Steven Landsburg | The Big Questions: Tackling the Problems of Philosophy with Ideas from Mathematics, Economics, and Physics » Math

The Big Surprise

Steve Landsburg — Thu, 15 Dec 2011 07:01:29 +0000

Back in the 1930’s, Kurt Godel proved two amazing facts about arithmetic: First, there are true statements in arithmetic that can’t be proven. Second, the consistency of arithmetic can’t be proven (at least not without recourse to logical methods that are on shakier ground than arithmetic itself).

Yesterday, I showed you Gregory Chaitin’s remarkably simple proof, of Godel’s first theorem. Today, I’ll show you Shira Kritchman and Ron Raz’s remarkably simple (and very recent) proof of Godel’s second theorem. If you work through this argument, you will, I think, have no trouble seeing how it was inspired by the paradox of the surprise examination.

Start with this list of statements:

It takes more than 10000 characters to specify the number 1.
It takes more than 10000 characters to specify the number 2.
It takes more than 10000 characters to specify the number 3.

and so forth.

Obviously, statements 1, 2 and 3 are all false, since it only takes a single character (namely “1″) to specify the number 1, and a different single character (namely “2″) to specify the number 2. But if you continue this list long enough, you’ll eventually get to some true statements. Yesterday we saw a proof that none of those true statements is provable.

But suppose that, undeterred by the proof, we are determined to identify a specific true statement on the list. Here’s our strategy:

First we write down the first gazillion statements on the list, where a gazillion is some number so big that we’re sure to have included some truths.
Next we go down the list, eliminating false statements. Notice that if we’re willing to work long enough, every false statement eventually gets eliminated — we just keep trying shorter-than-10000-character prescriptions and crossing off the numbers they describe. This leaves us a list of candidates.
Suppose there’s just one true statement on the list. Then eventually we cross all the others off, and learn that the one remaining candidate is true (because the list was so long there had to be at least one true statement). In fact, we’ve just proved (by process of elimination) that this statement is true. But we learned yesterday that no such proof is possible. Conclusion: There can’t be just one true statement on this list.
Suppose there are just two true statements on the list. Then eventually we’ll cross all the others off, leaving two candidates, at least one of which is true. But we’ve already proved that there can’t be just one true statement, so these must both be true. In fact, we’ve just proved they’re true. Which is impossible. Conclusion: There can’t be just two true statements on the list.
Do you see where this is going? There also can’t be just three true statements on the list, or four, or five, or any other number. Yet we know there are true statements on the list! Something’s wrong here!!!!!

Okay, what went wrong? Answer:

Our argument relies on the fact that every statement on our list must be unprovable. Why should we believe that? Well, we proved it yesterday; that’s why!
Actually, our argument relies on a bit more — it relies not just on the fact that every statement on our list is unprovable, but that we can prove their unprovability. But again, we did that yesterday.
The only way out of this is to conclude that there’s some gap in yesterday’s proof.
But there’s only one thing we used yesterday without proving it: We began from the assumption that our arithmetical reasoning is consistent.
So that must be where the gap is — and it must be impossible to fill that gap! In other words, it must be impossible to prove that arithmetic is consistent.
Tada!

There’s just one other way out: If arithmetic actually is inconsistent, then all bets are off and we can prove its consistency — but in that case, of course, our conclusion will be wrong. In any event, there are very few people who think this is a contingency worth worrying about.

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Berry Interesting

Steve Landsburg — Wed, 14 Dec 2011 07:01:46 +0000

Today, I’m going to give you a short, simple proof of Godel’s First Incompleteness Theorem — the one that says there are true statements in arithmetic that can’t be proven. The proof is due to Gregory Chaitin, and it is far far simpler than Godel’s original proof. A bright high-schooler can grasp it instantly. And it’s wonderfully concrete. At the end, we’ll have an infinite list of statements, all easy to understand, and none of them provable — but almost all of them true (though we won’t know which ones).

Chaitin’s proof, like Godel’s, is inspired by a classical paradox — in this case, Berry’s Paradox as opposed to the Liar Paradox (both of which I described yesterday).

We start by observing that some numbers are more complicated than others. The 10,000 digit number that starts off 10101010101010101….. and continues the same way is somehow less complicated than a random 10,000 digit number.

Kolmogorov formalized this notion by defining the “complexity” of a number as the length of the shortest prescription (in some fixed language) for writing it down. The number above has a 48-character prescription: “Write down a 1, then a 0, then repeat 5000 times”. That makes it pretty simple.

