Howard Pattee said [November 20, 2010 7:19 PM]:
This point is well-known in physics by the two models of a gas. Microscopically the dynamic model of an ideal box of atoms is reversible (time-symmetric) while the thermodynamic model is irreversible. Clearly these models are formally contradictory and therefore neither model can be derived from (or reduced to) the other. As Max Planck noted: “For it is clear to everybody that there must be an unfathomable gulf between a probability, however small, and an absolute impossibility . . . Thus dynamics and statistics cannot be regarded as interrelated.”
Thanks Howard for pointing this out. Although I've taught classical thermodynamics at the undergraduate level and statistical thermodynamics at the graduate level, there seems to be a gap between the two that is not really paid attention to in engineering programs. Although the Maxwell Demon paradox is mentioned, the logical implications are not explored. We simply teach that the microscopic and macroscopic are related by the formula for change in entropy equaI to Plancks constant times the natural log of the ratio of thermodynamic probabilities of the macrostates. I did some research after your post and days of hard thinking. My gut tells me that the assumption of sign reversal in the classical mechanics description being equivalent to reversibility has something wrong with it. One thing I find fascinating is that this paradox and its cousins are the stimulus for your semiotic approach to evolution. Like all other paradoxes, it doesn't seem to matter whether one actually finds the "true" answer. What matters is that the stimulation can lead to new ideas like your semiotic approach. Look at all the mathematical progress that has its roots in Zeno's Paradox.
Joel, I have always been bothered by the transition from one or two particle deterministic models to N-particle statistical models. Bohr gave the story about the boy who goes into a candy store and asks for a penny’s worth of mixed sweets. The store owner says I don’t have any mixed sweets, but here are three sweets, you can mix them yourself.
ReplyDeleteWhether reality is deterministic, as Einstein and Ira believe, or whether it is stochastic, as most quantum physicists believe, is a metaphysical difference that does not make any practical or empirically testable difference in our models.
However, conceptually it can’t be both. Either determinism is just the illusion of very good statistics, or chance is just the illusion of imprecise measurements.
I think the Monty Hall problemrelates to the conceptual difficulty of small N statistics.
"How quaint the ways of Paradox!
ReplyDeleteAt common sense she gaily mocks!" [W.S.Gilbert, Pirates of Penzance]
As Howard taught me, ALL models are, by their very nature, incomplete representations of something that is (hopefully at least a little bit) true, else they would be the actual thing modeled. Paradox inevitably arises when any model, equation, or -especially- statistical construct is taken to be more than casually related to reality.
The cock crows in the morning because, in his mental model, he is responsible for the Sun rising! And, they (the cock and the Sun) do so day after day, confirming the value of the model :^)
As Howard quotes Planck: "...there must be an unfathomable gulf between a probability, however small, and an absolute impossibility ... Thus dynamics and statistics cannot be regarded as interrelated."
Unless the biosphere is simulated reality in some kind of computer (ala The Matrix movie or Fredkin's digital physics), it is misleading to say or believe that some physical reality is governed by some Law or formula or any other human conceived model. We are so quick to forget that it is the model that is follows reality.
One of my favorite examples is two reliable observers looking at the exact same physical object, and one says it is a circular disk and the other that it is a rectangle. Both are correct, but neither model captures the reality in any meaningful way. The object is a can of soda, and one observer is looking directly at the top while the other sees only the side.
The real world is far more intricate and complex than a can of soda. Terribly so! We and other living things survive and reproduce more or less successfully only because our competition is also using incomplete models. Evolution and natural selection works because, on average, favored genotypes express themselves as the fittest phenotypes that tend to survive and reproduce, even though some very fit phenotypes get unlucky and die early and some unfit ones get lucky and survive and reproduce. No one has to be perfect, just a bit better than the others.
Of course, evolution and natural selection is, itself, a simplified model. Each individual bacteria, plant, or animal has its own story and adventure and reason for surviving (or not) and reproducing (or not).
The same is true for each atom and molecule in a thermodynamic situation. Each has its own story but there are so many of them we cannot keep track. Nor do we need to because we only need an approximate model to solve real problems like maintaining a pleasant temperature in our house or efficient combustion in a car engine. So we make models at the micro- or macro-level. We reduce the interaction of millions or billions of atoms and molecules to the absurdly simplified model of a box of idealized atoms (as if they were so many steel balls) or to the equally absurdly simplified equations of entropy and gas dynamics. Why are we surprised when these models are as contradictory as the disk and rectangle of my soda can example?
Ira Glickstein
I followed Howard's link to the Monty Hall Problem and found this incredible statistic: "...when the Monty Hall problem appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine claiming the published solution ('switch!') was wrong."
ReplyDeleteSo, about 10% of those who wrongly say "switch" is the wrong strategy, claim to be PhDs. Since only about 1% of us actually have PhDs, this statistic has several interpretations:
(1) PhDs are about ten times more likely to be wrong than ordinary people.
(2) PhDs are about ten times more likely to write to Parade magazine about Marilyn vos Savant's column than ordinary people.
