Flaws in constitution

 Cropped from book page.

Rebecca Goldstein is the author of the book Incompleteness: The Proof and Paradox of Kurt Gödel. She obtained her PhD in Philosophy from Princeton University, and has also written several novels set in academia, including The Mind-Body Problem and Properties of Light: A Novel of Love, Betrayal and Quantum Physics. The latter draws on the life and concerns of the physicist David Bohm.

Today Ken and I wish to talk about Kurt Gödel’s journey in getting his USA citizenship, and his journey since then in the interpretation and implications of his research.

Gödel’s citizenship interview happened on Thursday the 5th of December, 1947—over fifty years ago. Even over sixty years ago, come to think of it. Past a certain age it becomes better to focus on the wider part of the calendar than the four-digit number at the top.

I bought Goldstein’s book years ago and started to read it. But somehow the initial few pages were not that compelling, or I was distracted by doing something else. In any event, I recently had a long plane flight and took the book—yes I still read hard-copy printed books—along. Partially because it was small, partially because it was on Gödel, and partially by randomness.

It turns out the book is a mixed bag. It was a fun read, with many interesting insights into the life of Gödel. It was also filled with strange errors that I easily noticed, even flying at 36,000 feet without any access to Google search. Yet I did enjoy the book, and am sorry I had not read it before. Well not completely—without it the plane flight would have been longer, since reading helps shrink the time of a flight.

## The Story

Here is the story, according to Goldstein, of the day Gödel went to Trenton to get sworn in as an American citizen. Gödel had prepared well for his hearing, and had further discovered that the U.S. Constitution has a flaw that could allow it to become a dictatorship.

Oskar Morgenstern and Albert Einstein drove Gödel to Trenton for his hearing before the judge. On the car ride Einstein tried to distract Gödel with jokes: “Well, are you ready for your next-to-the-last test?” Gödel answered “What do you mean, ‘next-to-the-last’?” Einstein aded, “Very simple. The last will be when you step into your grave.”

Einstein continued on till they reached the court where the judge was Philip Forman, who was a friend of Einstein besides having administered Einstein’s own citizenship oath. The judge moved them quickly into his private chambers. Einstein and the judge chatted while Gödel sat mute. Finally the judge said to Gödel, “Up to now you have held German citizenship.” Gödel corrected him: Austrian citizenship. The judge added, “In any case, it was under an evil dictatorship. Fortunately, that is not possible in America.”

As Goldstein says, this was what Gödel was waiting for. Gödel started to explain how it could happen here because of the flaw in the Constitution. The judge interrupted and said “You needn’t go into all that.” The rest when smoothly and after the oath Gödel become a US citizen. Later in a letter to his mother, Gödel remarked that Forman was a “very sympathetic person.”

## The Lost Story

It was known that Morgenstern had written an account of that day, but when his widow was interviewed in 1983 by John Dawson, she had been unable to locate it. Dawson used her recollections in his 1997 biography of Gödel. In 2006 the Institute for Advanced Study hailed the centennial of Gödel’s birth in its spring newsletter. This included a sidebar titled “Gödel, Einstein, and the Immigration Service,” later reproduced on their Gödel page, but with a story quite different from what Dawson had heard. Moreover, the IAS gave the year as 1948. Perhaps they followed my advice about calendars.

Mathematician and author Jeffrey Kegler, who based a novel on Gödel’s two lost notebooks, tells the full story on a neat page with links to all sources, including his own blog posts. While editing Wikipedia’s Gödel page in November 2008, he found another account that “rang true” more than the existing hearsay accounts, and resembled the IAS version. He was convinced the latter had to be based on a true original. He contacted Dawson, who in turn prompted the Institute to find and release it.

Morgenstern in fact mentions only the year 1946. Here is part of what he wrote:

…[Gödel] rather excitedly told me that in looking at the Constitution, to his distress, he had found some inner contradictions and that he could show how in a perfectly legal manner it would be possible for somebody to become a dictator and set up a Fascist regime… I tried to persuade him that he should avoid bringing up such matters at the examination before the court in Trenton, and I also told Einstein about it: he was horrified that such an idea had occurred to Gödel, and he also told him he should not worry about these things nor discuss that matter.

Many months went by and finally the date for the examination in Trenton came. … While we were driving, Einstein turned around a little and said, “Now Gödel, are you really well prepared for this examination?” Of course, this remark upset Gödel tremendously, which was exactly what Einstein intended and he was greatly amused when he saw the worry on Gödel’s face. …

When we came to Trenton, we were ushered into a big room, and while normally the witnesses are questioned separately from the candidate, because of Einstein’s appearance, an exception was made and all three of us were invited to sit down together, Gödel, in the center. The examiner first asked Einstein and then me whether we thought Gödel would make a good citizen. We assured him that this would certainly be the case, that he was a distinguished man, etc. And then he turned to Gödel and said,

“Now, Mr. Gödel, where do you come from?”

