SATURDAY, MAY 3, 2014
Incompleteness incompletely explained: As we noted a few weeks back, we’ve conducted a long love affair with bad explanation.
The affair may have begun in September 1965, when we enrolled in Professor Nozick’s “Problems in Philosophy” class.
Who are these “problems” problems for, we found ourselves brightly wondering.
Years later, we began to suspect that the course had been designed to ensure that no one would major in philosophy. In fairness, though, that was pure conjecture.
Whatever! In recent weeks, we’ve been spending our spare time with Professor Goldstein’s 2005 book, Incompleteness: The Proof and Paradox of Kurt Godel. We’re going to make a lover’s confession:
We’re fascinated by the book’s mastery of bad explanation.
Who the Sam Hill is Kurt Godel, you ask. Early on, Professor Goldstein links him to Einstein and Heisenberg as a revolutionary thinker of the last century.
“His work was, in its own way, as revolutionary as Einstein’s,” Professor Goldstein writes (page 21). Godel’s work should “be grouped along the small set of the last century’s most radical and rigorous discoveries, all with consequences seeming to spill far beyond their respective fields, percolating down into our most basic preconceptions.”
According to Professor Goldstein, Godel’s incompleteness theorem (or theorems) “is the third leg, together with Heisenberg’s uncertainty principle and Einstein’s relativity, of that tripod of theoretical cataclysms that have been felt to force disturbances deep down in the foundations of the ‘exact sciences.’ ”
As far as we know, these are highly conventional judgments—hence the professor’s book. If Godel’s incompleteness theorem is as significant as is commonly said, it makes sense to write an accessible book explaining what he discovered, devised or invented.
That’s what Professor Goldstein set out to do. Her book is designed for non-specialists, for rubes like you and us. Indeed, the three blurbs on the back on the book stress how “accessible” it is:
Professor Pinker says that Professor Goldstein “offers us...a lucid exposition of Godel’s brainchild.” Professor Greene, of “Einstein made easy” fame, credits the book with “a detailed yet remarkable accessible account [Godel’s] most stunning breakthrough.”
Professor Lightman makes it three, crediting Professor Goldstein with a “penetrating, accessible, and beautifully written book.”
After all that, you open the book and start reading. But we don’t think that what you find is “remarkably accessible” at all!
According to us, you find something quite different—you find a fascinating compilation of wonderfully bad explanations. And, as we’ve already confessed, we’ve conducted a love affair with this literary form for perhaps four decades now.
Within the academy, Godel is famous, revered, regarded as deeply important. Does Professor Goldstein’s book offer “a lucid exposition” of his thought? Are her presentations “remarkably accessible?”
To the contrary! We’d say her work is fascinatingly muddled. In line with our true confessions, this attracts us all the more.
In our spare time, we’ll be doing some posts on Professor Goldstein’s book. In part, here’s why:
Our national discourse is virtually defined by bad explanation. Journalists rarely seem to notice, and professors like Professor Goldstein rarely intervene.
The professors leave us to twist in the wind. Left to themselves, they produce brilliantly muddled work which gets praised by gangs of their colleagues.
Before you know it, parents are paying tuition fees to expose teenagers to their work. And we know this! It happened to us!
The anthropologist in us regards this as a major discovery, a discovery concerning a form of life. Occasional weekend posts will follow, probably starting tomorrow.
Are you just learning that blurbs on the backs of books are often wrong? :)ReplyDelete
There are loads of popular books on the Incompleteness Theorems. Not all of them are good. The proof is very technical, and it's easy to go overboard about the implications.
Presumably, the book uses Einstein and Heisenberg because they also showed that there were absolute limits in the universe - we can't travel faster than the speed of light (Einstein), and the Uncertainty Principle.
Mathematicians have been bumping up against provable limits for a few centuries. Since Euclid, people tried to figure out how to trisect an angle with a compass and straight edge alone. In the 19th century, mathematicians proved it wasn't possible. For ages, mathematicians tried to prove Euclid's fifth postulate from the other four, and failed, until the 18th century, when it was proven that it couldn't be done. These are different in kind from the incompleteness theorem, but they were the start of this ability to see that our tools are limited. Einstein and Heisenberg found these limits in the laws of the universe, Godel found them in the world of logic and mathematics itself.
Well said, Thomas. I had planned to write a comment something like yours. Although I studied lots of math, I don't consider Gödel's result comparable to Einstein's and Heisenberg, because the latter two discovered things about the real world. Gödel merely proved something about our system of logic and numbers.Delete
Having taken a course in number theory, I wasn't surprised that there are unprovable theorems. There are infinitely many statements that could be made about the natural numbers, any of which might or might not be true. (Most of them are of no interest.) Anyhow, it's not surprising that some of them might be unprovable. On the contrary, I find it surprising when some number theorem statement is proved or disproved. E.g., it was impressive that Fermat's theorem was ever proved, although it took untold effort by scores of mathematicians over many decades to prove this one result.
