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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Scientists at CERN have found apparent evidence that neutrinos can travel faster than light.

Suppose that tomorrow historians at Harvard find apparent evidence that the South won the American Civil War — not in some metaphorical “they accomplished their goals” sense, but in the literal sense that it was actually Grant who handed his sword to Lee at Appomatox and not the other way around.

Question: Of which conclusion would you be more skeptical?

Of course your answer might depend on exactly what this new “apparent evidence” consists of. So let me reword: As of this moment, which do you think is more likely — that neutrinos can travel faster than light, or that the South won the Civil War?

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A little over a week ago, HP Research Scientist Vinay Delalikar claimed he could settle the central problem of theoretical computer science. That’s not the momentous part. The momentous part is what happened next.

Deolalikar claimed to prove that P does not equal NP. This means, very roughly, that in mathematics, easy solutions can be difficult to find. “Difficult to find” means, roughly, that there’s no method substantially faster than brute force trial-and-error.

Plenty of problems — like “What are the factors of 17158904089?” — have easy solutions that seem difficult to find, but maybe that’s an illusion. Maybe there’s are easy solution methods we just haven’t thought of yet. If Deolalikar is right and P does not equal NP, then the illusion is reality: Some of those problems really are difficult. Math is hard, Barbie.

So. Deolalikar presented (where “presented” means “posted on the web and pointed several experts to it via email”) a 102 page paper that purports to solve the central problem of theoretical computer science. Then came the firestorm. It all played out on the blogs.

Dozens of experts leapt into action, checking details, filling in logical gaps, teasing out the deep structure of the argument, devising examples to illuminate the ideas, and identifying fundamental obstructions to the proof strategy. New insights and arguments were absorbed, picked apart, reconstructed and re-absorbed, often within minutes after they first appeared. The great minds at work included some of the giants of complexity theory, but also some semi-outsiders like Terence Tao and Tim Gowers, who are not complexity theorists but who are both wicked smart (with Fields Medals to prove it).

The epicenter of activity was Dick Lipton’s blog where, at last count, there had been been 6 posts with a total of roughly 1000 commments. How to keep track of all the interlocking comment threads? Check the continuously updated wiki, which summarizes all the main ideas and provides dozens of relevant links!

I am not remotely an expert in complexity theory, but for the past week I have been largely glued to my screen reading these comments, understanding some of them, and learning a lot of mathematics as I struggle to understand the others. It’s been exhilarating.

Why is this momentous? In some ways, there’s nothing new about any of this. It’s not terribly uncommon for a serious looking paper to address a major outstanding problem, and it’s de rigeur for experts to comb through those papers, searching simultaneously for new paradigms, irreparable flaws, and salvageable insights. We call it peer review.

But what’s new is that this played out in public, and that it took a week instead of the usual months or years, and that the dozens of conversations taking place all over the world were melded into one giant conversation where every idea was available for everyone to hear—and for everyone to shoot down. Many very smart people said very smart things that turned out to be wrong, and in the world before the Internet, they might have gone on believing those things for weeks or months. The Net made it very difficult to believe wrong things for more than an hour.

It is a cliche to say that the Internet has changed everything, and in particular that it’s changed the way we do science. But what I saw this week seemed to me to be a whole new grand leap forward. The icing on the cake is the growing public record of everything that’s been said. Most of that record is a monument to the passion and dedication of a community obsessed with finding truth. In those thousand or so comments, I see almost nothing that smacks of self-aggrandizement, almost no instances where the proponent of an idea fails to back off instantly in the face of a better idea. Sometimes we in the academic community lose sight of how extraordinary are the high standards we routinely demand of ourselves and each other. Sometimes those outside of academics have no concept of how high those standards are. It’s inspiring to be reminded.

This week, there’s been no better inspiration—and no better education, and no better entertainment—than to read Dick Lipton’s blog.

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Not only is Æternitatis a great writer; he’s also a gracious colleague who (after I introduced myself by email) agreed to let me reprint one of his incisive commentaries as a guest post here. So without further ado:

Al Gore’s Revealed Beliefs A Guest Post by Sub Specie Æternitatis

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Before I explain the problem, let’s talk a little about tic-tac-toe. As you probably figured out long ago, intelligent players of ordinary tic-tac-toe (on a 3 by 3 board) will invariably battle to a draw. But, as you probably also figured out, not every game ends in a draw, because not every player is intelligent.

On the other hand, if we blacken out the three squares on the main diagonal and don’t allow anyone to play there (so the game ends when the remaining six squares are filled, then every game is sure to end in a draw. There’s simply no way to get three in a row when you’re not allowed to play on the diagonal:

Okay. Now let’s play tic-tac-toe in three dimensions, with three ordinary boards stacked on top of each other (giving you a total of 27 places to place your X.) How many of those 27 squares would I have to blacken to insure that winning is flat-out impossible? The answer, it turns out, is 9—as long as you choose the right 9. And if we go to 4 dimensions? Now there are 81 squares, and if you want to prevent any possibility of winning, you’re going to have to blacken at least 29 of them. In 5 dimensions, you’ve got to blacken at least 93.

(See yesterday’s post if you’re puzzled about how to play tic-tac-toe in four dimensions).

The density Hales Jewett Theorem says that as you go to higher and higher dimensions, the number of squares you must black out to prevent a win gets arbitrarily close to 100% of the squares available. In some high enough dimension, you’ll have to black out at least 90% of the squares; in some higher dimension, you’ll have to black out 95%, and then 99% and 99.9999%. If you’re not sure why anybody would care about such a thing, take my word for it—there are many applications to other areas of mathematics.

(My statement of the theorem glosses over some minor technicalities; the actual theorem is slightly stronger than what I’ve quoted here.)

