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Here’s a heartwarming story of religious reconciliation in Israel, one that puts the lie to those cynics who thought such ecumenism impossible. It seems that large portions of Jerusalem’s Orthodox Jewish, Muslim, and Christian communities have finally set aside their differences, and joined together to support a common goal: threatening the marchers in a Gay Pride parade with death.

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According to my usage statistics, of the people who come to scottaaronson.com via a search engine, about 5% do so by typing in one of the following queries:

biggest number in the world
the biggest number in the world
what is the largest number
largest number in the world
what is the biggest number

These people are then led to my big numbers essay, which presumably befuddles them even more.

So, let me satisfy the public’s curiosity once and for all: the biggest number in the world is a million billion gazillion. But stay tuned: even as I write, Space Shuttle astronauts are combing the galaxy for an even bigger number!

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So said my brother David (MIT math major), on forwarding me this animation of the inner life of a cell.

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O Achilles of Arkansas, O bane of Foxes and Roves, O solitary warrior among Democrats: dasher of hopes, prince of platitudes, felatee of Jewesses, belated friend of Tutsis, toothless tiger of climate change, greatest of all living Americans: how shall we summon thee back?

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So, it seems the arXiv is now so popular that even Leonhard Euler has contributed 25 papers, despite being dead since 1783. (Thanks to Ars Mathematica for this important news item, as well as for the hours of procrastination on my part that led to its rediscovery.) Since I’d long been curious about the mathematical research interests of the nonliving, I decided to check out Leonhard’s most recent preprint, math.HO/0608467 (“Theorems on residues obtained by the division of powers”). The paper starts out slow: explaining in detail why, if a mod p is nonzero, then a² mod p, a³ mod p, and so on are also nonzero. By the end, though, it’s worked out most of the basics of modular arithmetic, enough (for example) to analyze RSA. Furthermore, the exposition, while “retro” in style, is sufficiently elegant that I might even recommend acceptance at a minor theory conference, even though the basic results have of course been known for like 200 years.

Oh — you say that Mr. E’s papers were as difficult and abstract for their time as Wiles and Perelman’s papers are for our own time? BULLSHIT. Reading the old master brings home the truth: that, for better and worse, math has gotten harder. Much, much harder. And we haven’t gotten any smarter.

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If you hadn’t been reading the comments on my last post, you might not know that my old chum Mahmoud Ahmadinejad had launched his own blog on Sunday. Along with a rambling autobiography, this exciting new blog (which I’ve added to my linklog on the right) also includes a poll:

Do you think that the US and Israeli intention and goal by attacking Lebanon is pulling the trigger for another word [sic] war?

When I first visited, only 5% had voted “yes”, though it’s now up to 50%.

But wait, it gets better: if Mahmoud’s site identifies your IP address as coming from Israel, then it tries to install a virus on your computer by exploiting an Internet Explorer vulnerability. (Thanks to an anonymous commenter for bringing this to my attention.)

I suppose we should grateful that, at least for now, defending oneself against the modern-day Hitler is as simple as installing Firefox.

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A commenter on a previous post writes:

A lot of great discoveries came from non-scientific losers. E=MCC. Airplanes. America. Someone discovered how to make an airplane by playing with a box. Physics is mostly theoretical. America, I guess, is the most scientific discovery. They applied the scientific method to determine its existence, but they used no control group, and no placebo. For that, America’s existence is not yet proven. There seem to be other ways of establishing truth than just the scientific method. Scientists are contemporary soothsayers. They should use every means possible of proving a fact.

Despite its insightfulness and coherence, the above argument raises some immediate questions:

1. What does it have to do with anything I said?
2. E=MCC?
3. What would mean to use a placebo or control group to test America’s existence? Would it mean sending a ship in a different direction, and checking that it didn’t also reach America? Would it mean verifying that America can’t be reached from Europe by foot — since if it could, then it wouldn’t be America, but rather part of Eurasia?
4. Has England’s existence been scientifically proven? What about France’s?
5. Where do so many people get the cockamamie idea that there’s such a thing as a “scientific method” — that science is not just really, really, really careful thinking? (I blame the school system.)

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From these lecture notes by Harvey Friedman comes one of the best metamathematical anecdotes I’ve ever heard (and yes, I’ve heard my share). Apparently Friedman was attending a talk by the “ultra-finitist” Alexander Yessenin-Volpin, who challenged the “Platonic existence” not only of infinity, but even of large integers like 2¹⁰⁰. So Friedman raised the obvious “draw the line” objection: in the sequence 2¹,2²,…,2¹⁰⁰, which is the first integer that Yessenin-Volpin would say doesn’t exist?

Yessenin-Volpin asked Friedman to be more specific.

“Okay, then. Does 2¹ exist?”

Yessenin-Volpin quickly answered “yes.”

“What about 2²?”

After a noticeable delay: “yes.”

“2³?”

After a longer delay: “yes.”

It soon became clear that Yessenin-Volpin would answer “yes” to every question, but would take twice as long for each one as for the one before it.

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According to ancient complexity lore, at a saddle point high in the mountains of Oberwolfach lies buried a single flask of a mystical elixir known as Websbane, or the Hammer of Firefox. Some say that the productivity-enhancing potion was brewed from the sweat of Erdös and the toenail clippings of Euler; others that it was mixed, condensed, and extracted for the Prophesied One centuries hence who will derandomize BPP. Yet all agree on the tonic’s awesome efficacy: it is said that one drop would furnish lifelong protection against Slate and Salon; a teaspoonful would lift Wikipedia’s stranglehold on the soul. He who once imbibed would neither reread Onion articles from dusk till dawn, nor follow hyperlinks till scarcely a blue word remained amidst the purple, nor while away a Thursday googling a Montreal-born singer-songwriter mentioned in an email of de Wolf. Papers would get finished – books written – reimbursement forms turned in – blog entries posted without delay.