Now suppose we want to find a more complicated number — one that requires at least, say, 60 characters to prescribe. We know there must be many such numbers, but we’d like to find a specific example — together with a proof that our example can’t be prescribed in less than 60 characters. So we write a computer program, called Finder60, which searches for examples-with-proofs. Ideally, the program outputs something like: “I have found a proof that the number 2834932709472398472328923478902342903848927189374901742309842398742 cannot be prescribed with fewer than 60 characters.”. Finder60 searches systematically, so if there’s such a number/proof combination to be found, Finder60 will find it. Otherwise, it keeps on running forever.

We can also, of course, write programs called Finder90, Finder120, Finder10000 and so on, to find ever-more-complicated numbers together with proofs that they are complicated.

Once you’ve written the code for Finder60, you only have to tweak it slightly to get the code for Finder10000 — and the code gets only slightly longer in the process. Basically, you just change all the 60’s to 10000’s, replacing two-digit strings with five-digit strings. Not much difference.

So if the code for Finder60 is, say, 5000 characters long, then the code for Finder10000 is just a bit more than that — say 5200 characters.

Now suppose M is any number that provably requires more than 10000 characters in its prescription. Then Finder10000 will find it and print it out. Which means we have the following less-than-10000-character prescription for M:

Run the following program: [insert the 5200-character code for Finder10000 here]

Uh oh! If prescrbing M provably requires more than 10000 characters, then M can be prescribed in fewer than 10000 characters. Contradiction!

Conclusion: There must be no such M. That is, no number can be proved to require more than 10000 characters in its prescription.

But surely some numbers do require more than 10000 characters; after all there are only finitely many ways to prescribe a number with fewer than 10000 characters, which leaves infinitely many numbers left over.

So what’s a true statement that can’t be proven? Answer: Any true statement of the form “The number M cannot be prescribed in fewer than 10000 characters”. No matter what M you plug in , this statement must be unprovable. If you plug in M=1 or 2, you’ll get a false statement. But for most values of M (though we don’t know which values!) you’ll get a statement that’s true, but unprovable.

Isn’t that about the coolest trick ever? Well, it turns out that you can add another twist to make it even cooler. That’s where the surprise examination paradox comes in. I’ll explain it all one day later this week, though you won’t know which day till you log in that morning.

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A Tale of Three Paradoxes

Steve Landsburg — Tue, 13 Dec 2011 07:00:09 +0000

This is a tale of three paradoxes and why they matter.

First, the ancient Liar Paradox: “This sentence is false”. If this sentence is true, it must be false. If it’s false, it must be true.
Next, the century-old Berry Paradox: Call a phrase “short” if it contains fewer than 13 words. The English language contains a finite number of words, and hence a finite number of short phrases. Hence there must be some natural numbers that can’t be described by any short phrase. Among these natural numbers, there must be a smallest. What is that natural number? Why, it’s the smallest natural number that can’t be described by any short phrase, of course. Except that this number is in fact described by the short phrase in boldface.
Finally, the more modern Paradox of the Surprise Examination (or the Unexpected Hanging), which we discussed yesterday.

The paradoxes are slippery, because they are stated in the imprecise language of English. But each of them has inspired a precise mathematical counterpart that is central to a brilliant argument in mathematical logic.

Start with the liar: “This sentence is false” can’t be true, or it would be false — and can’t be false, or it would be true. This tells us that there’s such a thing as an English sentence that’s neither true nor false, which comes at first as a considerable surprise, but isn’t devastating.

One of Kurt Godel’s great insights was that you can go a lot deeper by considering a slightly different sentence: “This sentence is not provable”. If that statement is false, then it’s provable. But surely no false statement should be provable! So maybe the statement is true. In that case, it’s true but not provable, which says something about the limits of logic. It says that not every true statement can be proved.

At one level, this is still just wordplay. What makes it profound is Godel’s discovery of a code that converts certain English sentences into statements of pure arithmetic (that is, statements of the form “Every number is the sum of four squares” or “Every prime number is divisible by 2″) in such a way that true statements are matched with true statements and false statements are matched with false statements. The code is cleverly constructed so that there’s a statement in pure arithmetic (say, for illustration, that it’s the statement “every even number is the sum of two primes”) that corresponds to the English sentence “The statement that every even number is the sum of two primes cannot be proven.” These statements are either both false, in which case it’s possible to prove a false statement, which we believe (and hope to God!) is not the case — or they’re both true, in which case we’ve found a true statement in pure arithmetic that can’t be proven.

Much brilliant work goes into constructing the code, but the brilliant idea is to adapt the Liar Paradox to a context where you can’t just say “Well, I suppose it’s neither true nor false” — because statements like “Every even number is the sum of two primes” must be either true or false.