Didn't Godel also supply a way of identifying in advance which theorems can be proven and which not? That seems considerably more useful than just stating that there are some unprovable theorems and I think that is his larger contribution.Delete
Why would scientists be puzzled by the notion that some things are impossible?Delete
Didn't Godel also supply a way of identifying in advance which theorems can be proven and which not?Delete
I don't think so. He cleverly created a statement which could not be proved within a system, yet which could be shown to be true. He showed that for any reasonably rich set of axioms, there would always be true, unprovable theorems.
I think you are wrong. His invention of Godel numbering allowed him to identify which theorems are provable. On Godel Numbering, Wikipedia says: Gödel specifically used this scheme at two levels: first, to encode sequences of symbols representing formulas, and second, to encode sequences of formulas representing proofs. This allowed him to show a correspondence between statements about natural numbers and statements about the provability of theorems about natural numbers, the key observation of the proof.Delete
Godel numbering allowed Godel to construct essentially self-referential statements that are true but not provable within the axiom system. This says nothing about whether there are other types of such statements. When faced with a conjecture in the system that hasn't been proved, no one knows whether it's true or false, and if it's the former whether there's a proof or not.Delete
He definitely did *not* figure out a way to decide whether had a theorem had a proof or not.Delete
Godel numbering lets you enumerate all proofs, but that doesn't mean you can figure out which theorem's are provable and which are not. The fact that you can enumerate all proofs, and hence all provable theorems, is an essentially useless and trivial result, because you can't enumerate all unprovable theorems.
Say you want to figure out if Fermat's Last Theorem is provable. You start a computer which tries all proofs. When do you know if it is unprovable? We don't.
As for why mathematicians would be surprised, I'm not sure if "surprised" is the word I'd use. Instead, I think they reached a moment where they could find mechanisms for the big "unsolvable" questions. There were loads of known unsolvable problems before - finding a rational square root of 2, (finitely) listing all primes.
But the fifth postulate question required an ability to step out of geometric intuition. Though Euclid's postulates were an attempt to formalize geometry, he failed by modern standards, and accidentally uses a lot of results that are 'intuitive' but don't follow from his axioms. Mathematics up to the 18th century was not as "axiomatic" as modern mathematics is. The formalisms just didn't exist before.
Nice explanation, Thomas. Consider another example:Delete
"The decimal expansion of pi has an infinite number of 9's"
Presumably, either this statement is true, or its contradiction (the expansion has only a finite number of 9's) is true. But, you can't prove either of these by simply working out pi to more and more digits.
Because this phenomenon (difficulty of explanation) seems to happen so frequently, you might begin to wonder whether the problem isn't with the professors but with the nature of explanation and the difficulties of thought and language.ReplyDelete
When someone wishes to fully understand a discipline, they start with an introductory course and continue to more advanced ones that take up more difficult concepts. How can someone expect a single book to cover that territory, without the mental effort needed to acquire the foundational understandings?
I do not understand why everyone expects hard things to be easy. I do not understand why people think there must exist some magic shortcut that will eliminate the work involved in acquiring a skill, or knowledge. I agree the book sounds oversold, but why the expectation that it should be able to deliver on such promises?
I don't like philosophy because of the mental effort involved (and because it seems to attract noisy males who think arguing is the same as shouting). Math too seems to involve a lot of effort for very little reward. But I never expected either to be easy to follow.
There's another principle here -- if you don't like the book written by an author, you may not like the books that same author recommends. Pinker and Lightman were clues. Why waste time reading something you don't like, much less complaining about it. There are a lot of bad books in the world crying out for reviews and Amazon now lets you write and post them. What a huge waste of time for everyone involved! People can even write bad books and publish them online and idiots can spend time reading them and discussing why they are so bad. We all need better hobbies, in my opinion.
you might begin to wonder whether the problem isn't with the professors but with the nature of explanation and the difficulties of thought and language.
And before long, you're the next Wittgenstein, intoning "Whereof one cannot speak thereof one must be silent" while refusing to shut up.
You have mistaken TDH's complaint as one of esthetics, namely that he has spotted what he considers badly written books and wishes to scold the authors on their bad taste in language. But TDH's charges are political: bad writing undermines our "national discourse" and professors like Goldstein rarely stop to help, presumably because they're too busy writing the next bad book.
