(Larger version here.)

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Over the course of the evening, I proposed a variety of mutually contradictory health care policies; my intent was not to endorse any one of them, but to demonstrate that all of them were preferable to Professor Brown’s pet proposals. I cribbed some important ideas from David Goldhill’s Atlantic Monthly article, and some important ideas and facts from the always insdispensable Bill Easterly. The format did not lend itself to citations or hyperlinks, but I’m glad to make amends here.

We got the whole thing on video, and I’m posting it all here. (If you prefer to read text, I’ve already blogged many of my thoughts on this matter.) The final segment shows the handout that Professor Brown distributed and referred to during his initial remarks.

Professor Brown’s opening (length 23:11):
(Length: 23:11).

My response. (length 19:34):

His re-response (length 10:14):

My re-re-respnse (length 2:29):

Question period (length 30:17):

I also have a brief video of the flyer that Professor Brown distributed during his opening remarks:

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Today—and I literally mean today—Henry Molaison’s brain is being sliced and diced at the U.C. San Diego Brain Observatory in furtherance of neurological research. And starting soon, you can watch the slicing live via webcam. The process started yesterday, and they’re currently (as of 12:20PM eastern time on Thursday, December 3) on break, but a note on the site says that “cutting will resume shortly”.

The World Wide Web is a strange and wonderful place.

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There are many popularizations of Godel’s original argument, of which at least one (by Nagel and Newman) is superb. They do a marvelous job of boiling the argument itself and the surrounding issues down to a little over a hundred sparse pages. But I can boil it down further, into a single blog post. I can do this via the magic of one of my favorite expository techniques, a technique I call “lying”. I will lie to you throughout this post, by sweeping important technical details under the rug and (slightly) corrupting some important ideas to make them easier to grasp—but without, I think, sacrificing the flavor and the gist of the argument. At the end, once we’re clear on the big picture, I’ll ‘fess up to some of those lies.

Here, then, is (more or less) how Godel proved that arithmetic contains true but unprovable statements:

Step One: Start by dividing all numbers into two categories. We’ll call a number “good” if it can be written as the difference of two primes and “bad” if it can’t. (Here a “number” means a positive integer.)

Step Two: Next make a list of all possible statements about arithmetic. Something like this:

1. 3 is a prime number.
2. 2 plus 3 equals 7.
3. Every number is the sum of at most four squares.
4. x is a prime number.
5. 2 plus 3 equals x.

Some of these statements, (like numbers 1 and 3) are true. Others (like number 2) are false. Still others (like numbers 4 and 5) contain variables and hence are neither true nor false unless the value of the variable is specified.

Step Three: When we make our list, let’s be sure to follow this rule:

Rule A: Every provable statement should go next to a good number and every non-provable statement should go next to a bad number.

The list above violates this rule. After all, 2 is the difference of two primes (it’s 7 minus 5), so 2 is a good number, so statement number 2 should be provable. Since we’re pretty confident that “2 plus 3 equals 7” isn’t provable, we should move it someplace else on the list, say to line 7. Revise the list as needed so Rule A is always obeyed.

Step Four: After we’ve rearranged our list in accordance with Rule A, let’s peruse it a bit. Suppose—just suppose—that we happen to find a line like this:

24537. 24537 is a bad number.

That is, statement number 24537 just happens to mention the number 24537, and declares that it is a bad number.

Given this, we can figure out whether 24537 is good or bad. Suppose first that it’s good. Then (because of Rule A), statement #24537 must be provable, and hence true. But statement #24537 says that 24537 is a bad number. So if 24537 is a good number, then it’s a bad number. That makes no sense, and we conclude that it can’t be good.

Okay, then, 24537 is bad. So the statement “24537 is a bad number” is true. But that very statement is written next to a bad number, which according to Rule A, means it’s unprovable. Conclusion: “24537 is a bad number” is both true and unprovable. Our quest for a true but unprovable statement is complete.

Step Five: Of course it would be just as good to find a line like

198764. 198764 is a bad number.
or
2381212. 2381212 is a bad number.
or
37999999. 3799999 is a bad number.

Any one of these lines would be enough to demonstrate that there is a true statement in arithmetic that is not provable. But of course, maybe there simply are no such lines.

Step Six: Let’s revise our list again. In addition to following Rule A, we’ll follow one more Rule:

Rule B: If a statement contains a variable, and if that variable is replaced by the statement’s own number, the number on the resulting statement should be the square of the number on the original statement.