(The above intentionally sacrifices a little precision in the interest of readability; the linked post is more carefully worded.)

So that’s why mathematicians care about the Liar Paradox. More on the Berry Paradox and the Surprise Exam (and how all three tie together) as the week goes on.

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The Surprise Exam, and More Surprises

Steve Landsburg — Mon, 12 Dec 2011 06:01:13 +0000

If you’re the sort of person who reads this blog, you’re likely to be familiar with the paradox of the unexpected hanging, which has been floating around since 1943 but achieved popular notoriety around 1969 through the writing of Martin Gardner. But you’re less likely to be aware that the unexpected hanging plays a central role in a wonderful new piece of serious mathematics related to algorithmic complexity, Godel’s theorems, and the gap between truth and provability.

The unexpected hanging might as well be a surprise examination, and that’s the form in which I present this paradox to my students every year: In a class that meets every weekday morning, the professor announces that there will be an exam one day next week, but that students won’t know exactly which day until the exams are handed out.

The students, of course, immediately start trying to guess the day of the exam. One student (call him Bob) observes that the quiz can’t be on Friday — because if it is, the students will know that by Thursday afternoon. After all, if Monday, Tuesday, Wednesday and Thursday mornings have all passed by, only Friday remains. A Friday exam can’t be a surprise exam.

A more thoughtful student (call her Carol) observes that this means the quiz must be on one of Monday, Tuesday, Wednesday or Thursday — and that if it’s on Thursday, they’ll know that by Wednesday night. After all, Friday’s ruled out, so if Monday, Tuesday and Wednesday have passed by, then only Thursday remains. That rules out a surprise exam on Thursday.

Another student (call him Ted) observes that thanks to Bob and Carol, we know the exam must be on one of the first three days of the week — which means that if it’s not on Monday or Tuesday, it must be on Wednesday. Therefore if it’s on Wednesday, they’ll know this by Tuesday night. Scratch Wednesday from the list of possibilities.

Now Ted’s girlfriend Alice points out that the exam can’t be on Tuesday either. Whereupon Bob concludes that the exam must be on Monday. But wait a minute! Carol points out that if they know the exam will be on Monday, it can’t be a surprise. Therefore no surprise exam is possible.

The students, relieved, decide not to study. But they’re awfully surprised when they show up in class the following Tuesday and the professor hands out an exam.

Where did the students go wrong? There is no consensus among the many philosophers and logicians who have considered this problem. The great Willard Van Orman Quine believed that Bob went wrong at the very beginning when he ruled out Friday. (According to Quine, Bob’s argument fails to distinguish between a proof that the exam can’t be on Friday and a proof that the students will know that the exam can’t be on Friday.) Other deep thinkers have accepted Bob’s argument (agreeing that the exam can’t be on Friday) but refused to accept Carol’s (thus refusing to rule out Thursday). You can, if you wish, read a pretty comprehensive survey of this literature here. But even among those who think Bob (or Carol) is mistaken, there is little agreement about exactly why they are mistaken.

Now, I happen to think the surprise examination paradox is pretty interesting as a pure intellectual exercise. But it’s also got important applications. I use it in the classroom to illuminate our discussion of the underlying “backward induction” technique, which economists (and especially game theorists) use all the time in serious arguments. Much more recently, the surprise examination has been used to illuminate some key concepts in mathematical logic, which I alluded to back in the first paragraph of this post. That’s the coolest part of all, and I’ll tell you all about it later in the week.

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Beauty, Truth and Symmetry

Steve Landsburg — Tue, 25 Oct 2011 06:01:18 +0000

Today is the 200th birthday of Evariste Galois, who did not live to celebrate his 21st, but found time in his short 20 years to develop a circle of ideas that permeate modern mathematics. We know of these ideas because Galois spent the night of May 30, 1832 scribbling them furiously in a letter to a friend, in advance of the fatal duel he would fight the following morning. According to the great mathematician Hermann Weyl, “This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind.”

(If this were a less serious post, I might suggest that this famous letter was the first example of a Galois Correspondence.)

Now, two centuries later, every first year graduate student in mathematics spends a semester studying Galois Theory, and many devote their subsequent careers to its extensions and applications. Many of the greatest achievements of modern mathematics (for example, the solution to Fermat’s Last Theorem) are, at their core, elucidations of Galois’s 200-year-old insight.

As every high school student knows (or should know), a quadratic equation (like, say, x² – 4x – 1 = 0) can be solved by applying the quadratic formula (which, in this case, gives x = 2 ± √5). The quadratic formula uses only addition, subtraction, multiplication, division, and the extraction of square roots.

