The Long View 2003-09-2003: Gödel Incarnate

John's prediction here that the Supreme Court would find itself stripped of some of its powers by the other branches of government has not yet come to pass. At present, the highest court is still seen as a political prize to be won, rather than an impediment to be overcome.

The speculation about Hiliary Clinton running for President in 2008 was a pretty good guess, especially since in 2003 Barack Obama was still a member of the Illinois State Senate.


Gödel Incarnate

 

Can there be Gödel sentences in real life? Those are the kind of statements that crazy old Kurt Gödel identified, the ones whose truth value cannot be determined within the system in which they are expressed. I see that the character Neo in the Matrix series has been identified as a Gödel sentence in the Matrix system, but that's a movie. Reality is not nearly so pretentious. Nonetheless, it does sometimes do similar tricks.

* * *

Consider, for instance, the recent opinion by a panel of the 9th Circuit Court of Appeals, which held that California's gubernatorial recall election would have to be postponed until next March, so that some counties in that state could replace their paper-punch ballot machines with more modern ones. That decision, perhaps mischievously, relied in part on Bush v. Gore, the US Supreme Court decision that ended the presidential election of 2000. (We now know that the opinion did not decide the election: the recount the Florida Supreme Court had ordered, and which the US Supreme Court stopped, would still have produced a win for Bush; but that's another story.)

The interesting thing about using Bush v. Gore is that the per curiam opinion says you're not supposed to. The majority said:

The recount process, in its features here described, is inconsistent with the minimum procedures necessary to protect the fundamental right of each voter in the special instance of a statewide recount under the authority of a single state judicial officer. Our consideration is limited to the present circumstances, for the problem of equal protection in election processes generally presents many complexities.

Indeed it does, not the least of which is how to read a holding that says not to read it. Of course, it's not unusual for appeals court opinions to lack precidential value. There are so many that only a fraction are selected for the official case reports, because they deal with issues of general importance. The other opinions just decide the issue between the parties. Lawyers can cite unreported opinions, if they can find them. However, like articles in legal journals, those opinions are presented merely for their persuasive value; they are not precedents. The odd thing about Bush v. Gore is that the text seems to suggest the opinion should not be used even for that purpose in the future.

An expanded panel of the 9th Circuit is going to rehear the recall case in a few days. Maybe they will make the logical problem go away. If not, California could disappear into a Mobius strip.

* * *

Speaking of the US Supreme Court, the October issue of First Things has several pieces on that eminent tribunal and its wicked ways. The keynote article, "The Supreme Court Rules," by Michael M. Uhlmann of Claremont Graduate University, usefully highlights the fundamentally arbitrary nature of the Court's practice of according different levels of scrutiny to different classes of Constitutional questions; since the Court took to defining rights out of thin air, the "Scrutiny Game" has become wholly capricious. For our purposes, though, the relevant bit comes in the summary at the end:

The second flaw in living constitutionalism is that, if the Constitution is an endlessly changing document, it is unclear why its provisions authorizing judicial power should be considered sacred and permanent. In its aggressive assertions on behalf of a living Constitution, the Court runs the risk of undermining the principle basis of its own authority. It may find, as Professor [Alexander] Bickel warned long ago, that it has no ground on which to stand.

There is a bit of magic in this kind of analysis. The heroine defeats Rumpelstiltskin by identifying him. Captain Kirk unhinges misguided computers by proving that they are the sort of thing they were designed to destroy. Of course, we find these motifs in stories because we also meet them in the light of day. The Supreme Court has lost its supremacy more than once in American history. Once again, as in the days before the Civil War and during the Depression, the Court has become a June bug in search of a windshield.

* * *

Then there are sentences that generate all the confusion of Gödel sentences, but without the logical rigor. Consider this one from another First Things piece, "Scandal and the Constitution," by L. Marin Nussbaum. It appears in a grand jury's report on the affairs of the Diocese of Rockville Centre in New York State. The grand jury failed to find anything indictable, but it did opine "that the conduct of certain diocesan officials would have warranted criminal prosecution but for the fact that existing statutes are inadequate." In other words, the officials would have been criminals but for the fact they committed no crimes. This is very close to what is called "an Irish bull," of which a fair sample is this: "And now the only animals that live on the farm are the birds that fly over it." Yogi Berra is the master of this sort of expression, except that, unlike the grand jury, Berra isn't stupid.