(3) Most of the PhDs who wrote have degrees in English or Art History or Environmental Activism or Political "Science", and have therefore been educated beyond their intelligence.
(4) People, including a disproportionate sampling of PhDs, misinterpret the problem as a pure "random guess" statistical exercise, missing the point that Monty Hall deterministically KNOWS which curtain holds the "goat" and therefore there is nothing "random" about which curtain he opens before offering the contestant the chance to "switch". When he exercises his superior knowledge, he transforms a one-out-of three "random" choice to a one-out-of two "random" choice, effectively resetting the problem. The contestant's original choice is only 33.33% likely (1 in 3). Therefore, the other curtain must be 100 - 33.33 = 66.67% likely, which is why the contestant should "switch".
Spinoza, Einstein, and Ira hope (4) is the correct explanation.
Ira Glickstein
A certain professor of thermodynamics was known to give the same final exam every year, always consisting of just the single question: “What is entropy?” One day an assistant suggested that it might be better to ask a different question now and then, so the students wouldn’t know in advance what they would be asked. The professor said not to worry. “It’s always the same question, but every year I change the answer.”
ReplyDeleteJoel, your anecdote has historical support. Shannon asked von Neumann if he should call information entropy. Von Neumann told him "You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, nobody knows what entropy really is, so in a debate you will always have the advantage.”
ReplyDeleteThanks Howard. Another important technique for understanding problems is exaggeration or the limiting case. In the Monte Hall problem that Ira brought up, it's much easier to see the answer if one exaggerates the odds. Suppose there are 100,000 doors. The chance of choosing the door with the prize is almost zero. The chance that the door that Monte selects has the prize is almost one. Hence, it obviously pays to switch to Monte's door.
ReplyDeleteI'm trying to get hold of this article, but my account at the University of Hawaii library doesn't seem to afford me access or I'm doing something wrong at
ReplyDeletehttp://bjps.oxfordjournals.org/?code=phisci&homepage.x=129&homepage.y=6&.cgifields=code.
Is Classical Mechanics Really Time-reversible and Deterministic?
Br J Philos Sci (1993) 44(2): 307-323
Joel wrote: "Suppose there are 100,000 doors. The chance of choosing the door with the prize is almost zero. The chance that the door that Monte selects has the prize is almost one. Hence, it obviously pays to switch to Monte's door."
ReplyDeleteIt appears you have not watched "Lets Make a Deal" lately! (See http://en.wikipedia.org/wiki/Monty_Hall_problem for more details.)
Let me clarify how Monty Hall ran his game show and then I'll get to your large number of doors example.
There are three curtains. Behind one is a valuable prize, such as a car. Behind another is a moderate prize, such as a toaster. Behind the remaining door is a zero value prize, such as a goat.
The contestant, who obviously wants to win the car and avoid the goat, picks one of the three curtains. (Chance he picked the car is 1 in 3 = 33.33%)
Before revealing what is behind the curtain the contestant picked, Monty reveals the least desirable of the two remaining curtains, which will either be the goat or, if the contestant happened to already have picked the goat curtain, the toaster. Now the contestant has the choice to "stick" with his original choice (33.33% it is the car) -or- "switch" to the other available curtain (100% - 33.33% = 66.67% it is the car.) Obviously, the odds of getting a car are double if you "switch".
Given a large number of curtains, as Joel suggests, again with Monty revealing the lowest value curtain (or the next lowest if the contestant has picked the lowest value curtain) it still makes sense to "switch", but not as compellingly. Say there were ten curtains with one being very valuable (like a car), eight being much less valuable (like toasters), and one of zero value. There is a 10% chance the contestant has picked the car. When Monty reveals the "goat", there are still nine curtains available to "switch" to. Thus, if the contestant does a "switch" his changes are (100 - 10)/8 = 11.25%, only slightly improved over his initial 10%
Given Joel's 100,000 curtains, "switch" is still the way to go, but only microscopically better.
Ira Glickstein
Once again my attempt to be terse jumps up and bites me in the butt. Using a limiting case is a method of underscoring an important element that is otherwise neglected by the listener. The reason so many people who think they understand probability are thrown off is that they neglect to allow for the fact that Monte has perfect knowledge and intervenes in the process.
ReplyDeleteAssume 100,000 doors, one with the car, and all the rest with negligible prizes. In the end, the excitement is about whether or not the contestant will win the car. The odds that the contestant will choose the car at the outset is one in 100,000. In the next step, Monte, with perfect knowledge, eliminates all of the doors but one (the reason is just show business excitement). The contestant is asked whether they want to switch from their initial choice (success nearly zero) to the single door that Monte has selected (success nearly 100%). The important factor illustrated here is that Monte must choose to pair the grand prize door with the contestant's no-prize door, if the contestant chose the no-prize door to begin with or vice-versa. Raising the number of initial doors and the number of eliminated doors serves to underscore for the "listener" that Monte's perfect knowledge is an intervening agent which inverts the odds. By selecting "Monte's door" we have gone from almost no chance of success to almost no chance of failure. Once the listener gets this point, he or she is ready for a rigorous analysis of the three door problem.