What about a cubic equation, like, say, x³ + 2 x² – 5 x – 3 = 0 ? The less well-known cubic formula finds the solutions, using only addition, subtraction, multiplication, division, and the extraction of square and cube roots.

And, yes, there’s a quartic formula, for equations of degree 4. But it stops there. Galois’s contemporary Niels Abel (who, unlike Galois, survived to the ripe old age of 26) showed that no formula can consistently solve equations of degree 5 using only addition, subtraction, multiplication, division, and the extraction of roots.

On the other hand, some equations of degree 5 and higher can be solved by such formulas. Call those equations solvable. Galois figured out how to identify the solvable equations. It all comes down to understanding symmetry. Galois was the first to see clearly that the solutions to any equation satisfy certain symmetries. (For example, the solutions 2-√5 and 2+√5 are symmetric under the interchange of the plus and minus signs on the square root.) The nature of those symmetries differs from equation to equation, and dictates whether the equation is solvable. This in turn leads to a much deeper appreciation of the importance of symmetries throughout the theory of equations and throughout algebra more generally.

Today, algebraists take it for granted that understanding an equation, or a system of equations, entails understanding its symmetries. The development of that instinct was a key advance in the history of thought. After almost two centuries, we still use it to discover new insights and to solve old problems every single day.

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Big News

Steve Landsburg — Tue, 04 Oct 2011 06:01:18 +0000

Last week, the highly distinguished Princeton Professor Ed Nelson announced a proof that the Peano axioms for arithmetic are inconsistent — and hence so is arithmetic itself. If true, this would be much bigger news than faster-than-light neutrinos. It would be bigger news than a discovery that the South had won the American Civil War. It would be far, far bigger news than a discovery that all life on Earth was intelligently designed.

There are, after all, multiple proofs that Peano Arithmetic (that is, the fragment of arithmetic described by the Peano axioms) is consistent. Among those, the simplest and most convincing (to the overwhelming majority of mathematicians) is this: The axioms of Peano Arithmetic, and therefore the theorems of Peano Arithmetic, are all true statements about the natural numbers — and a set of true statements cannot contradict itself.

Ed Nelson rejects that argument because (exempting himself from that overwhelming majority) he doesn’t believe in the set of natural numbers — or perhaps even in individual numbers when those numbers are very large. (How do you know that 8¹⁰⁰⁰⁰ exists? Have you ever counted to it?)

Needless to say, this announcement — and the announcement of a forthcoming book providing details — generated more than a flurry of excitement on the math blogs — including one of my very favorite blogs, the n-Category Cafe. After Fields Medalist Terry Tao raised a specific technical objection to Nelson’s argument, Nelson showed up in the comments section to defend himself — and then Tao showed up to expand on his objections. Nelson responded, Tao re-responded, and then Nelson posted:

You are quite right, and my original response was wrong. Thank you for spotting my error.

I withdraw my claim.

Just to be clear, here: That’s Ed Nelson cheerfully acknowledging that the book-length argument he’s been painstakingly constructing for (probably) years, and which was intended to shake the mathematical world to its foundations, doesn’t work. This says so many good things about the culture of mathematics, and so many good things about the Internet, and so many good things about the way they interact (see here and here for more examples), and it says those things so eloquently, that I see no further need for comment.

(On the other hand, if you’re hungry for additional comments, the philosopher Catarina Dutilh Novaes provides some good ones here.)

The Internet’s impact on mathematics is a huge huge thing. Not quite as huge as an inconsistency in Peano Arithmetic, but huge enough to count as a marvel.

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Happy Birthday

Steve Landsburg — Fri, 30 Sep 2011 06:01:56 +0000

MathOverflow turns two years old this week — a milestone in the transformation of mathematical research into a massively collaborative endeavor. It’s happening on blogs, it’s happening on mailing lists, and it’s happening in a big way on MathOverlow, where mathematicians ask and answer the sorts of questions that might come up in the faculty lounge — if the faculty lounge were populated by hundreds of experts pooling their expertise.

If you’re interested in mathematics at the research level, MathOverflow is a place to learn something new and fascinating every single day. (If you are not doing mathematics at a research level, feel free to read but please don’t feel free to join the fray; questions at anything below about a second-year graduate level should be directed to MathStackExchange, another massively collaborative site aimed, roughly, at the college level — which reminds me that it’s not just mathematical research, but also mathematical education, that is being revolutionized before our eyes.)

It has been absolutely fascinating to watch MathOverflow and its sister sites develop. The mathematical content is awesome, but so is the sociological phenomenon — problems solved in hours instead of months via virtual brainstorming among some of the smartest and most knowledgable people in the world.