Now until a few months ago, the only known proof of the density Hales Jewett theorem was extremely difficult. But Tim Gowers, a Fields-Medal winning Cambridge mathematician, thought there ought to be an easier proof. So he did what everyone with an opinion about anything does nowadays; he posted his opinion on his blog. He also did what no mathematician had ever done before, and invited the entire world to collaborate with him in proving his opinion correct. Following an initial post asking “Is Massively Collaborative Mathematics Possible?”, he posted a description of the problem and invited his readers to have at it in comments.

Commenters leapt in. In response to a couple of dozen blog posts by Gowers and others, roughly spanning the calendar year 2009, commenters continued to build on each others’ ideas until they produced the (relatively) simple proof Gowers had been hoping for. Along the way, they accomplished a lot more—for example, we now know that in 5 and 6 dimensions, you’ve got to black out exactly 93 and 279 squares (again, glossing over some minor technicalities); these numbers were not known before the blogging project. For any single mathematician—or team of mathematicians—this would have been a singular accomplishment. It’s not clear it would ever have happened in a world without blogs.

Gowers believes this could be the beginning of a whole new way of doing mathematics, allowing hundreds or thousands of mathematicians to contribute to the solution of a single problem. Of course this raises all sorts of questions about rewards and incentives, many of which are addressed (but not, or course, settled) in Gowers’s “Massive Collaboration” post. Still, I have an inkling that this is a big freaking deal.

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John and Mary live in an isolated village where they have no access to reference materials, no contact with the outside world, and nobody to talk to except each other. One day an anthropologist arrives in this village, sits down for coffee with John and Mary, and quizzes them about their knowledge of the world. John says he’s sure that men have walked on the moon; Mary says she’s sure they haven’t. Never having discussed this issue before, each of them is astonished and flabbergasted by the others’ apparent ignorance. Rather than risk losing all respect for each other, John and Mary agree never to speak of the subject again. But the anthropologist mentions that she’ll be stopping by once a day from now on, and will be glad to know if either of them ever has a change of mind on this topic. If so, the anthropologist will inform the other. Otherwise, the anthropologist will never bring it up either.

The next day (a Monday) nobody’s mind has changed, and therefore the subject is not discussed. The same thing happens on Tuesday, Wednesday and Thursday. Can this go on forever?

The surpising answer, under quite general assumptions about the way people learn, is that eventually, if John and Mary care about the truth (as opposed to, say, winning a debating point) then somebody’s mind must change.

But how can this be? If nobody’s mind changed on Tuesday, Wednesday or Thursday, and if no new information or argument ever gets brought into the picture, how can Friday be different?

Here’s how: On Monday, all John knows is that when it comes to moon landings, Mary disbelieves. By Tuesday, he knows that her disbelief is strong enough to stand up in the face of his belief—which is something he didn’t know on Monday. By Wednesday, he knows that her disbelief is strong enough to stand up in the face of his continued belief in the face of her continued disbelief—which is something he didn’t know on Tuesday. And so forth.

Each time a day goes by with no minds changed, John and Mary learn something new about the strength of each others’ beliefs—and therefore find it harder to maintain their own beliefs, because, after all, there’s always a chance the other guy is right. And the surer the other guy is, the better that chance—at least on the assumption that the other guy is at least trying to arrive at the truth.

The fact that John or Mary must eventually break down in the face of this barrage takes some work to prove; the first round of that work was done by the Nobel prize winning economist Robert Aumann, who took the first step toward showing that it’s essentially impossible for honest truthseekers to “agree to disagree”. Other economists, especially the always innovative Robin Hanson, then took up the baton and pushed these results much further. You can read all about it in Chapter 8 of The Big Questions.

Of course, in the real world, John and Mary never change their minds; they just go off to sulk, become increasingly embittered and eventually stop talking to each other altogether. The disturbing implication of Aumann’s theorem is that therefore John and Mary cannot both be honest truthseekers. Nor can almost anyone else who’s ever been party to an ongoing disagreement. Which, most disturbingly of all, would seem to include me.

It occurs to me that this brain teaser has more than a little in common with a brain teaser that recently got a lot of play over on the blog of the Fields-medal winning mathematician Terence Tao:

There is an island upon which a tribe resides. The tribe consists of 1000 people, with various eye colours. Yet, their religion forbids them to know their own eye color, or even to discuss the topic; thus, each resident can (and does) see the eye colors of all other residents, but has no way of discovering his or her own (there are no reflective surfaces). If a tribesperson does discover his or her own eye color, then their religion compels them to commit ritual suicide at noon the following day in the village square for all to witness. All the tribespeople are highly logical and devout, and they all know that each other is also highly logical and devout (and they all know that they all know that each other is highly logical and devout, and so forth).

Of the 1000 islanders, it turns out that 100 of them have blue eyes and 900 of them have brown eyes, although the islanders are not initially aware of these statistics (each of them can of course only see 999 of the 1000 tribespeople).

One day, a blue-eyed foreigner visits to the island and wins the complete trust of the tribe.

One evening, he addresses the entire tribe to thank them for their hospitality.

However, not knowing the customs, the foreigner makes the mistake of mentioning eye color in his address, remarking “how unusual it is to see another blue-eyed person like myself in this region of the world”.

What effect, if anything, does this faux pas have on the tribe?

Once again the key to the brain teaser is that silence conveys information, silence in the face of silence conveys even more information, and silence in the face of silence in the face of silence conveys even more. One difference is that Tao’s brain teaser is self-contained; you ought to be able to figure it out without reference to journal articles. The agree-to-disagree problem lies deeper, but is also, I suspect, more profound.

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