Posted on 12/06/2003 9:14:26 AM PST by John W
Edited on 04/29/2004 2:03:32 AM PDT by Jim Robinson. [history]
CAPE CANAVERAL, Florida (AP) -- NASA is relying on Russian-made thrusters to steer the international space station following a new malfunction with the U.S. motion-control system, officials said Friday.
Flight controllers detected spikes in current and vibration in one of the station's three operating gyroscopes on November 8. Last week, when the gyroscopes were used again to shift the position of the orbiting outpost, all three worked.
(Excerpt) Read more at cnn.com ...
I reject any assertion of 'astronomical odds' without some minimally rigorous way of computing such odds.
We have actually no idea how complicated the irreducible set of physical laws is. But take a look at quantum mechanics. It's reducible to a very limited state of fundamental postulates. Add conservation of energy, Maxwell's laws, and the Coulomb equation, and you have enough to construct every molecule in the universe, from H2 to human chromosome 23.
Unimaginable complexity can come from a very sparse set of initial conditions. It might indeed be true that only a limited set of such initial conditions can lead to complexity, but at present we don't know how limited the set is, since we're biased by living in a set of conditions that permits our own complexity.
if an accident is responsible for universal laws, then why could not another accident come along and wipe out the product of the first accident, and set up new laws?
This is not an area I'm expert in, but I gather people theorize that the present set of conditions was set at the time of the big-bang, or very shortly thereafter, by a phenomenon called 'spontaneous symmetry breaking'. In general, as you lower the energy of a system, its symmetry tends to drop; you see this when you cool isotropic gases to better-ordered liquids and finally to crystals with highly ordered lattices. So, if you start with a empty universe, the most minimal way to reduce its symmetry is to create a point particle. A point particle creates what is in essence a one-dimensional space, since the only meaningful coordinate is distance from the particle. The symmetry can then be lowered further by splitting the single particle into many particles.
I gather cosmologists have a far more complex representation of this. One isn't just lowering the symmetry of space, but also the symmetry of particles and of forces. But that, they believe, is how you get order from nothingness.
Under the right conditions, stuff - almost any stuff - spontaneously orders. Trillions of snowflakes can't be wrong.
Consider the photon (or any other item you want to consider). I don't know where photons come from, or why they are the way they are, and presumably neither does anyone else. Not yet. But wherever they come from and whatever a photon may be, it is something.
What does that mean? It means that it's whatever a photon is, and not something else. To be something is to be something specific, and not any-old-thing. To put it another way, it is exactly what it is. (Sounds like "I am that I am.") It has the specific characteristics that it has, and not some other characteristics.
Ultimately, I think you're asking: "Why does anything exist?" Because once you have things that exist, you have things with specific characteristics. And their characteristics are inherent in the fact of their existence.
I don't know the specific cause of those characteristics. But if anything exists, it will, of necessity, have some specific characteristics. If you're asking: "Why is a photon a photon?" I guess the only answer anyone can offer you at this time is: "Because it's a photon."
Not very satisfactory, is it? But that's all I've got. I suspect it's all anyone has. We'll have to deal with it. That's the world. Love it or leave it. (That last sentence is my poor attempt at humor.)
Real numbers form a group under most arithmetical operations - that is, multiply, add, divide, square, subtract real numbers, and you still have a real number. Just don't take a square root. :-) When you say a physical quantity can be described as a real number, what you're postulating is what group theorists call an isomorphism; the set of real positive numbers can be mapped one on one with the possible masses of an object, for example. So mass is a real, positive quantity.
Complex numbers have more interesting and inclusive properties than the group of real numbers. You can take square roots of complex numbers and the answer will always be a complex number, for example. And many physically meaningful quantities are isomorphous with the complex numbers - refractive indexes, alternating currents, etc.. There is an even more general group, discovered by Hamilton, called the quaternions. And all of these, in modern math, comes under the category of group theory, which tells you what possible kinds of number sets and algebras can self-consistently exist.
So, in standard physics, time is a real number. But any real number is by definition a complex number with a zero imaginary component; the set of real numbers is a subset of the complex numbers, which is a subset of the quaternions, which... It's an interesting question to ask (in my very humble opinion, in this case) what would happen to physics if the imaginary component of time were non zero, that is, if time were more complicated than we think it is. It's an interesting question I can't answer, but maybe Hawking can.
Not different from any other mathematical entity. Perhaps calling them 'imaginary' put an unfortunate color to them. They are real and valid, not a metaphysical hypostatization, but a logical rather than psychological extension to our primitive 'count' numbers.
Lead on, MacDuff!
As a first step, one might accept the moon to be more useful than a big empty rock in space. But then that would require imagination, not to mention knowledge.
PatrickHenry: I take that to mean that it's "unsatisfactory" to have a problem with no solution. So they strive for further understanding. That's how science reacts to any problem. It's not necessary to assume a theological -- or rather, anti-theological -- intent on such curiosity.
The purpose of Hawking’s lecture is ”to discuss whether time itself has a beginning”.
Time is troubling to the metaphysical naturalist worldview simply because it means there is not an infinity of opportunity. In infinity, one can always use the “it was an accident” argument because in infinity we may presume ”that anything that can happen, will.”
Hawking’s imaginary time speculation looks just as kluged to me as Einstein’s cosmological constant, i.e. faced with inconvenient evidence, what formula or factor can be used to make it “go away.” That approach is like putting the cart before the horse. A Platonist would insist that the evidence be followed to its conclusion and that whatever it is makes sense.
I should clarify that I am very supportive of space exploration. We are a curious people and exploration of space will reveal more of the history (and likely future) of our solar system and the universe.
Laboratory experiments become meaningless at astronomical scales where other factors such as gravity enter the picture. And there are practical experiments and observations which cannot be performed on earth.
