than a machine. That the theorems can be deduced does not seem to me as interesting as how evocative they are. Something about the incompleteness theorems invites one to wonder if the mind is not more than a machine.3 Berlinski quotes Gödel twice on this subject, demonstrating that even Gödel could not escape the question. In fact, Gödel’s answer seems more like an application of his own result, making him appear undecided on this point. But the point is that something in the theorems leads us to consider the matter.4
Is it true that, as John Lucas and Roger Penrose have suggested, we can detect truths that the formal system is unable to decide? Are we able to see symbols, and go beyond them, in a way that computers cannot?5 No matter what the answers, by the simple fact that something in the theorems induces us to think beyond mechanism, the modern ideal is already the loser.
Tekel
A popular belief is that mathematics is devoid of paradoxes and possible contradictions. In reality, the discovery of paradoxes has been ongoing for centuries. When Bertrand Russell and Alfred North Whitehead published Principia Mathematica, they sought to free math from such paradoxes.6 The spirit of the times played a role here too. Modernity exalted reason above all things and logical reasoning therefore needed to be reliable. The Enlightenment was contrasted with the obscurantist Middle Ages. Auguste Comte, Russell, Ludwig Wittgenstein, the Vienna Circle, and David Hilbert all contributed to logical positivism.
Knowledge, to be accorded as such, should be subject to reason, and this with strictly logical criteria. With his theorems, Gödel demolishes this ideal: there are propositions that we realize are true but that are beyond the formal system. In this respect, Lucas and Penrose are right. If it is true, as Putnam has asserted, that the FIT when looked at from the inside says nothing, it is also true that when we look at it from the outside, we see true propositions that the system cannot identify. Perhaps it was Berlinski’s intention to explain the path to the proof in detail until he reached Gödel’s forty-sixth definition, ‘Bew,’ because as he says: “Peering into his own film, the director is now able to see himself peering into 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 opens the door wide to consider it as something more. It also puts an end to any pretense of subjecting all knowledge to a series of logical steps. In all its forms, positivism loses. Modern science needed to show conclusively that the mind is reducible to a mechanism. Gödel has not only not demonstrated it, but has led us to question it. The FIT leaves a bitter taste in the mouth: it states that the formal system is either consistent or complete, but not both, no matter how desirable that might be. But if having both is impossible, it is preferable to sacrifice completeness in order to have consistency.
Tao and Nelson
In 2011, Edward Nelson, a renowned mathematician at Princeton University and a former member of the Institute for Advanced Studies, announced that he had demonstrated the inconsistency of arithmetic. The matter filtered down into the social networks after John Baez wrote about the claim on the blog The n-Category Café. Baez claimed that Nelson’s result was too technical to follow. That same day, a fascinating conversation between Nelson and Terence Tao unfolded in the comments section.
Perhaps what most caught the attention of those following this event was that Tao, a mathematician considered by many to be a modern-day Carl Friedrich Gauss, had stepped in to clarify the issue. His intervention indicated the importance of the subject being discussed.
At this point, a not-so-small part of my own thinking began shifting from fascination to foreboding. What would happen if Nelson’s result was sustained? What would be the implications at a mathematical level? What from arithmetic would remain standing? Which of our rational arguments would continue to be sustained?
This gloomy outlook did not last long. As the conversation between Tao and Nelson continued, Tao found a flaw in his test, and Nelson ended up retracting his work. It became clear to me that although the great majority believed, without having read the proof, that Nelson was wrong, the possibility that he might be right remained open. The sense of bewilderment many felt that day spoke volumes. Baez wrote:
Most logicians don’t think the problem is “making a consistent arithmetic”—unlike Nelson, they believe they 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.7
In the end, Baez was right. Nelson had made a mistake in his demonstration. Although another attempt at the problem in 2011 also failed, Nelson continued working on the matter until his death in 2014. By then, he had produced two works, “Inconsistency of Primitive Recursive Arithmetic” and “Elements,” both of which were posthumously uploaded to arXiv.8 Each paper included an epilogue written by Tao and Sam Buss. “We of course believe that Peano arithmetic is consistent,” the pair wrote, “thus we do not expect that Nelson’s project can be completed according to his plans.”
In the excerpt above, Baez uses the verb “believe” twice. Tao and Buss use it, too. The choice could not be more appropriate. Given the impossibility of showing that a formal system that satisfies the conditions of incompleteness theorems is both complete and consistent, the only solution left is to accept consistency by faith—one of the most important tools available to mathematicians and logicians.9 It makes no sense for a mathematician to develop mathematics if he believes that the system is inconsistent. But this consistency is something he cannot know. At most he must believe that it is true.
In Closing
At one point, modern science had to move from relying on religion to relying on logical knowledge. Gödel’s second incompleteness theorem (SIT)—which states that if the system is consistent, it does not have to prove its own consistency—leads us to believe that mathematics, our most formal form of knowledge, cannot be sustained by logic. Rather, continuing to accept it requires faith. … Poor Comte.
“No formal system explains itself. It cannot say anything and we cannot say everything,” writes Berlinski in his conclusion. He stops at the FIT, in undecidability.10 Berlinski did not have to go further to the STI, a result he only mentions once in passing. But it would have been useful for him to reinforce the fact that we cannot say everything. The hope of consistency has become uncertainty. What was sinister in the FIT has become bitter in the SIT. The best we can hope for is to be uncertain of consistency. When we prove that the system is consistent, or that it is not, we have only demonstrated its inconsistency.
Indeed, the formal system cannot explain itself. Moreover, as Berlinski notes, it cannot say anything. Put another way, again from the perspective of the SIT, it can say many things, but none of them will be definitive. How do we know that another Edward Nelson will not appear in the future with an effective demonstration that arithmetic is inconsistent?
Arturo Pérez-Reverte, a member of the Royal Spanish Academy, recently published the following three tweets, which I translate here:
Before going to sleep (I just returned from a trip) I leave you with, or propose, an idea that I have had in my head for a long time: the perfect and impossible novel, in case any of you is a genius (which some of you will be) and has the courage to write it.
Write a novel whose last page is identical to the first and forces you to return to that first page; so that the new reading of the book, in the light of what has already been read, provides a different reading.And dedicate the novel to [Jorge Luis] Borges.
Goodnight.11
The novel of which Pérez-Reverte dreams will necessarily be based on Gödel’s incompleteness theorems. After all the tangle of modernism, Gödel left us as at the beginning: it is not merely that we cannot make a determination, but that even our most formal systems require faith—just like before modernism began. The last page of the story does not differ from the first, but it does force us to read it afresh. The incompleteness theorems “have changed how things are seen,” writes Berlinski. Before modernism, we sensed that we had no way of founding reason outside of faith. Now we know.
There is only one novel. All others are just derivations of Don Quixote. A tribute to Borges.
