Author: Glen

we have no tacos for sale at Chi Alpha.

One of my favorite web­sites, Church Mar­ket­ing Sucks, has hit the nail on the head with the Dr. Dob­son vs Sponge­bob brouha­ha.

There’s an absolute­ly fas­ci­nat­ing series of respons­es by lead­ing sci­en­tists and intel­lec­tu­als to the ques­tion, “WHAT DO YOU BELIEVE IS TRUE EVEN THOUGH YOU CANNOT PROVE IT?”

Like this fun­ny dia­log by Stan­ford prof Leonard Susskind.

Con­ver­sa­tion With a Slow Stu­dent

Stu­dent: Hi Prof. I’ve got a prob­lem. I decid­ed to do a lit­tle prob­a­bil­i­ty experiment—you know, coin flipping—and check some of the stuff you taught us. But it did­n’t work.

Pro­fes­sor: Well I’m glad to hear that you’re inter­est­ed. What did you do?

Stu­dent: I flipped this coin 1,000 times. You remem­ber, you taught us that the prob­a­bil­i­ty to flip heads is one half. I fig­ured that meant that if I flip 1,000 times I ought to get 500 heads. But it did­n’t work. I got 513. What’s wrong?

Pro­fes­sor: Yeah, but you for­got about the mar­gin of error. If you flip a cer­tain num­ber of times then the mar­gin of error is about the square root of the num­ber of flips. For 1,000 flips the mar­gin of error is about 30. So you were with­in the mar­gin of error.

Stu­dent: Ah, now I get if. Every time I flip 1,000 times I will always get some­thing between 970 and 1,030 heads. Every sin­gle time! Wow, now that’s a fact I can count on.

Pro­fes­sor: No, no! What it means is that you will prob­a­bly get between 970 and 1,030.

Stu­dent: You mean I could get 200 heads? Or 850 heads? Or even all heads?

Pro­fes­sor: Prob­a­bly not.

Stu­dent: Maybe the prob­lem is that I did­n’t make enough flips. Should I go home and try it 1,000,000 times? Will it work bet­ter?

Pro­fes­sor: Prob­a­bly.

Stu­dent: Aw come on Prof. Tell me some­thing I can trust. You keep telling me what prob­a­bly means by giv­ing me more prob­a­blies. Tell me what prob­a­bil­i­ty means with­out using the word prob­a­bly.

Pro­fes­sor: Hmmm. Well how about this: It means I would be sur­prised if the answer were out­side the mar­gin of error.

Stu­dent: My god! You mean all that stuff you taught us about sta­tis­ti­cal mechan­ics and quan­tum mechan­ics and math­e­mat­i­cal prob­a­bil­i­ty: all it means is that you’d per­son­al­ly be sur­prised if it did­n’t work?

Pro­fes­sor: Well, uh…

Or the some­what more heady essay on the lim­it­ed util­i­ty of for­mal proofs by Stan­ford math­emeti­cian Kei­th Devlin

Before we can answer this ques­tion we need to agree what we mean by proof. (This is one of the rea­sons why its good to have math­e­mati­cians around. We like to begin by giv­ing pre­cise def­i­n­i­tions of what we are going to talk about, a pedan­tic ten­den­cy that some­times dri­ves our physi­cist and engi­neer­ing col­leagues crazy.) For instance, fol­low­ing Descartes, I can prove to myself that I exist, but I can’t prove it to any­one else. Even to those who know me well there is always the pos­si­bil­i­ty, how­ev­er remote, that I am mere­ly a fig­ment of their imag­i­na­tion. If it’s rock sol­id cer­tain­ty you want from a proof, there’s almost noth­ing beyond our own exis­tence (what­ev­er that means and what­ev­er we exist as) that we can prove to our­selves, and noth­ing at all we can prove to any­one else.

Math­e­mat­i­cal proof is gen­er­al­ly regard­ed as the most cer­tain form of proof there is, and in the days when Euclid was writ­ing his great geom­e­try text Ele­ments that was sure­ly true in an ide­al sense. But many of the proofs of geo­met­ric the­o­rems Euclid gave were sub­se­quent­ly found out to be incorrect—David Hilbert cor­rect­ed many of them in the late nine­teenth cen­tu­ry, after cen­turies of math­e­mati­cians had believed them and passed them on to their students—so even in the case of a ten line proof in geom­e­try it can be hard to tell right from wrong.

When you look at some of the proofs that have been devel­oped in the last fifty years or so, using incred­i­bly com­pli­cat­ed rea­son­ing that can stretch into hun­dreds of pages or more, cer­tain­ty is even hard­er to main­tain. Most math­e­mati­cians (includ­ing me) believe that Andrew Wiles proved Fer­mat’s Last The­o­rem in 1994, but did he real­ly? (I believe it because the experts in that branch of math­e­mat­ics tell me they do.)

In late 2002, the Russ­ian math­e­mati­cian Grig­ori Perel­man post­ed on the Inter­net what he claimed was an out­line for a proof of the Poin­care Con­jec­ture, a famous, cen­tu­ry old prob­lem of the branch of math­e­mat­ics known as topol­o­gy. After exam­in­ing the argu­ment for two years now, math­e­mati­cians are still unsure whether it is right or not. (They think it “prob­a­bly is.”)

Or con­sid­er Thomas Hales, who has been wait­ing for six years to hear if the math­e­mat­i­cal com­mu­ni­ty accepts his 1998 proof of astronomer Johannes Keplers 360-year-old con­jec­ture that the most effi­cient way to pack equal sized spheres (such as can­non­balls on a ship, which is how the ques­tion arose) is to stack them in the famil­iar pyra­mid-like fash­ion that green­gro­cers use to stack oranges on a counter. After exam­in­ing Hales’ argu­ment (part of which was car­ried out by com­put­er) for five years, in spring of 2003 a pan­el of world experts declared that, where­as they had not found any irrepara­ble error in the proof, they were still not sure it was cor­rect.