Gödel: “Where I come from? Austria.”

The Examiner: “What kind of government did you have in Austria?”

Gödel: “It was a republic, but the constitution was such that it finally was changed into a dictatorship.”

The Examiner: “Oh! This is very bad. This could not happen in this country.”

Gödel: “Oh, yes, I can prove it.”

So of all the possible questions, just that critical one was asked by the Examiner. Einstein and I were horrified during this exchange; the Examiner was intelligent enough to quickly quieten Gödel… and broke off the examination at this point, greatly to our relief. …

Then off to Einstein’s home again, and he turned back once more toward Gödel, and said, “Now, Gödel, this was your last-but-one examination;” Gödel: “Goodness, is there still another one to come?” and he was already worried. And then Einstein said, “Gödel, the next examination is when you step into your grave.” Gödel: “But Einstein, I don’t step into my grave.” and then Einstein said, “Gödel, that’s just the joke of it!” and with that he departed. I drove Gödel home. Everybody was relieved that this formidable affair was over; Gödel had his head free again to go about problems of philosophy and logic.

## The Lost Flaw

Maddeningly left out is what exactly the “inner contradictions” were. There have been various speculations, even a paper, most revolving around the Constitution’s providing the power to amend itself. Kegler has his own hypothesis.

Here I—Ken writing this—must confess I am unable to locate the webpage with what I took to be the flaw when I did background reading for our first “interview” with Gödel two years ago. I’ve alas never picked up the index-card habit. What struck my memory, however, was the source’s reference to the Senate and the judiciary.

Trying to reconstruct it, I think the path to dictatorship Gödel feared starts with something like this: The President of the Senate declares that a rules issue is a Constitutional question. This enables a bare majority, exploiting the gaps in Article I, to rewrite the rules of the Senate. Such a rule change can enable the uncontested appointment of Federal judges. Those judges in turn can… Well, anyway, nothing like that would ever actually happen.

Back to Dick and to Goldstein’s book, which to be fair, came out a few months before the IAS newsletter with Morgenstern’s account.

## The Book

The book is—as I stated already—a mixed-bag, at best. I liked the history and insights into Gödel’s life. Yet it has many errors—both small errors that were almost just typos, and major errors. I had my thoughts, but Ken found the tough review by Solomon Feferman, so let’s quote that:

As to the core of Goldstein’s book, anyone familiar with Gödel’s work has to flinch. Dozens of errors could have been avoided by an expert vetting of the manuscript. At the very least we would not have had ‘Kreisl’ for ‘Kreisel,’ ‘Kline’ for ‘Kleene,’ and ‘Tannenbaum’ for ‘Teitelbaum’ (the birth surname of Alfred Tarski, the great logician, whose significant interaction with Gödel barely merits Goldstein’s notice).

In the air, flying way above the clouds, I certainly wondered if I was dreaming when I saw the reference to “Kreisl.” At first I wondered did she mean someone other than the famous logician Georg Kreisel? I could not believe that there could be another. Kreisel worked on proof theory and is known for many things including this amazing conjecture:

Suppose that Peano Arithmetic (PA) proves ${A(S^{n}(0))}$ in ${O(1)}$ steps for all ${n}$, then PA proves ${\forall x A(x)}$.

Note: ${S^{n}(0)}$ is the successor function applied to ${0}$ a total of ${n}$ times: it is ${n}$ in unary. There is some evidence for and against it; the latter two papers are by the same author, Pavel Hrubeš.

Errors aside, the book does have some interesting bits of history about Gödel and other mathematicians of his era. Many of the stories are known, perhaps well known. The book is much more about people and their history than a primer of the Incompleteness Theorems. One story that I knew but l like a lot is about Einstein’s salary negotiation with the head of IAS:

Einstein asked for a salary of $3,000, and the head “countered” with an offer of$16,000.

A very interesting example of negotiation. Quoting Feferman again:

What she does very well is to provide a vivid biographical picture of Gödel, beginning mid-stream with his touching relationship with Albert Einstein at the Institute for Advanced Study in Princeton, where, over a period of 15 years until Einstein’s death in 1955, they were often seen walking and talking together.

But he ends with:

Those who are fascinated by Gödel’s theorems—and the general idea of limits to what we can know—may still hunger for a more universal view of their possible significance. But they should not be satisfied with Goldstein’s ‘vast and messy’ goulash; hers is not a recipe for true understanding.

