The idea of assigning a probability to such a thing might be, I think, what Nassim Nicholas Taleb calls the "Ludic fallacy" (see http://en.wikipedia.org/wiki/Ludic_fallacy). Alternatively, as I see it, to do such a thing, we need to start with some sort of paradigm by which we know how to set probabilities and do something useful with them. Poker is a good example, and not coincidentally, "Ludic" is from the Latin ludus, meaning "play, game, sport, pastime." It is no accident that so many introductory examples in statistics are about things like coin tossing.
Can we start by asking "Of all the universes I've known, in how many of them does the Atomic Theory apply?" Maybe this is frequentist, and Bayes will offer some plausible approach, but does anyone have a clue how to attack it that way?
Taleb can seem like a curmudgeonish contrarian at times, but at other times he seems to me at least like he's onto some deep ideas. I need to read and think a lot more before I can possibly make up my mind, but at least feel very motivated to do just that.
The paradigm I'm currently looking at this is, generally, the accumulation of evidence over long periods. In the year 1800, not even Dalton had published his (wrong) results about the mass of oxygen; there was no particular evidence /to/ believe in the atomic theory.. In 1900, Einstein had yet to publish his work on Brownian motion; there was still a small but reasonable possibility that somebody would come up with a non-atomic theory that made better predictions. In 2000, atomic theory was so settled that few people even bothered calling it a 'theory' any...
How much confidence do you place in the scientific theory that ordinary matter is made of discrete units, or 'atoms', as opposed to being infinitely divisible?
More than 50%? 90%? 99%? 99.9%? 99.99%? 99.999%? More? If so, how much more? (If describing your answer in percentages is cumbersome, then feel free to use the logarithmic scale of decibans, where 10 decibans corresponds to 90% confidence, 20 to 99%, 30 to 99.9%, etc.)
This question freely acknowledges that there are aspects of physics which the atomic theory does not directly cover, such as conditions of extremely high energy. This question is primarily concerned with that portion of physics in which the atomic theory makes testable predictions.
This question also freely acknowledges that its current phrasing and presentation may not be the best possible to elicit answers from the LessWrong community, and will be happy to accept suggestions for improvement.
Edit: By 'atomic theory', this question refers to the century-plus-old theory. A reasonably accurate rewording is: "Do you believe 'H2O' is a meaningful description of water?".