I do agree with Nicolo Dallaporta (paraphrased) that science has become so large that we have lost our deep thinkers - scientists are forced to specialize and make progress in baby steps.
Nevertheless, I suspect that the scientists are capable of the innovation required for future space exploration - which we will need whether or not we are graced with a deep thinker in our generation.
Next I hope to hear your appeal for autonomy (or symbolism) in biological self-organizing complexity. If that day comes, maybe we can talk again about biological information content and how it may have originated.
String theories are classified according to whether or not the strings are required to be closed loops, and whether or not the particle spectrum includes fermions. In order to include fermions in string theory, there must be a special kind of symmetry called supersymmetry, which means for every boson (particle that transmits a force) there is a corresponding fermion (particle that makes up matter). So supersymmetry relates the particles that transmit forces to the particles that make up matter.
I can easily see the theological implications of the issue, but in principle it's not all that different from the issue of the origin of life on earth. If a natural cause of life can be demonstrated, we all know that there are theological concerns. Ditto for the Big Bang itself. But I still suggest that those who seek to understand such phenomena aren't necessarily motivated by an anti-theological purpose. There's always the time-honored motive of plain old curiosity. Some people are simply attracted to unsolved problems.
If there is the possibility of a comprehensible, natural cause for the Big Bang, wouldn't you really like to know what it is? Notwithstanding the theological implications? I suspect that you would. And that perfectly human curiosity doesn't mean you have atheistic tendencies.
The Platonist would be a bad scientist. We don't dump otherwise valuable theories because of a few discordant observations; we make some sort of ad hoc assumption, and in the meantime work very hard to find out what's really going on. A classic example is the neutrino. Initial observations of radioactive beta decay indicated that neither energy nor angular momentum was conserved. The scientific community wasn't willing to dump two well-established conservation laws because of this one discordant phenomenon, so they hypothesized the neutrino, a nearly undetectable particle which carried away the momentum and energy. Had they left it at that, it would have been extremely unsatisfactory; however, they did not. Having predicted the existence of the particle, experiments were done to try to detect it, and it was finally observed.
Einstein's cosmological constant has likewise been the subject of a half-century of research; and Hawking's 'imaginary time', if it passes the tests of consistency with well-established observations, and self consistency, will undoubtedly be tested and tested again.
I don't know why that would be surprising; perhaps it's that I tend not to post when I agree with something; I focus on those matters I disagree with.
When a system is far from equilibrium, all sorts of order can be created; as long as you can dump some heat into the surroundings to increase their entropy, the second law will let you decrease the system entropy up to the limit set by the amount of heat dissipated.
So sayeth Ilya Prigogine.....
;-)
No more so than integers or differential equations.
Complex (real and imaginary joined) numbers have several useful properties. For an example, if one solves x^2=2 (going back to Pythagoras), one finds that there is no rational p/q equal to x, so x is of a different kind than fractions. However, x^2=3 needs a new number, just attaching x from x^2=2 doesn't help solve x^2=3. Attaching all the (infinitely many) x^2=p for all prime integers p, doesn't solve the system x^3=2. It gets worse, attaching all the roots of all the integers gives neither pi, nor e. Eudoxus (and Dedekind and Cantor) gave complete discriptions of the real numbers, but these descriptions are rather long and technical.
With imaginary numbers, things are a bit different. One introduces the solution to x^2+1=0, (named i for this note.) Then the solutions to x^3+1=0, x^245+24x^44=0, and all polynomials in x have a solution involving i. Even equations with complex coefficients such as (3x+2i)x^5+7=0 have solutions in complex numbers. This is the Fundamental Theorem of Algebra.
Complex numbers are not just pairs of reals. They are pairs of reals with special rules for arithmetic. They form a field. The rules are (a+bi)+(c+di)=(a+b+(c+d)i) and (a+bi)*(c+di)=(ab-cd+(ad+bc)i). One gets the same result by just using i^2=1 and treating i as a "number."
These complex numbers describe electrical and magnetic fields and are useful in Fourier series because of the identity e^(ix)=Cos(x)+i*Sin(x) (Euler's relation.) Quantum mechanics requires complex numbers for its descriptions. For example, probabilities occur in QM as squares of numbers (actually squares of averages over wave functions) but the numbers are complex (the "square" of a complex number (a+bi) is actually taken as (a+bi)*(a-bi)= a^2+b^2, a-bi is called the conjugate of a+bi.) When two "waves" add in quantum mechanics, the complex numbers describing are added and the square of the result taken to give probabilities. (Classically, one would just add the probabilities.)
Complex numbers are essential in analyis. For example, take a simple function f(x)=1/(1+x^2) with real x. A power series for this function will only converge if |x| < 1. This is because there is a zero at x=i (or x=-1.) Things happening in the complex numbers are intruding onto the real line.
Imaginary numbers are just as "real" (in the ontolgical sense) as real numbers (or just as imaginary.) The terms are unfortunate as ordinary language uses the words "real" and "imaginary" differently. (Not surprisingly terms like: group, field, radical, or, implies, if, then, else, not, and, root, function, etc. have technical meanings that don't correlate well with ordinary usage.)
Very striking insight, Right Whale. Is it reasonable to draw the following corollary therefrom: A close correspondence exists between the modalities of human thought and the structure of the universe? (Thus perhaps suggesting a reason why they serve as such good "tools?")
There wouldn't be any reason to make that connection, except that thoughts of rational minds, human or otherwise would be a natural function of the universe like combining two Hs and an O to make a water molecule. Logic is not founded on modes of rational thought.
Disclaimer: Opinions posted on Free Republic are those of the individual posters and do not necessarily represent the opinion of Free Republic or its management. All materials posted herein are protected by copyright law and the exemption for fair use of copyrighted works.