Isotropy
Wednesday, May 19, 2004
What's the Big Idea?
Before I get too scattered, let me rein in and make a list of main topics for this blog (hey, Blogger! Topic labels? I suppose I could modify the template myself....). These are the topics I'm trying to teach myself, with varying degrees of success:
It is possible this list is too ambitious for a weekend hobby.
Tuesday, May 18, 2004
Zombies
Here's a chilling little philosophical argument:
Zombies are exactly like us in all physical respects but have no conscious experiences: by definition there is ‘nothing it is like’ to be a zombie. Yet zombies behave like us, and some even spend a lot of time discussing consciousness. This disconcerting fantasy helps to make the problem of phenomenal consciousness vivid, especially as a problem for physicalism.
The question arises if zombies are even "conceivable". It is apparently not enough (I'm not a philosopher) to say "we're talking about them, aren't we?" "Conceivable" looks like a term of art that requires some form of logical consistency - but this doesn't feel right to me. My ignorance is peeking out....
The rest of the entry is pretty interesting, but most interesting was this: if zombies can exist and be exactly like us in all physical respects - undetectably - then it's hard to see how consciousness can have any effect on the physical world. On the other hand, if zombies can't exist, so that any creature that physically matches us in all respects must be conscious as we are, then consciousness is a purely physical phenomenon.
Ugh...more thought needed....and more brains...brains...brains....
Thursday, May 13, 2004
A Noether Thing
I've read up a little on Noether's theorem and whether it applies to scale-invariant Lagrangians. The answer seems to be either "yes" or "no".
The Feynman Lectures Vol. I, section 52, talks about symmetry and scaling. Feynman points out that Galileo discussed the physical problems of scaling in Two New Sciences. After reading a bit more about that, I think we are robbing students of a proper knowledge of Galileo's contributions by spending as much time as we do talking about his heresy trial. It makes it seem like he was little more than a prominent supporter of Copernicus, and it badly shortchanges him.
If you were a geeky child like me, at some point you probably ran into the question "If ants can carry fifty times their own weight, wouldn't it ROCK if we could make an ant the size of a car?" The soul-crushing answer is, no, it wouldn't - because a huge ant would be squished by its own weight. Specifically, its strength scales quadratically with body length, but its weight scales cubically, so it soon grows too heavy to walk, let alone to ferry little boys and girls around town, catching up miscreant adults in its pincers. This analysis (although explained in terms of Ariosto's poetic giants) was first proposed by G. It turns out that Galileo not only did not welcome them, but actually saved us from the looming prospect of Insect Overlords.
Be that as it may, Feynman explanation is Galileo's - scale-invariant physics doesn't work essentially because of the problems keeping ratios lined up (e.g., volume and surface area change at different rates, so air resistance is higher for smaller bodies.)
So that's the "no" answer. More on the "yes" answer when I understand it better....
Wednesday, May 12, 2004
Irreversible Evolution
Here's my candidate for an irreversible evolutionary step. In Adam's Curse, Bryan Sykes talks about the whiptail lizard, one species of which reproduces asexually - there are no male lizards of this type, only egg-laying females that produce genetic replicas of themselves.
This is an example of sexual reproduction disappearing as a species adapts. Since the ancestors of these lizards reproduced sexually, you would have to have a male one to step backward, as Dawkins wants to do. But there aren't any males left. To start putting those notches back on the bedpost, the whiptail needs to make two simultaneous backward steps - one being the production of a male hatchling, and the other the production of a female hatchling that can reproduce with it. But that's not the single step forward that was taken in the first place.
This doesn't mean that you can't evolve any species into any other - it simply means that argument from walking back up the evolutionary tree to the common ancestor doesn't quite cut it. Something more sophisticated is required (what, I don't know yet.)
Tuesday, May 11, 2004
Reading The Extended Phenotype, Richard Dawkins....
and he said something on page 3 that I don't get. His thought experiment: Can an ant evolve into a red-tailed deer? His answer is "Yes", because (paraphrasing) all we need to do is exhibit a sequence of nearly-identical animals leading from the ant to the deer. You get this sequence by going up the evolutionary tree from the ant to the common ancestor, then back down to the deer.
