possible worlds
Epistemic logic is a delusional subject. The notion of ignorance can not be formalized as a uniform probability distribution over the future possible states of the world, because the notion of future possible states does not make any sense. For instance, how do these states differ from each other? You need to have a theory of everything (TOE) at your disposal to answer that question. (In other words, an ignorant person is supposed to have perfect knowledge of what is possible but no knowledge of what is coming next. What an insane proposition!) Unfortunately we do not have such a theory at the moment. In fact, the reality may be inarticulable in a formal language. So there is no reason to suppose that a TOE exists. Moreover, if it exists, it may be extremely complex and inaccessible. Even if it exists and is accessible, the totality of all its future possible states may be humanly incomprehensible.
The notion of future possible states serves an instrumental purpose in finance and cosmology, and should not be taken really seriously. Even insurance companies are aware of the entirely groundless human tendency to think of the future as a model-dependent variation of the past. That is why they always use a God clause to contractually protect themselves from the totally unforeseeable surprises which the reality is always capable of producing.
While there is no model-independent way of legitimately speaking about the future possible worlds, there is absolutely no legitimate way of speaking about all possible worlds!
Even if we had a TOE, we would never know which aspects of it are arbitrary and therefore could have been different. In cosmology, one is often encouraged to engage in thought experiments where certain fundamental constants are imagined to be different than their current values. This is not a caricature example of what we will be able to do with a TOE. This sort of tweaking is exactly all we will be able to do! Of course, a TOE will be more than just a bunch of constants. But what is arbitrary about the rest of the structure will be a complete mystery. We will just have to sit down and wait for an anomaly to take place.
- Wait. What sort of an anomaly are you talking about? The TOE is supposed to explain everything!
That is my point. There are so many things wrong with the idea of a TOE! We can not rule out the possibility of a genuinely surprising future development. (Yes, I can legitimately use the word "possible" without invoking the language of possible worlds. It is an expression of my epistemological limitations.) Similarly, what is today taken as a constant may later be recognized to be a time-dependent variable. (No, a TOE will not be a timeless theory. We can explain only what we experience.) In fact, it may even be recognized to be space-dependent, as the horizon of our observable universe expands with the passage of time.
For instance, physicists have started to take seriously the idea that the fine structure constant may not be a constant after all:
Until quite recently, all attempts to evaluate what happens to the universe if the fine-structure constant changes were unsatisfactory. They amounted to nothing more than assuming that alpha became a variable in the same formulas that had been derived assuming it is a constant. This is a dubious practice. If alpha varies, then its effects must conserve energy and momentum, and they must influence the gravitational field in the universe. In 1982 Jacob D. Bekenstein of the Hebrew University of Jerusalem was the first to generalize the laws of electromagnetism to handle inconstant constants rigorously. The theory elevates alpha from a mere number to a so-called scalar field, a dynamic ingredient of nature. His theory did not include gravity, however. Four years ago one of us (Barrow), with Håvard Sandvik and João Magueijo of Imperial College London, extended it to do so.
This theory makes appealingly simple predictions. Variations in of a few parts per million should have a completely negligible effect on the expansion of the universe. That is because electromagnetism is much weaker than gravity on cosmic scales. But although changes in the fine-structure constant do not affect the expansion of the universe significantly, the expansion affects alpha. Changes to alpha are driven by imbalances between the electric field energy and magnetic field energy. During the first tens of thousands of years of cosmic history, radiation dominated overcharged particles and kept the electric and magnetic fi elds in balance. As the universe expanded, radiation thinned out, and matter became the dominant constituent of the cosmos. The electric and magnetic energies became unequal, and alpha started to increase very slowly, growing as the logarithm of time. About six billion years ago dark energy took over and accelerated the expansion, making it difficult for all physical influences to propagate through space. So alpha became nearly constant again.
In 2003 Barrow and David F. Mota of Cambridge calculated that alpha may behave differently within the galaxy than inside emptier regions of space. Once a young galaxy condenses and relaxes into gravitational equilibrium, alpha nearly stops changing inside it but keeps on changing outside. Thus, the terrestrial experiments that probe the constancy of suffer from a selection bias... No spatial variations of alpha have yet been seen... If alpha is susceptible to change, however, other constants should vary as well, making the inner workings of nature more fickle than scientists ever suspected.
