Physics and Incompleteness

Responding to new-age spewage the other day -- at greater length than was probably warranted -- I asserted in passing that Godel's (first) incompleteness theorem probably didn't apply to "theories of everything" in physics because such theories need not include Peano arithmetic, which is one of the hypotheses in Godel's proof. I'm going to think aloud about this; comments would be appreciated.

Let me remind you briefly of how Godel's proof works. You start with an axiomatic system (i.e., a list of allowed characters -- a language -- and a list of axioms in that language) that includes all the usual operations involving natural numbers. Because of the unique prime factorization property, you can assign each string in the language a unique number (called its Godel number) as follows: first assign each allowed character a number, then encode the string as 2^(number of first character) x 3^(number of second character) x 5^(...). E.g. if the language were English, you could encode the string "babe" as 2^2 x 3^1 x 5^2 x 7^5 = some large number. You could recover the word from the number by counting the 2's in its (unique) prime factorization to get your first character, then counting the 3's for your second character, and so on. Now you can start making self-referential statements within your language, you can say something like "the string with Godel number N has certain properties," which would mean "the string 'blah' has certain properties." You could drop "the string" as that's implied by the quotes.

It turns out that one of these sentences is essentially Quine's version of the liar's paradox:

“Yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation.

(The stuff in quotes is to be encoded as a Godel number, so formally the statement would look like "The Godel number G has certain properties, and is thus possible to construct in the language of Peano arithmetic.)

Now the point is that none of this would have worked without the uniqueness of prime factorization. And it's hard to see why a physical theory of everything -- if it were axiomatizable -- would have prime factorization in it. Consider the following simple theory of everything, which is of course incorrect but is probably structurally similar to what the real one would look like -- it has a finite list of things, 1 through N, each with a charge and a mass; the forces acting on each of them in any given configuration follow from Maxwell's equations, and how they react to the force is determined by Newton's laws of motion. (Throw in gravitation as well, it doesn't matter.) The point is that you now have all the prerequisites for an axiomatic system in which one can ask all sorts of questions about what the velocities, positions, position-velocity correlations, etc. of the particles are at any time, i.e., this is Laplace's universe. The key point is that the minimal set of mathematical rules you need to make sense of this theory are rules about arithmetical operations on the real numbers. Although the reals include the naturals, they are infinitely simpler because everything divides everything else. Natural numbers do not enter the language of the theory at all -- yes, you need natural numbers to list the particles while constructing the language of the theory, but that's in the meta-language. Ergo no prime factorization, ergo no self-referential statements, ergo no incompleteness theorem. While we do not know quite what the theory of everything would be it seems almost certain to be structurally rather like the one I just described. (You might say, well, isn't quantum mechanics "grainy" and doesn't it therefore have something to do with the integers; the graininess doesn't enter into the fundamental equations, but even if this were the case, you only add and never multiply quanta so you wouldn't need all of Peano arithmetic so you wouldn't have an incompleteness theorem.)

I think this is all true so far as it goes but there's a potential loophole in the argument, which is that one might want to ask questions about the world that are not phraseable in the language of the theory. For example one might want to ask questions about thermodynamics, involving macroscopic quantities like pressure and temperature in my toy axiomatic system. I could, in principle, define pressure in terms of a sum over all the velocities of the particles near the walls divided by the number of particles (or something like that), but really what I want to talk about when I'm talking about pressure is a property that becomes well-defined only for systems, and I can't even construct my toy theory of everything for an infinite system. Therefore, what I'm really doing when I construct thermodynamics is setting up an entirely new axiomatic system by a process of taking limits over larger and larger versions of the original theory of everything.

Basically what this says is that a theory of everything in the ordinarily understood sense is not a theory in which a lot of the questions that physicists in practice work on would be well-defined. Could these "effective" theories, or the union of them, be strong enough to include PA? Maybe. In particular, a lot of natural-number arithmetic comes in from the topological properties of wavefunctions, the spaces they live in, etc. I'm still doubtful that you'd ever have to multiply natural numbers, but it's not beyond the bounds of possibility.

Dept. of complicated rhyme schemes

Brad Leithauser (I think) says this poem has the most intricate rhyme scheme of any really successful poem. BTW flush/flash is what's called pararhyme and was an invention of Wilfred Owen's. It's neither dulce nor decorum but it's kind of a nice effect. Interestingly most of the formal inventions in 20th century English poetry have been by relatively minor writers -- Owen, Marianne Moore, Kay Ryan, R.S. Thomas, Louis MacNeice, H.D. etc. -- and someone like Auden was much more of an adopter than an inventor.

from Five Songs
W.H. Auden

That night when joy began
Our narrowest veins to flush,
We waited for the flash
Of morning's leveled gun.