With the idea of proof so shaky—in practice—even in math­e­mat­ics, answer­ing this year’s Edge ques­tion becomes a tricky busi­ness. The best we can do is come up with some­thing that we believe but can­not prove to our own sat­is­fac­tion. Oth­ers will accept or reject what we say depend­ing on how much cre­dence they give us as a sci­en­tist, philoso­pher, or what­ev­er, gen­er­al­ly bas­ing that deci­sion on our sci­en­tif­ic rep­u­ta­tion and record of pre­vi­ous work. At times it can be hard to avoid the whole thing degen­er­at­ing into a slang­ing match. For instance, I hap­pen to believe, firm­ly, that sta­ples of pop­u­lar-sci­ence-books and breath­less TV-spe­cials such as ESP and mor­phic res­o­nance are com­plete non­sense, but I can’t prove they are false. (Nor, despite their repeat­ed claims to the con­trary, have the pro­po­nents of those crack­pot the­o­ries proved they are true, or even worth seri­ous study, and if they want the sci­en­tif­ic com­mu­ni­ty to take them seri­ous­ly then the onus if very much on them to make a strong case, which they have so far failed to do.)

Once you rec­og­nize that proof is, in prac­ti­cal terms, an unachiev­able ide­al, even the old math­e­mati­cians stand­by of GÏdel’s Incom­plete­ness The­o­rem (which on first blush would allow me to answer the Edge ques­tion with a state­ment of my belief that arith­metic is free of inter­nal con­tra­dic­tions) is no longer avail­able. GÏdel’s the­o­rem showed that you can­not prove an axiomat­i­cal­ly based the­o­ry like arith­metic is free of con­tra­dic­tion with­in that the­o­ry itself. But that does­n’t mean you can’t prove it in some larg­er, rich­er the­o­ry. In fact, in the stan­dard axiomat­ic set the­o­ry, you can prove arith­metic is free of con­tra­dic­tions. And per­son­al­ly, I buy that proof. For me, as a liv­ing, human math­e­mati­cian, the con­sis­ten­cy of arith­metic has been proved—to my com­plete sat­is­fac­tion.

So to answer the Edge ques­tion, you have to take a com­mon sense approach to proof—in this case proof being, I sup­pose, an argu­ment that would con­vince the intel­li­gent, pro­fes­sion­al­ly skep­ti­cal, trained expert in the appro­pri­ate field. In that spir­it, I could give any num­ber of spe­cif­ic math­e­mat­i­cal prob­lems that I believe are true but can­not prove, start­ing with the famous Rie­mann Hypoth­e­sis. But I think I can be of more use by using my math­e­mati­cian’s per­spec­tive to point out the uncer­tain­ties in the idea of proof. Which I believe (but can­not prove) I have.

Or Seth Lloyd from MIT:

I can­not prove that elec­trons exist, but I believe fer­vent­ly in their exis­tence. And if you don’t believe in them, I have a high volt­age cat­tle prod I’m will­ing to apply as an argu­ment on their behalf. Elec­trons speak for them­selves.

And the appar­ent­ly lone the­ist, David Myers:

As a Chris­t­ian monothe­ist, I start with two unproven axioms:

1. There is a God.

2. It’s not me (and it’s also not you).

Togeth­er, these axioms imply my surest con­vic­tion: that some of my beliefs (and yours) con­tain error. We are, from dust to dust, finite and fal­li­ble. We have dig­ni­ty but not deity.

And that is why I fur­ther believe that we should

a) hold all our unproven beliefs with a cer­tain ten­ta­tive­ness (except for this one!),

b) assess oth­ers’ ideas with open-mind­ed skep­ti­cism, and

c) freely pur­sue truth aid­ed by obser­va­tion and exper­i­ment.

This mix of faith-based humil­i­ty and skep­ti­cism helped fuel the begin­nings of mod­ern sci­ence, and it has informed my own research and sci­ence writ­ing. The whole truth can­not be found mere­ly by search­ing our own minds, for there is not enough there. So we also put our ideas to the test. If they sur­vive, so much the bet­ter for them; if not, so much the worse.

With­in psy­chol­o­gy, this “ever-reform­ing” process has many times changed my mind, lead­ing me now to believe, for exam­ple, that new­borns are not so dumb, that elec­tro con­vul­sive ther­a­py often alle­vi­ates intractable depres­sion, that Amer­i­ca’s eco­nom­ic growth has not improved our morale, that the auto­mat­ic uncon­scious mind dwarfs the con­scious mind, that trau­mat­ic expe­ri­ences rarely get repressed, that most folks don’t suf­fer low self-esteem, and that sex­u­al ori­en­ta­tion is not a choice.

Any­way, inter­est­ing stuff.

Here’s a fas­ci­nat­ing lit­tle snap­shot of mod­ern sci­ence:

A team of US physi­cists has proved a the­o­rem that explains how our objec­tive, com­mon real­i­ty emerges from the sub­tle and sen­si­tive quan­tum world…

The Los Alam­os team define a prop­er­ty of a sys­tem as ‘objec­tive’, if that prop­er­ty is simul­ta­ne­ous­ly evi­dent to many observers who can find out about it with­out know­ing exact­ly what they are look­ing for and with­out agree­ing in advance how they’ll look for it.

The full arti­cle Nat­ur­al Selec­tion Acts On The Quan­tum world is just plain cool (and acces­si­ble to non-sci­en­tists).