Monday, January 30, 2012

2012 U.S. Figure Skating Championships' men on ice

SAN JOSE, CA - JANUARY 29: Adam Rippon, Jeremy Abbott, Ross Miner and Armin Mahbanoozadeh pose for photographers after the Men's Competition during the 2012 Prudential U.S. Figure Skating Championships at the HP Pavilion on January 29, 2012 in San Jose, California. (Photo by Matthew Stockman/Getty Images)


video link as available

Senior Men
1 - Jeremy Abbott, Detroit SC
2 - Adam Rippon, SC of NY
3 - Ross Miner, SC of Boston
4 - Armin Mahbanoozadeh, WA FSC
5 - Douglas Razzano, Coyotes SC of AZ
6 - Stephen Carriere, SC of Boston
7 - Keegan Messing, AK Assoc of Figure Skaters
8 - Max Aaron, Broadmoor SC
9 - Jason Brown, Skokie Valley SC
10 - Scott Dyer, All Year FSC
11 - Jonathan Cassar, All Year FSC
12 - Grant Hochstein, St. Clair Shores FSC
13 - Richard Dornbus, All Year FSC
14 - Brandon Mroz, Broadmoor SC
15 - Alexander Johnson, Broadmoor SC
16 - Joshua Farris, Broadmoor SC
17 - Wesley Campbell, Colonial FSC
18 - William Brewster, Detroit SC
19 - Alexander Aiken, Atlanta FSC
20 - Daniel Raad, FL Everblades FSC

Junior Men
1 - Nathan Chen, Oval FSC
2 - Timothy Dolensky, Atlanta FSC
3 - Philip Warren, All Year FSC
4 - Harrison Choate, SC of Boston
5 - Lukas Kaugars, Broadmoor SC
6 - Timothy Koleto, Broadmoor SC
7 - Jay Yostanto, All Year FSC
8 - Troy Tomasello, Strongville SC
9 - Ryan Hartley, Queen City FSC
10 - Andrew Nagode, University of DE FSC
11 - David Wang, All Year FSC
12 - Emmanuel Savary, SC of NY

U.S. Figure Skating Championships (other results)
AP: Jeremy Abbott wins 3rd U.S. crown

Thursday, January 19, 2012

Wednesday, January 18, 2012

Tuesday, January 17, 2012

Curry on ice

John Curry
Innsbruck Olympics 1976

John Curry
Scheherazade 1980

Men's rankings

Friday, January 13, 2012

Weir on ice

Johnny Weir
2004 Nationals (his first title)

Men's rankings

Thursday, January 12, 2012

Monday, January 9, 2012

Rippon on ice

Adam Rippon

Men's rankings

on ice

This blog is being put on ice for the next couple of weeks while I edit two collections of my poetry to be published this month.

So in the interlude, I am posting some of my favorite figure skaters — on ice!

Sunday, January 8, 2012

Closing Plato's Cave

Jan Mycielski

February 7, 2012 will be Jan Mycielski's 80th birthday. He is known perhaps to topologists and graph theorists for his work in those fields, but I know about him for his work in mathematical logic and his interpretation of mathematics, intentionalism (Mathematics is "a description of finite structures consisting of finitely many individually imagined objects"), and a series of papers ("Locally Finite Theories" <>, ...) as covered by Shaughan Lavine's Understanding the Infinite. I wrote about that in Mathematica materialis, or How not to be lured into Plato's cave.

What is this all about?

Mathematics as generally taught in schools is thought to commit one to the existence of sets with an infinite number of elements.

Take one of the axioms of arithmetic of natural numbers:

(forall (m) (if (natural m) (exists (n) (and (natural n) (succ m n)))))

Here I'm using a formulation of the axioms with S-expressions for the object language. (succ m n) says n is the successor of m.

(forall list-of-variables expression)
(exists list-of-variables expression)

are interpreted in the standard way as the universal and existential quantifiers.

What Jan Mycielski does is to add a domain of quantification:

(forall P list-of-variables expression)
(exists Q list-of-variables expression)

where P and Q are the domains over which the variables range.