To understand what that mouthful means, suppose that statement 6 just happens to be:

6. x is prime.

Then Rule B requires that statement 36 must be

36. 6 is prime.

Likewise if statement 7 happens to be:

7. 2 plus 3 equals x

then Rule B requires that statement 49 is:

49. 2 plus 3 equals 7.

Step 7:Once we’ve revised our list according to Rule B, let’s scan up and down the list for the following statement:

x² is a bad number

This statement has to be somewhere on our list, because every statement about arithmetic is somewhere on our list. Let’s just suppose it happens to be statement number 13:

13. x² is a bad number.

Then Rule B tells us what’s on line 169 (169 is the square of 13):

169. 13² is a bad number.

Or in other words:

169. 169 is a bad number,

So we have found exactly the kind of entry that we agreed in Steps Four and Five provides us with a true but non-provable statement, namely:

169 is a bad number

or, eliminating our value-laden terminology:

169 is not the difference of two primes.

The existence of this true but unprovable statement is a necessary consequence of our ability to make a list that obeys Rules A and B.

Step Seven: Are we done? Not quite. How do you know we can make a list that obeys both Rule A and Rule B? We needed to revise our list so it follows Rule A, and then revised it again so it follows Rule B. But when you revise your list to implement Rule B, you might screw up Rule A. When you re-revise to make sure you’re following Rule A, you might screw up Rule B.

Here’s what Godel did: He gave an explicit recipe for building a list that obeys both Rule A and Rule B. Once you’ve got that list you’re done.

Confession: Actually, that’s a lie. In order to make things work, Godel needed a considerably more complicated definition of “good” and “bad” numbers; our definition in terms of differences of primes doesn’t quite work. And in Rule B, instead of squaring, he needed a more complicated rule to give the number of the new statement that results when you plug a statement’s number into itself. But this more complicated rule still has a purely arithmetical description, which is all we need to make the argument work.

(In fact, Godel went much further. His recipe constructs a list where every grammatical relationship between statements is reflected by an arithmetical relationship between their corresponding numbers. Rule B—or something like Rule B—is one very special case of that phenomenon.)

The Bottom Line. Our true but unprovable statement turned out to be 169 is not the difference of two primes. This is a bogus example, because our Rules A and B are simplifications of the rules Godel actually implemented. But the real example has exactly the same flavor; it says that a particular number fails to have a particular (purely arithmetical) property. The number itself, and the description of the property, are too gigantic to squeeze into this post, but are no less significant for being large.

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There have been several reports of individual stores failing to honor this commitment. My publisher will be having a chat about this with the Barnes and Noble folks in a couple of days, and prior to that discussion it would be useful to know just how widespread the problem is.

So—if you happen to be going past a Barnes and Noble (or a Borders) in the next couple of days, I’ll be most grateful to know whether you found The Big Questions out on the front table where it ought to be. You can comment here or email me at “questions at landsburg dot com”. Please include the address of the store, or the street it’s on, or the town it’s in —whatever you’ve got. Thanks for your help!

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Are the laws of mathematics inherent in our universe, and therefore really a product of physics (and not the other way around), or are they supra-universal?

This question, of course, plays a starring role in The Big Questions , where I’ve explained why I believe that the supra-universality of mathematics (thanks for that word, Al!) gives the most coherent explanation of why anything exists at all.

The same issue arose—with many comments well worth rereading—on the What Are You Surest Of? thread. There, Bill T. raises a provocative question: If (as I’ve suggested in The Big Questions), the physical world is in some sense “made of mathematics”, why can’t we take this one step further and speculate that mathematics is “made of logic”? The answer is that Godel’s theorems make that a very difficult step to take. Starting with the standard axioms of arithmetic, and armed with the full power of logic, there remain true statements about arithmetic that cannot be proven. This means that arithmetic must be more than just logic.

(Logicians in the audience will want to quibble about what constitutes the “full power of logic”; I will duck that question to avoid a long technical digression right now.)

I offered the first in a planned series of posts on attempting to see Darwin through 19th century eyes; Snorri Godhi checked my math carefully and made a valuable correction. And we got into a debate about whether cellphones really would have resolved the plot tension in a substantial fraction of 20th century movies.

Thanks for the fun and the enlightenment. I’ll be back on Monday with more.

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