There a couple of points in the Nussbaum article I'm not altogether happy about. For one thing, I don't think that grand jury went beyond its authority by investigating the diocese. On the other hand, I would agree that the investigation was probably an instance of prosecutorial abuse. I was on a grand jury once, and I think everything they do is prosecutorial abuse.

* * *

Finally, we come to the significance of the newly announced candidacy of retired NATO commander, General Wesley Clark, for the Democratic presidential nomination. The timing of the announcement, and the fact that so many of Clark's campaign staff are members of the Clinton machine, have led to some very entertaining speculation.

Some say Clark is really running for vice president, anybody's vice president. In other versions, Clark is just the last in a series of no-hope candidates whom Bill Clinton has encouraged to run. The idea is to ensure the Democrats lose in 2004, thus leaving Hillary Clinton the obvious candidate for 2008. The more elaborate scenarios make Clark a "stalking horse" for Hillary. She will announce in due course; then Clark will turn over his campaign to her, with himself as her running mate.

There is a story about Charles DeGaulle and stalking horses. During the 1950s, if you were writing about French politics, you could make a modest living by speculating about when DeGaulle was coming back to power. Fourth Republic governments changed every few months, and every new government had one or more members who could plausibly be characterized as a stalking horse for him. DeGaulle did come back, in 1959. Then one wire-service reporter, who had been filing stalking-horse stories in a robotic fashion for years, filed a story speculating that DeGaulle was a stalking horse for DeGaulle.

The reporter was fired immediately. DeGaulle was fired in 1969.  

Copyright © 2003 by John J. Reilly

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The Long View: Gödel: A Life of Logic

Kurt GödelI like to think that I am re-posting all of John's blog as some sort of service to humanity, but really I just enjoy rediscovering gems like this one. John's review of a biography of Kurt Gödel has been definitive in shaping my opinions about AI and computation.

In short, I don't think strong AI is possible, and this is the true explanation why computer scientists have spent the last seventy years looking for it without finding it.

Roger Penrose famously criticized strong AI in his book The Emperor's New Mind. Wikipedia's summary claims that so many eminent scientists have criticized Penrose's position that it is effectively refuted, to which one might reply, "OK, when where are all the AIs?"

I think Penrose truly fails by looking for the mind in physics. He is really just embodying the spirit of the age, but it is a sad thing to see when he was perceptive enough to notice that the essence of thinking, abstraction, is not algorithmic.

There really is a similarity between what minds do and what computers do, but the real similarity makes AI less likely instead of more. Computers are the instantiation of the immaterial forms of Plato [thereby proving Aristotle right]. Ross's conference presentation also illustrates the dangers of treading outside one's field. I do it, I like to do it, but I am always aware that I can sound just as silly to others as they sometimes sound to me. Ross makes an off-hand comment in his presentation about the deadliness of dioxin, which was quite the trendy toxin for a while. Then the Russians tried to poison the Ukrainian presidential candidate Viktor Yushchenko wtih dioxin [a preview of the recent unpleasantness], presumably under the impression that it was exceptionally deadly, only to find that all it did was give him a bad case of acne. Oops. Maybe that was just a clever counter-intelligence ploy in the wilderness of mirrors, like the time we sold the Russians faulty gas equipment.

I'm not an expert in toxicology, but I at least need to know enough to be able to accurately communicate with the experts so I can demonstrate the products I design are safe. Dioxin isn't nice stuff, but the dangers were wildly overblown.

This was also the beginning of the end of my interest in Neal Stephenson's books. His environmental thriller Zodiac featured a plucky band of environmental crusaders who thwarted a plot to dump dioxin in Boston Harbor. I already knew that dioxin wasn't all it was cracked up to be, and once I noticed one that was a little off, I started to notice a lot of things that were a little off. Oh well.