Then he says the iffy thing: presumably the right set of environmental pressures can be applied to produce this sequence of evolutionary steps. So any animal can evolve into any other. He claims this is so unlikely it will never happen, but it's possible.
Huh? Take genotype A and genotype B, differing only by a mutation. Just because the comparative fitness of A against B depends on the environment, that doesn't mean there have to be two distinct environmental regimes - one where A is better, and another where B is better. There doesn't seem to be any a priori reason to think that, but it's necessary for his statement of reversability to be true.
Maybe things will get cleared up by page 5....
Monday, May 10, 2004
Strange Long Knots
Here's another thing I've been thinking about (just this afternoon): consider a strange attractor of a dynamical system - the Lorenz attractor is a good one. Here's a Java-enabled picture.
For the purposes of my question, all we need to consider is that the trajectory never intersects itself, and is parameterized by times t.
A knot K can also be treated as a t-parameterized curve in three-space. However, t lies in some finite interval, or you can think of K as repeatedly traced out over all time. There are lots and lots of nice algebraic and combinatorial invariants you can compute with knots, like Vassiliev invariants, the Jones polynomial, the Alexander polynomial, and so on.
Is there anything useful that can be sussed out about strange attractors by trying to extend what we know about knot invariants to trajectories of dynamical systems?
Sunday, May 09, 2004
More Ado About Noether
I didn't say exactly how Noether connects e.g. conservation of energy to time-isotropy. Here's John Baez's explanation of Noether's theorem for the Lagrangian (a nice quantity tracking every position and velocity in a physical system): Noether's Theorem in a Nutshell
Baez gives the formula for the conserved quantity in a one-particle system:
where p(t) is the momentum at time t, q(t) the position, and s is the parameter of the symmetry - s measures how much the system was rotated or slid. If the Lagrangian has an s-isotropy, dC/ds = 0.
Let's throw caution to the wind (since we're laymen here) and talk about a Lagrangian that is invariant under changes of scale. Take s > 0 to be any real positive number, the scaling factor - this acts as our "symmetry parameter". For instance, if s = 2, we're doubling the size of the universe. The new position is s times the old position:
What's C?
Well, that was easier than I hoped - and interesting. The conserved quantity in our scale-invariant system is the product of position and momentum. I wonder what that means....
Saturday, May 08, 2004
Here's something I've always wanted to know the answer to. There's a result in mathematical physics by Emmy Noether about symmetry and conservation: generally, if a physical system has a symmetry built into in its description, there's a corresponding conservation law. Physical law looks the same today as it will tomorrow - it's isotropic in time. Physical law is the same if you move up or down a few miles, or left or right a few feet - it's isotropic in space. Where there's isotropy, there's a conservation law - in the case of time, conservation of energy, and for space, conservation of momentum.
In geometry, symmetries preserve volumes. Let's throw that out and talk about scaling maps. And lets suppose that we don't already know that physics looks different on the small scale (I'm talking very loosely here - I mean, let us consider classical mechanics.) If materials are homogenous at all scales, so there are no atomic effects and no limit to divisibility, we can take any mechanical system and shrink or magnify it arbitrarily - and it ought to work the same way, intuitively. Does it or doesn't it? Possibly not, because features like mass, and surface tension would scale differently. So that would be odd, right? You could detect that change experimentally. But if the physics is identical at all length scales, then that shouldn't happen - you should be able to magnify every distance in the universe uniformly and not detect it experimentally.
Let's pretend that's true, although I think the scaling issue kills it. If so (here's the real question), can we weaken Noether's theorem to get a new conservation law? What would that conservation law (conservation because of scale invariance?) look like?
This is where I get in touch with my inner crackpot. There are entirely too many silly ideas knocking around in my head, so if I get them out there, perhaps they'll let me alone and bother some other people.
When I say "crackpot", I'm not entirely unserious. I had some training in math and physics a while ago, and I can still follow a paper readily enough. But I'm not going to work very hard to rigorize these questions. Sometimes that's going to mean they're nothing but mush - amalgams of misinterpreted fragments I've picked up here and there. If so, and you're feeling patient, feel free to set me straight. I want to know the answers, and I freely admit I'm a bear of very little brain - not wedded to my mistakes.