But morning let us pass,
And day by day relief
Outgrows his nervous laugh,
Grown credulous of peace,

As mile by mile is seen
No trespasser's reproach,
And love's best glasses reach
No fields but are his own.

Pundits on kids

E.D. Hearse takes on the most widely read morons in America:

She also challenges the claim by economists, and taken up by influential writers such as Malcolm Gladwell and Nicholas Kristof, that if only students were exposed to excellent teachers for three to five years in a row, the achievement gap between advantaged and disadvantaged students would be overcome. It is true that some math teachers have been able to make up as much as 33 percent of the lag for some students in a single year—whence the economists’ extrapolation to the three-to-five-year figure.

Not, of course, that one can assume either Gladwell or Kristof knows what an extrapolation is. I generally recommend the article -- a review of Diane Ravitch's [a.k.a. Dan Ravage] new book on how school choice has failed -- as it is sporadically sensible and has a lot of interesting factoids. E.g. here's Richard Hofstadter on child nature (on which see also Grobstein):

To believe that Dewey’s synthesis was successful required a certain credulity about the pre-established harmony between child nature and democratic culture which not everyone could share. It seemed…that one would have to give up either the emphasis on child nature or the emphasis on educating for democracy.

I would however part ways with both Hearse and Ravage when they rhapsodize about the "neighborhood school" as a pillar of the community. (I also find Hearse's whining about international league tables a little tiresome. Bloody awful, isn't it, that other countries educate their kids?) Most of the real problems with the school system have to do with schools for the poor; these tend to be in areas where there isn't a meaningful sense of community -- whose residents are in any case largely transient -- and the approach of the successful, no-excuses sort of inner-city schools has been to give kids an alternative social anchor from their "communities." Beyond that I don't think K-12 education can be sensibly addressed in isolation from the Big Sort -- it's a good thing to get middle-class kids into the public school system as their parents function as watchdogs, but it's not sufficient because it doesn't deal with the districts where most parents are too illiterate and/or busy to find out what their kids are learning at school.

Another bit of information that I think is crucial to this entire picture is this study (I'm quoting from an email I wrote a while ago):

[...] there's Richard Elmore's paper on high-performing schools. What Elmore found was that high-performing inner-city schools were different from both suburban schools and crap inner-city schools, which were rather like each other. In suburban schools, the instructors assumed (correctly) that parents would play the primary role in the education of students, and were hands-off; in crap inner-city schools, the instructors (incorrectly) made the same assumption, and let the kids smoke crack in class; in good inner-city schools, instructors realized that they had to make up for the lack of useful parenting, and put a huge amount of work into making sure the students were kept engaged.

Geezers in Freezers

A Winter Ode to the Old Men of Lummus Park, Miami, Florida

Donald Justice

Risen from rented rooms, old ghosts
Come back to haunt our parks by day,
They crept up Fifth Street through the crowd,
Unseeing and almost unseen,
Halting before the shops for breath,
Still proud, pretending to admire
The fat hens dressed and hung for flies
There, or perhaps the lone, dead fern
Dressing the window of a small
Hotel. Winter had blown them south--
How many? Twelve in Lummus Park
I counted, shivering where they stood,
A little thicket of thin trees,
And more on benches, turning with
The sun, wan heliotropes, all day.

O you who wear against the breast
The torturous flannel undervest
Winter and summer, yet are cold,
Poor cracked thermometers stuck now
At zero everlastingly,
Old men, bent like your walking sticks
As with the pressure of some hand,
Surely they must have thought you strong
To lean on you so hard, so long!

Birthers and calories

There's this well-known paper by Brendan Nyhan and Jason Reifler that shows -- I haven't really read it in detail but the analysis is simple and seems pretty airtight, and it is widely cited -- that political misinformation is very hard to correct; in particular, people tend to "double down," and lend more credence, to myths that line up with their political beliefs after they are debunked. (Empirical evidence for Credo quia absurdum est.) This is, of course, a broader phenomenon than just politics; it appears to have been behind some of the hysterical reaction last year (was it last year?) when someone suggested that mammograms were unnecessary for certain age groups. Via Nyhan's blog, here's an article describing how evidence-based medicine is ineffectual when it comes to getting people to give up their false beliefs re medication. (The article also has some pretty interesting stuff about breast cancer.)