 

Gödel
A Life of Logic

by John Casti and Werner DePauli
Perseus Publishing, 2000
210 Pages, US$25
ISBN 0-7382-0274-6

 

Kurt Gödel (1906-1978) was the mathematician and logician whose now famous incompleteness theorem easily ranks among the most uncanny products of the notoriously uncanny first half of the European 20th century. This very brief book by two computer scientists does try to fit Gödel into the world of scientific Vienna in the1920s and 30s. (The book started life as a program for Austrian television: there is a great deal of talk about mysteriously undecidable recipes for Sachertorte pastry.) The authors are more concerned, however, to explain the theorem itself, its relationship to the idea of computability, and the connection all these things have to such questions as the feasibility of artificial intelligence and time travel. This is an unmanageable amount of ground to cover, and the treatment is uneven. Still, simply addressing all these topics between two covers is an accomplishment. The authors provide a blessedly brief, ten-item reading list for those who want to look more deeply into the separate areas covered.

Gödel was born in the town of Brno, in what is now the Czech Republic, to a family that had grown wealthy from textile manufacturing. The Gödels were German-speaking. The authors tell us they were not Jewish, but we learn no more about confessional affiliation, beyond the fact Kurt was anti-Catholic all his life. Gödel entered the University of Vienna to study physics, but switched to mathematics after a few years. He soon became a member of the Vienna Circle, the influential group that sought to reduce all philosophical questions to problems of language.

Like Karl Popper and Ludwig Wittgenstein, who were more loosely associated with the Circle, Gödel's membership probably helped him most by providing fodder for criticism. Indeed, few thinkers have ever been less interested than was Gödel in closing down metaphysics. If mathematical Platonism were a religion, Gödel would have been Billy Sunday, his American evangelist contemporary. For Gödel, mathematical objects were as "given" as lumber. They are just another kind of semantic content of sentences. What Gödel did in his proof, the first published version of which appeared in 1931, was to show the weakness of syntax, the system by which semantic content is ordered. The incompleteness theorem shows that there are propositions that we know to be true, but that are nevertheless logically unprovable. A slightly more rigorous formulation is that any logical system at least as complicated as arithmetic will be incomplete, because it will be able to produce statements that cannot be proven or dispoven within the terms of the system. The natural languge versions of the "Liar Paradox" are of this nature.

While Gödel was thinking these deep thoughts, the politics and economy of the German-speaking world were going to hell in a hand-basket. The failure of the Austrian bank, the Credit-Anstalt, in the same year as the publication of the theorem is usually blamed for blowing up the already stressed European financial system. Austria's First Republic, created when the Habsburg empire disintegrated after the First World War, collapsed into rule-by-decree in 1933. Nazi Germany annexed Austria in 1938. (This happened, it must be said, with the approval of most Austrians.) The Second World War began in 1939.

Gödel divided his time in those years between Vienna and the Institute for Advanced Studies at Princeton, New Jersey. The Institute, acting in large part under the influence of John von Neumann, served through most of the'30s as a haven for scientific refugees from Europe. Gödel was not neglected. He was offered and took several temporary appointments, but he kept going back to the University of Vienna. Although too unworldly to have ever engaged in politics, he did lose his license to lecture, the Privat Docent, because of his connection with the Vienna Circle, which the Nazis regarded as too Leftist and too Jewish. However, he applied for and actually received a new license as a Docent of the New Order. It was only in 1940, when it was apparent he would be drafted, that he left Austria for good. He and his wife traveled east by train across the Soviet Union, then to Japan, then to the West Coast of the United States, and then to Princeton. His wife, Adele, did not like New Jersey, but they stayed permanently.

There are many legends about Gödel's antics at Princeton. This book gives us only a few of the best-known ones, such as how Einstein himself had to help calm Gödel down when the latter went to take the oath of citizenship. (It seems that Gödel had found a logical flaw in the federal constitution that would permit the creation of a dictatorship, and he insisted on telling the judge.) The most surprising thing to me, however, was that Gödel was actually a conscientious faculty member. His flaw was that he tended to obsess about the work of any committee on which he sat.

Although Gödel continued to produce significant mathematical results during his time at Princeton, he was never again as productive as he had been at Vienna. (His wife called the Institute "an old-folks' home," and she may have had a point.) In any case, his interests turned increasingly to philosophy. Gödel famously constructed an ontological proof of the existence of God (he was a great admirer of Leibniz, who had a proof of the same type), and an independent proof of personal immortality. (Karl Popper had one of these too, by the way.) We are told that Gödel was also interested in "the occult," but are given no specifics.