When MathOverflow first came on line, I thought it would be a superfluous addition to the many electronic resources already available. I couldn’t have been more wrong. I now suspect that like all other toddlers, it will be awesome at age five in ways that are only dimly imaginable at age two. If you like mathematics, these are very good times to live in.

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Cats, Dogs and Quantum Mechanics

Steve Landsburg — Tue, 13 Sep 2011 06:01:20 +0000

The game of Cats and Dogs works like this: You and your teammate are placed in separate rooms and forbidden to communicate. You are each asked a randomly chosen question: Either “Do you like cats?” or “Do you like dogs?” (Each of your questions is determined by a separate fair coin flip.)

You win if your answers agree — unless you were both asked the “cats” question, in which case you win if your answers disagree.

A little reflection should convince you that if you are allowed to meet with your partner and plot strategy before the game, then the best you can do is agree to always agree — say by both always answering “yes”. That way, you win 75% of the time, and there’s no way to do better. In particular, there’s nothing to be gained by randomizing your answers.

That, at least, is true, in a world governed by the laws of classical physics and probability theory. But in a world governed by the laws of quantum mechanics — which is to say, in the world we live in — you can in principle do better. Namely: You each carry with you one of a pair of entangled “quantum coins” (actually elementary particles, but I prefer to think of them as coins, since you’re going to use them as randomizing devices).

Because these coins are very small, you need special apparatus to see whether they’re heads-up or tails-up. Before making your measurement, you can rotate your apparatus through either of two angles — call them C and D. The two coins agree 85% of the time — unless both you and your partner’s apparatus have been rotated through angle C, in which case they disagree 85% of the time.

Now you and your partner can each adopt this strategy: If you get the cat question, rotate your apparatus through angle C; if you get the dog question, rotate through angle D. Then examine your coin, and answer “yes” if it’s heads, or “no” if it’s tails. If you both follow this strategy, you’ll win 85% of the time.

Moral of the story: Quantum technology can improve your performance in strategic situations.

(You can read more about this in Chapter 15 of The Big Questions.)

Some time ago, Gordon Dahl and I wrote a paper that explores the implications of quantum mechanics for the cat/dog game and similar (more economically interesting) strategic interactions. Early versions of this paper have floated around for a while, but we’ve just completed a substantial rewrite that I both hope and believe is substantially more readable. We’ll be very glad for feedback from readers who have a taste for this sort of thing. Click here to read it!

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Friday Solution

Steve Landsburg — Fri, 19 Aug 2011 07:01:17 +0000

Re yesterday’s puzzle, you’ll find answers in the comments. (We are blessed with some very smart commenters here at The Big Questions!!)

Commenter Roger Schlafly pointed this Wikipedia article where I was surprised and delighted to see a reference to a paper co-written by my old friend Dave Rusin. I did not remember that Dave had anything to do with this problem, but in retrospect I bet I knew this at one time.

I managed to dig out some notes I jotted down on this subject many many years ago. I have not doublechecked these results, and I can’t completely vouch for the careful accuracy of my younger self, so take these for what they’re worth. But here’s what I once claimed to have proved:

The reason there is exactly one pair of nonstandard six-sided dice is that six is the product of two distinct primes. For the same reason, there is exactly one pair of nonstandard n-sided dice when n is 10, or 15, or 21, or …. For any product of three distinct primes, there are at most 40 nonstandard pairs.

I also found (in what appears to be my handwriting) this chart, which I reproduce with the same caveats:

Number of sides	Number of nonstandard pairs
1	0
2	0
3	0
4	1
5	0
6	1
7	0
8	3
9	1
10	1
11	0
12	at most 13
13	0
14	1
15	1
16	9
17	0
18	at most 13
19	0
20	at most 13
21	1
22	1
23	0
24	at most 94
25	1
26	1
27	3
28	at most 13
29	0
30	at most 40
31	0
32	25

Corrections welcome!

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Thursday Puzzle

Steve Landsburg — Thu, 18 Aug 2011 07:01:39 +0000

I love this problem, which I found on the Internet many years ago. I suppose you could find a solution by Googling, but that’s of course no fair.

A standard pair of six-sided dice induce a probability distribution on the outcomes 1 through 12: The probability of rolling a 1 is 0, of rolling a 2 is 1/36, of rolling a 3 is 1/18, etc. Is there any nonstandard pair of six-sided dice that induces exactly the same probability distribution? If so, how many such pairs are there?

(A non-standard pair of six-sided dice might have, say, the numbers 1,2,2,3,8,9 on one cube and the numbers 2,3,4,4,4,4 on the other.)

What about the same problem for seven-sided dice, or eight-sided, or n-sided?

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