Archive for the 'Truthseeking' Category

Big News

Published
by
Steve Landsburg
on October 4, 2011
in Logic, Math, Progress and Truthseeking
. 51 Comments

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?)

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Neutrinos and Appomattox

Published
by
Steve Landsburg
on September 26, 2011
in Current Events, Musings, Physics, Science and Truthseeking
. 56 Comments

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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O Brave New World!

Published
by
Steve Landsburg
on August 16, 2010
in Math, Progress and Truthseeking
. 8 Comments

complexitySomething momentous happened this week. Of this I feel certain.

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.

Continue reading ‘O Brave New World!’

From an Eternal Perspective

Published
by
Steve Landsburg
on May 13, 2010
in Blogging and Truthseeking
. 55 Comments

My favorite new blogger is the pseudonymous Sub Specie Æternitatis, who I discovered when he left a particularly thoughtful comment on the Fair and Balanced thread here at The Big Questions. A little Google-stalking later, I was immersed in his blog. Before much longer, I was in love with it.

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

It is reported that former Vice President Al Gore just purchased a villa in Montecito, California for $8.875 million. The exact address is not revealed, but Montecito is a relatively narrow strip bordering the Pacific Ocean. So its minimum elevation above sea level is 0 feet, while its overall elevation is variously reported at 50ft and 180ft.

At the same time, Mr. Gore prominently sponsors a campaign and award-winning movie that warns that, due to Global Warming, we can expect to see nearby ocean-front locations, such as San Francisco, largely under water. The elevation of San Francisco is variously reported at 52ft up to high of 925ft.

There being very little reason to suppose that the Pacific Ocean would (or could) rise much less in Montecito than in San Francisco, it follows that Mr. Gore just paid nearly $9 million for property, which according to his professed beliefs, will likely soon be literally under water and hence worthless both as a residence and for resale.

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Blogging, Tic Tac Toe and the Future of Math

Published
by
Steve Landsburg
on April 8, 2010
in Blogging, Innovations, Math and Truthseeking
. 25 Comments

Blogging, as you might have heard, is changing the face of the media. It may also be changing the face of mathematical research. For the first time ever, a substantial mathematical problem has been solved via an accumulation of blog comments, all building on each other. Could this be the future of mathematical research?

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:

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Brain Teaser

Published
by
Steve Landsburg
on November 12, 2009
in Puzzles and Truthseeking
. 37 Comments

Here’s a brain teaser I wish I’d invented in time to include it in The Big Questions:

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?

Continue reading ‘Brain Teaser’