Gödel was paranoid, convinced that someone was trying to poison him. He therefore always made a great fuss about eating. When he died of what his doctor called "malnourishment and inanition," he weighed just 60 pounds. On the other hand, he also suffered throughout his life from some obscure gastro-intestinal disorder, so it is possible that an underlying basis for this behavior was simply never diagnosed.

Why should we care about crazy old Kurt and his annoying theorem? For one thing, it's immensely practical. The theorem, and Alan Turing's related Halting Problem that was developed at about the same time, are key to our understanding of what computer programs can and cannot do.

Perhaps the most interesting such question, covered at length in this book, is whether it is possible to construct an artificial computer intelligence. In "The Emperor's New Mind" (1989), Roger Penrose revived an argument based on Gödel's theorem against the possibility of an algorithmic machine mind. To put it briefly, Penrose pointed out that people can spot "Gödel sentences," true but unprovable propositions, that computer programs cannot detect. Thus, he reasoned, whatever else the human mind is doing when it spots these sentences, it is not computing. Refutations of Penrose are usually variations on the idea that Gödel's theorem applies only to consistent systems, and human beings clearly do not think consistently.

I should note that I find this argument mysterious. If human minds are being inconsistent when they do advanced mathematics, then how do we manage to reach the same conclusions consistently? In any case, even the most committed Artificial Intelligence believers have mostly abandoned the idea that a program for an artificial intelligence can be written. Now they hope to create Darwinistic, self-programming systems that will organize an intelligent entity spontaneously. Good luck.

Gödel's theorem serves in popular culture as a symbol of the supposed irrationality of reality. As the authors note, the theorem tends to be dragged out these days to "hit people over the head" with. The authors are too polite to point out that this most subtle of logical arguments is often employed by persons who cannot make any logical argument at all. Nonetheless, it is clear that the theorem and the body of study it make possible are philosophically important, though people differ on just why. For me, the theorem is good evidence that the limits of language are not the limits of knowledge, or even of reason, broadly construed. This suggests that the world is objectively knowable. Surely this is a good thing.

 


Copyright © 2000 by John J. Reilly


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The Long View 2002-05-02: Bill Clinton in 2005!

Another Constitutional law post from John. Since I'm not a lawyer, I'll refrain from commenting on the technical merits of his proposal other than to say it seems plausible to this non-specialist. I also think I remember a joke making the rounds a while ago about how W. was still eligible to run for re-election in 2008 since he wasn't really elected the first time.

Bill Clinton in 2005!

They tell a wonderful story about Kurt Gödel, the greatest of 20th century logicians. He fled Europe during World War II, and when he went to take the oath of U.S. citizenship before a federal judge, Albert Einstein himself came along as a witness. The judge chatted with his prominent visitors before the ceremony, unfortunately. Alluding to the collapse of law in Nazi Germany, the judge remarked that the Constitution prevented anything like that from happening in the United States. "Not true!" Gödel replied, and explained that he had found a logical flaw in the Constitution that could be used to found a dictatorship. It took Einstein two hours to calm him down.

Say what you like about the Clinton Administration, it did at least provide an eight-year tutorial in aspects of constitutional law that almost no one had ever heard of before. Indeed, the Clinton's still have that effect, even though they left the White House almost a year and a half ago. Liz Smith, the gossip columnist, aired an argument in her column of May 7 for the proposition that Bill Clinton really could serve a third term. The notion is that, if Bill Clinton were elected vice president, presumably as number two on a Hillary ticket, he could succeed her if she did not serve out her term. Liz Smith has no pretensions to constitutional scholarship, and it is not clear who suggested the idea to her. Nonetheless, the argument is plausible.

The Twenty-second Amendment to the Constitution was ratified in 1951, in the aftermath of the administration of Franklin Delano Roosevelt, the only president to break with the tradition of the two-term maximum. (He sought and won four terms in office.) Common knowledge has it that the Constitution now prohibits anyone from serving as president for more than two terms. However, the Amendment does not quite say that:

Section 1. No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once. But this Article shall not apply to any person holding the office of President when this Article was proposed by Congress, and shall not prevent any person who may be holding the office of President, or acting as President, during the term within which this Article becomes operative from holding the office of President or acting as President during the remainder of such term.