The above axiom becomes

(forall P (m) (if (natural m) (exists Q (n) (and (natural n) (succ m n)))))

and the interpretation is that when domains of quantification are nested, then the domain of the inner quantification is a superset of the domain that encloses it. (The inner-more you go, the bigger the sets get.) And not only that, but all domains of quantification are finite! (When the domains are the same as in standard logic one is committed to an infinite set of natural numbers, but this is not the case for nesting domains.)

If one has an entire mathematical theory (expressed as a collection of S-expressions) then one is talking about variables only ranging over finite sets, but one gets bigger and bigger finite sets as needed in any practical application of the theory. Lavine's book (discussed in the previous post) likens this to getting bigger and bigger bags of beans when needed. (I guess it helps to be a bean counter.)

So whatever Platonic infinities mathematicians hold dear can be dispensed with (with the Mycielskian change of quantification, of course).

Plato's Cave is closed.

2012/02/29: I came across a more recent paper, "A System of Axioms for Set Theory for the Rationalists", Notices of the AMS, February 2006 (the infinite sets and universes of pure mathematics are not actually but only potentially infinite). Again, infinite is replaced by only as large as needed.

Note: There is a comment in the above AMS paper of Mycielski that "it seems actual infinity exists in physical reality" (the supposed space-time continuum of Penrose's The Road to Reality), but I don't think there is a sufficient basis for that.

Friday, January 6, 2012

Politics, be a sport

Of politics, the sport I view,
like others watch the balls
that fly above the fields run through
the dulled suburban sprawls:

It's back and forth, dialectic'ly,
and people are its pawns.
It's making myth and cunning pleas,
and language is its brawn.

The others: frivolous delights,
but do not have the heft
of history-made human plights
and all that we have left.

The team who wins will change our lives
much more than people know,
the people who themselves deprive
of watching Maddow's show.

Tuesday, January 3, 2012

It's about time

The beginning of a new year (calendrically speaking) is a timely time to think about time, whether it's pinning down the details of the difference between a year and the time it takes for the Earth to orbit the Sun, or reading about what philosophers have to say about it. But with all the December news of the results coming from the Large Hadron Collider, I thought about time from a quantum particle's perspective.

We (humans) remember the past, but wonder about the future. (Curiously, it's not the reverse.) This has to do with a complicated explanation that involves the second law of thermodynamics, statistical mechanics, and boundary conditions set at the origin of the big bang (Huw Price, Time's Arrow and Archimedes' Point, We "perceive" an arrow of time.

At the level of quantum particles, there is no time arrow as such. "The behavior of the microworld appears paradoxical only when we insist on applying to it concepts from the macroworld that have no meaning at the elementary level" (Victor Stenger, The Unconscious Quantum, "A quantum system can thus be viewed as being influenced by its future as well as its past. The final condition defines all the possible outcomes, with a quantum mechanical probability calculated for each. One of these outcomes happens in accordance with these probabilities. As long as the dice are being tossed to determine the outcome, that is, we do not have deterministic hidden variables, then the macroworld can develop with a future that is not already written in the stars."

There is a wonderful story in the Huw Price book (mentioned above) about an imaginary planet, Ypiaria. The following comes from an edited version of a post on a board on The Straight Dope.
JasonFin 10-08-2004, 07:22 PM

By modern standards the criminal code of Ypiaria (pronounced "E-P-R-ia") allowed its police force excessive powers of arrest and interrogation. Random detention and questioning were accepted weapons in the fight against serious crime. This is not to say the police had an entirely free hand, however. On the contrary, there were strict constraints on the questions the police could address to anyone detained in this way. One question only could be asked, to be chosen at random from a list of three:

(1) Are you a murderer?
(2) Are you a thief?
(3) Have you committed adultery?

Detainees who answered "yes" (Y) to the chosen question were punished accordingly, while those who answered "no" (N) were immediately released. (Lying seems to have been frowned on, but no doubt was not unknown.)

To ensure that these guidelines were strictly adhered to, records were required to be kept of every such interrogation. Some of these records have survived, and therein lies our present concern. The records came to be analyzed by the psychologist Alexander Graham Doppelganger, known for his work on long distance communication. Doppelganger realized that among the many millions of cases in the surviving records there were likely to be some in which the Ypiarian police had interrogated both members of a pair of twins. He was interested in whether in such cases any correlation could be observed between the answers given by each twin.