Today’s topic is what we can do until the Websbane is unearthed from its resting-ground. I offer four suggestions below; any additions are welcome.

• Use the embryo strategy. Whenever you’re procrastinating on something, someone is bound to tell you “divvy it up into smaller chunks, then tackle ’em one at a time.” I’ve found that to be terrible advice. When I’m starting a project, I have no idea how to divvy it up. I might commit myself to writing chapters on A, B, and C, only to realize later that A and C are trivial and that everything worth saying pertains to B. Or I might start the introduction, then freeze for days because I can’t decide what belongs in the introduction and what belongs in the “meat” until I’ve already written them.What I’ve found to be more effective is what I’ll call the “embryo strategy.” Here you simplify your project so dramatically that you can finish the entire thing (more or less) in one afternoon. For example, if before your goal was to write a ten-page popular article about quantum computing, now your goal is to write two paragraphs. Then, once you’ve finished something, you progressively add layers to it. This seems to be the approach taken by most successful software projects, not to mention by Nature herself. The advantages are twofold: firstly, everything is built around one initial idea. This changes what the end product looks like, but I think for the better. And secondly — here’s the real beauty — at no point are you ever working on something that will take “unimaginably long,” compared to the amount of time you’ve already spent. (Give or take a small additive constant.)

• Exploit the “quantum Zeno effect.” One to keep a quantum state from drifting uncontrollably is just to measure it over and over in some fixed basis. Roughly speaking, the mere fact that you’re looking means that the state “can’t try anything funny.” Similarly, I’ve taken to having my girlfriend spend the night with me when I need to finish a paper. What ensues is a long, romantic evening, wherein I sit at my computer and do my work, and Kelly sits at her computer and does her work. Interestingly, her mere presence often has the effect of projecting me onto a non-procrastinating subspace. (Kelly reports a similar effect on her as well.)

• Don’t eat. When you’re trying to prove theorems about quantum complexity classes, hunger is your friend and linguini-induced sleepiness your enemy. As obvious as that sounds, it took me almost a decade fully to understand its importance. These days I usually eat only one meal per day — my “brinner” — and don’t even try to work till three or four hours after it. (Does anyone know the physiological reason why humans seem unable to multitask between brains and stomachs?)

• Find yourself a “boss.” When I was at Berkeley, Umesh was my boss. That doesn’t mean he told me what to work on (he didn’t); it means that I got a warm fuzzy feeling from eliciting his opinion of what I had worked on. Since graduating, to stay productive I’ve had to seek out a succession of new “bosses” — from Avi Wigderson at IAS, to collaborators like Greg Kuperberg and Daniel Gottesman. Indeed, if you get a long, technical email from me, it’s not necessarily for your benefit. Mathematicians might be machines for turning coffee into theorems, but the fuel I run on is feedback.

Follow these rules, and you might someday become as disciplined and productive as I am.

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I took a lot of flak for expressing wrong musical opinions last week. Since I so enjoy the role of human flamebait, I’ve decided to have another go at clarifying my views about Art in general. See, until a few years ago, I was intimidated by art and music snobs, by the sort of person who recently deposited the following on Lance Fortnow’s blog:

man, the ignorance displayed here is taken to new levels. your ph.d. in computer science qualifies you as nothing musically, dumbass. ever heard of dynamic range? go look it up.

A bit uncivil, perhaps, but doesn’t this anonymous fount of musical wisdom have a point? After all, spouting off about quantum computers, entanglement, or Gödel’s Theorem without studying them first would certainly qualify you as a dumbass. So if I don’t think the same about music, then aren’t I a big fat hypocrite?

Ah, but consider the following. If — as the snob would be first to affirm — the purpose of art is not to assert or argue anything as a research paper would, but simply to produce an emotional response in the viewer or listener, then what does it even mean to be unqualified to voice that response? Presumably one person’s emotional response is as valid as another’s. Indeed, the difficulty with the snob is that he wants it both ways. “What made Picasso the greatest artist of the twentieth century is ineffable, indescribable — and I’m the one who knows enough to describe it to you.” “This opera is astounding because it induces a visceral, gut response in the audience — and if you don’t have that response, your gut must be mistaken.” The point is that, once you’ve declared something to be nonscientific, emotional, subjective, you have to allow that someone else’s subjective reaction might differ from yours.

So on this day, let us celebrate our freedom from the tyranny of pretending to like stuff we don’t. I’ll start the honesty ball rolling by dividing the world’s artistic output into three categories, then giving examples of each (not representative, just the first things that popped into my head).

Art that’s stirred my soul

The Simpsons
Futurama
South Park
Shakespeare (comedies especially)
Tom Sawyer and Huck Finn
The Mind-Body Problem by Rebecca Goldstein
Everything by Pixar
Arcadia by Tom Stoppard

Art that maybe hasn’t moved me, but that I can nevertheless agree is quite impressive, based not on what other people say but on my own experience of it

The Sistine Chapel (indeed, pretty much everything in Rome)
Them big paintings in the Louvre
Them big Buddhist temples in Kyoto
Beethoven
Mozart
The Beatles
Jazz improv
Jimi Hendrix
Early Woody Allen

Art in neither of the two above categories

Late Woody Allen
Everything in the MoMA
Picasso
Van Gogh
Weird indie films where nothing happens
Anything by David Lynch or M. Night Shyamalan
Rap (except MC Hawking)
“Experimental” music

PS. There’s really no need to flame me if you have different tastes, since I won’t take it as a moral failing on your part. (Except with regard to M. Night Shyamalan.)

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