Note that this text does not address how long one may be president, but simply how one becomes president. It forbids anyone to be "elected" more than twice. Succeeding to the office is another matter, however. This provision does not, by its terms, forbid someone who has already been elected president twice from becoming president if the incumbent should die or resign. I might also remark that not only vice presidents can succeed to the presidency; a two-term president emeritus might be anywhere in the line of succession.

The Twelfth Amendment defines the operation of the Electoral College and how Congress should choose a president if the College does not give any candidate a majority. A seeming objection to the possibility of a president-for-life is offered by the last sentence of the Amendment, which says:

But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President.

At the time the Twelfth Amendment was ratified, the terms of eligibility in question were clearly those set out in Article II, Section 1, Paragraph 5, which require that the president be a natural-born citizen, at least 35 years old, and a US resident for at least 14 years. The Twelfth Amendment adds a further requirement that the president and vice president not be "inhabitants" of the same state. Did the addition of the Twenty-second Amendment add to the eligibility requirements?

Not by the letter of the text. The Twelfth Amendment is about how presidents are elected, not about who can serve. All we are told about "eligibility" is that it is the same for the vice president as for the president. If succession by a two-term president is possible under the Twenty-second Amendment, the Twelfth does nothing to change matters. But might the Twelfth Amendment make a former president ineligible to run for vice president? Probably not, because no provision of the Constitution makes someone who has been twice elected president "ineligible for the office of President." The Constitution simply forbids such a person to be elected yet again. If there is no such ineligibility for a president, then there is none for a vice president.

Even if my interpretation of the text were the only possible one, that would not settle the issue. A look at the statutory history of the Twenty-second Amendment might show that its drafters and the legislators who voted for it were all intend on ensuring that no one would ever again be president for more than eight years. In that case, a court asked to apply the Twenty-second Amendment would probably look to the intent of the Amendment, rather than to its literal terms. Of course, legislative history might also show that the drafters and ratifiers meant to leave open the possibility that an experienced gray head could serve again as president, presumably in some emergency when the government had been decapitated. When they spoke of "election," maybe that is what they meant.

The only place to look for precedents would be the states. I am not a great fan of term limits in any form, but many states have them. It is quite possible that just the question we have been considering has arisen before. State court decisions interpreting such statutes would not be binding on the federal judiciary, of course, but they might be persuasive. From what little I recall about the subject, I believe that the states have tended to interpret term limits narrowly rather than broadly. In other words, if an incumbent makes a plausible argument for why a term limit should not apply, the courts will usually accept it.

I doubt that the particular anomaly we have been considering is the one that Kurt Gödel was thinking about. I am also pretty sure that Bill Clinton has no intention of running for vice president in 2004, or in any other year. Still, it may someday be important that the rules for succession to the presidency are looser than those for election. Constitutional law is full of surprises.


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Physicists' notoriously casual attitude toward mathematics

Alexander Pruss, a philosopher at Baylor, comments on physicists' notoriously casual attitude towards mathematics and notation. This is right on. Like the first commenter, in school I remember the snide remarks the math and physics profs would direct at each other on this subject. As a physicist at heart, I pretty much adopted the more casual, plug-n-chug attitude of the physics professors. I suppose engineers are even worse.

This brings the unreasonable effectiveness of mathematics to a whole new level. Math still works even when you are not really doing it right.

h/t The Fourth Checkraise

Content Logic vs Formal Logic

In an earlier post, I discussed the merits of an Aristotelian logic as term logic: we begin by simply noting that things exist, and we know them. Only then can we move on to construct propositions and then chains of reasoning.

Aristotelian logic is also content logic. Content logic may be distinguished from formal logic by its dependence on the terms within it. Formal logic analyzes propositions solely by their form, its primary tool is the truth table. For a content logic, on the other hand, it makes all the difference in the world what we are actually talking about, because we are often interested in arguments that are not true by necessity. Formal logic can tell us whether an argument must always be true, may sometimes be true, or can never be true. A content logic is most useful in the middle case, because we may need to know just how often, or how much, sometimes really is. Especially in matters of natural science, we often deal with contingent being, things that could be otherwise.

One modern attempt at reintroducing the full range of logic was David Stove's Rationality of Induction. A brief teaser of this is available from William Briggs. Most of the examples provided therein are textbook cases of an undistributed middle. I have not read Stove's work, but a more robust logic like that of Aristotle can make use of formal logic while avoiding the kinds of mistakes highlighted in the teaser.