As we now know, Doppelganger's interest was richly rewarded. He uncovered the two striking and seemingly incompatible correlations now known collectively as Doppelganger's Twin Paradox . He found that

(8.1) When each member of a pair of twins was asked the same question, both always gave the same answer;

and that

(8.2) When each member of a pair of twins was asked a different question, they gave the same answer on close to 25 percent of such occasions.

It may not be immediately apparent that these results are in any way incompatible. But Doppelganger reasoned as follows: 8.1 means that whatever it is that disposes Ypiarians to answer Y or N to each of the three possible questions 1, 2, and 3, it is a disposition that twins always have in common. For example, if YYN signifies the property of being disposed to answer Y to questions 1 and 2 and N to question 3, then correlation 8.1 implies that if one twin is YYN then so is his or her sibling. Similarly for the seven other possible such states: in all, for the eight possible permutations of two possible answers to three possible questions. (The possibilities are the two homogeneous states YYY and NNN, and the six inhomogeneous states YYN, YNY, NYY, YNN, NYN, and NNY.)

Turning now to 8.2, Doppelganger saw that there were six ways to pose a different question to each pair of twins: the possibilities we may represent by 1:2, 2:1, 1:3, 3:1, 2:3, and 3:2. (1:3 signifies that the first twin is asked question 1 and the second twin question 3, for example.) How many of these possibilities would produce the same answer from both twins? Clearly it depends on the twins' shared dispositions. If both twins are YYN, for example, then 1:2 and 2:1 will produce the same response (in this case, Y) and the other four possibilities will produce different responses. So if YYN twins were questioned at random, we should expect the same response from each in about 33 percent of all cases. And for homogeneous states, of course, all six posible question pairs produce the same result: YYY twins will always answer Y and NNN twins will always answer N.

Hence, Doppelganger realized, we should expect a certain minimum correlation in these different question cases. We cannot tell how many pairs of Ypiarian twins were in each of the eight possible states, but we can say that whatever their distribution, confessions should correlate with confessions and denials with denials in at least 33 percent of the different question interrogations. For the figure should be 33 percent if all the twins are in inhomogeneous states, and higher if some are in homogeneous states. And yet, as 8.2 describes, the records show a much lower figure.

Doppelganger initially suspected that this difference might be a mere statistical fluctuation. As newly examined cases continued to confirm the same pattern, however, he realized that the chances of such a variation were infinitesimal. His next thought was therefore that the Ypiarian twins must generally have known what question the other was being asked, and determined their answer partly on this basis. He saw that it would be easy to explain 8.2 if the nature of one's twin's question could influence one's own answer. Indeed, it would be easy to make a total anticorrelation in the different question cases be compatible with 8.1—with total correlation in the same question cases.

Doppelganger investigated this possibility with some care. He found, however, that twins were always interrogated separately and in isolation. As required, their chosen questions were selected at random, and only after they had been separated from one another. There therefore seemed no way in which twins could conspire to produce the results described in 8.1 and 8.2. Moreover, there seemed a compelling physical reason to discount the view that the question asked of one twin might influence the answers given by another. This was that the separation of such interrogations was usually spacelike in the sense of special relativity; in other words, neither interrogation occurred in either the past or the future light cone of the other. (It is not that the Ypiarian police force was given to space travel, but that light traveled more slowly in those days. The speed of a modern carrier pigeon is the best current estimate.) Hence according to the principle of the relativity of simultaneity, there was no determinate sense in which one interrogation took place before the other.

This is the problem posed by the EPR experiment in a nutshell, but instead of twins we are talking about entangled particles and instead of answers to questions we are talking about measurements of the particles' "spin" along their 3 axes (there is an uncertainty relation between these spins). As the great physicist Richard Feynman said, "Nobody understands quantum mechanics ... do not keep saying to yourself, if you can possibly avoid it, 'But how can it be like that?' because you will go 'down the drain' into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that." The weirdness of QM, and the difficulty with imagining "how can it be like than" in a way consistent with the view that reality exists before we observe it, extends to other famous experiments and thought-experiments, like the Double-slit experiment and the Schroedinger's Cat thought-experiment. (Is the cat "really" alive or dead before it is measured?) But Bell's theorem and the EPR experiment show most clearly what the basic problem here is for a realist.