Indeed Feferman most loudly criticizes her signing on to the “view [t]hat Gödel’s theorems were designed to refute the formalist program of David Hilbert.” Both Ken and I have been careful to portray Gödel in harmony with Hilbert, and even as compressing rather than expanding the implications of his own theorems. Of course we have conjured our own fictionalizations of Gödel, and however well sourced, they may have errors. If so, we will amend them. Scrupulousness even made this post a day late.

## How Many Unprovable Statements Are There?

While we are talking about Gödel’s Incompleteness Theorems, Tim Gowers has raised a question about unprovable statements in mathematics. In essence it is: Why do we as practicing theorem provers seem to be able to avoid the unprovability issues of Gödel? Or do we?

I have an answer that I am sure Gowers saw, but thought I would share. Consider all true statements ${\phi}$ in Peano Arithmetic of size ${N}$ in some standard encoding. I claim that there is a positive ${\delta}$ so that at least a ${\delta}$ fraction of these true statements cannot be proved in PA. The proof is quite simple. Pick any single unprovable statement ${A}$. Then consider the set of statements of the form:

$\displaystyle \phi = A \wedge B$

for any true ${B}$. None of these are provable in PA, and they form a positive fraction of all the true statements of length ${N}$. Statements ${A \vee B}$ where ${A}$ is provable yield a similar upper bound separated from ${1}$ on the proportion of unprovable statements.

## Open Problems

A natural question is: in the limit are there more unprovable than provable statements of size ${N}$ as ${N}$ goes to infinity? This depends on encoding details but should be a robust enough question under reasonable conditions. Is it clear that there is a limit? Of course the above construction leads to many uninteresting statements. So the second question might be: can we sharpen the question, for instance by associating to a provable ${\phi}$ the idea of minimizing the size of ${\psi}$ such that ${\psi \longrightarrow \phi}$ has a “trivial” proof?

December 6, 2013 5:39 pm

Good question!
If Landauer’s principle holds…a “trivial” proof from physics’ principle

December 6, 2013 10:52 pm

Thanks for mentioning my work on Gödel. By the way, I was one of Prof. Lipton’s students in the late 1970’s at Yale. Dick taught my Theory of Computation course. In succeeding years, I’ve come to realize just how good the job he did was. In those days there was no general textbook on Theory, which makes organizing a course much, much harder.

• December 6, 2013 11:34 pm

Great, thanks! Part of the day’s delay alluded to in the post was to summarize your discovery story correctly in yea-few words. Have you seen the recent buzz about coding the “Ontological Proof” for machine verification? Time has been one factor keeping us from touching that, but it may lend itself to technical treatment along lines of the “Graph of Math” post. To begin, allocate a node for each formula, and put an edge between f and Not-f, identifying Not-Not-f with f etc. If f forces g, add an edge from f to Not-g, and note that this is the same edge you get from the contrapositive. Now Gödel is talking about 2-colorings of this graph…so what does the rest say in this (decidedly non-sensational) vein?

December 7, 2013 10:55 am

That series did a great job of capturing Gödel’s “voice”, especiallly if you assume that he & Einstein have resumed their walks, and that’s he’s been putting in a lot of work on his one-liners.

• December 7, 2013 2:56 pm

Thanks again. About the graph in my other comment, on fourth thought, I might need 3 colors after all, which could put the task into NP-hardness territory…

December 8, 2013 1:07 am

Wrt the Benzmüller and Paleo machine proof, they admitted they chose the Ontological proof because it would make the headlines. So the real news would be their verification approach, which I haven’t had time to check out. As Jordan Sobel points out, the Ontological (aka God) Proof is Kurt Gödel cleaned up by Dana Scott and, looking at it purely as an exercise in logic, things don’t get much better attested than that,

December 7, 2013 6:45 pm

Chaitn’s Omega number holds the answer to the question of how many unprovable statements there are in math. Check out Chaitin’s book Meta-math, which can be found on arxiv.

December 10, 2013 5:55 am

A very good book indeed, but Chaitin’s constant is uncomputable except for a finite number of its digits. I guess that’s a consequence of the undecidability of Pip’s open problem…

December 13, 2013 11:14 am

From Chaitin’s work, we can conclude the following:

1) For any mathematical problem, the bits of Omega, when Omega is expressed in binary, completely determine whether that problem is solvable or not.

2) The bits of Omega are random, and only a finite number of them are even possible to know. (Therefore, an infinite number of them are impossible to know.)

3) Hence, most mathematics problems are impossible to solve, and most mathematical facts are impossible to prove.