This also applies to stuff like listing the calorie content of stuff on menus. Although salad dressing is unhealthy, it's hard to debunk the myth that salads are healthy no matter what. (If you don't like this example -- I don't know how true it is; frankly, food is not something I've ever been that interested in -- there must be other cases of deeply held misperceptions.) In general, this research on myth-busting suggests why (as e.g. Yglesias says here in passing) calorie labeling has been ineffectual so far, and will continue to be ineffectual, at getting people to form accurate beliefs about what's healthy. Of course, it might still be good for other things, like reminding people of how unhealthy the stuff they already consider unhealthy is.

snarXiv

In its own words:

The snarXiv is a ran­dom high-energy the­ory paper gen­er­a­tor incor­po­rat­ing all the lat­est trends, entropic rea­son­ing, and excit­ing mod­uli spaces. The arXiv is sim­i­lar, but occa­sion­ally less ran­dom.

The abstracts are utterly compelling. I have absolutely seen crazier-sounding stuff than this on the arxiv:

The SUSY CP Problem

Authors: D. Nelson, R. E. Fermi

Comments: 7 pages, minor corrections

Subjects: High Energy Physics - Phenomenology (hep-ph)

Some work has been done in recent papers on models of bubbles. We take a holomorphic approach to a primordial resolution of the fine-tuning problem, and find that observables follow from adjoint Matrix Models surrounded by a stack of (p,q) 7- branes wrapped on an Euclidean symmetric space. We conjecture that a surface defect is present as realized in the MSSM. When constructing a solution to the U(1) problem, we discover that, in the Clebsch-Gordon decomposition case, a (p,q) 7- brane reduction of superconformal TQFTs is gauge mediated.

T.S. Eliot, without the text

Eliot's book of essays on Elizabethan drama is available online; I strongly recommend it, as much for the quotations as anything else. (The pdf renders best, I think; just right for an iPad.) I'd read most of the essays in this book a long time ago -- 2004 or 2005, I think; I remember being impelled to spend a summer reading Webster and the lesser Elizabethans, who by and large did not live up to Eliot's hype -- and had forgotten a lot of the good stuff in here. A few quotations that jumped out:

Now comes my lover tripping like a roe,
And brings my longings tangled in her hair.
(Peele)

I pass'd, methought, the melancholy flood,
With that grim ferryman which poets write of,
Unto the kingdom of perpetual night.
The first that there did greet my stranger soul,
Was my great father-in-law, renowned Warwick;
Who cried aloud, "What scourge for perjury
Can this dark monarchy afford false Clarence?"
(Shakespeare)

Go hurtless souls, whom mischief hath oppressed
Even in first porch of life but lately had,
And fathers fury -- go, unhappy kind,
O little children, by the way full sad
Of journey knowen.
Go see the angry kings.
(Jasper Heywood, translating Seneca re the dead children of Hercules -- I found the Latin helped me parse the English here:

ite ad Stygios, umbrae, portus
ite, innocues, quas in primo
limine vitae scelus oppressit
patriusque furor;
ite, iratos visite reges.)

If thou wilt stay,
Leap in mine arms; mine arms are open wide;
If not, turn from me, and I'll turn from thee;
For though thou hast the heart to say farewell,
I have not power to stay thee.
(Marlowe)

The rawish dank of clumsy winter ramps
The fluent summer's vein; and drizzling sleet
Chilleth the wan bleak cheek of the numb'd earth,
While snarling gusts nibble the juiceless leaves
From the nak'd shuddering branch
(Marston)

Mineralnye Vody

Earlier today, some airport-related link reminded me of this marvelous old Economist piece about Russian airports (gated version; Dec 19 2006). My favorite part:

Mineralnye Vody: To reach this airport, in the north Caucasus, passengers pass through a series of military roadblocks, where documents and the boots of cars are checked by slouching policemen, looking for weapons or terrorists. But a sensible terrorist would leave his weapons at home and buy new ones at the airport, where a wide selection of enormous knives and ornamental Caucasian swords is on sale. There are also embossed Caucasian drinking horns, and a large number of Brezhnev-era copies of the Kama Sutra.

Mineralnye Vody airport is a lower circle of hell. In Soviet times, before the region that the airport serves was desolated by separatist insurgencies, blood feuds and government brutality, the nearby mineral spas were popular holiday resorts. The building is incongruously large for a part of Russia that today, for all its macho hospitality and merriment, feels more African than European in its violence, poverty and corruption. It is weirdly cold inside. Feral cats have been sighted. The floor has not been cleaned since perestroika; the toilets are hauntingly squalid. On the wall there are arrival and departure boards that no longer work, and a big, proud map of the Soviet Union.

The article is also memorable for coining the phrase "incremental babushka."

And now for sth completely different... The Telegraph has a pic of a lemur playing with an Easter egg.