A number of different "interpretations" of quantum-mechanical weirdness have emerged over the years, with none yielding any new physical predictions (and thus being experimentally indistinguishable) but each offering a different way to conceptualize what's "really" going on in these sorts of experiments.

1. The Copenhagen Interpretation

Basically, the Copenhagen interpretation says that we shouldn't worry about how what's really going on in the first place—science can only deal with correlating and predicting the results of various measurements, but it can't tell us anything about what goes on when we're not looking. This is basically a logical positivist perspective, and it was preferred by Bohr.

2. "Objective Collapse" Interpretation

Here the wave-particle duality is taken literally—the world exists as a wavelike potential when it's not being observed, but somehow measurements periodically "collapse" the wavefunction into a definite state. Some versions of this suppose that it's consciousness that does the collapsing, others suppose that an entangled system collapses once it reaches a certain limit in mass. Unlike the other interpretations, these might actually be expected to yield different predictions than orthodox QM—so far, there's no evidence for anything like this though.

3. The Bohm-de Broglie Interpretation

Bell's theorem shows that no local hidden variable theory can explain the results of the EPR experiment, but that leaves open the possibility of a nonlocal hidden variables theory where particles can communicate faster than light. This is the route taken by Bohm and de Broglie's interpretation. In the Ypiarian story, this would be like the twins having a psychic link which allows one to know what question the other was asked, and adjust his own answer accordingly.

4. The Many-Worlds Interpretation

This interpretation takes the mathematical formalism of QM literally and proposes that the wavefunction is all there is. This means that when I measure the state of a particle that's in superposition, instead of "collapsing" it into a definite state, I just become entangled with it and enter into a superposition myself; basically, I "split" into two versions of myself, one of whom observes one state and another of whom observes another. In popular accounts this is sometimes explained in terms of the entire universe splitting into parallel histories all the time, but it's a bit more subtle than that, since different "worlds" can interfere with each other and cannot be viewed as totally "parallel", although thermodynamics may explain the appearance of splitting through a phenomenon called decoherence. For technical reasons this interpretation preserves locality, and it's also 100% deterministic to boot (although it suggests an odd kind of subjective indeterminacy in which my first-person experience randomly chooses which split copy to become—hence a variation of this interpretation is the many-minds interpretation which deals with this issue a little more explicitly). This interpretation is seen as being the most theoretically elegant one by a number of physicists, and it seems that it is sometimes implicitly assumed in quantum cosmology, although physicists are often agnostic about whether other worlds/histories are actually "real." Many-worlds could also make sense of quantum computation, which some physicists believe can be understood in terms of the quantum computer performing different computations in different worlds and then combining the results through interference.

5. Time symmetric [or Advanced action] Interpretation

The EPR experiment can also be explained if you assume the future can affect the past, so that the particle's original properties are affected by the measurements that will be made on them later, once they are separated. In the Ypiarian story, this would mean that the twin's choices to commit or not commit various crimes would be affected by which questions they would be asked much later when they're interrogated. This isn't as strange as it sounds, since all the laws of physics we currently know of are time-symmetric (they look the same forwards as they do backwards) and apparently the apparent "arrow of time" emerges solely from statistical mechanics, perhaps because the universe started off in a very low-entropy state. Huw Price's book Time's Arrow and Archimedes' Point, which I quoted from above, deals with this problem, and he favors a version of this interpretation.

So, those are the various interpretations. As I said, the main problem is that none of them really gives any new testable predictions, which is a bit unsatisfying. There's some good reason to think that a theory of quantum gravity would transform our understanding of QM somewhat, so perhaps such a theory will depend on a modified version of one of these interpretations that is testable in some way. In any case, Bell's inequality shows definitively than no classical, realist picture of the world can explain the EPR results, so whatever the truth turns out be, it's guaranteed to violate our cherished assumptions in one way or another (... faster-than light signaling, parallel universes‚ the future affecting the past — take your pick!)

So there it is. I pick the last one, naturally: Unlike a human person, a quantum particle both remembers and wonders about its past and future in equal measure!

added 2011/01/23: Does Time-Symmetry Imply Retrocausality? How the Quantum World Says "Maybe"