Barrow & Webb - Inconstant
Taking the set of all possible mathematical theories as the set of all possible TOEs does not resolve the issue neither. It just pushes the problem to a one step higher level. Now we will be at a loss, because it is not clear which type of logic is the right setting for these theories. (There are many different candidates out there!)
What about the set of all possible mathematical theories in all possible logics? That will not solve the problem neither. While one can legitimately consider mathematical statements within a certain logic, one can not legitimately consider the set of all logics. What will be the parameters of your variation?
P.S. I just love the insane examples that metaphysicists come up with. They literally make me burst into laughterous tears. Here is one example where a mentally challenging concept is illustrated via a physically handicapping example:
"The second type of apparent modal truth, however, is more challenging. Julius Caesar could have had a sixth right finger which was never burnt but which could have been burnt. This involves a nested possibility, which is troublesome to actualist representationism. To reveal the nesting clearly, let us articulate the possibility in question in a more pedantic and rigorous way. The following is possible: Julius Caesar had a sixth right finger such that (a) it was never burnt, and (b) the following is possible: It was burnt." (Source)
Update (November 2011): There now seems to be some evidence for spatial variation of the fine structure constant. Here is an extract from Michael Murphy's website. (He is one of the scientists involved in the research.)
Our results are by no means conclusive. We have thoroughly searched for other possible explanations for our results. Research science is constantly plagued by the problem of "systematic errors". These errors mimic your result or somehow destroy it. They are notoriously hard to identify. And this is what we've been looking for in our results. But we still can't find anything that explains our results besides a varying alpha! The new "alpha dipole" adds a new twist as well, making it even more difficult to understand how a systematic effect (or several) might have mimicked what we found.
The best approach in science is to always check (and re-check if necessary) your results using different equipment and analyses. We continue to analyse new, large datasets observed with the Very Large Telescope and Keck telescope. But, as mentioned above, there's a famous saying in science: "Extraordinary claims require extraordinary evidence.". Though we are claiming something quite extraordinary here, the evidence, though very strong, is not extraordinary enough. Yet. No one should really believe that constants are varying until another type of experiment confirms the results. Possibilities for other types of experiments include making very precise measurements of the fluctuations seen on the Cosmic Microwave Background sky -- the radiation left over from the big bang. Another possibility is to measure very accurately the abundances of the elements that were produced in the big bang. But these methods have their own problems and systematic errors. But we're hoping this will improve soon!
Probably the best chance of confirming our results is to search for varying alpha in the laboratory somehow, perhaps by comparing the ticking rates of ultra-precise atomic clocks. The current best precision is not quite high enough but atomic clock technology is improving extremely rapidly, so we may know for sure sooner rather than later.
... I mentioned that alpha is made up of three other constants: alpha = e2/hc where c is the speed of light, e is the charge of an electron and h is Planck's constant. Both laymen and scientists alike always ask whether we have any idea whether it's c, e or h that varies. This frequently asked question has a subtle and often misunderstood answer.
In fact, one can never experimentally distinguish between a varying c or e because these quantities are always measured in some arbitrary units like meters, kilograms, seconds etc. Consider measuring the time it takes light to travel between you and me on Monday and then again on Tuesday. Imagine that the two answers were different. What does this tell you? You might conclude that the speed of light, c, has changed between Monday and Tuesday or, equally well, you could conclude that time has slowed/accelerated or that your measuring rods (i.e. meter rules) have changed length. These three conclusions are all equally valid and can not be distinguished by an experiment! But alpha is special because it is a dimensionless combination of other constants: alpha is just a number, i.e. no units! We can therefore measure changes in alpha unambiguously.
Some confusion has arisen recently in the literature about this question. The problem is that there exist well defined theories called "Varying Speed of Light" (VSL) and "Varying Electric Charge" (VEC) theories. For example, in VSL theories, it is indeed the speed of light that is considered to vary. But this is just a mathematical convenience: one could easily convert any VSL theory into a VEC theory! The only reason one chooses to label one particular theory a VSL or VEC theory is because that theory might look simpler (mathematically and intuitively) when considering a varying c or e. Essentially, the confusion is that the (arbitrary) names given to these theories mask their inherent duality (or triality if you include h!).
