Our first pass at Godel’s theorem: In her 2005 book, Incompleteness: The Proof and Paradox of Kurt Godel, Professor Goldstein states a fairly standard view concerning Godel’s role in twentieth century thought.
(For yesterday's post on this topic, click here.)
“His work was, in its own way, as revolutionary as Einstein’s,” she writes on page 21 of her book, which is intended for non-specialists. 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.”
That’s a fairly standard assessment. But what did Godel say or do in his revolutionary mathematical work? This brings us to the professor’s first attempt to explain Godel’s incompleteness theorem.
What is the incompleteness theorem? What does it assert?
Here at The Howler, we’ve conducted a love affair with bad explanation for three or four decades now. In our view, we can learn a lot about incoherence and bad explanation from Professor Goldstein’s first pass at this question.
What is the incompleteness theorem? As she starts, Professor Goldstein attempts to reassure us general readers. We’ll be able to understand Gödel's theorem or theorems, she seems to say:
PROFESSOR GOLDSTEIN (page 23): Unlike most mathematical results, Godel’s incompleteness theorems are expressed using no numbers or other symbolic formalisms. Though the nitty-gritty details of the proof are formidably technical, the proof’s overall strategy, delightfully, is not. The two conclusions that emerge at the end of all the formal pyrotechnics are rendered in more or less plain English. The Encyclopedia of Philosophy’s article “Godel’s Theorem” opens with a crisp statement of the two theorems:If we’re reading that passage correctly, we’ve been told that Godel rendered his theorem (or theorems) “in more or less plain English.” And not only that—the Encyclopedia of Philosophy offers “a crisp statement of the two theorems.”
At this point, Professor Goldstein block-quotes the following text from that encyclopedia. We'll present her block-quote in italics.
We can learn a lot about bad explanation from perusing this text:
PROFESSOR GOLDSTEIN (continuing directly):According to Professor Goldstein, that’s “a crisp statement” of conclusions which were originally “rendered in more or less plain English.” After she presents her block-quote, she refers to “this terse statement of the incompleteness theorems.”
In any formal system adequate for number theory there exists an undecidable formula—that is, a formula that is not provable and whose negation is not provable. (This statement is occasionally referred to as Godel’s first theorem.)
A corollary to the theorem is that the consistency of a formal system adequate for number theory cannot be proved within the system. (Sometimes it is this corollary that is referred to as Godel’s theorem; it is also referred to as Godel’s second theorem.)
We won’t even say that the professor is technically wrong. But it’s also true that no non-specialist will have any idea what that crisp, terse statement means.
From here on in, let’s consider the first statement alone:
“In any formal system adequate for number theory there exists an undecidable formula—that is, a formula that is not provable and whose negation is not provable.”
Is that statement of incompleteness rendered in more or less plain English? On its face, that may be how it looks.
As Professor Goldstein notes, there are no “symbolic formalisms” in that statement. And there’s no language which, on its face, seems excessively technical.
“Undecidable” is an uncommon word; according to Nexis, it has appeared in the New York Times just once in the past five years. But it’s built from a very common word, and every other word in that statement will be very familiar.
Everyone is familiar with words like “system,” “theory” and “formula.” On first glance, non-specialists won’t see the type of complex vocabulary which may throw them for a loop or signal that they have no idea what is being said.
But uh-oh! In that statement, familiar words are combined in unfamiliar ways:
Everyone knows what “numbers” are, and everyone is familiar with “theories.” But non-specialists will not be able to explain what “number theory” is. The same problem obtains with the unfamiliar phrase “formal system.”
Everyone can describe a “formal” occasion. Similarly, everyone has heard of the Social Security “system.” But non-specialists will have no idea what a “formal system” is, even though each word in the two-word phrase would be completely familiar in many other contexts.
Pity the poor general reader! He doesn’t know what “number theory” is—and he doesn’t have the slightest idea what a “formal system” is. Surely, he will have no idea what it is to have a “formal system” that is “adequate for” number theory, even though “adequate,” in other contexts, is an everyday word.
As such, this reader won’t have to try to figure out what an “undecidable formula” is. Before he meets that unusual term, he will be lost in a maze of concepts which are not part of “plain English” in any way at all.
You don’t need multisyllabic, Latinate words to depart from the realm of “plain English!” For the general reader, that statement of Godel’s first theorem may as well be rendered in Sanskrit.
This doesn’t mean there’s anything “wrong” with Godel’s first theorem, although that’s always possible too. It does mean there’s something a little bit strange about Professor’s Goldstein’s upbeat description of that first account of Gödel’s famous theorem.
It also means that we’ve wandered into the realm of bad explanation. No general reader of Goldstein’s book will have any idea what that statement of Godel’s theorem actually means.
Let’s be fair to Professor Goldstein! Three pages later, she seems to acknowledge the fact that we have a semi-problem here:
PROFESSOR GOLDSTEIN (page 26): Godel’s theorems, then, appear to be that rarest of rare creatures: mathematical truths that also address themselves—however ambiguously and controversially—to the central question of the humanities: what is involved in our being human? They are the most prolix theorems in the history of mathematics. Though there is disagreement about precisely how much, and precisely what, they say, there is no doubt that they say an awful lot and that what they say extends beyond mathematics, certainly into metamathematics and perhaps even beyond. In fact, the mathematical nature of the theorems is intimately linked with the fact that the Encyclopedia of Philosophy stated them in (more or less) plain English. The concepts of “formal system,” “undecidable,” and “consistency” might be semi-technical and require explication (which is why the reader should not worry if the succinct statement of the theorems yielded little understanding); but they are metamathematical concepts whose explication (which will eventually come) is not rendered in the language of mathematics...For the record, we looked up the meaning of “prolix.” We were puzzled by what we found, given various things the professor had already said.
But in this passage, the professor belatedly notes the fact that the concepts which appear in that first statement of Godel’s theorem are in fact “semi-technical.” They may “require explication,” we are belatedly told.
Three pages earlier, we were given no suggestion of this. Instead, it was suggested that we were delightfully looking at renderings which were composed in plain English.
Alas! Concepts like “formal system” and “number theory” aren’t semi-technical at all. For the non-specialist, these concepts may as well come from the dark side of the moon.
For the general reader, those concepts are completely “technical.” This means that Professor Goldstein’s first pass at Godel’s theorem wasn’t “accessible” at all.
Three pages later, the reader is told not to worry about that. Explication will eventually come, he is told—and indeed, Professor Goldstein does return to these basic questions, about a hundred pages later.
When she does, the incoherence continues, a point we’ll examine next weekend. For now, though, we’ve gained two key understandings:
Complete incoherence—bad explanation—can be rendered using language which looks extremely familiar. And also this:
As part of “what is involved in our being human,” we’re constantly asked to swallow incoherent presentations from our most honored professors. Inevitably, other professors will praise their lucidity on their books’ dust jackets.
Our culture runs on bad explanation. This is where it starts.