In search of GodelThink: Kurt Godel is one of the greats, or so you’ll routinely be told. The leading authority on his work says it went something like this:
Kurt Friedrich Gödel (1906-1978) was an Austrian, and later American, logician, mathematician, and philosopher. Considered with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell, A. N. Whitehead, and David Hilbert were pioneering the use of logic and set theory to understand the foundations of mathematics.He’s right up there with Frege—and with Aristotle, who “defined motion as the actuality of a potentiality as such.” (For additional details, click here.)
Godel is considered one of the greats. In her 2005 book, Incompleteness: The Proof and Paradox of Kurt Godel, Professor Goldstein ranks Einstein, Heisenberg and Godel as the great revolutionary thinkers of the last century.
We’re not saying Professor Goldstein is wrong! We’re merely wondering what it is that Godel determined, devised or discovered. Why is Godel one of the greats? This is where our love affair with bad explanation comes in.
Why is Godel one of the greats? And can it be explained to us rubes? As we noted in yesterday's post, Professor Goldstein’s book was aimed as non-specialists—and it was hailed by three other professors for being “remarkably accessible.”
Is Professor Goldstein able to explain what Godel said or did? In a series of posts, we plan to let you be the judge.
What the heck did Godel say, determine or discover? Early in her book, Professor Goldstein’s explanation starts as shown below.
She pictures Godel (“the logician”) in one of his conversations with Einstein, the friend of his later life. The two men often strolled through the streets of Princeton:
GOLDSTEIN (pages 20-21): The topics of their daily conversations range over physics and mathematics, philosophy and politics, and in all of these areas the logician is likely to say something to startle Einstein in its originality or profundity, naivete or downright outlandishness. All his thinking is governed by an “interesting axiom,” as Ernst Gabor Straus, Einstein’s assistant from 1944 to 1947, once characterized it...An “interesting axiom” governed Godel’s thought, we’re told as our journey begins.
The reader leans forward, expectant. But here’s what the reader gets next:
GOLDSTEIN: The topics of their daily conversations range over physics and mathematics, philosophy and politics, and in all of these areas the logician is likely to say something to startle Einstein in its originality or profundity, naivete or downright outlandishness. All his thinking is governed by an “interesting axiom,” as Ernst Gabor Straus, Einstein’s assistant from 1944 to 1947, once characterized it. For every fact, there exists an explanation as to why that fact is a fact; why it has to be a fact...Below, we’ll provide the full text of this lengthy paragraph. But in our view, our long day’s journey into bad explanation rather plainly starts here.
In that passage, Professor Goldstein endorses the view that Godel’s thought stemmed from “an interesting axiom.” She then presents two formulations of that axiom.
One of her formulations seems completely mundane. The other is quite hard to parse.
This is how bad explanation starts! Let’s consider the two formulations which constitute Godel’s “interesting axiom,” at least as explained by Professor Goldstein.
(1) “For every fact, there exists an explanation as to why that fact is a fact.”
On its face, that seems to be the most mundane assertion on earth. On its face, that claim would seem startling, controversial or insightful to almost no one.
Almost everyone is familiar with the idea that factual claims must be supported. You can’t simply make a factual claim. You have to back it up.
For every fact, there exists an explanation as to why that fact is a fact? On its face, it’s hard to see how this could possibly constitute “an interesting axiom,” let alone serve as the foundation for revolutionary thought.
On its face, that seems like a mundane statement. Let’s move on to Professor Goldstein’s second formulation:
(2) “For every fact, there exists an explanation as to why that fact has to be a fact.”
For every fact, there exists an explanation as to why it has to be a fact? Do you have any idea what that means?
Frankly, we do not. Consider:
“Boise is the capital of Idaho.” Most people would regard that as a statement of fact.
You could explain why it’s a fact. But could you explain why it has to be a fact? Would you even have any idea what such a request would mean?
Frankly, we would not. For ourselves, we have no idea what that second formulation means.
At this point, we’re flirting with bad explanation! Professor Goldstein has given us two formulations of the “interesting axiom” which lies at the heart of Godel’s revolutionary thinking.
One formulation seems highly mundane; the other seems incoherent. But Professor Goldstein doesn’t seem to notice this problem. She simply moves ahead in this, her complete, rather flowery paragraph:
GOLDSTEIN: The topics of their daily conversations range over physics and mathematics, philosophy and politics, and in all of these areas the logician is likely to say something to startle Einstein in its originality or profundity, naivete or downright outlandishness. All his thinking is governed by an “interesting axiom,” as Ernst Gabor Straus, Einstein’s assistant from 1944 to 1947, once characterized it. For every fact, there exists an explanation as to why that fact is a fact; why it has to be a fact. This conviction amounts to the assertion that there is no brute contingency in the world, no givens that need not have been given. In other words, the world will never, not even once, speak to us in the way that an exasperated parent will speak to her fractious adolescent: “Why? I’ll tell you why. Because I said so!” The world always has an explanation for itself, or as Einstein's walking partner puts it, Die Welt is vernunftig, the world is intelligible. The conclusions that emanate from this rigorously consistent application of the “interesting axiom” to every subject that crosses the logician’s mind—from the relationship between the body and soul to global politics to the very local politics of the Institute for Advanced Study itself—often and radically diverge from the opinions of common sense. Such divergence, however, counts as nothing for him. It is as if one of the unwritten laws of his thought processes is: If reasoning and common sense should diverge, then so much the worse for common sense! What, in the long run, is common sense, other than common?Die Welt is vernunftig, Gödel said. He hadn’t read this book!
If you’re an obedient student or reader—if you’re in the (very bad) habit of deferring to intellectual authority—you may simply say all those words to yourself and just continue reading. You’ll fail to note that you have no idea what this professor is talking about, if she knows herself.
If you are one of her fellow professors, you will agree to compose a back-of-book blurb about how “beautifully written” or “artfully written” this “remarkably accessible” book is (Professors Lightman and Greene). If you’re Professor Pinker, you’ll blurb that “this book is a gem,” that it has been “written with grace and passion” by a “gifted novelist and philosopher.”
If you’re less inclined to defer to authority, less flattering thoughts may enter your head. You may consider the possibility that this long paragraph is an example of pure argle-bargle, composed by a gifted novelist who may not be real clear as to what she’s talking about.
You may note that this large bouquet of flowery language emerges from that one highlighted passage, in which the novelist attributes two ideas to Godel—one of which seems completely mundane, one of which seems incoherent.
Let’s be fair! In our view, Professor Goldstein swings and misses as she tries to explain the “interesting axiom” at the heart of GodelThink.
But in fairness, her book has barely started. Speaking of incompleteness, Professor Goldstein’s “Introduction” starts on page 13; pages 1-12 are unaccounted for. The professor’s explanation is just getting started as this passage appears.
As advocates of fairness, we didn’t stop reading at this point. Generously, we moved ahead, hoping for clarity.
We were quickly rewarded. On page 23, Professor Goldstein quotes an account of Godel’s seminal incompleteness theorem in which, she says, that theorem is “rendered in more or less plain English.”
In our next post, we’ll show you that rendering. For ourselves, we would say that the rendering in question doesn’t involve “plain English” at all. The fact that Professor Goldstein thinks it does may teach a key lesson about a key topic—where bad explanation comes from.
In one area after another, our American national discourse is built around bad explanations. In a slightly more rational world, ranking professors would push back, skillfully and hard, against that state of affairs.
In our world, professors hand us work like this; other professors praise such work for its remarkable clarity. When our ranking professors function this way, we are all lambs in the end.