My issue with TDH is not just that he expects "hard things" to be easy; he expects authors to make these things clear to him from first principles, so that he needn't make the slightest effort to understand the basics of the field under discussion. Certainly this has been his habit when he discusses physics. I expect the same when he confronts mathematical logic.
By the way, I expect you mean that those bad boys think shouting is the same as arguing, but perhaps I have ventured into one of those whereof/thereof areas.
Right, you get me.Delete
Clearly, Anonymous, you are mistaken. And to prove how provable that theorem is, Professor Whereof (deadrat) legibly states his issue by paraphrasing your comment. Too clever by half! Needless to say, enlightenment, while not achieved, was certainly abstractly defined. Looking forward to other kinds of statements from the deceased rodent is an existential fallacy because his offerings will be limited to self-reverential vacuous truths.Delete
"Our national discourse is virtually defined by bad explanation."ReplyDelete
In order for this to be more than feckless ranting, we should see an example of a nation and a period that was not like this. It would also be helpful to see a pathway described for healing the broken culture. Somehow, daily character assaults on a standard roster of bad people in our "tribe" doesn't seem like one.
In politics, taking complex issues that are being poorly explained and explaining them properly seems like a better use for a website than complaining about the explanations.ReplyDelete
The catch is that it requires a specific skill to be able to do that. One must first understand the issue, second understand the points of confusion, third know how to straighten them out and present the points more clearly, fourth be sufficiently succinct to keep readers involved, and fifth have the marketing skills to get someone to read what you've invested such effort into creating. Journalists are supposed to be taught all this. I don't think that is happening any more -- so I'm on Somerby's side about that. I think the reason is that our society doesn't value clear explanation any more and there is no pay for this skill, no market for it either. People want to be entertained not informed and politicians and bureaucrats realize that an uninformed or misinformed public is more pliable, so politics is now sound-bites.
I don't think Somerby has the skills to do this himself. I do think he has good critical thinking skills and is good at pointing out the problems and that he is performing a useful function that way. I'm not sure how often the recognition of a problem and its solution lie in the same person, in any domain.
In which case, Bob is batting 0 for 5.Delete
Or you are on the corresponding reading-for-comprehension scale.Delete
Professor Goldstein's husband, Professor Pinker, writes clear explanations. Ditch her books, read his.ReplyDelete
His books are controversial within psychology because he generalizes beyond the data and expresses opinion as fact.Delete
So he's full of shit but at least he's clear about it?Delete
Pinker's next book, to be published September 30, is The Sense of Style: The Thinking Person's Guide to Writing in the 21st Century.Delete
The discussion here put me in mind of a recent post by Paul Waldman, from which I quote:ReplyDelete
"One of the most persistent and defining features of American public opinion is that as a whole, the electorate is what political scientists call "symbolic conservatives" and "operational liberals." That is, when you ask them abstract questions they sound like conservatives expressing a dislike of big government. But when you ask them specific questions they sound like liberals, expressing support (and wanting to increase funding) for just about everything government does. The parties understand that, which is why Republicans tend to talk about principles and Democrats tend to talk about programs."
As some above have noted, explanation of complex issues is difficult in large part because the issues being explained are difficult for anyone to engage, must less grasp. I'd add that different explanations, or kinds of explanation, work for different people (a fundamental of teaching -- it's why, for instance, college students need to study with different professors in sequential courses). And everyone needs explanation more than once (even if that involves nothing more than re-reading, studying the same explanation), and reinforcement of points or methods already grasped, to gain some level of mastery.
This is why it is not fair to select a few journalists to go after -- and by "go after" I mean in part to cherry-pick from their work, to seize every perceived weak point in their arguments and ignore much of what they write or speak about, and then have a hissy fit.
Philosophers will never be kings, let us hope.
Btw, if anyone is looking for a very accessible and important book written by a professor (yes, a professor!) that might put some of the discussions at this site about race into a different perspective, I highly recommend Craig Wilder's Ebony and Ivy.
I like this theme of Bad Explanation, Bob. And it's definitely a fascinating and amusing phenomenon, if discouraging.ReplyDelete
This post and most of the comments read like the irrelevant, ivory tower ramblings of our nation's esoteric and useless professors.ReplyDelete
Then go away.Delete
You left out the treehouse.Delete
Occasional weekend posts will follow, probably starting tomorrow.ReplyDelete
Threat or menace?
I had a dream I was being chased by a cat. A nightmare! Woke up in a sweat and had to come here to post that I (pretend to) know something about philosophy, math, and physics. Also that I (pretend to) have a cheeky dialogue with Papa Bear Bob.ReplyDelete